North Carolina Math 2 - Walch

Custom Program Overview

North Carolina Math 2

1 2 3 4 5 6 7 8 9 10

ISBN 978-0-8251-9022-3

Copyright © 2020

J. Weston Walch, Publisher

Portland, ME 04103

www.walch.com

Printed in the United States of America

© Common Core State Standards. Copyright 2010. National Governor’s Association Center for Best Practices and

Council of Chief State School Officers. All rights reserved.

The classroom teacher may reproduce these materials for classroom use only.The reproduction of any part for an entire school or school system is strictly prohibited.

No part of this publication may be transmitted, stored, or recorded in any formwithout written permission from the publisher.

This program was developed and reviewed by experienced math educators who have both academic and professional backgrounds in mathematics. This ensures: freedom from mathematical errors, grade level

appropriateness, freedom from bias, and freedom from unnecessary language complexity.

Developers and reviewers include:

Joyce Hale

Vanessa Sylvester

Jasmine Owens

Ruth Estabrook

Shelly Northrop Sommer

Joanne Whitley

Robert Leichner

Michelle Adams

Marie Vrablic

Kaithlyn Hollister

Carrisa Johnson-Scott

Joseph Nicholson

Samantha Carter

Tiffany Fele

Zachary Lien

Valerie Ackley

Laura McPartland

Cameron Larkins

Frederick Becker

Jennifer Blair

Mike May, S.J.

James Quinlan

Lenore Horner

Pamela Rawson

Nancy Pierce

Jane Mando

Kim Brady

Lynze Greathouse

Martie

Sticky Note

Marked set by Martie

© Walch Educationiii

North Carolina Math 2 Custom Teacher Resource

© Walch Educationiii

Table of Contents for Instructional Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vIntroduction to the Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Correspondence to Standards for Mathematical Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Correspondence to NCTM Principles to Actions Teaching Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Unit Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Standards Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Conceptual Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Station Activities Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Digital Enhancements Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Standards for Mathematical Practice Implementation Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Instructional Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Graphic Organizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GO-1Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-1Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-1

iii

Contents of Program OverviewPROGRAM OVERVIEW

Martie

Sticky Note


Martie

Sticky Note


Martie

Sticky Note


Martie

Sticky Note


Martie

Sticky Note


Martie

Sticky Note


Martie

Sticky Note


Martie

Sticky Note


© Walch Educationv


Table of ContentsPROGRAM OVERVIEW

v

Unit 1: TransformationsUnit 1 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Lesson 1.1: Proving the Vertical Angles Theorem (G–CO.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-1Lesson 1.2: Proving Theorems About Angles in Parallel Lines

Cut by a Transversal (G–CO.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-35Lesson 1.3: Proving the Interior Angle Sum Theorem (G–CO.10) . . . . . . . . . . . . . . . . . . . . . . . U1-70Lesson 1.4: Proving Theorems About Isosceles Triangles (G–CO.10) . . . . . . . . . . . . . . . . . . . . U1-103Lesson 1.5: Proving the Midsegment of a Triangle (G–CO.10) . . . . . . . . . . . . . . . . . . . . . . . . . U1-135Lesson 1.6: Function Notation and Evaluating Functions (F–IF.2) . . . . . . . . . . . . . . . . . . . . . . U1-175Lesson 1.7: Translations on the Plane (G–CO.2, G–CO.4, G–CO.5) . . . . . . . . . . . . . . . . . . . . . U1-199Lesson 1.8: Reflections on the Plane (G–CO.2, G–CO.4, G–CO.5) . . . . . . . . . . . . . . . . . . . . . . U1-227Lesson 1.9: Rotations on the Plane (G–CO.2, G–CO.4, G–CO.5) . . . . . . . . . . . . . . . . . . . . . . . U1-257Lesson 1.10: Applying Lines of Symmetry (G–CO.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-284Lesson 1.11: Dilations on the Plane (G–SRT.1a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-311Lesson 1.12: Defining Congruence in Terms of Rigid Motions (G–CO.6) . . . . . . . . . . . . . . . . U1-344Lesson 1.13: Transformations and Rigid Motions (G–CO.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-383Lesson 1.14: Perpendicular Bisectors and Angle Bisectors (G–CO.9) . . . . . . . . . . . . . . . . . . . . U1-409Lesson 1.15: Triangle Congruency (G–CO.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-410Lesson 1.16: Explaining ASA, SAS, and SSS (G–CO.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-445

Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-483

Station ActivitiesSet 1: Parallel Lines and Transversals (G–CO.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-497Set 2: Corresponding Parts, Transformations, and Proof (G–CO.2, G–CO.5, G–CO.6,

G–CO.7, G–CO.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U1-509Set 3: Rotations and Reflections (G–CO.3, G–CO.4, G–CO.5) . . . . . . . . . . . . . . . . . . . . . . . . . . U1-521

Mid-Unit Assessment and Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MA-1

End-of-Unit Assessment and Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EA-1

vi© Walch EducationNorth Carolina Math 2 Custom Teacher Resource

PROGRAM OVERVIEWTable of Contents

vi

Unit 2: Graphing Quadratic FunctionsUnit 2 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Lesson 2.1: Identifying the Domain and Range of a Quadratic Function (F–IF.4•) . . . . . . . . . . U2-1Lesson 2.2: Identifying Terms, Factors, and Coefficients (A–SSE.1a•) . . . . . . . . . . . . . . . . . . . . U2-22Lesson 2.3: Interpreting Complicated Expressions (A–SSE.1b•) . . . . . . . . . . . . . . . . . . . . . . . . U2-44Lesson 2.4: Quadratic Modeling (A–CED.2•, F–BF.1•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-67Lesson 2.5: Interpreting Key Features of Quadratic Functions (F–IF.4•) . . . . . . . . . . . . . . . . . . U2-93Lesson 2.6: Interpreting Various Forms of Quadratic Functions (F–IF.8) . . . . . . . . . . . . . . . . . U2-122Lesson 2.7: Comparing Properties of Quadratic Functions

Given in Different Forms (F–IF.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-159Lesson 2.8: Replacing f(x) with f(x) + k and f(x + k) (F–BF.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-180Lesson 2.9: Replacing f(x) with k • f(x) and f(k • x) (F–BF.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-210 Lesson 2.10: Interpreting Quadratic Functions (F–IF.4•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-244Lesson 2.11: Quadratic Regression (A–CED.2•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-282

Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-305

Station ActivitiesSet 1: Graphing Quadratic Equations (F–IF.7•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-323Set 2: Quadratic Transformations in Vertex Form (F–IF.7•, F–IF.8•, F–BF.3) . . . . . . . . . . . . . U2-337



Unit 3: Solving Quadratic EquationsUnit 3 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Lesson 3.1: Rational and Irrational Numbers and Their Properties (N–RN.3) . . . . . . . . . . . . . . U3-1Lesson 3.2: Adding and Subtracting Polynomials (A–APR.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-27Lesson 3.3: Multiplying Polynomials (A–APR.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-47Lesson 3.4: Factoring (A–REI.4b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-69Lesson 3.5: Simplifying Radicals with Numbers (N–RN.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-91Lesson 3.6: Defining Complex Numbers, i, and i 2 (N–CN.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-108Lesson 3.7: Solving Quadratics by Completing the Square (A–REI.4a) . . . . . . . . . . . . . . . . . . U3-127Lesson 3.8: Applying the Quadratic Formula (A–REI.4b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-149Lesson 3.9: Solving Systems Algebraically (A–REI.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-169Lesson 3.10: Solving Systems Graphically (A–REI.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-195

Martie

Sticky Note


Martie

Sticky Note


© Walch Educationvii



vii

Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-225

Station ActivitiesSet 1: Operations with Complex Numbers (N–CN.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-237Set 2: Operations with Polynomials (A–APR.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-244Set 3: Factoring (A–SSE.3•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-257Set 4: Solving Quadratics (A–REI.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U3-265



Unit 4: Square Root and Inverse Variation FunctionsUnit 4 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Lesson 4.1: Evaluating Negative Exponents (N–RN.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U4-1Lesson 4.2: Working with Radicals and Properties of Real Numbers (N–RN.1, N–RN.2) . . . U4-16Lesson 4.3: Operating with Rational Exponents and

Converting with Radicals (N–RN.1, N–RN.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U4-47Lesson 4.4: Solving Radical Equations (A–REI.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U4-69Lesson 4.5: Inverse Variation (F–BF.1•, A–REI.2, A–REI.11•) . . . . . . . . . . . . . . . . . . . . . . . . . U4-101Lesson 4.6: Radical Functions (F–IF.7•, F–IF.9, F–BF.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U4-123Lesson 4.7: Inverse Variation Functions (F–IF.4•, F–IF.7•, F–IF.9) . . . . . . . . . . . . . . . . . . . . . U4-164

Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U4-165



Unit 5: Similarity, Right Triangles, and CongruenceUnit 5 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Lesson 5.1: Defining Similarity (G–SRT.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U5-1Lesson 5.2: Applying Similarity Using the Angle-Angle (AA) Criterion (G–SRT.3) . . . . . . . . . U5-36Lesson 5.3: Proving Triangle Similarity Using Side-Angle-Side (SAS) and

Side-Side-Side (SSS) Similarity (G–SRT.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U5-61Lesson 5.4: Working with Ratio Segments (G–SRT.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U5-92Lesson 5.5: Proving the Pythagorean Theorem Using Similarity (G–SRT.4) . . . . . . . . . . . . . . U5-119Lesson 5.6: Special Right Triangles (G–SRT.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U5-157

viii© Walch EducationNorth Carolina Math 2 Custom Teacher Resource


viii

Lesson 5.7: Defining Trigonometric Ratios (G–SRT.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U5-176

Lesson 5.8: Calculating Sine, Cosine, and Tangent (G–SRT.8•) . . . . . . . . . . . . . . . . . . . . . . . . U5-215

Lesson 5.9: Applying the Pythagorean Theorem (G–SRT.8•) . . . . . . . . . . . . . . . . . . . . . . . . . . U5-250Lesson 5.10: Problem Solving with the Pythagorean Theorem

and Trigonometry (G–SRT.8•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U5-270

Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U5-307

Station ActivitiesSet 1: Similarity and Scale Factor (G–SRT.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U5-315Set 2: Sine, Cosine, and Tangent Ratios, and Angles of Elevation

and Depression (G–SRT.6, G–SRT.8•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U5-328



Unit 6: ProbabilityUnit 6 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Lesson 6.1: Describing Events (S–CP.1•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-1Lesson 6.2: Using Simulation (S–IC.2•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-39Lesson 6.3: The Addition Rule (S–CP.7•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-75Lesson 6.4: Conditional Probability and the General Multiplication Rule (S–CP.8•) . . . . . . . . U6-95Lesson 6.5: Introducing Conditional Probability (S–CP.3•, S–CP.5•, S–CP.6•) . . . . . . . . . . . U6-114Lesson 6.6: Using Two-Way Frequency Tables (S–CP.4•, S–CP.5•, S–CP.6•) . . . . . . . . . . . . . . U6-149Lesson 6.7: Understanding Independent Events (S–CP.3b•) . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-177

Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-209

Station ActivitiesSet 1: Probability (S–CP.1•, S–CP.3•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-215



© Walch Education1


IntroductionThe North Carolina Math 2 Custom Teacher Resource is a complete set of materials developed around the North Carolina Standard Course of Study (NCSCOS) for Mathematics. Topics are built around accessible core curricula, ensuring that the North Carolina Math 2 Custom Teacher Resource is useful for striving students and diverse classrooms.

This program realizes the benefits of exploratory and investigative learning and employs a variety of instructional models to meet the learning needs of students with a range of abilities.

The North Carolina Math 2 Custom Teacher Resource includes components that support problem-based learning, instruct and coach as needed, provide practice, and assess students’ skills. Instructional tools and strategies are embedded throughout.

The program includes:

• More than 150 hours of lessons

• Essential Questions for each instructional topic

• Vocabulary

• Instruction and Guided Practice

• Problem-based Tasks and Coaching questions

• Step-by-step graphing calculator instructions for the TI-Nspire and the TI-83/84

• Station activities to promote collaborative learning and problem-solving skills

Purpose of Materials

The North Carolina Math 2 Custom Teacher Resource has been organized to coordinate with the North Carolina Math 2 content map and specifications from the NCSCOS. Each lesson includes activities that offer opportunities for exploration and investigation. These activities incorporate concept and skill development and guided practice, then move on to the application of new skills and concepts in problem-solving situations. Throughout the lessons and activities, problems are contextualized to enhance rigor and relevance.

PROGRAM OVERVIEW

Introduction to the Program

2© Walch EducationNorth Carolina Math 2 Custom Teacher Resource

PROGRAM OVERVIEWIntroduction to the Program

This program includes all the topics addressed in the North Carolina Math 2 content map. These include:

• Transformations

• Graphing Quadratic Functions

• Solving Quadratic Equations

• Square Root and Inverse Variation Functions

• Similarity, Right Triangles, and Congruence

• Probability

The eight Standards for Mathematical Practice are infused throughout:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Structure of the Teacher Resource

The North Carolina Math 2 Custom Teacher Resource materials are completely reproducible. The Program Overview is the first section. This section helps you to navigate the materials, offers a collection of research-based Instructional Strategies along with their literacy connections and implementation suggestions, and shows the correlation between the NCSCOS for Mathematics and the district-specfic content map and course requirements.

The remaining materials focus on content, knowledge, and application of the six units in the North Carolina Math 2 custom program: Transformations; Graphing Quadratic Functions; Solving Quadratic Equations; Square Root and Inverse Variation Functions; Similarity, Right Triangles, and Congruence; and Probability. The units in this program are designed to be flexible so that you can mix and match activities as the needs of your students and your instructional style dictate.

© Walch Education3


PROGRAM OVERVIEWIntroduction to the Program

The Station Activities correspond to the content in the units and provide students with the opportunity to apply concepts and skills, while you have a chance to circulate, observe, speak to individuals and small groups, and informally assess and plan.

Each unit includes a mid-unit assessment and an end-of-unit assessment. These enable you to gauge how well students have understood the material as you move from lesson to lesson and to differentiate as appropriate.


PROGRAM OVERVIEW

How Do Walch Integrated Mathematics Resources Address the Standards for Mathematical Practice?Walch’s mathematics courses employ a problem-based model of instruction that supports and reinforces the eight Standards for Mathematical Practice. Although the following table focuses on Problem-Based Tasks, Walch’s full programs also include hundreds of additional problems in warm-ups and practices. The Implementation Guides for selected PBTs highlight SMPs to focus on during implementation and discussion.

Standards for Mathematical Practice

Relevant Attributes of Walch Integrated Math Resources

1 Make sense of problems and persevere in solving them.

Each lesson is built around a Problem-Based Task (PBT) that requires students to “make sense of problems and persevere in solving them.”

2 Reason abstractly and quantitatively.

Each PBT uses a meaningful real-world context that requires students to reason both abstractly about the situation/relationships and quantitatively about the values representing the elements and relationships.

3 Construct viable arguments and critique the reasoning of others.

Since the PBT provides opportunities for multiple problem-solving approaches and varied solutions, students are required to construct viable arguments to support their approach and answer. This, in turn, provides other students the opportunity to analyze and critique their classmates’ reasoning.

4 Model with mathematics.

Each PBT represents a real-world situation and requires students to model it with mathematics.

5 Use appropriate tools strategically.

PBTs require students to make choices about using appropriate tools, such as calculators, spreadsheets, graph paper, manipulatives, protractors, and compasses. The tasks do not prescribe specific tools, but instead provide opportunities for their use.

6 Attend to precision. The real-world contexts of the PBTs require students to be precise in their solutions, both in the ways that the solutions are stated, labeled, and explained, and in the degree of precision necessary given the context (e.g., tripling chili for a crowd vs. machining a part for an airplane engine).

7 Look for and make use of structure.

The PBTs present students with complicated scenarios that must be analyzed to discern patterns and significant mathematical features.

8 Look for and express regularity in repeated reasoning.

PBTs require multiple steps, providing opportunities for students to note repeated calculations, monitor their process, and continually evaluate reasonableness of intermediate results before arriving at a solution.

Correspondence to Standards for Mathematical Practice

© Walch Education5


PROGRAM OVERVIEW

How Do Walch Integrated Mathematics Resources Address the NCTM Principles to Actions Mathematics Teaching Practices?Walch’s mathematics programs were designed by experienced educators and curriculum developers, informed by best-practice research, and refined through an iterative process of implementation and feedback. Together with professional development, these materials support and sustain good teaching practices.

NCTM Mathematics Teaching Practices

Relevant Attributes of Walch Integrated Resources

Establish mathematics goals to focus learning.

Each lesson in Walch’s programs addresses specified standards which can be used as goals to focus learning. Essential Questions offer further focus.

Implement tasks that promote reasoning and problem solving.

Each lesson in Walch’s programs is built around a Problem-Based Task (PBT), set in a meaningful real-world context and designed to promote reasoning and problem solving. The courses include dozens of PBTs as well as warm-up and practice problems.

Use and connect mathematical representations.

Walch’s Integrated programs make frequent use of, and connections among and between, equations, tables, and graphs. PBTs often require students to use and connect two or more of these representations, and the representations are modeled through guided practice.

Facilitate meaningful mathematical discourse.

Several features of the programs support mathematical discourse, including warm-up debriefs with connections to upcoming lessons, implementation guides and optional coaching questions for the PBTs, and discussion guides for Station Activities. Explanations of PBT solutions are another opportunity for discourse. Please note: Mathematical discourse is an important topic for professional development, in conjunction with implementation of these materials.

Pose purposeful questions.

The implementation guides, coaching questions and discussion guides provide samples of purposeful questions. Note that this is another important topic for professional development.

Build procedural fluency from conceptual understanding.

The programs develop conceptual understanding through modeling, guided practice, and application, and then provide additional opportunities to practice and develop fluency.

Support productive struggle in learning mathematics.

The PBTs require “productive struggle;” implementation guides include suggestions for facilitation and monitoring, and coaching questions provide an option for additional support as appropriate, allowing students to proceed through the task and ensuring that the struggle remains productive rather than too frustrating.

Elicit and use evidence of student thinking.

Various discussions and PBTs require students to display their thinking. Implementation guides offer specific prompts and suggestions for eliciting and responding to student thinking. Professional development supports teachers in using that evidence to respond in instructionally appropriate ways.

Correspondence to NCTM Principles to Actions Teaching Practices


PROGRAM OVERVIEW

All of the instructional units have common features. Each unit begins with a list of all the standards addressed in the lessons; Essential Questions; vocabulary (titled “Words to Know”); a list of recommended websites to be used as additional resources, and one or more conceptual activities.

Each lesson begins with a warm-up, followed by a list of identified prerequisite skills that students need to have mastered in order to be successful with the new material in the upcoming lesson. This is followed by an introduction, key concepts, common errors/misconceptions, guided practice examples, a problem-based task with coaching questions and sample responses, a closure activity, and practice. Each unit includes a Mid-Unit Assessment and an End-of-Unit Assessment to evaluate students’ learning.

All of the components are described below and on the following pages for your reference.

North Carolina Standard Course of Study for the Unit

All standards that are addressed in the entire unit are listed.

Essential Questions

These are intended to guide students’ thinking as they proceed through the unit. By the end of each unit, students should be able to respond to the questions.

Words to Know

A list of vocabulary terms that appear in the unit are provided as background information for instruction or to review key concepts that are addressed in the lesson. Each term is followed by a numerical reference to the lesson(s) in which the term is defined.

Recommended Resources

This is a list of websites that can be used as additional resources. Some websites are games; others provide additional examples and/or explanations. (Note: Links will be monitored and repaired or replaced as necessary.) Each Recommended Resource is also accessible through Walch’s cloud-based Curriculum Engine Learning Object Repository as a separate learning object that can be assigned to students.

Conceptual Activities

Conceptual understanding serves as the foundation on which to build deeper understanding of mathematics. In an effort to build conceptual understanding of mathematical ideas and to provide more than procedural fluency and application, links to interactive open education and Desmos resources are included. (Note: These website links will be monitored and repaired or replaced as necessary.) These and many other open educational resources (OERs) are also accessible through the Learning Object Repository as separate objects that can be assigned to students.

Warm-Up

Each warm-up takes approximately 5 minutes and addresses either prerequisite and critical-thinking skills or previously taught math concepts.

Unit Structure

© Walch Education7


PROGRAM OVERVIEWUnit Structure

Warm-Up Debrief

Each debrief provides the answers to the warm-up questions, and offers suggestions for situations in which students might have difficulties. A section titled Connection to the Lesson is also included in the debrief to help answer students’ questions about the relevance of the particular warm-up activity to the upcoming instruction. Warm-Ups with debriefs are also provided in PowerPoint presentations.

Identified Prerequisite Skills

This list cites the skills necessary to be successful with the new material.

Introduction

This brief paragraph gives a description of the concepts about to be presented and often contains some Words to Know.

Key Concepts

Provided in bulleted form, this instruction highlights the important ideas and/or processes for meeting the standard.

Graphing Calculator Directions

Step-by-step instructions for using a TI-Nspire and a TI-83/84 are provided whenever graphing calculators are referenced.

Common Errors/Misconceptions

This is a list of the common errors students make when applying Key Concepts. This list suggests what to watch for when students arrive at an incorrect answer or are struggling with solving the problems.

Scaffolded Practice (Printable Practice)

This set of 10 printable practice problems provides introductory level skill practice for the lesson. This practice set can be used during instruction time.

Guided Practice

This section provides step-by-step examples of applying the Key Concepts. The three to five examples are intended to aid during initial instruction, but are also for individuals needing additional instruction and/or for use during review and test preparation.

Enhanced Instructional PowerPoint (Presentation)

Each lesson includes an instructional PowerPoint presentation with the following components: Warm-Up, Key Concepts, and Guided Practice. Selected Guided Practice examples include GeoGebra applets. These instructional PowerPoints are downloadable and editable.


PROGRAM OVERVIEWUnit Structure

Problem-Based Task

This activity can serve as the centerpiece of a problem-based lesson, or it can be used to walk students through the application of the standard, prior to traditional instruction or at the end of instruction. The task makes use of critical-thinking skills.

Optional Problem-Based Task Coaching Questions with Sample Responses

These questions scaffold the task and guide students to solving the problem(s) presented in the task. They should be used at the discretion of the teacher for students requiring additional support. The Coaching Questions are followed by answers and suggested appropriate responses to the coaching questions. In some cases answers may vary, but a sample answer is given for each question.

Recommended Closure Activity

Students are given the opportunity to synthesize and reflect on the lesson through a journal entry or discussion of one or more of the Essential Questions.

Problem-Based Task Implementation Guide

This instructional overview, found with selected Problem-Based Tasks in each unit, highlights connections between the task and the lesson’s key concepts and SMPs. The Implementation Guide also offers suggestions for facilitating and monitoring, and provides alternative solutions.

Printable Practice (Sets A and B) and Interactive Practice (Set A)

Each lesson includes two sets of practice problems to support students’ achievement of the learning objectives. They can be used in any combination of teacher-led instruction, cooperative learning, or independent application of knowledge. Each Practice A is also available as an interactive Learnosity activity with Technology-Enhanced Items.

Answer Key

Answers for all of the Warm-Ups and practice problems are provided at the end of each unit.

Station Activities

Each unit includes a collection of station-based activities to provide students with opportunities to practice, reinforce, and apply mathematical skills and concepts. The debriefing discussions after each set of activities provide an important opportunity to help students reflect on their experiences and synthesize their thinking.

Mid-Unit and End-of-Unit Assessments

A mid-unit assessment and an end-of-unit assessment offer multiple-choice questions and extended-response questions that incorporate critical thinking and writing components. These can be used to document the extent to which students grasped the concepts and skills of each unit.

© Walch Education9


Standards CorrelationsPROGRAM OVERVIEW

Each lesson in this program was written specifically to address the North Carolina Standard Course of Study (NCSCOS) for Mathematics. Each unit lists the standards covered in all the lessons, and each lesson lists the standards addressed in that particular lesson. In this section, you’ll find a comprehensive list mapping the lessons to the NCSCOS.

As you use this program, you will come across a star symbol (★) included with the standards for some of the lessons and activities. This symbol is explained below.

Symbol: ★

Denotes: Modeling Standards

Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★).From http://www.walch.com/CCSS/00003


PROGRAM OVERVIEWStandards Correlations

NORT

H CA

ROLI

NA M

ATH

2 ST

ANDA

RDS

CORR

ELAT

IONS

Uni

t 1: T

rans

form

atio

ns

Less

onTi

tle

Stan

dard

(s)

Page

s

1.1

Prov

ing

the

Ver

tical

Ang

les T

heor

emG

–CO

.9U

1-1

1.2

Prov

ing

Theo

rem

s Abo

ut A

ngle

s in

Para

llel L

ines

Cut

by

a Tr

ansv

ersa

lG

–CO

.9U

1-35

1.3

Prov

ing

the

Inte

rior

Ang

le S

um T

heor

emG

–CO

.10

U1-

70

1.4

Prov

ing

Theo

rem

s Abo

ut Is

osce

les T

rian

gles

G–C

O.1

0U

1-10

3

1.5

Prov

ing

the

Mid

segm

ent o

f a T

rian

gle

G–C

O.1

0U

1-13

5

1.6

Func

tion

Not

atio

n an

d Ev

alua

ting

Func

tions

F–IF

.2U

1-17

5

1.7

Tran

slat

ions

on

the

Plan

eG

–CO

.2, G

–CO

.4,

G–C

O.5

U1-

199

1.8

Refle

ctio

ns o

n th

e Pl

ane

G–C

O.2

, G–C

O.4

, G

–CO

.5U

1-22

7

1.9

Rota

tions

on

the

Plan

eG

–CO

.2, G

–CO

.4,

G–C

O.5

U1-

257

1.10

App

lyin

g Li

nes o

f Sym

met

ryG

–CO

.3U

1-28

4

1.11

Dila

tions

on

the

Plan

eG

–SRT

.1a

U1-

311

1.12

Def

inin

g Co

ngru

ence

in T

erm

s of R

igid

Mot

ions

G–C

O.6

U1-

344

1.13

Tran

sfor

mat

ions

and

Rig

id M

otio

nsG

–CO

.6U

1-38

3

1.14

Perp

endi

cula

r Bis

ecto

rs a

nd A

ngle

Bis

ecto

rsG

–CO

.9U

1-40

9

1.15

Tria

ngle

Con

grue

ncy

G–C

O.7

U1-

410

1.16

Expl

aini

ng A

SA, S

AS,

and

SSS

G–C

O.8

U1-

445

© Walch Education11



NORT

H CA

ROLI

NA M

ATH

2 ST

ANDA

RDS

CORR

ELAT

IONS

Uni

t 2: G

raph

ing

Qua

drat

ic F

unct

ions

Less

onTi

tle

Stan

dard

(s)

Page

s

2.1

Iden

tifyi

ng th

e D

omai

n an

d Ra

nge

of a

Qua

drat

ic F

unct

ion

F–IF

.4•

U2-

1

2.2

Iden

tifyi

ng T

erm

s, F

acto

rs, a

nd C

oeffi

cien

tsA

–SSE

.1a•

U2-

22

2.3

Inte

rpre

ting

Com

plic

ated

Exp

ress

ions

A–S

SE.1

b•U

2-44

2.4

Qua

drat

ic M

odel

ing

A–C

ED.2

•, F

–BF.

1•U

2-67

2.5

Inte

rpre

ting

Key

Fea

ture

s of Q

uadr

atic

Fun

ctio

nsF–

IF.4

•U

2-93

2.6

Inte

rpre

ting

Var

ious

For

ms o

f Qua

drat

ic F

unct

ions

F–IF

.8U

2-12

2

2.7

Com

pari

ng P

rope

rtie

s of Q

uadr

atic

Fun

ctio

ns G

iven

in

Diff

eren

t For

ms

F–IF

.9U

2-15

9

2.8

Repl

acin

g f(x

) with

f(x)

+ k

and

f(x +

k)F–

BF.3

U2-

180

2.9

Repl

acin

g f(x

) with

k •

f(x) a

nd f(

k • x)

F–BF

.3U

2-21

0

2.10

Inte

rpre

ting

Qua

drat

ic F

unct

ions

F–IF

.4•

U2-

244

2.11

Qua

drat

ic R

egre

ssio

nA

–CED

.2•

U2-

282

Martie

Sticky Note


Martie

Sticky Note




NORT

H CA

ROLI

NA M

ATH

2 ST

ANDA

RDS

CORR

ELAT

IONS

Uni

t 3: S

olvi

ng Q

uadr

atic

Equ

atio

ns

Less

onTi

tle

Stan

dard

(s)

Page

s

3.1

Ratio

nal a

nd Ir

ratio

nal N

umbe

rs a

nd T

heir

Pro

pert

ies

N–R

N.3

U3-

1

3.2

Add

ing

and

Subt

ract

ing

Poly

nom

ials

A–A

PR.1

U3-

27

3.3

Mul

tiply

ing

Poly

nom

ials

A–A

PR.1

U3-

47

3.4

Fact

orin

gA

–REI

.4b

U3-

69

3.5

Sim

plify

ing

Radi

cals

with

Num

bers

N–R

N.2

U3-

91

3.6

Def

inin

g Co

mpl

ex N

umbe

rs, i

, and

i2N

–CN

.1U

3-10

8

3.7

Solv

ing

Qua

drat

ics b

y Co

mpl

etin

g th

e Sq

uare

A–R

EI.4

aU

3-12

7

3.8

App

lyin

g th

e Q

uadr

atic

For

mul

aA

–REI

.4b

U3-

149

3.9

Solv

ing

Syst

ems A

lgeb

raic

ally

A–R

EI.7

U3-

169

3.10

Solv

ing

Syst

ems G

raph

ical

lyA

–REI

.7U

3-19

5

Uni

t 4: S

quar

e R

oot a

nd In

vers

e V

aria

tion

Fun

ctio

ns

Less

onTi

tle

Stan

dard

(s)

Page

s

4.1

Eval

uatin

g N

egat

ive

Expo

nent

sN

–RN

.2U

4-1

4.2

Wor

king

with

Rad

ical

s and

Pro

pert

ies o

f Rea

l Num

bers

N–R

N.1

, N–R

N.2

U4-

16

4.3

Ope

ratin

g w

ith R

atio

nal E

xpon

ents

and

Con

vert

ing

with

Rad

ical

sN

–RN

.1, N

–RN

.2U

4-47

4.4

Solv

ing

Radi

cal E

quat

ions

A–R

EI.2

U4-

69

4.5

Inve

rse

Var

iatio

nF–

BF.1

•, A

–REI

.2,

A–R

EI.1

1•U

4-10

1

4.6

Radi

cal F

unct

ions

F–IF

.7•

, F–I

F.9,

F–B

F.3

U4-

123

4.7

Inve

rse

Var

iatio

n Fu

nctio

nsF–

IF.4

•, F

–IF.

7•,

F–IF

.9U

4-16

4




NORT

H CA

ROLI

NA M

ATH

2 ST

ANDA

RDS

CORR

ELAT

IONS

Uni

t 5: S

imila

rity

, Rig

ht T

rian

gles

, and

Con

grue

nce

Less

onTi

tle

Stan

dard

(s)

Page

s

5.1

Def

inin

g Si

mila

rity

G–S

RT.2

U5-

1

5.2

App

lyin

g Si

mila

rity

Usi

ng th

e A

ngle

-Ang

le (A

A) C

rite

rion

G–S

RT.3

U5-

36

5.3

Prov

ing

Tria

ngle

Sim

ilari

ty U

sing

Sid

e-A

ngle

-Sid

e (S

AS)

and

Si

de-S

ide-

Side

(SSS

) Sim

ilari

tyG

–SRT

.4U

5-61

5.4

Wor

king

with

Rat

io S

egm

ents

G–S

RT.4

U5-

92

5.5

Prov

ing

the

Pyth

agor

ean

Theo

rem

Usi

ng S

imila

rity

G–S

RT.4

U5-

119

5.6

Spec

ial R

ight

Tri

angl

esG

–SRT

.12

U5-

157

5.7

Def

inin

g Tr

igon

omet

ric

Ratio

sG

–SRT

.6U

5-17

6

5.8

Calc

ulat

ing

Sine

, Cos

ine,

and

Tan

gent

G–S

RT.8

•U

5-21

5

5.9

App

lyin

g th

e Py

thag

orea

n Th

eore

mG

–SRT

.8•

U5-

250

5.10

Prob

lem

Sol

ving

with

the

Pyth

agor

ean

Theo

rem

and

Tr

igon

omet

ryG

–SRT

.8•

U5-

270



NORT

H CA

ROLI

NA M

ATH

2 ST

ANDA

RDS

CORR

ELAT

IONS

Uni

t 6: P

roba

bilit

y

Less

onTi

tle

Stan

dard

(s)

Page

s

6.1

Des

crib

ing

Even

tsS–

CP.1

•U

6-1

6.2

Usi

ng S

imul

atio

nS–

IC.2

•U

6-39

6.3

The

Add

ition

Rul

eS–

CP.7

•U

6-75

6.4

Cond

ition

al P

roba

bilit

y an

d th

e G

ener

al M

ultip

licat

ion

Rule

S–CP

.8•

U6-

95

6.5

Intr

oduc

ing

Cond

ition

al P

roba

bilit

yS–

CP.3

•, S

–CP.

5•,

S–CP

.6•

U6-

114

6.6

Usi

ng T

wo-

Way

Fre

quen

cy T

able

sS–

CP.4

•, S

–CP.

5•,

S–CP

.6•

U6-

149

6.7

Und

erst

andi

ng In

depe

nden

t Eve

nts

S–CP

.3b•

U6-

177



Conceptual ActivitiesPROGRAM OVERVIEW

Use these interactive open education and/or Desmos resources to build conceptual understanding of mathematical ideas. (Note: Activity links will be monitored and repaired or replaced as necessary.)

Unit 1

• Desmos. “Lines, Transversals, and Angles.”

http://www.walch.com/ca/01033

In this activity, students explore the relationship of angles formed by a transversal and a system of two lines. In particular, students consider what happens when the two lines are parallel versus when they are not.

• Desmos. “Polygraph: Angle Relationships.”


This activity is designed to spark vocabulary-rich conversations about angle relationships. Key vocabulary that may appear in student questions includes: parallel, transversal, adjacent, opposite, alternate interior, corresponding, alternate exterior, vertical, and right.

• Desmos. “Polygraph: Figure It Out.”


This activity is intended as an introduction to geometric notation and vocabulary. Depending on prior knowledge, students could use the following to distinguish figures: points, lines, rays, segments, parallel, perpendicular, angles, congruence, midpoints, bisectors, betweenness, collinearity, and more.

Unit 2

• Desmos. “Card Sort: Parabolas.”


In this activity, students will find the shape of a parabola by using its form to reveal its characteristics. The activity begins with a review of both the characteristics and forms of a parabola. Later, students will determine characteristics of the graph of a parabola given in standard form, vertex form, or intercept form.

• Desmos. “Free-Range Functions.”


This activity challenges students to strengthen their ideas about the range of quadratic functions.


PROGRAM OVERVIEWConceptual Activities

• Desmos. “Marbleslides: Parabolas.”


In this activity, students will rapidly become fluent in transforming, translating, and restricting the domains of parabolas by manipulating their symbolic form to cause marbles to slide down the parabola through stars.

• Desmos. “Polygraph: Parabolas.”


In this paired game, one student picks a single parabola from a collection and the other student asks yes/no questions in an attempt to guess which parabola was chosen. The game provides students with a reason for noticing important features of parabolas, which in turn provides a need for words to name the features. “Does your graph cross the x-axis twice?” is a common question during play that leads to the naming roots, for example.

• Desmos. “Will It Hit the Hoop?”


In this activity, students predict whether various basketball shots will go through the hoop, and then model these shots with parabolas to check their predictions.

Unit 4• Desmos. “Polygraph: Square Root Functions.”


This activity is designed to spark vocabulary-rich conversations about transforming square root functions. Key vocabulary terms that may appear in student questions include translation, reflection, intercept, and quadrant.

Unit 5• Mathematics Assessment Resource Service, University of Nottingham. “Discovering the

Pythagorean Theorem.”


This sample lesson begins with students working individually to find the area of a tilted square. The lesson then moves to a class discussion that explores different methods for solving similar problems. Suggested time needed ranges from 80 to 120 minutes.

Unit 6• Mathematics Assessment Resource Service, University of Nottingham. “Representing

Probabilities: Medical Testing.”


This sample lesson begins with students working individually to determine different probabilities in a medical testing worksheet. The lesson then moves to a class discussion that explores different methods for solving similar problems. Suggested time needed ranges from 110 to 150 minutes.



Station Activities GuidePROGRAM OVERVIEW

IntroductionEach unit includes a collection of station-based activities to provide students with opportunities to practice and apply the mathematical skills and concepts they are learning. You may use these activities in addition to the instructional lessons, or, especially if the pre-test or other formative assessment results suggest it, instead of direct instruction in areas where students have the basic concepts but need practice. The debriefing discussions after each set of activities provide an important opportunity to help students reflect on their experiences and synthesize their thinking. Debriefing also provides an additional opportunity for ongoing, informal assessment to guide instructional planning.

Implementation Guide The following guidelines will help you prepare for and use the activity sets in this section.

Setting Up the Stations

Each activity set consists of four or five stations. Set up each station at a desk, or at several desks pushed together, with enough chairs for a small group of students. Place a card with the number of the station on the desk. Each station should also contain the materials specified in the teacher’s notes, and a stack of student activity sheets (one copy per student). Place the required materials (as listed) at each station.

When a group of students arrives at a station, each student should take one of the activity sheets to record the group’s work. Although students should work together to develop one set of answers for the entire group, each student should record the answers on his or her own activity sheet. This helps keep students engaged in the activity and gives each student a record of the activity for future reference.

Forming Groups of Students

All activity sets consist of four or five stations. You might divide the class into four or five groups by having students count off from 1 to 4 or 5. If you have a large class and want to have students working in small groups, you might set up two identical sets of stations, labeled A and B. In this way, the class can be divided into eight groups, with each group of students rotating through the “A” stations or “B” stations.


PROGRAM OVERVIEWStation Activities Guide

Assigning Roles to Students

Students often work most productively in groups when each student has an assigned role. You may want to assign roles to students when they are assigned to groups and change the roles occasionally. Some possible roles are as follows:

• Reader—reads the steps of the activity aloud

• Facilitator—makes sure that each student in the group has a chance to speak and pose questions; also makes sure that each student agrees on each answer before it is written down

• Materials Manager—handles the materials at the station and makes sure the materials are put back in place at the end of the activity

• Timekeeper—tracks the group’s progress to ensure that the activity is completed in the allotted time

• Spokesperson—speaks for the group during the debriefing session after the activities

Timing the Activities

The activities in this section are designed to take approximately 10 minutes per station. Therefore, you might plan on having groups change stations every 10 minutes, with a two-minute interval for moving from one station to the next. It is helpful to give students a “5-minute warning” before it is time to change stations.

Since each activity set consists of four or five stations, the above time frame means that it will take about 50 to 60 minutes for groups to work through all stations.

Guidelines for Students

Before starting the first activity set, you may want to review the following “ground rules” with students. You might also post the rules in the classroom.

• All students in a group should agree on each answer before it is written down. If there is a disagreement within the group, discuss it with one another.

• You can ask your teacher a question only if everyone in the group has the same question.

• If you finish early, work together to write problems of your own that are similar to the ones on the activity sheet.

• Leave the station exactly as you found it. All materials should be in the same place and in the same condition as when you arrived.



PROGRAM OVERVIEWStation Activities Guide

Debriefing the Activities

After each group has rotated through every station, bring students together for a brief class discussion. At this time, you might have the groups’ spokespersons pose any questions they had about the activities. Before responding, ask if students in other groups encountered the same difficulty or if they have a response to the question. The class discussion is also a good time to reinforce the essential ideas of the activities. The questions that are provided in the teacher’s notes for each activity set can serve as a guide to initiating this type of discussion.

You may want to collect the student activity sheets before beginning the class discussion. However, it can be beneficial to collect the sheets afterward so that students can refer to them during the discussion. This also gives students a chance to revisit and refine their work based on the debriefing session. If you run out of time to hold class discussions, you might want to have students journal about their experiences and follow up with a class discussion the next day.


Digital Enhancements GuidePROGRAM OVERVIEW

IntroductionWith this program, you have access to the following digital components, described here with guidelines and suggestions for implementation.

Digital Instruction PowerPoints (Presentations)

These optional versions of the Warm-Ups, Warm-Up Debriefs, Introductions, Key Concepts, and Guided Practices for each lesson run on PowerPoint. (Please note: Computers may render PowerPoint images differently. For best viewing and display, use a PowerPoint Viewer and adjust your settings to optimize images and text.)

Each PowerPoint begins with the lesson’s Warm-Up and is followed by the Warm-Up Debrief, which reveals the answers to the Warm-Up questions.

In the notes section of the last Warm-Up slide, you will find the “Connections to the Lesson,” which describes concepts students will glean or skills they will need in the upcoming lesson. The “Connections” help transition from the Warm-Up to instruction.

GeoGebra Applets (Interactive Practice Problems)

One or two interactive GeoGebra applets are provided for most lessons. The applets model the mathematics in the Guided Practice examples for these lessons. Links to these applets are also embedded within the Instructional PowerPoints. With an Internet connection, simply click on the “Play” button slide that follows selected examples.

Once you’ve accessed the GeoGebra applet, please adjust your view to maximize the image. Each applet illustrates the specific problem addressed in the Guided Practice example. The applets allow you to walk through the solution by visually demonstrating the steps, such as defining points and drawing lines. Variable components of the applets (usually fill-in boxes or sliders) allow you to substitute different values in order to explore the mathematics. For example, “What happens to the line when we increase the amount of time?” or “What if we cut the number of students in half?” This experimentation and discussion supports development of conceptual understanding.

GeoGebra for PC/MAC

GeoGebra is not required for using the applets, but can be downloaded for free for further exploration at the following link:

http://www.geogebra.org/cms/en/download

GeoGebra Applet Troubleshooting

If you are experiencing any difficulty in using the applets in your browser, please visit the following link for our troubleshooting document.

http://www.walch.com/applethelp



PROGRAM OVERVIEWDigital Enhancements Guide

Curriculum Engine Item Bank

Walch’s Curriculum Engine comes loaded with thousands of curated learning objects that can be used to build formative and summative assessments as well as practice worksheets. District leaders and teachers can search for items by standard and create assessments or worksheets in minutes using the three-step assessment builder.

For more information about the Curriculum Engine Item Bank, or for additional support, please contact Customer Service at (800) 341-6094 or [email protected].


Standards for Mathematical Practice Implementation Guide

PROGRAM OVERVIEW

IntroductionThe eight Standards for Mathematical Practice describe features of lesson design, teaching pedagogy, and student actions that will lead to a true conceptual understanding of the mathematics standards. Walch’s lessons, practice problems, and Problem-Based Tasks lend themselves to teaching through this framework. When the Walch resources are combined with high-level questioning and engaging teacher decisions in the classroom, it will lead to high-level math instruction and student achievement.

Here is a brief description of the SMPs and how they can be applied in the classroom:

SMP 1: Make sense of problems and persevere in solving them.

Students will read, interpret, and understand complicated mathematical and real-world problems, and they will be willing to try multiple methods with the ultimate goal of determining the correct answer. Strategies such as annotation and student discourse can lead to improvement on this standard. Presenting students with higher-level problems is essential to ensuring students achieve maximum understanding. Teacher prompts that can enhance this standard include:

• What is the problem asking you to solve?

• What are some (other) strategies you could use to solve this problem?

• Compare your answer with a classmate’s answer. Who is correct? Why?

SMP 2: Reason abstractly and quantitatively.

Mathematical reasoning with numbers and variables is essential to understanding the connections among the standards. Students must be able to discover and formalize general rules using numbers and variables, and apply them to determine numerical quantities in other situations. Teacher prompts that can enhance this standard include:

• Substitute realistic numbers into the situation.

• What operation/strategy would you use?

• Will your strategy work for any number?

• For which categories of numbers (negative integers, all real numbers, etc.) will your strategy work?



PROGRAM OVERVIEWStandards for Mathematical Practice Implementation Guide

SMP 3: Construct viable arguments and critique the reasoning of others.

Many students are most concerned with the “what” aspects of mathematics, i.e. “what” do we do or “what” is the answer. However, math educators must develop the “why” of mathematics. Students must learn to question algorithms, challenge answers, and justify their reasoning in order to truly understand the concepts behind their answers. Teacher prompts that can enhance this standard include:

• How did you determine your answer?

• Why did you choose that strategy?

• Defend your answer based on a real-world situation.

SMP 4: Model with mathematics.

An important goal of mathematics instruction is for students to be able to apply mathematics to the world around them. Students should be able to link a real problem to a mathematical concept, identify quantities that are modeled well with mathematics, and use mathematics to find a solution. Emphasizing this standard will help students represent and interpret information using physical, visual, and abstract models. Encourage students to use any or all of their learning experiences to gain a deep and flexible understanding of mathematics. Teacher prompts that can enhance this standard include:

• Can you represent this situation with a visual model?

• How will it help you solve the problem?

• What information is needed to solve this problem?

• Is there another way to solve this problem?

• While working to solve this problem, what do you notice/wonder?

SMP 5: Use appropriate tools strategically.

There are many available tools suitable for mathematics, such as calculators, manipulatives, formulas, rulers, computers, and developed mathematical strategies. Choosing and using the correct tool to work through a problem is an important skill for mathematicians. Teacher prompts that can enhance this standard include:

• Can you graph this equation in the calculator to see a relationship?

• What formula or strategy might help you determine the answer to this question?

• How can you represent the situation using handheld tools (rulers, protractors, etc.) to determine an answer?


PROGRAM OVERVIEWStandards for Mathematical Practice Implementation Guide

SMP 6: Attend to precision.

When using mathematics to solve problems, an answer can be considered correct only if it is sufficiently precise and accurate for the situation to which it pertains. When applying mathematics, it is vital to clearly define the question, the reasoning, the answer, and the explanation. Vocabulary, units, numerical responses, and pictures must be represented precisely in questions and answers to ensure that the mathematical solutions represent the true answer to a question. Teacher prompts that can enhance this standard include:

• What does your answer represent in a real-world context?

• Is your answer reasonable based on your initial estimate?

• What units of measure help describe your numerical answer?

SMP 7: Look for and make use of structure.

Structure, whether geometric, algebraic, statistical, or numerical, is an important aspect of mathematical reasoning that students often overlook. Teachers often explicitly refer to geometric and other visual structures as explanations of mathematical concepts, but algebraic and numerical structures can often be just as important in analyzing and interpreting mathematical situations. These structures yield clues as to the meaning of expressions, equations, graphs, and other representations. As students interpret these structures, they will gain a greater understanding of the mathematical concepts. Teacher prompts that can enhance this standard include:

• What do the characteristics of the graph tell us about the situation?

• What do each of the variables and numbers in the equation/formula represent?

• How are these situations the same and different based on their representations?

SMP 8: Look for and express regularity in repeated reasoning.

Just as patterns appear in real life, patterns appear throughout the subject of mathematics. Recognizing and applying these patterns, and applying the reasoning contained within, is one of the most important skills teachers can instill in their students. Rather than teaching isolated algorithms to determine answers, have students discover relationships, create their own algorithms, and apply the reasoning to other situations. These skills can be applied throughout their education and will enrich their lives after high school. Teacher prompts that can enhance this standard include:

• What relationship do you notice in the graph/table/numbers?

• Why did you choose to use this process to solve this word problem/equation?

• How can you apply this process in other situations?



Instructional StrategiesPROGRAM OVERVIEW

Ensuring Access for All StudentsIntroduction

The increased focus on literacy in math instruction can help some students navigate mathematical contexts, but for struggling readers, it can further complicate calculations. English language learners struggle to master difficult mathematical concepts while simultaneously processing a new language. Students with learning and behavioral disabilities struggle with the math concepts in their own contexts. This is where teachers and the strategies they select for their classrooms become essential.

The strategies presented here can help all students succeed in math, literacy, school, and, ultimately, in life. These instructional strategies provide teachers with a wide range of instructional support to aid English as a Second Language (ESL) students, students with disabilities (SWD), and struggling readers. These strategies provide support for the Mathematics Standards and the Standards of Mathematical Practice (SMP), English Language Development (ELD) Standards, English Language Arts Standards, and WIDA English Language Development Standards.

Within each lesson throughout this course, you will find suggested instructional strategies. These instructional strategies are research-based strategies and best practices that work well for all students.

The instructional strategies detailed here fall into four main categories: Literacy, Mathematical Discourse, Annotation, and Graphic Organizers. These strategies provide teachers with research-based strategies to address the needs of all students.

• Close Reading• Text to Speech• Concept-Picture- Word Wall• Novel Ideas

• Reverse Annotation

• CUBES Protocol

• Frayer Model

• Table of Values

• Sentence Starters

• Small Group DiscussionLiteracy

Strategies

MathematicalDiscourseStrategies

AnnotationStrategies

GraphicOrganizerStrategies

Source

• WIDA: https://www.wida.us/standards/eld.aspx


PROGRAM OVERVIEWInstructional Strategies: Literacy

Understanding the Language of Mathematics: Literacy Mathematics has its own language consisting of words, notations, formulas, and visuals. In education, the language of mathematics is often regarded solely in the context of word problems and articles. This neglects the vocabulary and other mathematical representations students must be able to interpret. The strategies presented here help students navigate the language of mathematics so that they can understand text and feel confident speaking in and listening to mathematical discussions. For students with disabilities, the stress on repetition and different representations in this approach is essential to their ability to grasp the math concepts. For ESL students, repetition and different representations can strip out some of the English language barriers to understanding the language of mathematics, as well as provide multiple means of accessing the content. Literacy strategies include Close Reading, Text-to-Speech, Concept-Picture-Word Walls, and Novel Ideas.

LiteracyStrategies







Literacy Strategies

??Close Reading with Guiding Questions

What is Close Reading with Guiding Questions?

Close Reading with Guiding Questions is a process that allows students to preview mathematical reading and problems by answering questions related to the text in advance and reviewing their responses during and/or after reading. Multiple reading protocols can be used in conjunction with guiding questions to enhance their effectiveness.

How do you implement Close Reading with Guiding Questions in the classroom?

When utilizing a textbook, task, or article in a math class, literacy struggles are often a strong barrier to entry into the mathematical ideas. Asking students to answer accessible questions before and/or as they read can lead them to the key information.

Prior to implementation, the teacher should determine the most important information students need to obtain from a text, whether it is a math problem to solve, a task to complete, or an informational lesson or article to read. Then, the teacher should come up with some questions to guide students before they read. These questions can:

• assess and relate prior knowledge

• define key vocabulary words

• discuss non-mathematical concepts in the text

The teacher should also prepare some questions to guide students as they read. These questions can:

• point out key concepts within the text

• relate the text and concepts to future learning

• assist students in identifying key facts in the text

• highlight the importance of text features (graphics, headings, etc.) in the text

To ensure the questions are accessible for students and to encourage reflection and debate after reading, many of these questions should be designed as either “True/False” or “Always True/Sometimes True/Never True.” Students can represent their reasoning for their answer in writing, numbers, or graphic/pictorial representations. Students should complete the guiding questions and reading individually, with discussion to follow.

After students complete the reading, they should be given some time to individually evaluate their initial answers. Then, in partners or in groups, they can discuss their answers and come to final conclusions that will help them find the important information initially identified by the teacher. After deciphering the text through close reading, students will be able to complete the given activity.



When would I use Close Reading with Guiding Questions in the classroom?

Close Reading with Guiding Questions can be used for any activity in which literacy could be a barrier to learning or demonstrating mastery of mathematical concepts. The number of questions and length of the discussions can be altered based on the length, importance, and difficulty of the text and concept. As students become more accustomed to mathematical literacy, the text complexity can be increased, but the adherence to close reading strategies must be maintained to ensure students can access the mathematical concepts. The length of time spent on the literacy aspect can be shortened as students become more skilled, but the questioning and discussions must occur to ensure students are properly interpreting the text in the mathematical context.

How can I use Close Reading with Guiding Questions with students needing additional support?

For struggling readers, including ESLs, Close Reading with Guiding Questions can help make an intimidating lesson, word problem, or task much more accessible. Questions focusing more on Tier 2 and Tier 3 vocabulary, text features, and real-world concepts can help struggling readers relate to the text and learn how to decipher the text in context. Discussions around the questions will help students grasp the math concepts.

Allowing struggling readers to explain their answers using words, numbers, or graphics/pictures ensures that they can express their opinion and rationale despite a potential lack of vocabulary. Through these representations and the ensuing discussion, students will begin to learn the necessary vocabulary to be successful.

What other standards does Close Reading with Guiding Questions address?

Standards of Mathematical Practice:

• SSMP.1

• SSMP.6

WIDA English Language Development Standards:

• ELD Standard 3

Language Arts Standards:

• ELA–LITERACY.WHST.9–10.4


• ELA–LITERACY.SL.9–10.4

• ELA–LITERACY.RST.9–10.3






Sources

• Anne Adams, Jerine Pegg, and Melissa Case. “Anticipation Guides: Reading for Mathematics Understanding.”

https://www.nctm.org/Publications/mathematics-teacher/2015/Vol108/Issue7/Anticipation-Guides-Reading-for-Mathematics-Understanding/

• Diane Staehr Fenner and Sydney Snyder. “Creating Text Dependent Questions for ELLs: Examples for 6th to 8th Grade.”

http://www.colorincolorado.org/blog/creating-text-dependent-questions-ells-examples-6th-8th-grade-part-3



Literacy Strategies Text-to-Speech Technology

What is Text-to-Speech Technology?

Text-to-Speech Technology is an adaptive technology that reads text aloud from a text source for students. It is usually accessed through an application or program on a computer, smartphone, or tablet. Some new programs utilize Mathematical Markup Language (MathML) to read mathematical notation in a common, understandable manner for students. Many programs also highlight the words and notation on the screen as the audio plays, which helps students relate the written representation to the words they hear. The use of Text-to-Speech Technology allows students who struggle with literacy to hear the words and notation and access the text in a different way.

How do you implement Text-to-Speech Technology?

A classroom community focused on everyone’s learning and a growth mindset is the first step in implementing Text-to-Speech Technology. One of the main barriers to implementation is encouraging students to use the program. Once they do, they will realize how the audio can help them understand the difficult mathematical texts and interpret the math content within them. After students realize the benefits of Text-to-Speech Technology, it can become part of the regular routine for group and independent work.

The use of headphones can be very important for effective use of Text-to-Speech Technology. Students can use the technology to listen to lessons and texts at their own pace. Extra noise from other students working or other students listening at different paces can confuse students attempting to use Text-to-Speech Technology, and headphones can help mitigate these distractions. Many teachers are nervous about the potential disruption headphones can cause in class. However, well-managed use of headphones can help students successfully utilize the technology to learn.

When would I use Text-to-Speech Technology in the classroom?

Text-to-Speech Technology can be used at any time throughout the year, and if the program speaks in MathML, it can be used with any lesson. Without MathML, effective use could be limited to word problems without unusual notation. For example, if x2 is read as “x-two” instead of “x-squared” or “x to the second power,” that could confuse students more.

During a lesson or small group discussion, Text-to-Speech Technology could detract from students’ ability to listen, question, and process information. However, during warm-ups, independent work, or assessments, Text-to-Speech Technology can help students process the information and access the activity. It can become a routine for students to automatically listen to the question, problem, or directions first, and then attempt the activity.




How can I use Text-to-Speech Technology with students needing additional support?

Text-to-Speech Technology is an important adaptation and accommodation for struggling readers. Students who have read-aloud accommodations sometimes don’t receive them because they are either embarrassed to accept them or because of staffing restrictions. These students can use Text-to-Speech Technology to supplement their math instruction by having text automatically read to them in a manner in which they can process it.

Additionally, for ESL students, hearing the English mathematical language, especially referring to mathematical representations and notation, can help put English words to the ideas they see. Some Text-to-Speech Technology can translate written and mathematical text into other languages, so students can hear the text in their natural language and see the English highlighted on the screen as they hear it. In this way, students are learning English vocabulary as well as learning the mathematical content in a language they can understand.

What other standards does Text-to-Speech Technology address?


• SMP.1

• SMP.6


• ELD Standard 3








Source • Steve Noble. “Using Mathematics eText in the Classroom: What the Research Tells Us.”

http://scholarworks.csun.edu/bitstream/handle/10211.3/133379/JTPD201412-p108-118.pdf;sequence=1



Literacy Strategies more> thanConcept-Picture-Word Wall

What is a Concept-Picture-Word Wall?

A Concept-Picture-Word Wall is a classroom display, often a bulletin board or a set of posters, that exposes students to important vocabulary words they will use in math class.

Posting vocabulary words in class helps reinforce the words students will see in textbooks, videos, websites, and test questions on math concepts. These Tier 3 vocabulary words are often not used in everyday language, and the exposure to the words visually through Concept-Picture-Word Walls can help students connect them to the math content.

How do you implement Concept-Picture-Word Walls in the classroom?

Just seeing the vocabulary on a Concept-Picture-Word Wall by itself will help students; more importantly, referring to the words as the teacher uses them in class helps students connect the visual to the application. A simple gesture to the wall makes a very explicit reference to the word as it is used and allows students to connect the unfamiliar word to its meaning in context. Additionally, students can be taught to refer to the wall as they use the words in class, and they can be asked to make sure they say at least 3 words from the wall during each class period in small-group discourse or as answers to whole-class questions. The comfort gained from using these Tier 3 words will help students to use appropriate math vocabulary while solving problems and will help students connect concepts more explicitly.

Postings on the Concept-Picture-Word Wall can be arranged strategically to connect concepts, units of study, or groups of words where appropriate. Having three sections of the Concept-Picture-Word Wall—for example, an “In the Future” section, a “Live in the Present” section, and a “Remember the Past” section–—can help students see and remember the vocabulary throughout the entire course. Even without regular use of some words, just seeing the words before a unit can help instill a familiarity with the vocabulary. Leaving the words on the Concept-Picture-Word Wall after a unit is taught can help students connect “old” concepts to the current lesson and ensure that students still have access to the vocabulary.

When would I use Concept-Picture-Word Walls in the classroom?

Concept-Picture-Word Walls can be used for the entire year. The actual words might have to change, or at least be moved to different areas of the Concept-Picture-Word wall. The more exposure students have to the words, the more familiar and comfortable they will become. The constant exposure to the math context is beneficial for students throughout the entire course, especially for words with multiple meanings (bias, tangent, etc.) that could exist as Tier 2 words in everyday conversation but are Tier 3 words in the math classroom.




How can I use Concept-Picture-Word Walls with students needing additional support?

For all students learning mathematics, knowing and using the math vocabulary is often a major barrier. This is a problem especially for ESL students, who are learning the English language along with math content. If teachers try to simplify the words too much for students, it does them a disservice as they seek out information from other teachers, textbooks, and online sources that use the proper vocabulary. Most tests, especially state tests, will expect students to have knowledge of the Tier 3, math-specific vocabulary. The more students see these words, the more familiarity they will have when they apply them.

Concept-Picture-Word Walls can also be written in multiple languages. Especially for students who are on-grade-level in their native language, a multi-lingual Concept-Picture-Word Wall can help students connect the content they already know in another language to the English vocabulary necessary for success on English-language math activities and tests.

This website can help you get started on an English-Spanish Concept-Picture-Word Wall: http://math2.org/math/spanish/eng-spa.htm

What other standards do Concept-Picture-Word Walls address?


• SMP.1

• SMP.6


• ELD Standard 3








Source

• Janis M. Harmon, Karen D. Wood, Wanda B. Hedrick, Jean Vintinner, and Terri Willeford. “Interactive Word Walls: More Than Just Reading the Writing on the Walls.”

http://citeseerx.ist.psu.edu/cdownload;jsessionid=A250AF8A870B13B40B2934 BA515FEC9?doi=10.1.1.690.6740&rep=rep1&type=pdf



Literacy Strategies !!!Novel Ideas

What is Novel Ideas?

Novel Ideas is a classroom activity that explores students’ understanding of important Tier 2 vocabulary words they will use in math class. Instead of asking students to look up vocabulary words in the dictionary, Novel Ideas allows students to have conversations with their peers about vocabulary words in class. This reinforces the mathematical vocabulary students will see in textbooks, videos, websites, and test questions. These Tier 2 vocabulary words are often used in everyday language, but have specific meaning in mathematics. Exposure to the words through Novel Ideas can help students connect them to the math content.

How do you implement Novel Ideas in the classroom?

While building a rich representation of math content words and connecting the words to other words and concepts has inherent merit, it is more important to consider that pre-teaching the words before they are used in class helps students connect to the application. The understanding gained from discussing these Tier 2 words will help students apply them in a mathematical context to solve problems and connect concepts.

Here is a step-by-step process for implementing Novel Ideas:

1. Students separate into groups of four.

2. Students copy the teacher generated prompt/sentence starters and number their papers 1–8.

3. One student offers an idea, another echoes it, and all write it down.

4. After three minutes, students draw a line under the last item in the list.

5. All students stand, and the teacher calls one student from a group to read the group’s list.

6. The student starts by reading the prompt/sentence starters, “We think a ______ called ______ may be about … ,” and then adds whatever ideas the team has agreed on.

7. The rest of the class must pay attention because after the first group has presented all their ideas, the teacher asks them to sit down and calls on a student from another team to add that team’s “novel ideas only.” Ideas that have already been presented cannot be repeated.

8. As teams complete their turns and sit down, each seated student should record novel ideas from other groups below the line that marks the end of his or her team’s ideas.




When would I use Novel Ideas in the classroom?

Novel Ideas can be used for the entire year. The more students are exposed to mathematical vocabulary, the more familiar and comfortable they become, leading to increased usage of these math terms in their conversation and writing. Using math vocabulary in context is beneficial for students throughout the entire course, especially for words with multiple meanings (bias, tangent, etc.) that could exist as Tier 2 words in everyday conversation but are Tier 3 words in the math classroom.

How can I use Novel Ideas with students needing additional support?

Most tests, especially state tests, will expect students to have knowledge of the Tier 3, math-specific vocabulary. The more students use these words in conversation, the more familiarity they will have when they apply them. Understanding Tier 2 words also helps students avoid misconceptions in mathematics. Twice a week before the start of a lesson, allow students to use sentence starters in small groups that include all students. Prepare the sentence starter “When I hear the word ______, I think about ______” to share out with whole class. This will allow students who know the vocabulary words to share their knowledge, and will allow other students to hear the meaning of the vocabulary words. This strategy is particularly helpful for ESL students.

What other standards does Novel Ideas address?


• SMP.1

• SMP.6


• ELD Standard 3








Sources

• Colorín Colorado. “Selecting Vocabulary Words to Teach English Language Learners.”

http://www.colorincolorado.org/article/selecting-vocabulary-words-teach-english-language-learners

• Elsa Billings and Peggy Mueller, WestEd. “Quality Student Interactions: Why Are They Crucial to Language Learning and How Can We Support Them?”

http://www.nysed.gov/common/nysed/files/programs/bilingual-ed/quality_student_interactions-2.pdf



Novel Ideas Sentence StartersSlope

• When I hear the word climb, I think about …

• When I hear the word steep, I think about …

Volume

• When I hear the word filling, I think about …

Equations

• When I hear the word balance, I think about …

• When I hear the word equal, I think about …

Graphing

• When I hear the word grid, I think about …

• When I hear the word graph, I think about …

Scatter Plots

• When I hear the word scattered, I think about …



PROGRAM OVERVIEWInstructional Strategies: Annotation

Understanding Mathematical Content: Annotation

≅Σ± ÷≤ ∞θ

∅f (x)

2πr2Understanding mathematical content is an extremely important skill, both in the math classroom and in life. When students read word problems, articles, charts, graphs, equations, tables, or other forms of mathematical text, they must be able to decode and extract meaning from the text. Annotation can help. The strategies presented here help students identify and focus on key characteristics and facts from various forms of text while ignoring the non-essential information. For students with disabilities, many of whom struggle with the distractions inherent in many high-school level texts, making notes and drawing pictures to explain a problem can help them focus. ESL students will be pointed to certain Tier 3 vocabulary words and determine which Tier 2 vocabulary words they must learn to be proficient in math class and in the English language. Annotation strategies include Reverse Annotation and CUBES protocol.

LiteracyStrategies






Annotation Strategies Reverse Annotation Protocol

What is Reverse Annotation?

Reverse Annotation is a strategy that asks students to identify and write down key information from math problems. This is especially helpful for problems given on a computer or tablet, where students can’t annotate directly on the problem. A template is given at the end of this section.

How do you implement Reverse Annotation in the classroom?

Many annotation strategies ask students to write, underline, or mark directly on the text of a problem. While those forms of annotation are also beneficial, they are not always possible with technology. Whether the problem is given on paper or using technology, having students write the answers to these questions will ensure that they are thinking strategically and specifically about the strategies and information needed to solve the problem.

The three questions at the top of the Reverse Annotation template are the key to understanding mathematical problems. For every problem given in class, ask students:

1. What is the problem asking us to solve?

2. What key words tell us the mathematical steps we need to perform?

3. What information in the problem can help us figure it out?

After answering the initial questions, students should make a guess, or estimate, of what they think the answer will be. This helps grow their number sense, and provides an initial, reasonable solution to guide their work. Students can then use the strategies they selected to solve the problem and evaluate their solution using the questions at the bottom of the template.

When students first begin to use Reverse Annotation, the teacher should walk them through the steps individually to ensure they can accurately identify the question, key words, and important information. Teachers can also lead students through the estimation process, making a game out of which student has the closest estimate.

Work through each step individually for several “easy” problems first, so that difficult math doesn’t interfere with the process. Increase the problem difficulty incrementally as students begin to master the process. This may seem like a long process at first, but the ultimate result is worth the time investment.

When would I use Reverse Annotation in the classroom?

Reverse Annotation can be used to solve any math problem, and is especially helpful for word problems. When Reverse Annotation is initially implemented, the steps should be discussed in detail. As students become accustomed to Reverse Annotation and begin thinking about problems in this manner automatically, the individual steps become less important and can be scaffolded out to




improve efficiency. Students should reach the point where they immediately ask themselves the three initial questions when they first see a problem. However, the teacher should ensure that students are truly evaluating all the key information before routine discussions of the individual steps are removed.

How can I use Reverse Annotation with students needing additional support?

Annotation strategies can help students identify key information, even when certain vocabulary words are not known. As teachers introduce the content-specific Tier 3 vocabulary to their classes, annotation strategies such as reverse annotation can help students use these words to apply appropriate strategies while problem solving. Answering the three initial questions can help students organize the key facts and vocabulary, and the identification of key information can simplify the problem. This strategy is especially beneficial for ESL students.

Using reverse annotation with graphic organizers benefits ESL students by removing a lot of the confusing wording and allowing them to focus on the important pieces of a problem. When using Reverse Annotation, all students, including ESL students, will begin to think about problem solving in a way that encourages them to use the appropriate information to find a solution.

What other standards does the Reverse Annotation Protocol address?


• SMP.1

• SMP.2

• SMP.5

• SMP.6


• ELD Standard 3







Source

• Alliance for Excellent Education. “Six Key Strategies for Teachers of English Language Learners.”

https://uteach.utexas.edu/sites/default/files/files/SixKeyStrategiesELL.pdf



Reverse Annotation Template

Name: _________________________ Problem/Assignment: _________________________

Analyze the Problem

What is the problem asking us to solve?

What key words will tell us the mathematical steps we need to perform?

What information in the problem can help us figure it out?

Initial estimate of solution:

Work Space

Remember to box in your solution!




Name: _________________________ Problem/Assignment: _________________________

Check It Over

How close was your estimate?

Does your answer make sense? Is it reasonable? How do you know?

Did you perform the calculations correctly?

What does your answer mean in context?



Annotation Strategies CUBES Protocol

What is the annotation strategy CUBES?

CUBES is an annotation strategy in which students use different written designs to highlight the key aspects of word problems. It can help them choose the correct mathematical strategy to solve the problem accurately.

How do you implement CUBES in the classroom?

The steps for CUBES are:

1. C: Circle all the key numbers.

2. U: Underline the question.

3. B: Box in the key words that will determine the operation(s) necessary and write the mathematical symbol for the operation(s).

4. E: Evaluate the information given to determine the strategy needed. Eliminate any unnecessary information.

5. S: Solve the problem, show your work, and check your answer.

As students learn to use CUBES, walk them through the steps individually to ensure they can accurately identify the key numbers, question, key words, unnecessary information, and strategy. Work through each step individually for several “easy” problems first, so that difficult math doesn’t interfere with the process. Increase the problem difficulty incrementally as students begin to master the process. This may seem like a long process at first, but the ultimate result is worth the time investment.

A graphic organizer can help students master the process, especially when problems are given on a computer or tablet where students can’t always annotate directly on the problem. Students can write down the key numbers and circle them, write down the question and underline it, and so on. This will encourage students to truly think about the different pieces of the problem they are identifying, and how these pieces will guide the strategy and affect the solution.

When would I use CUBES in the classroom?

CUBES can be used to solve any math problem, and is especially helpful for word problems. When CUBES is initially implemented, the steps should be discussed in detail. As students become accustomed to using CUBES and begin thinking about problems in this manner automatically, the individual steps become less important and can be scaffolded out to improve efficiency. However, the teacher should ensure that students are truly evaluating all the key information before routine discussions of the individual steps are removed.




How can I use CUBES with students needing additional support?

Design features can help students identify key words and features, even when certain vocabulary words are not known. As teachers introduce the content-specific Tier 3 vocabulary to their classes, annotation strategies such as CUBES can help students use these words to apply appropriate strategies while problem solving. Using circles, underlines, and boxes can help students organize the key facts and vocabulary, and the elimination of unnecessary information can simplify the problem. This strategy is especially beneficial for ESL students.

Combining CUBES with graphic organizers also benefits ESL students by removing a lot of the confusing wording and allowing them to focus on the important facts of a problem. When using CUBES with a graphic organizer, all students, including ESL students, will begin to think about problem solving in a way that helps encourage them to use the appropriate information to find a solution.

What other standards does the CUBES Protocol address?


• SMP.1

• SMP.2

• SMP.5

• SMP.6


• ELD Standard 3







Source

• Margaret Tibbett. “Comparing the effectiveness of two verbal problem solving strategies: Solve It! and CUBES.”

https://rdw.rowan.edu/cgi/viewcontent.cgi?article=2633&context=etd


PROGRAM OVERVIEWInstructional Strategies: Graphic Organizers

Organizing Mathematical Content: Graphic Organizers Organizing mathematical content is a crucial skill for problem solving, exploring other possible methods for finding solutions, and managing math content. All students need strategies for organizing content to build conceptual understanding. For students with disabilities, visual representations and graphic organizers can help them clarify their thoughts and focus on the math. ESL students also benefit from visual representations and graphic organizers. Organizing mathematical knowledge with visuals can help ESL students navigate math content while learning the language. Graphic organizers include Frayer Models and Tables of Values.

LiteracyStrategies







Graphic Organizers Frayer Models

What is a Frayer Model?

A Frayer Model is a graphic organizer that can help students understand new vocabulary words and concepts by exploring their characteristics. A Frayer model lists the definition of a word or concept, describes some key facts, and gives examples and non-examples. Examples and non-examples can come from a mathematical or real-world context.

How do you implement Frayer Models in the classroom?

Students can learn to create Frayer Models the first week of school, and the process can be used throughout the year each time students experience a new word or concept.

While it is important for teachers to give students precise mathematical definitions with appropriate content vocabulary, it is maybe more important for students to understand the application of mathematical words and concepts in their own context. As students learn new information, small group discussions and think-pair-share activities are great ways for students to formulate their own definitions, review the characteristics and facts they have learned, and discuss examples and non-examples.

Discussions of the examples and non-examples can help lead to the mathematical definition. For example, if students use a Frayer Model to define a quadratic function, they would notice that all examples have a highest exponent of 2, and all non-examples would not have a highest exponent of 2. All examples would have parabolic graphs, and all non-examples would have other graphs. Through these comparisons, students will understand the definition of quadratics using different representations, and they will be able to apply it in different contexts.

When would I use Frayer Models in the classroom?

Frayer Models can be used at different points during instruction. They are appropriate as introductions to new concepts, summaries to ensure understanding of new concepts, or as note-organizers throughout the lesson for students to fill in as they learn new concepts. At first, students might need help figuring out how to list and differentiate between the definition, facts and characteristics, examples, and non-examples. As students adapt to the process, they will be able to categorize information on their own or in small groups. As they compare newer Frayer Models to previous models, they will also be able to see how concepts build upon each other.



How can I use Frayer Models with students needing additional support?

Frayer Models can be a point of reference for students as they progress throughout the year. As students determine their own definitions for math-specific words and concepts, and use the examples and non-examples to determine the key facts, they will be able to put them in their own context and apply them to solve complicated problems. As math concepts build upon each other both within a unit and throughout the year, the use of Frayer Models to remind students of their initial definitions of words or concepts can help solidify their understanding. Using Frayer Models as part of a Word Wall or Concept Wall, or having a consistent notebook process to reference past Frayer models, can help consistently reinforce learning.

What other standards do Frayer Models address?


• SMP.1

• SMP.2

• SMP.6


• ELD Standard 3








Source

• Deborah K. Reed. “Building Vocabulary and Conceptual Knowledge Using the Frayer Model.”

https://iris.peabody.vanderbilt.edu/module/sec-rdng/cresource/q2/p07/




Frayer ModelDefinition Characteristics

WORD

Examples from Life Non-Examples



Graphic OrganizersTables of Values

What is a Table of Values?

A Table of Values is an organized way to list numbers that represent different categories of values. These values can be represented as ordered pairs, graphs, word problems, or lists. Tables can help students see and compare values in a different way.

How do you implement Tables of Values in the classroom?

Tables can be used throughout the year to support various mathematical standards. Some standards mention tables specifically, and in others, tables can be an effective support to help students organize and understand the meaning and application of values.

Tables can be set up with numerical values in rows or columns. The key to understanding the values lies in the headings. The headings must be specific enough to show students the meaning and/or application of the numerical values, but not so wordy that they interfere with the clarity of the numbers in the table. For example:

x (year)y (population in millions)

1960 219

1970 230

1980 258

1990 312

2000 342

Mean (statistical average) 50 45

Median (middle value) 52 43

Quartile 1 (median of the lower 50%) 40 38

Quartile 3 (median of the upper 50%) 72 80

Range (difference of max and min values) 80 61

Interquartile Range (difference of quartiles) 32 42

Standard Deviation (measure of spread of data) 7.24 10.23




When would I use Tables of Values in the classroom?

Various mathematical topics can be represented by tables. For example:

• An (x, y) table of values to represent coordinates on a graph or independent and dependent variables for a given context

• A table to represent coefficients and/or constants in an equation

• A table to show different statistical measures when comparing sets of data

• A table to compare output values for the same input given different functions

Each time numbers or values are being listed, compared, or graphed, a table can help students differentiate between the values. Tables are easy to create, and students can be encouraged to create them as another representation to clarify and compare numbers for nearly any topic.

How can I use Tables of Values with students needing additional support?

Tables of Values can help students focus on numerical values and their meaning in context without distraction. They clarify what each number represents, what numbers can be compared, and what ordered pairs can be graphed to give a visual representation. Additionally, headings can be used to either highlight the relevant facts from a context or to describe mathematical vocabulary.

In general, graphic organizers benefit students by removing much of the confusing wording and focusing on the important facts and numbers of a problem.

What other standards do Tables of Values address?


• SMP.1

• SMP.2

• SMP.6


• ELD Standard 3








Source • Alliance for Excellent Education. “Six Key Strategies for Teachers of English Language Learners.”

https://uteach.utexas.edu/sites/default/files/files/SixKeyStrategiesELL.pdf


PROGRAM OVERVIEWInstructional Strategies: Mathematical Discourse

Communicating Mathematical Content: Mathematical Discourse Reading, writing, speaking, and listening are all important ways to learn and express information, but the last two ways are often slighted in the math classroom. The mathematical discourse strategies presented here promote speaking and listening in a math-focused literacy context. Working these strategies into the daily routine of a classroom can help students become comfortable speaking and listening in a mathematical context, which will help them become comfortable with the mathematical content. Routines and structures are essential to support students with disabilities, as they often benefit from following a routine. This can lead to developing capability in their mathematical skills. These strategies also remove the barrier to entry for many ESL students, as structure and routine can help them focus on the math content rather than English language deficiencies. Mathematical Discourse strategies include Sentence Starters and Small Group Discussion.

LiteracyStrategies







Mathematical Discourse StrategiesSentence Starters

What is a Sentence Starter?

A Sentence Starter is a common phrase or mathematical sentence frame that can help students begin and sustain academic conversations around mathematical content. It helps guide students through the discussion and bring out pertinent ideas that can lead to greater understanding.

How do you implement Sentence Starters in the classroom?

Many people view math class as a place to calculate solutions to math problems. However, to ensure the conceptual understanding and proper application of a math concept, students need to be able to explain the concepts and reasoning behind a solution to a problem. As many students are not accustomed to having academic conversations about math, sentence starters can help begin and continue these conversations in a productive manner.

There are two main types of sentence starters for mathematical discussions: discourse starters and math starters. For example, a poster with these or other sentence starters can be displayed from the beginning of the year, and the expectation can be set that any answer to a question or comment in a discussion should be framed using one of these starters. As students become accustomed to framing mathematical conversations in this way, they can expand on the given sentence starters and create some of their own. They will begin to realize how these statements ensure that their conversations revolve around math, enhance understanding of the concept, and force them not only to state, but also to explain their thinking. They will gain confidence from the ability to engage, as the first step has already been taken for them.

When would I use Sentence Starters in the classroom?

Sentence Starters can be used throughout the entire school year with any concept. However, they are most important to use at the beginning of the school year to build a mathematical community in the classroom centered on a comfort with mathematical discourse. Especially at the beginning of the year, students should be encouraged to use these sentence starters for every math statement. Appropriate settings include during small group discussion, while responding to whole class questions, and when writing explanations for problem solutions.

Modifications can be introduced so that students must use certain mathematical vocabulary within the sentences, or must use certain sentence starters at different points in conversations or for different conversation types and situations. However the starters are implemented, it is important for students to realize that these are intended to enhance and focus their conversations, not limit them.



How can I use Sentence Starters with students needing additional support?

Often, students are reluctant to talk about math concepts because they either lack confidence in their knowledge, are afraid to be “wrong,” or don’t know how to start or continue the conversation. Sentence starters can help students overcome this reluctance. The non-threatening, easy-to-interpret sentence starters remove the barrier to entry for students who don’t know how to engage, and the respectful, mathematical focus promoted by sentence starters can help build confidence and provide a structure so that students will not fear being wrong.

For ESL students specifically, sentence starters can provide the English language support to help students engage with and discuss the math. The support of sentence structure removes language barriers to entry for students who don’t fully understand English sentence structure.

Discourse Starters Math Starters

I agree/disagree with … because …

I understand/don’t understand …

First/Next/Finally I … because …

I noticed that …

I wonder …

My answer was … because …

The next step is … because …

I used (insert formula/equation/concept) because …

My answer is right/reasonable because …

What other standards do Sentence Starters address?

WIDA English Language Development Standards

• ELD Standard 3


• SMP.1

• SMP.3

• SMP.6








Source

• AVID. “Sentence Starters.”

https://sweetwaterschools.instructure.com/files/29100523/download?download_frd=1&verifier=CBvje9CPNKUe6IkN4TPBJDuXmZY3464aTTK1Fk2r math sentence starters research




Mathematical Discourse StrategiesSmall Group Discussion

What is Small Group Discussion?

Small Group Discussion is a structured way for students to verbalize their mathematical thinking in a comfortable setting to solve a problem, build conceptual understanding, or summarize a concept.

How do you implement Small Group Discussion?

Small Group Discussion in math class depends on a trusting relationship between the teacher and the students. From there, students can build trusting relationships among themselves. Once this trust has been built, students will feel free to explore mathematical topics in groups, take risks, and engage in a productive struggle toward understanding or a solution.

Once these relationships have been established, certain structures should be established for Small Group Discussion to be effective. Discussion norms can be set by the class to ensure discussions are respectful and productive, and discussions should have predetermined time limits. The group composition is also important and should be based on instructional measures. For different activities, homogeneous groups, heterogeneous groups, or groups based on specific data by standard could be appropriate. Students should always be aware that the groups were chosen to maximize their learning.

Another structure that can be effective for Small Group Discussion is assigning group roles. These roles can include group leader, note taker, timekeeper, resource manager, culture keeper, or other roles determined to be appropriate for the classroom context. During the discussion, assigning each student a letter within the group (A, B, C, D, etc.) can help structure the discussion. Different roles can specify certain time limits for talk, which sentence starters to use, or other structured aspects of the discussion.

When implementing a Small Group Discussion, the question or task should inspire students to think in different ways about a concept. Through the structured format of the discussion, students will compare their ideas and arrive at an answer or explanation of the concept. Within the trusting framework of the class and group, students can focus on the common goal of the discussion and develop their thinking around the math concept. These rich discussions will enhance their understanding.

When would I use Small Group Discussion in the classroom?

Small Group Discussion can be used for nearly any topic, and it can be used at a variety of times in the classroom. The questions and tasks may need to change depending on when it is used. Opening activities for lessons can be Small Group Discussions where students explore properties of new math concepts or review/build upon their prior learning. Turn and talks throughout the lesson can be structured as Small Group Discussions if a consistent framework is in place. At the end of class, a Small Group Discussion can be used to come to a common understanding about an essential question from the lesson.



Depending on when the Small Group Discussion is used in class, and what the goal of the discussion is, the discussion reporting may vary. For a warm-up, each group might be asked to share their thinking. For a guided practice, recording answers on chart paper and a gallery walk could be appropriate. For a closing activity, individual written responses to a question could be appropriate.

How can I use Small Group Discussion with students needing additional support?

As discussed in other Mathematical Discourse strategies, struggling students are reluctant to talk about math concepts because they lack confidence in their knowledge and don’t always have the needed vocabulary in their toolbox. Structured discussions with effective grouping can help students through these barriers. After a trusting and respectful classroom environment has been established, struggling students often feel more comfortable sharing their ideas with just a few classmates rather than the whole class. Additionally, adding structure can help students engage by providing the expectation that they participate in the process.

The intentional grouping of students can also help them succeed using Small Group Discussion. At times, heterogeneous groups could be appropriate so that stronger students can help struggling students, and at other times, homogeneous groups could be appropriate so the teacher can work with an entire group of struggling students. ESL students can be grouped with other students with the same dominant language to help remove the language barrier from the conversation.

What other standards does Small Group Discussion address? WIDA English Language Development Standards:

• ELD Standard 3


• SMP.1

• SMP.3

• SMP.6








Source

• Jessie C. Store. “Developing Mathematical Practices: Small Group Discussions.”

https://kb.osu.edu/dspace/bitstream/handle/1811/78055/OJSM_69_Spring2014_12.pdf

© Walch EducationGO-1


Graphic OrganizersPROGRAM OVERVIEW

OverviewGraphic organizers can be a versatile tool in your classroom. Organizers offer an easy, straightforward way to visually present a wide range of material. Research suggests that graphic organizers support learning in the classroom for all levels of learners. Gifted students, students on grade level, and students with learning difficulties all benefit from the use of graphic organizers. They reduce the cognitive demand on students by helping them access information quickly and easily. Using graphic organizers, learners can understand content more clearly and can take concise notes. Ultimately, learners find it easier to retain and apply what they’ve learned.

Graphic organizers help foster higher-level thinking skills. They help students identify main ideas and details in their reading. They make it easier for students to see patterns such as cause and effect, comparing and contrasting, and chronological order. Organizers also help students master critical-thinking skills by asking them to recall, evaluate, synthesize, analyze, and apply what they’ve learned. Research suggests that graphic organizers contribute to better test scores because they help students understand relationships between key ideas, and enable them to be more focused as they study.

Types of Graphic OrganizersThere are four main purposes for using graphic organizers in mathematics and a variety of tools within each category:

Purpose 1: Organizing,

Categorizing, and Classifying

Purpose 2: Problem Solving

Purpose 3: Understanding Mathematical Information

Purpose 4: Communicating

Mathematical Information

Tables

Flowcharts

Webs

Venn Diagrams

Number Lines

Geometric Drawings

Factor Trees

Attribute Tables

Cause and Effect Maps

Coordinate Plane

Probability Trees

Frayer Model

Semantic Map/ Concept Map

Compare-and-Contrast Diagram

Line Graphs

Bar Charts

GO-2© Walch EducationNorth Carolina Math 2 Custom Teacher Resource

PROGRAM OVERVIEWGraphic Organizers

Tables

A table is simply a grid with rows and columns. Tables are useful because information stored in a table is easy to find—much easier than the same information embedded in text.

Usually, a table has a row (horizontal) for each item being listed. The columns (vertical) provide places for details about the listed items—the things they have in common. The places where the rows and columns meet are called cells. In each cell, we write information that fits both the topic of the row (the thing being listed) and the topic of the column (the aspect being examined). To create a table, we make rows and columns to fit the number of items and attributes.

Flowcharts

Flowcharts are graphic organizers that show the steps in a process. Flowcharts can be very simple—just a series of boxes with one step in each box. However, there is also a more formal type of flowchart. These flowcharts use special symbols to show different things, such as starting and stopping points, or points where decisions must be made. These symbols make flowcharts especially useful for showing complicated processes.

Each step in a flowchart is written in a box. The boxes are connected by arrows to show the sequence of steps. The boxes aren’t all rectangular; different shapes are used to indicate different actions. The shapes and symbols are a kind of visual shorthand. Whenever a certain symbol is used, it always has the same meaning.

• Circles and ovals show starting and stopping points. They often contain the words start or stop. The “start” circle or oval has no arrows in and one arrow out. The “stop” circle or oval has one arrow in and no arrows out.

• Arrows show the direction in which the process is moving.

• Diamonds show points where a decision must be made or a question must be answered. The question can usually be answered either “yes” or “no.”

• Rectangles and squares show steps where a process or an operation takes place.

• Parallelograms show input or output, such as writing or printing a result or solution.




Webs

Webs are graphic organizers that help take notes, identify important ideas, and show relationships between and among pieces of information. In a web, the main idea is written in the center circle. Details are recorded in other circles with lines to connect related topics. Circles or lines can be added or deleted as necessary.

Number Lines

In its simplest form, a number line is any line that uses equally spaced marks to show numbers. Number lines are used to visualize equalities and inequalities, positive and negative numbers, and measurements of all kinds. They can “map” math problems, especially ones that involve negative numbers or distances.

Geometric Drawings

A geometric drawing is a representation on paper (or some other surface) of a geometric figure. The geometric drawings we make can never be as perfect as the geometric figures they represent, but as long as they are reasonably accurate, they can help us visualize the figures. In fact, it’s often impossible to solve a geometry problem without making a drawing.

Factor Trees

There are several ways to find factors. One that helps to visually keep track of all the factors is called a factor tree. This is a diagram with a tree-like shape. It uses “branches” to show the factors of a number.

All whole numbers other than 1 can be written as the product of factors. A prime number is a number that has only two factors, itself and 1. An example of a prime number is 13. Its only factors are 13 and 1. A composite number is a number that has more than two factors. An example of a composite number is 6. Its factors include 6, 3, 2, and 1. Prime factors are factors that are also prime numbers. The greatest common factor (GCF) of two numbers is the largest number that is a factor of both numbers.

Coordinate Plane

This is the plane determined by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis, intersecting at a point called the origin. A coordinate plane can be used to illustrate locations and relationships using ordered pairs of numbers.



Venn Diagrams

A set is a list of objects in no particular order. Items in a set can be numbers, but they can also be letters or words. Venn diagrams are a visual way of showing how sets of things can include one another, overlap, or be distinct from one another.

Venn diagrams are often used to compare and contrast things. But they are also a useful tool to sort and classify information. You can use Venn diagrams to take notes on material that shows relationships between things or ideas. You can also use them to solve certain types of word problems. When a word problem names two or three different categories and asks you how many items fall into each category, a Venn diagram can be a useful problem-solving tool.

A Venn diagram begins with a rectangle representing the universal set. Then each set in the problem is represented by a circle. Circles can be separate, overlapping, or one within another. When two circles overlap, it means that the two sets intersect. Some members of one set are also members of the other set.

Venn Diagrams AND Compare-and-Contrast Diagrams

The Venn diagram is an organizing device for planning comparisons and contrasts. A completed Venn diagram helps students categorize and organize similarities and differences, and provides a blueprint for a comparison-and-contrast exercise. The compare-and-contrast diagram provides a structure to identify or list similarities and differences between two objects.

Attribute Tables

To solve logic problems, you need a way to keep track of the subjects and which attributes they have or don’t have. An attribute table can help. This is a table with a row for each subject in the problem, and a column for each attribute. The rows and columns meet to form cells. Because the attributes in logic problems are usually exclusive, you can use Xs or check marks (4) to show which attribute belongs to which subject.

Cause and Effect Maps

Cause and effect maps help you work through information to make sense of it. Write each cause in the oval. Write all its effects in the boxes. Add or delete ovals and boxes as needed.

Frayer Model

The Frayer Model is a word categorization activity that helps learners to develop their understanding of concepts. Using this model, students provide a definition, list characteristics, and provide examples and non-examples of the concept.



Semantic Map

A semantic word map allows students to conceptually explore their knowledge of a new term or concept by mapping it with other related words, concepts, or phrases that are similar in meaning. Semantic maps portray the schematic relations that compose a concept. It assumes that there are multiple relations between a concept and the knowledge that is associated with the concept.

Line Graphs

Line graphs are often used to show how things change over time. They clearly show trends in data and can let you make predictions about future trends, too. Line graphs use two number lines, one horizontal and one vertical. The horizontal number line is called the x-axis. The vertical line is called the y-axis. The x-axis often shows the passage of time. The y-axis often shows a quantity of some kind, such as height, speed, cost, and so forth.

Bar Charts

Bar charts are useful when you want to compare things or to show how one thing changes over time. They are a good way to show overall trends. Bar charts use horizontal or vertical bars to represent data. Longer bars represent higher values. Different colors can be used to show different variables. When you look at a bar chart, it’s easy to see which element has the greatest value—the one with the longest bar.

Bar charts have an x-axis (horizontal) and a y-axis (vertical). If the graph is being used to show how something changes over time, the x-axis has numbers for the time period. If the graph is being used to compare things, the x-axis shows which things are being compared. The y-axis has numbers that show how much of each thing there is.

Probability Trees

When we have probability problems with many possible outcomes, or events that depend on one another, probability trees can help. Probability trees show all the possible outcomes of an event. Whenever a problem calls for figuring out how many possible outcomes there are, and the probability that any one of them will happen, a probability tree can be useful.




Table




Flowchart

Start



Web




Number Line

0

0



Geometric Drawing




Coordinate Plane

y

x-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

109876543210

-1-2-3-4-5-6-7-8-9

-10

-10 10



Venn Diagram




Venn Diagram

Same

Different Different



Compare-and-Contrast Diagram

Item 1 ____________________________ Item 2 ____________________________

How Alike?

How Different?

With Regard To

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________




Attribute Table


Cause and Effect Map

Cause Effect





Frayer Model

Definition Characteristics

WORD

Examples from Life Non-Examples



Semantic Map/Concept Map




Factor Tree



Line Graph

Axi

s ti

tle

____

____

____

____

____

__

Axis title ______________________

Graph title ______________________




Bar Chart/Histogram

Axi

s ti

tle

____

____

____

____

____

__

Axis title ______________________

Graph title ______________________

KEY



Probability Trees

Formulas

© Walch Education© Walch EducationF-1

Formulas

ALGEBRA

Symbols

≈ Approximately equal to

≠ Is not equal to

a Absolute value of a

a Square root of a

∞ Infinity

[ Inclusive on the lower bound

] Inclusive on the upper bound

( Non-inclusive on the lower bound

) Non-inclusive on the upper bound

Linear Equations

=−−

my y

x x2 1

2 1

Slope

y = mx + b Slope-intercept form

ax + by = c General form

y – y1 = m(x(x( – x

1) Point-slope form

Exponential Equations

= +

A Pr

n

nt

1 Compounded interest formula

Compounded…n (number of times per year)

Yearly/annually 1

Semi-annually 2

Quarterly 4

Monthly 12

Weekly 52

Daily 365

Functions

f( f( f x(x( )Function notation, “fnotation, “fnotation, “ of f of f x”

f –1f –1f (x(x( )Inverse function notation

f( f( f x(x( ) = mx + b Linear function

f( f( f x(x( ) = b x + k Exponential function

(f(f( + f + f g)(g)(g x)(x)( ) = f(f(f x(x( ) + g(x(x( ) Addition

(f(f( – f – f g)(g)(g x)(x)( ) = f(f(f x(x( ) – g(x(x( ) Subtraction

(f(f( • f • f g)(g)(g x)(x)( ) = f(f(f x(x( ) • g(x(x( ) Multiplication

=f

gx

f x

g x( )

( )

( )Division

−−

f b f a

b a

( ) ( ) Average rate of change

f(– f(– f x(–x(– ) = –f) = –f) = – (f(f x(x( ) Odd function

f(– f(– f x(–x(– ) = f(f(f x(x( ) Even function

= f x x( ) Floor/greatest integer function

= f x x( ) Ceiling/least integer function

= − +f x a x h k( ) ( )3 Cube root function

( ) ( )f x a x h kn= − + Radical function

= − +f x a x h k( ) Absolute value function

= ≠f xp x

q xq x( )

( )

( ); ( ) 0 Rational function

Formulas

F-2© Walch Education© Walch EducationFormulas

Quadratic Functions and Equations

=−

xb

a2Axis of symmetry

=+

xp q

2Axis of symmetry using the midpoint of the x-intercepts

− −

b

af

b

a2,

2Vertex

f( f( f x(x( ) = ax2 + bx + c General form

f( f( f x(x( ) = a(x(x( – h)2 + k Vertex form

f( f( f x(x( ) = a(x(x( – p)(x)(x)( – q) Factored/intercept form

b2 – 4ac Discriminant

+ +

x bxb

22

2

Perfect square trinomial

=− ± −

xb b ac

a

4

2

2

Quadratic formula

− = + −ax b ax b ax b( ) ( )( )2 2 Difference of squares

(x(x( – h)2 = 4p = 4p = 4 (y(y( – k) Standard form for a parabola that opens up or down

(y(y( – k)2 = 4p = 4p = 4 (x(x( – h) Standard form for a parabola that opens right or left

F(F(F h, k + p) Focus for a parabola that opens up or down

F(F(F h + p, k) Focus for a parabola that opens right or left

y = k – p Directrix for a parabola that opens up or down

x = h – p Directrix for a parabola that opens right or left

Formulas


Formulas

Exponential Functions

1 + r Growth factor

1 – r Decay factor

= +f t a r t( ) (1 ) Exponential growth function

= −f t a r t( ) (1 ) Exponential decay function

=f x abx( ) Exponential function in general form

Equations of Circles

(x (x ( – h)2 + (y + (y + ( – k)2 = r2r2r Standard form

x2 + y2 = r2r2r Center at (0, 0)

Ax2 + By2 + Cx + Dy + E = 0E = 0E General form

Properties of Exponents

Property General rule

Zero Exponent a0 = 1

Negative Exponent =−

bb

m

nm

n

1

Product of Powers • = +a a am n m n

Quotient of Powers = −a

aa

m

nm n

Power of a Power ( ) =b bm n mn

Power of a Product ( ) =bc b cn n n

Power of a Quotient

=a

b

a

b

m m

m

Properties of Radicals

•=ab a b

=a

b

a

b

Radicals to Rational Exponents

=a an n

1

=x xmnm

n

Imaginary Numbers

= −i 1

i2 = –1i3 = –ii4 = 1

Multiplication of Complex Conjugates

(a + bi)(a – bi) = a2 + b2

General

(x(x( , y) Ordered pair

(x(x( , 0) x-intercept

(0, y) y-intercept

Formulas


DATA ANALYSIS

Symbols

∅ Empty/null set

∩ Intersection, “and”

∪ Union, “or”

⊂ Subset

A Complement of Set A

! Factorial

Cn r Combination

Pn r Permutation

Rules and Equations

=P EE

( )# of outcomes in

# of outcomes in sample spaceProbability of event E

∪ = + − ∩P A B P A P B P A B( ) ( ) ( ) ( ) Addition rule

= −P A P A( ) 1 ( ) Complement rule

( )= ∩P B A

P A B

P A

( )

( )Conditional probability

( ) ( )• ( )∩ =P A B P A P B A Multiplication rule

( ) ( )• ( )∩ =P A B P A P B Multiplication rule if A and Bare independent

=−

Cn

n r rn r

!

( )! !Combination

=−

Pn

n rn r

!

( )!Permutation

n n n n! •( 1)•( 2) • • 1= − − Factorial

Formulas


Formulas

GEOMETRY

Symbols

ABC Major arc length

AB Minor arc length

∠ Angle

Circle

≅ Congruent� ��PQ Line

PQ Line segment� ��PQ Ray

Parallel

⊥ Perpendicular

• Point

Triangle

Parallelogram

′A Prime

° Degrees

θ Theta

φ Phi

π Pi

Pythagorean Theorem

a2 + b2 = c2

Trigonometric Ratios

θ =sinopposite

hypotenuseθ =cos

adjacent

hypotenuseθ =tan

opposite

adjacent

θ =cschypotenuse

oppositeθ =sec

hypotenuse

adjacentθ =cot

adjacent

opposite

Trigonometric Identities

sin cos(90 )θ θ= °−

cos sin(90 )θ θ= °−

θθθ

=tansin

cos

θθ

=csc1

sin

θθ

=sec1

cos

θθ

=cot1

tan

θθθ

=cotcos

sin

θ θ+ =sin cos 12 2

Pi Defined

circumference

diameter

circumference

2•radiusπ = =

Area

A = lw Rectangle

=A bh1

2Triangle

π=A r 2 Circle

= +A b b h1

2( )1 2

Trapezoid

Volume

=V lwhRectangular prism

V = V = V Bh Prism

V r h1

32π= Cone

=V Bh1

3Pyramid

π=V r h2 Cylinder

π=V r4

33 Sphere

Distance Formula

= − + −d x x y y( ) ( )2 12

2 12

Dilation

=D x y kx kyk ( , ) ( , )

Formulas


MEASUREMENTS

Length

Metric

1 kilometer (km) = 1000 meters (m)

1 meter (m) = 100 centimeters (cm)

1 centimeter (cm) = 10 millimeters (mm)

Customary

1 mile (mi) = 1760 yards (yd)

1 mile (mi) = 5280 feet (ft)

1 yard (yd) = 3 feet (ft)

1 foot (ft) = 12 inches (in)

Volume and Capacity

Metric

1 liter (L) = 1000 milliliters (mL)

Customary

1 gallon (gal) = 4 quarts (qt)

1 quart (qt) = 2 pints (pt)

1 pint (pt) = 2 cups (c)

1 cup (c) = 8 fluid ounces (fl oz)

Weight and Mass

Metric

1 kilogram (kg) = 1000 grams (g)

1 gram (g) = 1000 milligrams (mg)

1 metric ton (MT) = 1000 kilograms

Customary

1 ton (T) = 2000 pounds (lb)

1 pound (lb) = 16 ounces (oz)

Inverse Trigonometric Functions

Arcsin θ = sin–1θ

Arccos θ = cos–1θ

Arctan θ = tan–1θ

Circumference of a Circle

π=C r2 Circumference given the radius

π=C d Circumference given the diameter

Arc Length

θ=s r Arc length (θ in radians)

Converting Between Degrees and Radians

π=

radian measure degree measure

180

Midpoint Formula

+ +

x x y y

2,

21 2 1 2

Glossary

Glossary

English Unit/Lesson EspañolA

acute triangle a triangle in which all of the angles are acute (less than 90°)

1.3 triángulo agudo triángulo en el que todos los ángulos son agudos (menos de 90°)

Addition Rule If A and B are any two events, then the probability of A or B, denoted P(A or B), is given by: P(A or B) = P(A) + P(B) – P(A and B). Using set notation, the rule is P A B P A P B P A B( ) ( ) ( ) ( )∪ = + − ∩ .

6.3 Regla de la suma Si A y B son dos eventos cualquiera, entonces la probabilidad de A o B, que se indicacon P (A o B), está dada por: P(A o B) = P(A) + P(B) – P(A y B). Con el uso de notación de conjuntos, la regla es P A B P A P B P A B( ) ( ) ( ) ( )∪ = + − ∩ .

adjacent side the leg next to an acute angle in a right triangle that is not the hypotenuse

5.7 lado adyacente el cateto junto a un ángulo agudo en un triángulo rectángulo que no es la hipotenusa

alternate exterior angles angles that are on opposite sides of the transversal and lie on the exterior of the two lines that the transversal intersects

1.2 ángulos exteriores alternos ángulos en lados opuestos de la transversal que se sitúan en el exterior de las dos líneas que corta la transversal

alternate interior angles angles that are on opposite sides of the transversal and lie within the interior of the two lines that the transversal intersects

1.2 ángulos interiores alternos ángulos que están en los lados opuestos de la transversal y se ubican en el interior de las dos líneas que corta la transversal

altitude the perpendicular line from a vertex of a figure to its opposite side; height

5.5 altitud línea perpendicular desde el vértice de una figura hasta su lado opuesto; altura

Angle-Angle (AA) Similarity Statement If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

5.2 Criterio de semejanza ángulo-ángulo (AA) Si dos ángulos de un triángulo son congruentes con dos ángulos de otro triángulo, entonces los triángulos son similares.

angle-side-angle (ASA) if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent

1.15 ángulo-lado-ángulo (ASA) si dos ángulos y el lado incluido de un triángulo son congruentes con los dos ángulos y el lado incluido de otro triángulo, entonces los dos triángulos son congruentes

angle bisector a ray that divides an angle into two congruent angles

5.4 bisectriz del ángulo semirrecta que divide un ángulo en dos ángulos congruentes

© Walch EducationG-1

Glossary

Glossary

English Unit/Lesson Españolangle of depression the angle created

by a horizontal line and a downward line of sight to an object that is below the observer

5.10 ángulo de depresión ángulo creado por una línea horizontal y una línea de mira descendente en relación a un objeto que se encuentra por debajo del observador

angle of elevation the angle created by a horizontal line and an upward line of sight to an object that is above the observer

5.10 ángulo de elevación ángulo creado por una línea horizontal y una línea de mira ascendente en relación a un objeto que se encuentra por encima del observador

angle of rotation the angle through which a figure is rotated about a center point; the measure of the angle created by connecting the center of rotation to a point on the preimage and to the corresponding point on the image

1.9 ángulo de rotación el ángulo a través del cual se hace girar una figura alrededor de un punto central; la medida del ángulo creado por la conexión del centro de rotación a un punto de la imagen inversa y para el punto correspondiente en la imagen

arccosine the inverse of the cosine function, written cos–1θ or arccosθ

5.8 arcocoseno inversa de la función coseno; se expresa cos–1θ o arccosθ

arcsine the inverse of the sine function, written sin–1θ or arcsinθ

5.8 arcoseno inversa de la función seno; se expresa sen–1θ o arcsenθ

arctangent the inverse of the tangent function, written tan–1θ or arctanθ

5.8 arcotangente inversa de la función tangente; se expresa tan–1θ o arctanθ

axis of symmetry of a parabola

the line through the vertex of a

parabola about which the parabola is

symmetric. The equation of the axis of

symmetry is xb

a2=−

.

2.10 eje de simetría de una parábola

línea que atraviesa el vértice de una

parábola sobre la que la parábola

es simétrica. La≈ecuación del eje de

simetría es xb

a2=−

.

Bbase the quantity that is being raised

to an exponent in an exponential expression; in ax, a is the base; or, the quantity that is raised to an exponent which is the value of the logarithm, such as 2 in the equation log2 g(x) = 3 – x

1.4 4.1

base cantidad que es elevada a un exponente en una expresión exponencial; en ax, a es la base; o, la cantidad que se eleva a un exponente que es el valor del logaritmo, tal que 2 en la ecuación log2 g(x) = 3 – x

G-2© Walch Education

Glossary

Glossary

English Unit/Lesson Españolbase angle an angle formed by the base

and one congruent side of an isosceles triangle

1.4 ángulo base ángulo formado por la base y un lado congruente de un triángulo isósceles

binomial a polynomial with two terms 2.2 binomio polinomio con dos términos

C center of dilation a point through

which a dilation takes place; all the points of a dilated figure are stretched or compressed through this point

1.11 centro de dilatación punto a través del cual se produce una dilatación; todos los puntos de una figura dilatada se alargan o comprimen a través de este punto

clockwise rotating a figure in the direction that the hands on a clock move

1.9 sentido horario rotación de una figura en la dirección en que se mueven las agujas de un reloj

closure a system is closed, or shows closure, under an operation if the result of the operation is within the system

3.2 cierre un sistema es cerrado, o tiene cierre, en una operación si el resultado de la misma está dentro del sistema

coefficient the number multiplied by a variable in an algebraic expression

2.2 coeficiente número multiplicado por una variable en una expresión algebraica

collinear points points that lie on the same line

1.11 puntos colineales puntos que se ubican en la misma línea

complex number a number in the form a + bi, where a and b are real numbers, and i is the imaginary unit

3.6 número complejo número en la forma a + bi, donde a y b son números reales e i es la unidad imaginaria

complex number system all numbers of the form a + bi, where a and b are real numbers, including complex numbers (neither a nor b equal 0), real numbers (b = 0), and imaginary numbers (a = 0)

3.6 sistema de números complejos todos los números de la forma a + bi, donde a y b son números reales, incluidos los números complejos (ni a ni b son iguales a 0), reales (b = 0) e imaginarios (a = 0)

complement a set whose elements are not in another set, but are in some universal set being considered. The complement of set A, denoted by A , is the set of elements that are in the universal set, but not in A. The event does not occur. The probability of an event not occurring is 1 minus the probability of the event occurring, P A P A( )= −1 ( ) .

6.1 complemento conjunto cuyos elementos no se encuentran en otro conjunto, pero están en algún conjunto universal que se considera. El complemento del conjunto A, que se indica con A , es el conjunto de elementos que se encuentran en el conjunto universal, pero no en A. El evento no se produce. La probabilidad de que un evento no se produzca es 1 menos la probabilidad de que se produzca, P A P A( )= −1 ( ) .


Glossary

Glossary

English Unit/Lesson Españolcompression a transformation in which

a figure becomes smaller; compressions may be horizontal (affecting only horizontal lengths), vertical (affecting only vertical lengths), or both

1.11 1.12

compresión transformación en la que una figura se hace más pequeña; las compresiones pueden ser horizontales (cuando afectan sólo la longitud horizontal), verticales (cuando afectan sólo la longitud vertical), o en ambos sentidos

concave down a graph of a curve that is bent downward, such as a quadratic function with a maximum value

2.5 cóncavo hacia abajo gráfico de una curva que se inclina hacia abajo, tal como una función cuadrática con un valor máximo

concave up a graph of a curve that is bent upward, such as a quadratic function with a minimum value

2.5 cóncavo hacia arriba gráfico de una curva que se inclina hacia arriba, tal como una función cuadrática con un valor mínimo

concavity with respect to a curve, the property of being arched upward or downward. A quadratic with positive concavity will increase on either side of the vertex, meaning that the vertex is the minimum or lowest point of the curve. A quadratic with negative concavity will decrease on either side of the vertex, meaning that the vertex is the maximum or highest point of the curve.

2.5 concavidad con respecto a una curva, la propiedad de ser arqueado hacia arriba o hacia abajo. Una función cuadrática con concavidad positiva se incrementa-rá en ambos lados del vértice, lo que sig-nifica que el vértice es el punto mínimo o más bajo de la curva. Una función cuadrática con concavidad negativa disminuirá a cada lado del vértice, lo que significa que el vértice es el punto máximo o más alto de la curva.

conditional probability of B given A the probability that event B occurs, given that event A has already occurred. If A and B are two events from a sample space with P(A) ≠ 0, then the conditional probability of B given A, denoted P B A( ), has two equivalent

expressions: P B AP A B

P A

A( )= ( )( ) =and number of outcomes in andd

number of outcomes in

B

A

( )

P B AP A B

P A



B

A

( ).

6.5 probabilidad condicional de B dado A la probabilidad de que el evento B se produzca, dado que el evento A ya se ha producido. Si A y B son dos eventos de un espacio muestral con P(A) ≠ 0, entonces la probabilidad condicional de B dado A, indicado P B A( ) tiene dos expresiones equivalentes:

P B AP A B

P A



B

A

( )=

P A B

P A

( y )

( )

A B

A

numero de resultados en ( y )

numero de resultados en.


Glossary

Glossary

English Unit/Lesson Españolcongruency transformation

a transformation in which a geometric figure moves but keeps the same size and shape; a dilation where the scale factor is equal to 1

1.11 1.12

transformación de congruencia transformación en la cual una figura geométrica se mueve pero mantiene el mismo tamaño y la misma forma; dilatación en la que el factor de escala es igual a 1

congruent having the same shape, size, lines, and angles; the symbol for congruent is ≅

1.7 congruente tiene la misma forma, tamaño, líneas y anglos; el símbolo paracongruente es ≅

constant term a term whose value does not change

2.2 término constante término cuyo valor no cambia

converse of the Pythagorean Theorem If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle.

5.5 conversa del teorema de Pitágoras Si la suma de los cuadrados de las medidas de dos lados de un triángulo equivale al cuadrado de la medida del lado más largo, entonces el triángulo es rectángulo.

coordinate proof a proof that involves calculations and makes reference to the coordinate plane

1.5 prueba de coordenadas prueba que involucra cálculos y hace referencia al plano de coordenadas

corresponding angles angles in the same relative position with respect to the transversal and the intersecting lines

1.2 1.9

ángulos correspondientes ángulos en la misma posición relativa con respecto a las líneas transversal y de intersección

Corresponding Parts of Congruent Triangles are Congruent (CPCTC) if two or more triangles are proven congruent, then all of their corresponding parts are congruent as well

1.13 Las partes correspondientes de triángulos congruentes son congruentes (CPCTC) si se comprueba que dos o más triángulos son congruentes, entonces todas sus partes correspondientes son también congruentes

corresponding sides sides of two figures that lie in the same position relative to the figure. In transformations, the corresponding sides are the preimage and image sides, so AB and A B′ ′ are corresponding sides and so on.

1.9 1.11

lados correspondientes lados de dos figuras que están en la misma posición relativa a la figura. En las transformaciones, los lados correspondientes son los de preimagen e imagen, entonces AB y A B′ ′ son los lados correspondientes, etc.


Glossary

Glossary

English Unit/Lesson Españolcosecant the reciprocal of the sine ratio,

csc1

sinθ

θ= ; the cosecant of θ =

csc θ = length of hypotenuse

length of opposite side

5.7 cosecante razón inversa del seno,

θθ

=csc1

sen; la cosecante de θ =

csc θ = longitud de la hipotenusa

longitud del lado opuesto

cosine a trigonometric function of

an acute angle in a right triangle

that is the ratio of the length of the

side adjacent to the length of the

hypotenuse; the cosine of θ = cos θ = length of adjacent side

length of hypotenuse

5.7 coseno función trigonométrica de

un ángulo agudo en un triángulo

rectángulo que es la proporción de

la longitud de lado adyacente a la

longitud de la hipotenusa; el coseno de

θ = cos θ = longitud del lado adyacente

longitud de la hipotenusa

cotangent the reciprocal of tangent,

cot1

tanθ

θ= ; the cotangent of θ =

cot θ = length of adjacent side

length of opposite side

5.7 cotangente recíproco de la tangente,

cot1

tanθ

θ= ; la cotangente de θ =

cot θ = longitud del lado adyacente

longitud del lado opuesto

counterclockwise rotating a figure in the opposite direction that the hands on a clock move

1.9 en sentido antihorario rotación de una figura en la dirección opuesta a la que se mueven las agujas de un reloj

D decreasing the interval of a function for

which the output values are becoming smaller as the input values are becoming larger

2.5 decreciente intervalo de una función por el que los valores de salida se hacen más pequeños a medida que los valores de entrada se hacen más grandes

dependent events events that are not independent. The outcome of one event affects the probability of the outcome of another event.

6.7 eventos dependientes eventos que no son independientes. El resultado de un evento afecta la probabilidad del resultado de otro.

dilation a transformation in which a figure is either enlarged or reduced by a scale factor in relation to a center point

1.11 1.12

dilatación transformación en la que una figura se amplía o se reduce por un factor de escala en relación con un punto central


Glossary

Glossary

English Unit/Lesson Españoldiscriminant an expression whose

solved value indicates the number and types of solutions for a quadratic. For a quadratic equation in standard form (ax2 + bx + c = 0), the discriminant is b2 – 4ac.

3.8 discriminante expresión cuyo valor resuelto indica la cantidad y los tipos de soluciones para una ecuación cuadrática. En una ecuación cuadrática en forma estándar (ax2 + bx + c = 0), el discriminante es b2 – 4ac.

disjoint events events that have no outcomes in common. If A and B are disjoint events, then they cannot both occur. Disjoint events are also called mutually exclusive events.

6.3 eventos disjuntos eventos que no tienen resultados en común. Si A y B son eventos disjuntos, entonces no pueden producirse ambos. También se denominan eventos mutuamente excluyentes.

domain 1. the set of all inputs values for which a function is defined; the set of x-values that are valid for a relation or function 2. the set of all input values (x-values) that satisfy the given function without restriction

2.1 dominio 1. el conjunto de todos los valores de entradas para los que se define una función; el conjunto de valores x que son válidos para una relación o función 2. conjunto de todos los valores de entrada (valores de x) que satisfacen la función dada sin restricciones

Eelement an item in a set; also called a

member 6.1 elemento ítem en un conjunto; también

se denomina miembroempty set a set that has no elements,

denoted by ∅ . The empty set is also called the null set.

6.1 conjunto vacío conjunto que no contiene elementos, indicado con ∅ . También se denomina conjunto nulo.

enlargement a dilation of a figure where the scale factor is greater than 1

1.11 ampliación dilatación de una figura en la que el factor de escala es mayor que 1

end behavior the behavior of the graph as x approaches positive infinity and as x approaches negative infinity

2.5 comportamiento final el comporta-miento de la gráfica al aproximarse x a infinito positivo o a infinito negativo

equal sets sets with all the same elements

6.1 conjuntos iguales conjuntos con todos los mismos elementos

equiangular having equal angles 1.4 equiangular que tiene ángulos igualesequilateral triangle a triangle with all

three sides equal in length1.3 triángulo equilátero triángulo que tiene

los tres lados de la misma longitud


Glossary

Glossary

English Unit/Lesson Españolevent an outcome or set of outcomes of

an experiment. An event is a subset of the sample space.

6.1 evento resultado o conjunto de resultados de un experimento. Un evento es un subconjunto del espacio de muestral.

experiment a process or action that has observable results. The results are called outcomes.

6.1 experimento proceso o acción con consecuencias observables. Las consecuencias se denominan resultados.

exponent the number which tells how many times a number is multiplied by itself; in the expression a x, the exponent is x

4.1 exponente el número que indica cuántas veces un número se multiplica por sí mismo; en la expresión a x, x es el exponente

exterior angles angles that lie outside a pair of parallel lines

1.2 1.3

ángulos exteriores ángulos que están fuera de un par de líneas paralelas

extrema the minima or maxima of a function

2.5 2.10

extremos los mínimos o máximos de una función

F factor (noun) one of two or more

numbers or expressions that when multiplied produce a given product

2.2 3.5

factor uno de dos o más números o expresiones que al multiplicarse dan un producto determinado

factor (verb) to write an expression as the product of its factors

3.4 factorizar escribir una expresión como el producto de sus factores

factored form of a quadratic function the intercept form of a quadratic equation, written as f(x) = a(x – p)(x – q), where p and q are the x-intercepts of the function; also known as the intercept form of a quadratic function

2.6 forma factorizada de una función cuadrática forma de intercepto de una ecuación cuadrática, se expresa como f(x) = a(x – p)(x – q), en la que p y q son los interceptos de x de la función; también se conoce como la forma de intercepto de una función cuadrática

G greatest common factor (GCF)

the largest factor that two or more terms share

3.4 máximo común divisor (GCF) el factor más grande que comparten dos o más términos

H horizontal compression squeezing of

the parabola toward the y-axis 2.9 compresión horizontal contracción de

la parábola hacia el eje y


Glossary

Glossary

English Unit/Lesson Españolhorizontal stretch pulling of the parabola

and stretching it away from the y-axis2.9 estiramiento horizontal jalar de la

parábola y estirarla lejos del eje yhypotenuse the side opposite the vertex

of the 90° angle in a right triangle5.7 hipotenusa lado opuesto al vértice del

ángulo de 90° en un triángulo rectángulo

Iimage the new, resulting figure after a

transformation1.7 imagen nueva figura resultante después

de una transformaciónimaginary number any number of

the form bi, where b is a real number, i = −1 , and b ≠ 0

3.6 número imaginario cualquier número de la forma bi, en el que b es un número real, i = −1 , y b ≠ 0

imaginary unit, i the letter i, used to represent the non-real value, i = −1

3.6 unidad imaginaria, i la letra i, utilizada para representar el valor no real i = −1

included angle the angle between two sides

1.15 ángulo incluido ángulo entre dos lados

included side the side between two angles of a triangle

1.15 lado incluido lado entre dos ángulos de un triángulo

increasing the interval of a function for which the output values are becoming larger as the input values are becoming larger

2.5 creciente intervalo de una función para el que los valores de salida se hacen más grandes a medida que los valores de entrada también se vuelven más grandes

independent events events such that the outcome of one event does not affect the probability of the outcome of another event

6.7 eventos independientes eventos en los que el resultado de un evento no afecta la probabilidad del resultado de otro evento

integer a number that is not a fraction or a decimal

4.2 entero un número que no es una fracción ni un decimal

intercept the point at which a line intercepts the x- or y-axis

2.10 intercepto punto en el que una línea intercepta el eje x o y

intercept form of a quadratic function the factored form of a quadratic equation, written as f(x) = a(x – p)(x – q), where p and q are the x-intercepts of the function; also known as the factored form of a quadratic function

2.6 forma de intercepto de una función cuadrática forma factorizada de una ecuación cuadrática, expresada como f(x) = a(x – p)(x – q), donde p y q son los interceptos de x de la función; también se conoce como la forma factorizada de una ecuación cuadrática


Glossary

Glossary

English Unit/Lesson Españolinterior angles angles that lie between a

pair of parallel lines 1.2 1.3

ángulos interiores ángulos ubicados entre un par de líneas paralelas

intersection a set whose elements are each in both of two other sets. The intersection of sets A and B, denoted by A B∩ , is the set of elements that are in both A and B.

6.1 intersección conjunto cuyos elementos están todos en otros dos conjuntos. La intersección de los conjuntos A y B, indicada por A B∩ , es el conjunto de elementos que se encuentran tanto en A como en B.

irrational number a real number that

cannot be written as m

n, where both m

and n are integers and n ≠ 0; a non-

terminating or non-repeating decimal

3.1 4.2

número irracional un número real que

no puede ser escrito como m

n, donde

m y n son números enteros y n ≠ 0; un

no-terminación o no repetitivo decimal

irreducible radical a radical whose radicand contains no perfect square factors. In other words, the radical cannot be further reduced. For example, 7 is an irreducible radical because the radicand, 7, does not have any perfect square factors.

4.2 radical irredutíble un radical cuya radicand no contiene factores cuadrados perfectos. En otras palabras, el radical no puede ser más reducido. Por ejemplo, 7 es un radical irreducible porque la radicand, 7, no tiene ningún factor cuadrado perfecto.

isosceles triangle a triangle with at least two congruent sides

1.3 triángulo isósceles triángulo con al menos dos lados congruentes

isometry a transformation in which the preimage and image are congruent

1.7 isometría transformación en la que la preimagen y la imagen son congruentes

Kkey features of a quadratic function

the x-intercepts, y-intercept, where the function is increasing and decreasing, where the function is positive and negative, relative minimums and maximums, symmetries, and end behavior of the function used to describe, draw, and compare quadratic functions

2.5 características clave de una función cuadrática interceptos de x, intercepto de y, donde la función aumenta y disminuye, donde la función es positiva y negativa, máximos y mínimos relativos, simetrías y comportamiento final de la función utilizado para describir, dibujar y comparar las funciones cuadráticas


Glossary

Glossary

English Unit/Lesson EspañolL

leading coefficient the coefficient of the term with the highest power. For a quadratic equation in standard form ( y = ax2 + bx + c), the leading coefficient is a.

3.4 coeficiente líder coeficiente del término con la mayor potencia. En una ecuación cuadrática en forma estándar ( y = ax2 + bx + c), el coeficiente líder es a.

legs congruent sides of an isosceles triangle

1.4 catetos lados congruentes de un triángulo isósceles

like terms terms that contain the same variables raised to the same power

2.2 3.2

términos semejantes términos que contienen las mismas variables elevadas a la misma potencia

line of reflection the perpendicular bisector of the segments that connect the corresponding vertices of the preimage and the image

1.8 1.9

línea de reflexión bisectriz perpendicular de los segmentos que conectan los vértices correspondientes de la preimagen y la imagen

line of symmetry a line separating a figure into two halves that are mirror images

1.10 línea de simetría línea que separa una figura en dos mitades que son imágenes en espejo

line symmetry exists for a figure if for every point on one side of the line of symmetry, there is a corresponding point the same distance from the line of symmetry on the other side

1.10 simetría lineal la que existe en una figura si para cada punto a un lado de la línea de simetría, hay un punto correspondiente a la misma distancia de la línea de simetría en el otro lado

Mmaximum the largest y-value of a

quadratic equation2.10 máximo el mayor valor de y de una

ecuación cuadráticamember an item in a set; also called

an element6.1 miembro ítem en un conjunto; también

se denomina elementomidpoint a point on a line segment that

divides the segment into two equal parts1.5 punto medio punto en un segmento de

recta que lo divide en dos partes igualesmidpoint formula formula that states

the midpoint of a segment created by

connecting (x1, y1) and (x2, y2) is given

by the formula 2

,2

1 2 1 2+ +

x x y y

1.5 fórmula de punto medio fórmula

que establece el punto medio de un

segmento creado al conectar (x1, y1)

con (x2, y2) está dado por la fórmula

2,

21 2 1 2+ +

x x y y


Glossary

Glossary

English Unit/Lesson Españolmidsegment a line segment joining the

midpoints of two sides of a figure1.5 segmento medio segmento de recta que

une los puntos medios de dos lados de una figura

midsegment triangle the triangle formed when all three of the midsegments of a triangle are connected

1.5 segmento medio de un triángulo triángulo que se forma cuando los tres segmentos medios de un triángulo están conectados

minimum the smallest y-value of a quadratic equation

2.10 mínimo el menor valor de y en una ecuación cuadrática

monomial an expression with one term, consisting of a number, a variable, or the product of a number and variable(s)

2.2 monomio expresión con un solo término, que consiste en un número, una variable, o el producto de un número y una o más variables

mutually exclusive events events that have no outcomes in common. If A and B are mutually exclusive events, then they cannot both occur. Mutually exclusive events are also called disjoint events.

6.3 eventos mutuamente excluyentes eventos que no tienen resultados en común. Si A y B son eventos mutuamente excluyentes, entonces no pueden producirse ambos. También se denominan eventos disjuntos.

Nnon-rigid motion a transformation

done to a figure that changes the figure’s shape and/or size

1.11 1.12

movimiento no rígido transformación hecha a una figura que cambia su forma o tamaño

null set a set that has no elements, denoted by ∅ . The null set is also called the empty set.

6.1 conjunto nulo conjunto que no tiene elementos, indicado con ∅ . También se denomina conjunto vacío.

Oobtuse triangle a triangle with one angle

that is obtuse (greater than 90°)1.3 triángulo obtuso triángulo con un

ángulo que es obtuso (de más de 90°)

one-to-one a relationship wherein each point in a set of points is mapped to exactly one other point

1.7 unívoca relación en la que cada punto de un conjunto de puntos se corresponde con otro con exactitud

opposite side the side across from an angle

5.7 lado opuesto lado al otro lado de un ángulo

outcome a result of an experiment 6.1 resultado consecuencia de un experimento


Glossary

Glossary

English Unit/Lesson EspañolP

parabola the U-shaped graph of a quadratic equation; the set of all points that are equidistant from a fixed line, called the directrix, and a fixed point not on that line, called the focus. The parabola, directrix, and focus are all in the same plane. The vertex of the parabola is the point on the parabola that is closest to the directrix.

2.10 parábola gráfico de una ecuación cuadrática en forma de U; conjunto de todos los puntos equidistantes de una línea fija denominada directriz y un punto fijo que no está en esa línea, llamado foco. La parábola, la directriz y el foco están todos en el mismo plano. El vértice de la parábola es el punto más cercano a la directriz.

paragraph proof statements written out in complete sentences in a logical order to show an argument

5.5 prueba de párrafo declaraciones redactadas en oraciones completas en orden lógico para demostrar un argumento

parallel lines lines in a plane that either do not share any points and never intersect, or share all points; written as � ��

AB PQ

5.4 líneas paralelas líneas en un plano que no comparten ningún punto y nunca se cortan, o que comparten todos los puntos; se expresan como

� ��

AB PQ

perfect square the product of an integer and itself

3.5 4.2

cuadrado perfecto el producto de un número entero multiplicado por sí mismo

perfect square trinomial a trinomial

of the form x bxb

2

2

2+ +

that can be

written as the square of a binomial

3.7 trinomio cuadrado perfecto trinomio

de la forma x bxb

2

2

2+ +

que puede

expresarse como el cuadrado de

un binomiophi (φ) a Greek letter sometimes used to

refer to an unknown angle measure5.7 fi (φ) letra del alfabeto griego que se

utiliza a veces para referirse a la medida desconocida de un ángulo

plane a flat, two-dimensional figure without depth that has at least three non-collinear points and extends infinitely in all directions

1.1 plano figura plana, bidimensional, sin profundidad, que tiene al menos tres puntos no colineales y se extiende infinitamente en todas direcciones


Glossary

Glossary

English Unit/Lesson Españolpoint(s) of intersection the ordered

pair(s) where graphed functions intersect on a coordinate plane; these are also the solutions to systems of equations

3.10 puntos de intersección pares ordenados en los que se intersecan funciones representadas en gráficos en un plano de coordenadas; son también las soluciones a sistemas de ecuaciones

point of rotation the fixed location that an object is turned around; the point can lie on, inside, or outside the figure

1.9 punto de rotación ubicación fija en torno a la que gira un objeto; el punto puede estar encima, dentro o fuera de la figura

polynomial a monomial or the sum of monomials

2.2 polinomio monomio o suma de monomios

postulate a true statement that does not require a proof

1.13 1.15

postulado declaración verdadera que no requiere prueba

preimage the original figure before undergoing a transformation

1.7 preimagen figura original antes de sufrir una transformación

prime an expression that cannot be factored

3.4 número primo expresión que no puede ser factorizada

probability a number from 0 to 1 inclusive or a percent from 0% to 100% inclusive that indicates how likely an event is to occur

6.1 probabilidad número de 0 a 1 inclusivo o porcentaje de 0% a 100% inclusivo que indica cuán probable es que se produzca un evento

probability model a mathematical model for observable facts or occurrences that are assumed to be random; a representation of a random phenomenon

6.3 modelo de probabilidad modelo matemático para hechos o sucesos observables que se presumen aleatorios; representación de un fenómeno aleatorio

probability of an event E

denoted P(E), and is given by

P EE

( ) =number of outcomes in

number of outcomes iin the sample space

in a uniform probability model

6.3 probabilidad de un evento E se expresa

como P(E), y está dado por P(E) =

=P EE

( )número de resultados en

número de resultados en el espacio de muestreo

en un modelo de probabilidad uniformeProduct Property of Square Roots

for all real numbers a and b, • •a b a b=

3.5 Propiedad del Producto de las Raíces Cuadradas para todos los números reales a y b, • •a b a b=


Glossary

Glossary

English Unit/Lesson Españolproof a set of justified statements

organized to form a convincing argument that a given statement is true

5.3 prueba conjunto de declaraciones justificadas y organizadas para formar un argumento convincente de que determinada declaraciónes verdadera

proportional having a constant ratio to another quantity

5.1 proporcional que tiene una proporción constante con otra cantidad

Qquadratic equation an equation

that can be written in the form ax2 + bx + c = 0, where x is the variable, a, b, and c are constants, and a ≠ 0

2.2 ecuación cuadrática ecuación que se puede expresar en la forma ax2 + bx + c = 0, donde x es la variable, a, b, y c son constantes, y a ≠ 0

quadratic expression an algebraic expression that can be written in the form ax2 + bx + c, where x is the variable, a, b, and c are constants, and a ≠ 0

2.2 expresión cuadrática expresión algebraica que se puede expresar en la forma ax2 + bx + c, donde x es la variable, a, b, y c son constantes, y a ≠ 0

quadratic formula a formula that states

the solutions of a quadratic equation

of the form ax2 + bx + c = 0 are given

by xb b ac

a=− ± −2 4

2. A quadratic

equation in this form can have no real

solutions, one real solution, or two

real solutions.

3.8 fórmula cuadrática fórmula que

establece que las soluciones de una

ecuación cuadrática de la forma

ax2 + bx + c = 0 están dadas por

xb b ac

a=− ± −2 4

2. Una ecuación

cuadrática en esta forma tener ningún

solución real, o tener una solución real,

o dos soluciones reales.quadratic-linear system a system of

equations in which one equation is quadratic and one is linear

3.10 sistema lineal cuadrático sistema de ecuaciones en el que una ecuación es cuadrática y una es lineal

quadratic function a function that can be written in the form f(x) = ax2 + bx + c, where a ≠ 0. The graph of any quadratic function is a parabola.

2.10 función cuadrática función que puede expresarse en la forma f(x) = ax2 + bx + c, donde a ≠ 0. El gráfico de cualquier función cuadrática es una parábola.


Glossary

Glossary

English Unit/Lesson EspañolQuotient Property of Square Roots

for all real numbers a and b, a

b

a

b=

where b ≠ 0

3.5 Propiedad del Cociente de las Raíces Cuadradas para todos los números

reales a y b, a

b

a

b= con b ≠ 0

Rradical equation an algebraic equation

in which at least one term includes a radical expression

4.4 ecuación radical ecuación algebraica en la cual al menos un término incluye una expresión radical

radical expression an expression containing a root

4.2 4.4

expresión radical expresión que contiene una raíz

radicand in a radical expression, the number under the root sign; in the expression 5 , the radicand is 5

4.2 radicand en una expresión radical, el número bajo el signo de la raíz; en la expresión 5 , la radicand es 5

random sample a subset or portion of a population or set that has been selected without bias, with each item in the population or set having the same chance of being found in the sample

6.2 muestra aleatoria subconjunto o porción de población o conjunto que ha sido seleccionado sin sesgo, con cada elemento de la población o conjunto con la misma probabilidad de encontrarse en la muestra

range the set of all outputs of a relation or function; the set of y-values for which a function is defined

2.1 rango conjunto de todas las salidas de una función; conjunto de valores de y para el que se define una función

ratio the relation between two quantities; can be expressed in words, fractions, decimals, or as a percentage

5.7 proporción relación entre dos cantidades; puede expresarse en palabras, fracciones, decimales o como porcentaje

ratio of similitude a ratio of corresponding sides; also known as the scale factor

5.1 proporción de similitud proporción de lados correspondientes; se conoce también como factor de escala

rational number any number that can

be written as m

n, where m and n are

integers and n ≠ 0; any number that

can be written as a decimal that ends

or repeats

4.2 números racionales números que

pueden expresarse como m

n, en los

que m y n son enteros y n ≠ 0; cualquier

número que puede escribirse como

decimal finito o periódico


Glossary

Glossary

English Unit/Lesson Españolreal numbers the set of all rational and

irrational numbers4.2 números reales conjunto de todos los

números racionales e irracionales

reciprocal a number that when multiplied by the original number or algebraic expression has a product of 1

5.7 recíproco número que cuando se multiplica por el número original o la expresión algebraica tiene un producto de 1

reduction a dilation where the scale factor is between 0 and 1

1.11 reducción dilatación en la que el factor de escala está entre 0 y 1

reflection a transformation where a mirror image is created; also called a flip

1.7 1.8

reflexión transformación por la cual se crea una imagen en espejo

Reflexive Property of Congruent Segments a segment is congruent to itself; ≅AB AB

5.4 Propiedad reflexiva de congruencia de segmentos un segmento es congruente con él mismo; ≅AB AB

regular polygon a closed two-dimensional figure with all sides and all angles congruent

1.10 polígono regular figura bidimensional cerrada con todos los lados y todos los ángulos congruentes

relative frequency (of an event) the number of times an event occurs divided by the number of times an experiment is performed

6.3 frecuencia relativa (de un evento) cantidad de veces que un evento se produce dividido por la cantidad de veces que se realiza el experimento

remote interior angles interior angles that are not adjacent to the exterior angle

1.3 ángulos interiores remotos ángulos interiores que no son adyacentes al ángulo exterior

right triangle a triangle with one angle that measures 90°

1.3 triángulo rectángulo triángulo con un ángulo que mide 90°

rigid motion a transformation done to a figure that maintains the figure’s shape and size or its segment lengths and angle measures

1.9 1.11

movimiento rígido transformación que se realiza a una figura que mantiene su forma y tamaño o las longitudes de sus segmentos y las medidas de ángulos

root(s) solution(s) of a quadratic equation

3.8 raíces soluciones de una ecuación cuadrática

rotation a transformation that turns a figure around a fixed center point; also called a turn

1.7 1.9

rotación transformación que convierte una figura alrededor de un punto central fijo; también llamado un giro


Glossary

Glossary

English Unit/Lesson Español S

same-side exterior angles angles that lie on the same side of the transversal and are outside the lines that the transversal intersects; sometimes called consecutive exterior angles

1.2 ángulos exteriores del mismo lado ángulos que se ubican en el mismo lado de la transversal y están fuera de las líneas que corta la transversal; a veces se denominan ángulos exteriores consecutivos

same-side interior angles angles that lie on the same side of the transversal and are in between the lines that the transversal intersects; sometimes called consecutive interior angles

1.2 ángulos interiores del mismo lado ángulos que se ubican en el mismo lado de la transversal y están en medio de las líneas que corta la transversal; a veces se los denomina ángulos interiores consecutivos

sample space the set of all possible outcomes of an experiment

6.1 espacio de muestreo conjunto de todos los resultados posibles de un experimento

sampling bias errors in estimation caused by flawed (non-representative) sample selection

6.2 sesgo de muestreo errores de cálculo ocasionados por una selección defectuosa (no representativa) de la muestra

scale factor a multiple of the lengths of the sides from one figure to the transformed figure. If the scale factor is larger than 1, then the figure is enlarged. If the scale factor is between 0 and 1, then the figure is reduced.

1.11 1.12

factor de escala múltiplo de las longitudes de los lados de una figura a la figura transformada. Si el factor de escala es mayor que 1, entonces la figura se agranda. Si el factor de escala se encuentra entre 0 y 1, entonces la figura se reduce.

scalene triangle a triangle with no congruent sides

1.3 triángulo escaleno triángulo sin lados congruentes

secant the reciprocal of cosine,

sec1

cosθ

θ= ; the secant of θ = sec θ =


length of adjacent side

5.7 secante recíproco del coseno,

sec1

cosθ

θ= ; secante de θ = sec θ =


longitud del lado adyacente

Segment Addition Postulate If B is between A and C, then AB + BC = AC. Conversely, if AB + BC = AC, then B is between A and C.

5.4 Postulado de la suma de segmentos Si B está entre A y C, entonces AB + BC = AC. A la inversa, si AB + BC = AC, entonces B se encuentra entre A y C.


Glossary

Glossary

English Unit/Lesson Españolset a collection or list of items 6.1 conjunto colección o lista de elementosside-angle-side (SAS) if two sides and

the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent

1.15 lado-ángulo-lado (SAS) si dos lados y el ángulo incluido de un triángulo son congruentes con dos lados y el ángulo incluido de otro triángulo, entonces los dos triángulos son congruentes

Side-Angle-Side (SAS) Similarity Statement If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

5.3 Criterio de semejanza lado-ángulo-lado (SAS) Si las medidas de dos lados de un triángulo son proporcionales a las medidas de dos lados correspondientes de otro triángulo y los ángulos incluidos son congruentes, entonces los triángulos son similares.

side-side-side (SSS) if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent

1.15 lado-lado-lado (SSS) si los tres lados de un triángulo son congruentes con los tres lados de otro triángulo, entonces los dos triángulos son congruentes

Side-Side-Side (SSS) Similarity Statement If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar.

5.3 Criterio de semejanza lado-lado-lado (SSS) Si las medidas de los lados correspondientes de dos triángulos son proporcionales, entonces los triángulos son similares.

similar (or similar figures) two figures that are the same shape but not necessarily the same size. Corresponding angles must be congruent and sides must have the same ratio. The symbol for representing similarity is .

5.1 similar (o figuras similares) dos figuras que tienen la misma forma pero no necesariamente el mismo tamaño. Ángulos correspondientes deben ser congruentes y los lados deben tener la misma relación. El símbolo para representar similitud entre figuras es .

similarity transformation a dilation; a transformation that results in the size of a figure changing, but not the shape

5.1 transformación de semejanza dilatación; una transformación que resulta en el tamaño de un cambio de figura, pero no la forma

simple random sample a sample in which any combination of a given number of individuals in the population has an equal chance of selection

6.2 muestra aleatoria simple muestra en la cual cualquier combinación de una cantidad dada de individuos de la población tiene iguales posibilidades de selección


Glossary

Glossary

English Unit/Lesson Españolsine a trigonometric function of an acute

angle in a right triangle that is the ratio of the length of the opposite side to the length of the hypotenuse; the sine of

θ = sin θ = length of opposite side


5.7 seno función trigonométrica de un ángulo agudo en un triángulo rectángulo que es la proporción de la longitud del lado opuesto a la longitud de la hipotenusa; sen de θ = sen θ = longitud del lado opuesto


square root For any real numbers a and b, if a2 = b, then a is a square root of b. The square root of b is written using a radical: b .

4.2 raíz cuadrada Para cualquier número real a y b, si a2 = b, entonces a es la raíz cuadrada de b. La raíz cuadrada de b se expresa con un radical: b .

standard form of a quadratic function a quadratic function written as f(x) = ax2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term

2.5 2.7

2.10

forma estándar de función cuadrática función cuadrática expresada como f(x) = ax2 + bx + c, donde a es el coeficiente del término cuadrático, b es el coeficiente del término lineal, y c es el término constante

stretch a transformation in which a figure becomes larger; stretches may be horizontal (affecting only horizontal lengths), vertical (affecting only vertical lengths), or both

1.11 ampliación transformación en la que una figura se hace más grande; las ampliaciones pueden ser horizontales (cuando afectan sólo las longitudes horizontales), verticales (cuando afectan sólo las longitudes verticales), o en ambos sentidos

subset a set whose elements are in another set. Set A is a subset of set B, denoted by A ⊂ B, if all the elements of A are also in B.

6.1 subconjunto conjunto cuyos elementos están en otro conjunto. El conjunto A es un subconjunto del conjunto B, indicado por A ⊂ B, si todos los elementos de A se encuentran también en B.

substitution the replacement of a term of an equation by another term that is known to have the same value

3.9 sustitución reemplazo de un término de una ecuación por otro que se sabe que tiene el mismo valor

Symmetric Property of Congruent Segments If ≅AB CD , then ≅CD AB .

5.4 Propiedad simétrica de congruencia de segmentos Si ≅AB CD , entonces

≅CD AB .system of equations a set of equations

with the same unknowns3.10 sistema de ecuaciones un conjunto de

ecuaciones con las mismas incógnitasG-20

© Walch Education

Glossary

Glossary

English Unit/Lesson Español T

tangent a trigonometric function of an acute angle in a right triangle that is the ratio of the length of the opposite side to the length of the adjacent side; the tangent of θ = tan θ = length of opposite side

length of adjacent side

5.7 tangente función trigonométrica de un ángulo agudo en un triángulo rectángulo que es la proporción de la longitud del lado opuesto a la longitud del lado adyacente; tangente de θ =

tan θ = longitud del lado opuesto

longitud del lado adyacente

term a number, a variable, or the product of a number and variable(s)

2.2 término número, variable, o producto de un número y una o más variables

theorem a statement that is shown to be true

5.5 teorema declaración que se demuestra que es verdadera

theta (θ) a Greek letter commonly used to refer to unknown angle measures

5.7 teta (θ) letra griega que se utiliza por lo general para referirse a medidas de ángulos desconocidas

Transitive Property of Congruent Segments If ≅AB CD , and ≅CD EF , then ≅AB EF .

5.4 Propiedad transitiva de congruencia de segmentos Si ≅AB CD, y ≅CD EF , entonces ≅AB EF .

transversal a line that intersects a system of two or more lines

1.2 transversal línea que corta un sistema de dos o más líneas

transformation adding or multiplying a constant to a function that changes the function’s position and/or shape

1.7 2.8

transformación suma o multiplicación de una constante con una función que cambia la posición y/o forma de la función

translation transforming a function where the shape and size of the function remain the same but the function moves horizontally and/or vertically; adding a constant to the independent or dependent variable

1.7 2.8

traslación transformación de una función en la que la forma y el tamaño de la función permanecen iguales pero la función se traslada en sentido horizontal y/o vertical; suma de una constante a la variable independiente o dependiente

trigonometry the study of triangles and the relationships between their sides and the angles between these sides

5.7 trigonometría estudio de los triángulos y las relaciones entre sus lados y los ángulos entre ellos

trinomial a polynomial with three terms 2.2 trinomio polinomio con tres términos


Glossary

Glossary

English Unit/Lesson Españoltwo-column proof numbered

statements and corresponding reasons that show the argument in a logical order

5.5 prueba de dos columnas declaraciones numeradas y las razones correspondientes que muestran el argumento en orden lógico

two-way frequency table a frequency table that shows two categories of characteristics, one in rows and the other in columns. Each cell value is a frequency that shows how many times two different characteristics appear together, or how often characteristics are associated with a person, object, or type of item that is being studied.

6.6 tabla de frecuencia de dos vías tabla de frecuencia que muestra dos categorías de características, una en filas y la otra en columnas. Cada valor de celda es una frecuencia que demuestra cuántas veces dos características diferentes aparecen juntas, o con qué frecuencia las características se asocian con una persona, objeto, o tipo de elemento que se está analizando.

Uuniform probability model a

probability model in which all the outcomes of an experiment are assumed to be equally likely

6.3 modelo de probabilidad uniforme modelo de probabilidad en el que se presume que todos los resultados de un experimento son igualmente probables

union a set whose elements are in at least one of two other sets. The union of sets A and B, denoted by A B∪ , is the set of elements that are in either A or B or both A and B.

6.1 unión conjunto cuyos elementos están al menos en uno de otros dos conjuntos. La unión de los conjuntos A y B, indicada por A B∪ , es el conjunto de elementos que están en A o en B, o a la vez en A y B.

universal set a set of all elements that are being considered in a particular situation. In a probability experiment, the universal set is the sample space.

6.1 conjunto universal conjunto de todos los elementos que se consideran en una situación particular. En un experimento de probabilidad, el conjunto universal es el espacio de muestreo.

V variable a letter used to represent a value

or unknown quantity that can change or vary

2.2 variable letra que se utiliza para representar un valor o cantidad desconocida que puede cambiar o variar

Venn diagram a diagram that shows how two or more sets in a universal set are related

6.1 diagrama de Venn diagrama que muestra cómo se relacionan dos o más conjuntos en un conjunto universal

vertex angle angle formed by the legs of an isosceles triangle

1.4 ángulo vértice ángulo formado por los catetos de un triángulo isósceles


Glossary

Glossary

English Unit/Lesson Españolvertex form a quadratic function written

as f(x) = a(x – h)2 + k, where the vertex of the parabola is the point (h, k); the form of a quadratic equation where the vertex can be read directly from the equation

2.5 2.6

fórmula de vértice función cuadrática que se expresa como f(x) = a(x – h)2 + k, donde el vértice de la parábola es el punto (h, k); forma de una ecuación cuadrática en la que el vértice se puede leer directamente de la ecuación

vertex of a parabola the point on a parabola that is closest to the directrix and lies on the axis of symmetry; the point at which the curve changes direction; the maximum or minimum

2.10 vértice de una parábola punto en una parábola que está más cercano a la directriz y se ubica sobre el eje de simetría; punto en el que la curva cambia de dirección; el máximo o mínimo

vertical compression squeezing of the parabola toward the x-axis

2.9 compresión vertical contracción de la parábola hacia el eje x

vertical stretch pulling of the parabola and stretching it away from the x-axis

2.9 estiramiento vertical jalar y estirar la parábola lejos del eje x

W whole numbers the set of positive

integers and 0: {0, 1, 2, 3, ...}4.2 números enteros conjunto de enteros

positivos que incluye el 0: {0, 1, 2, 3, ...}wholly imaginary a complex number

that has a real part equal to 0; written in the form a + bi, where a and b are real numbers, i is the imaginary unit, a = 0, and b ≠ 0: 0 + bi

3.6 totalmente imaginario número complejo que tiene una parte real igual a 0; se expresa en la forma a + bi, donde a y b son números reales, i es la unidad imaginaria, a = 0, y b ≠ 0: 0 + bi

wholly real a complex number that has an imaginary part equal to 0; written in the form a + bi, where a and b are real numbers, i is the imaginary unit, b = 0, and a ≠ 0: a + 0i

3.6 totalmente real número complejo que tiene una parte imaginaria igual a 0; se expresa en la forma a + bi, donde a y b son números reales, i es la unidad imaginaria, b = 0, y a ≠ 0: a + 0i

Xx-intercept the point at which the graph

crosses the x-axis; written as (x, 0)2.10 intercepto de x punto en el que el gráfico

cruza el eje x; se expresa como (x, 0)

Yy-intercept the point at which the graph

crosses the y-axis; written as (0, y)2.10 intercepto de y punto en el que el gráfico

cruza el eje y; se expresa como (0, y)

ZZero Product Property If the product

of two factors is 0, then at least one of the factors is 0.

3.4 Propiedad de producto cero Si el producto de dos factores es 0, entonces al menos uno de los factores es 0.


CustomTeacher ResourceUnit 2: Graphing

Quadratic Functions

North CarolinaMath 2

© Common Core State Standards. Copyright 2010. National Governor’s Association Center for Best Practices and

Council of Chief State School Offi cers. All rights reserved.

1 2 3 4 5 6 7 8 9 10

ISBN 978-0-8251-9017-9

Copyright © 2020

J. Weston Walch, Publisher

Portland, ME 04103

www.walch.com

Printed in the United States of America

The classroom teacher may reproduce these materials for classroom use only.The reproduction of any part for an entire school or school system is strictly prohibited.

No part of this publication may be transmitted, stored, or recorded in any formwithout written permission from the publisher.

This program was developed and reviewed by experienced math educators who have both academic and professional backgrounds in mathematics. This ensures: freedom from mathematical errors, grade level

appropriateness, freedom from bias, and freedom from unnecessary language complexity.

Developers and reviewers include:

Joyce Hale

Vanessa Sylvester

Jasmine Owens

Ruth Estabrook

Shelly Northrop Sommer

Joanne Whitley

Robert Leichner

Michelle Adams

Marie Vrablic

Kaithlyn Hollister

Carrisa Johnson-Scott

Joseph Nicholson

Samantha Carter

Tiffany Fele

Zachary Lien

Valerie Ackley

Laura McPartland

Cameron Larkins

Frederick Becker

Jennifer Blair

Mike May, S.J.

James Quinlan

Lenore Horner

Pamela Rawson

Nancy Pierce

Jane Mando

Kim Brady

Lynze Greathouse

iiiNorth Carolina Math 2 Custom Teacher Resource

© Walch Education

Table of ContentsUnit 2: Graphing Quadratic Functions

Unit 2 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vLesson 2.1: Identifying the Domain and Range of a Quadratic Function (F–IF.4•) . . . . . . . . . . U2-1Lesson 2.2: Identifying Terms, Factors, and Coefficients (A–SSE.1a•) . . . . . . . . . . . . . . . . . . . . U2-22Lesson 2.3: Interpreting Complicated Expressions (A–SSE.1b•) . . . . . . . . . . . . . . . . . . . . . . . . U2-44Lesson 2.4: Quadratic Modeling (A–CED.2•, F–BF.1•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-67Lesson 2.5: Interpreting Key Features of Quadratic Functions (F–IF.4•) . . . . . . . . . . . . . . . . . . U2-93Lesson 2.6: Interpreting Various Forms of Quadratic Functions (F–IF.8) . . . . . . . . . . . . . . . . . U2-122Lesson 2.7: Comparing Properties of Quadratic Functions

Given in Different Forms (F–IF.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-159Lesson 2.8: Replacing f(x) with f(x) + k and f(x + k) (F–BF.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-180Lesson 2.9: Replacing f(x) with k • f(x) and f(k • x) (F–BF.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-210 Lesson 2.10: Interpreting Quadratic Functions (F–IF.4•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-244Lesson 2.11: Quadratic Regression (A–CED.2•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-282

Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-305

Station ActivitiesSet 1: Graphing Quadratic Equations (F–IF.7•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-323Set 2: Quadratic Transformations in Vertex Form (F–IF.7•, F–IF.8•, F–BF.3) . . . . . . . . . . . . U2-337



vNorth Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource

Unit 2 Resources© Walch Education© Walch Education

Unit 2 ResourcesInstruction

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS

North Carolina Math 2 Standards

A–CED.2 Create and graph equations in two variables to represent quadratic, square root and inverse variation relationships between quantities.★

A–SSE.1 Interpret expressions that represent a quantity in terms of its context.★

a. Identify and interpret parts of a quadratic, square root, inverse variation, or right triangle trigonometric expression, including terms, factors, coefficients, radicands, and exponents.

b. Interpret quadratic and square root expressions made of multiple parts as a combination of single entities to give meaning in terms of a context.

F–BF.1 Write a function that describes a relationship between two quantities by building quadratic functions with real solution(s) and inverse variation functions given a graph, a description of a relationship, or ordered pairs (include reading these from a table).★

F–BF.3 Understand the effects of the graphical and tabular representations of a linear, quadratic, square root, and inverse variation function flinear, quadratic, square root, and inverse variation function flinear, quadratic, square root, and inverse variation function with f with f k • k • k f • f • (f(f x(x( ), f(f(f x(x( ) + k, f, f, (f(f x(x( + x + x k) for specific values of k (both positive and negative).k (both positive and negative).k

F–IF.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: domain and range, rate of change, symmetries, and end behavior.★

F–IF.7 Analyze quadratic, square root, and inverse variation functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; maximums and minimums; symmetries; and end behavior.★

F–IF.8 Use equivalent expressions to reveal and explain different properties of a function by developing and using the process of completing the square to identify the zeros, extreme values, and symmetry in graphs and tables representing quadratic functions, and interpret these in terms of a context.

F–IF.9 Compare key features of two functions (linear, quadratic, square root, or inverse variation functions) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

viNorth Carolina Math 2North Carolina Math 2 Custom Teacher Resource Custom Teacher ResourceUnit 2 ResourcesUnit 2 Resources

© Walch Education© Walch Education

UNIT 2 • GRAPHING QUADRATIC FUNCTIONSUnit 2 Resources

InstructionEssential Questions

1. How are the domain and range of a quadratic function related?

2. How are quadratic expressions and quadratic equations alike? How are they different?

3. How does changing the value of a coefficient, constant, or variable in an expression change the value of the expression?

4. In what situations is it appropriate to use a quadratic model?

5. How do you find the vertex of a quadratic function?

6. What information can be gathered by analyzing the key features of a quadratic function?

7. What properties must be true for a function to be identified as odd, even, or neither?

8. How is a function f(f(f x(x( ) affected when multiplied by a constant k?

9. How is a function f(f(f x(x( ) affected when x is multiplied by a constant k?

10. What are the effects of k on the graph of f on the graph of f on the graph of (f(f x(x( + k) compared to f(f(f x(x( )?

11. What are the effects of k on the graph of f on the graph of f on the graph of (f(f x(x( ) + k compared to f(f(f x(x( )?

12. How are the graphs of quadratic functions related to their equations?

13. How do you use quadratic equations to model data?

axis of symmetry of a parabola (2.10)

binomial (2.2)

coefficient (2.2)

concave down (2.5)

concave up (2.5)

concavity (2.5)

constant term (2.2)

decreasing (2.5)

domain (2.1)end behavior (2.5)

extrema (2.5, 2.10)

factor (noun) (2.2)

factored form of a quadratic function (2.6)

horizontal compression (2.9)

horizontal stretch (2.9)

increasing (2.5)

intercept (2.10)

intercept form of a quadratic function (2.6)

key features of a quadratic function (2.5)

like terms (2.2)

maximum (2.10)

minimum (2.10)

monomial (2.2)

parabola (2.10)

polynomial (2.2)

quadratic equation (2.2)

quadratic expression (2.2)

quadratic function (2.10)

range (2.1)

standard form of a quadratic function (2.5, 2.7, 2.10)

term (2.2)

transformation (2.8)

translation (2.8)

trinomial (2.2)

variable (2.2)

vertex form of a quadratic function (2.5, 2.6)

vertex of a parabola (2.10)

vertical compression (2.9)

vertical stretch (2.9)

x-intercept x-intercept x (2.10)

y-intercept y-intercept y (2.10)

WORDS TO KNOW

viiNorth Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource

Unit 2 Resources© Walch Education© Walch Education


InstructionRecommended Resources

• ChiliMath. “Finding the Domain and Range of a Function.”

http://www.walch.com/rr/00112

This website provides a summary of finding the domain and range for various types of functions as well as practice problems. Illustrated examples walk through finding the domain and range for different situations.

• IXL Learning. “Characteristics of Quadratic Functions.”


Users can practice determining the key features of graphs of quadratic functions, including vertices, y-intercepts, maximums, and more. Instant feedback is provided along with explanations for incorrect answers.

• IXL Learning. “Write Linear, Quadratic, and Exponential Functions.”


This website provides practice and instant feedback on determining linear, exponential, or quadratic equations given a table of data.

• JamesRahn.com. “Rate of Change.”


This website provides a summary, practice, and an answer key for problems related to average rate of change. The intended audience is precalculus and calculus students; however, the summary is written so students of all levels feel comfortable exploring the concept.

• MathIsFun.com. “Algebra—Basic Definitions.”


This website gives an overview of the important vocabulary for this lesson. Color-coded expressions help users visualize the differences between similar terms.

• MathIsFun.com. “Even and Odd Functions.”


This website provides a summary and practice problems for even functions, odd functions, and functions that are neither odd nor even. The site also illustrates the differences in behavior for each type of function.

• PhET Interactive Simulations. “Equation Grapher 2.02.”


This website allows users to compare the graphs of various self-created equations.

viiiNorth Carolina Math 2North Carolina Math 2 Custom Teacher Resource Custom Teacher ResourceUnit 2 ResourcesUnit 2 Resources



Instruction

• West Texas A&M University Virtual Math Lab. “Graphs of Quadratic Functions.”


This tutorial offers a review and examples for writing and graphing quadratic functions in different forms, as well as practice problems with worked solutions for reference.

Conceptual Activities• Desmos. “Card Sort: Parabolas.”


In this activity, students will find the shape of a parabola by using its form to reveal its characteristics. The activity begins with a review of both the characteristics and forms of a parabola. Later, students will determine characteristics of the graph of a parabola given in standard form, vertex form, or intercept form.

• Desmos. “Free-Range Functions.”


This activity challenges students to strengthen their ideas about the range of quadratic functions.

• Desmos. “Marbleslides: Parabolas.”


In this activity, students will rapidly become fluent in transforming, translating, and restricting the domains of parabolas by manipulating their symbolic form to cause marbles to slide down the parabola through stars.

• Desmos. “Polygraph: Parabolas.”


In this paired game, one student picks a single parabola from a collection and the other student asks yes/no questions in an attempt to guess which parabola was chosen. The game provides students with a reason for noticing important features of parabolas, which in turn provides a need for words to name the features. “Does your graph cross the x-axis twice?” is a common question during play that leads to the naming roots, for example.

• Desmos. “Will It Hit the Hoop?”


In this activity, students predict whether various basketball shots will go through the hoop, and then model these shots with parabolas to check their predictions.

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–IF.4★

Lesson 2.1: Identifying the Domain and Range of a Quadratic Function

Name: Date:


Warm-Up 2.1Gina works in a video game store over her summer break. She earns $8 per hour plus commission on the video games and gaming consoles she sells. This month, her store offered an incentive of an extra day off to the employee who sells the most copies of a certain new game. She makes $3 in commission for each game sold.

1. Write a linear model to represent Gina’s take-home pay as a function of the number of games she sells in one 8-hour workday.

2. What is a reasonable domain for this function?

3. What does the domain represent?

U2-1© Walch Education North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource

2.1


Lesson 2.1: Identifying the Domain and Range of a Quadratic FunctionInstruction


Warm-Up 2.1 Debrief1. W rite a linear model to represent Gina’s take-home pay as a function of the number of games

she sells in one 8-hour workday.

y = 3x + 64, where 3 is the rate of change of Gina’s take-home pay, x represents the number of games she sells, and 64 is the total hourly pay she receives for working 8 hours.

2. What is a reasonable domain for this function?

A reasonable domain i s all whole numbers, since selling a negative number of games is impossible, as is selling fractional parts of games.

3. What does the domain represent?

The domain represents the possible number of games that Gina could sell in one 8-hour workday.

Connection to the Lesson

• Students will extend their knowledge of identifying and interpreting the domain of a linear function to the domain of a quadratic function.

North Carolina Math 2 Standard


U2-2© Walch EducationNorth Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource

2.1



IntroductionThe domain of a function is all input values that satisfy the function without restriction. Often, the domain is all real numbers; however, with application problems, there may be important restrictions. This, in turn, will affect the function’s range, which is the set of all outputs of the function. For example, negative or fractional input values may not make sense in the context of the problem, which could result in a range of only positive whole number values. In this lesson, you will practice identifying the domain and range of a quadratic function given an equation, a graph, or a real-world context.

Key Concepts

• When a quadratic function does not have a specified interval and is not an application of a real-world situation, its domain is all real numbers.

• In other words, the input values exist from negative infinity to infinity: −∞< <∞x .

• When the domain of a quadratic function is all real numbers, its range will be a subset of all real numbers.

• If a quadratic function f (f (f x(x( ) has a local maximum, M, the range will be all real numbers less M, the range will be all real numbers less Mthan or equal to M. In other words, the output values exist from negative infinity to M: –∞ < f (f (f x(x( ) ≤ M.

• If a quadratic function f (f (f x(x( ) has a local minimum, n, the range will be all real numbers greater than or equal to n. In other words, the output values exist from n to infinity: n ≤ f (f (f x(x( ) < ∞.

• If the domain of a quadratic function is restricted, the range will be restricted too. You can determine the range by evaluating the function.

Prerequisite Skills

This lesson requires the use of the following skills:

• using graphing technology to interpret quadratic functions (F–I F.4★)

• understanding the difference between domain and range (F–IF.1)


• identifying domain with output values rather than with input values


2.1



E xample 1

Describe the domain and range of the quadratic function g(x(x( ) = 1.5x2.

1. Sketch a graph of the function.

x

y

–3 –2 –1 0 1 2 3

2

4

6

8

2. Describe what will happen if the function continues.

Looking at the function, you can see that the function will continue to increase upward and the function will continue to grow wider.

Growing wider without end means that the domain of this function is all real numbers, or −∞< <∞x .

Notice that the graph’s lowest y-value is 0. Since the function continues to increase upward, the range of the function will be all real numbers greater than or equal to 0, or 0 ≤ g(x(x( ) < ∞.

Guided Practice 2.1


2.1



E xample 2

Describe the domain and range of the following function, f(f(f x(x( ).

x

y

–3 –2 –1 0 1 2 3

–8

–6

–4

–2

1. Describe what happens to the width of the parabola as x approaches positive and negative infinity.

The function continues to grow wider and wider as x approaches both positive and negative infinity.

2. Determine the domain of the function.

Growing wider without end means that the domain of this function is all real numbers, or −∞< <∞x .

3. Describe what happens to the y-values of the parabola as x approaches positive and negative infinity.

The function starts at 0 at x = 0, then becomes more negative at either end as x approaches both positive and negative infinity.


2.1



4. Determine the range of the function.

The function starts at 0 and grows more negative. This means the range of this function is –∞ < f(f(f x(x( ) ≤ 0.

Example 3

Amit is a diver on the swim team. Today he’s practicing by jumping off a 14-foot platform into the pool. Amit’s height in feet above the water is modeled by f(f(f x(x( ) = –16x2 + 14, where x is the time in seconds after he leaves the platform. About how long will it take Amit to reach the water? Describe the domain and range of this function.

1. Sketch a graph of the situation.

x

y

–1.5 –1 –0.5 0 0.5 1 1.5

2

4

6

8

10

12

14

2. Identify the x-intercepts of the function.

Using graphing technology, the x-intercepts occur when x ≈ –0.94 and x ≈ 0.94.


2.1



3. Determine a reasonable domain for this context.

The x-values in this problem represent the time in seconds after Amit leaves the platform. In this context, negative values of x do not make sense.

The positive x-intercept represents the moment Amit makes contact with the water. It will take him 0.94 second to reach the water.

A reasonable domain of this context is 0 to 0.94 seconds, or 0 ≤ x ≤ 0.94.

4. Determine the range of this function.

The domain of this function represents the time when Amit leaves the platform to the time he reaches the water. The corresponding range would be Amit’s height above the water as he jumps from the platform and lands in the water.

Amit’s starting height is 14 feet above the water, and when he lands in the water, the height is 0. A reasonable range is 0 to 14 feet, or 0 ≤ f(f(f x(x( ) ≤ 14.


2.1



Name: Date:

Problem-Based Task 2.1: Window WashersA window washer tossed a wet sponge from a height of 10 meters above ground level t o his coworker above him. The sponge reached its maximum height of 11.25 meters exactly 0.5 second later, but the coworker did not catch the sponge and it fell to the ground. After how many seco nds did the sponge fall to a height of 10 meters? What is the span of time that the sponge was in the air?


2.1

After how many seconds did

the sponge fall to a height of

10 meters?10 meters?1What is the

span of time that the sponge was in

the air?

SMP1 ✓ 2 ✓3 4 5 6 ✓7 8



Name: Date:

Problem-Based Task 2.1: Window Washers

Coachinga. What are the known values?

b. What are the unknown values?

c. What type of model is needed in order to find the unknowns?

d. What is the vertex form of a quadrati c function?

e. How can the known information be substituted into the vertex form of a quadratic function?

f. How can you find the value of a, and what is the quadratic model that represents this situation?

g. After how many seconds did the sponge fall to a height of 10 meters?

h. What is a reasonable domain of the function? H ow long was the spon ge in the air?


2.1



Problem-Based Task 2.1: Window WashersProblem-Based Task 2.1: Window Washers

Coaching Sample Responsesa. What are the known values?

We know the maximum height of the sponge, 11.25 meters.

The sponge reached the maximum height after 0.5 second.

These two values represent the vertex, which can be written as the point (0.5, 11.25).

We also know the starting point of the sponge, (0, 10).

b. What are the unknown values?

We do not know how long it took for the sponge to fall to a height of 10 meters, and we do not know the span of time that the sponge was in the air.

c. What type of model is needed in order to find the unknowns?

A quadratic model will help determine the unknown values.

d. What is the vertex form of a quadratic function?

The vertex form of a quadratic function is f(f(f x(x( ) = a(x(x( – h)2 + k, where (h, k) is the vertex, x represents the time in seconds, and f(f(f x(x( ) is the height of the sponge above the ground.

e. How can th e known information be substituted into the vertex form of a quadratic function?

Substitute the vertex, (0.5, 11.25), and the starting point, (0, 10), into the vertex form of a quadratic functio n.

f(f(f x(x( ) = a(x(x( – h)2 + k

(10) = a[(0) – (0.5)]2 + (11.25)

10 = 0.25a + 11.2 5

f. How can you find the value of a, and what is the quadratic model that represents this situation?

Solve the equation for a. The n, substitute the values of a, h, and k into f(f(f x(x( ) = a(x(x( – h)2 + k.

10 = 0.25a + 11.25

–1.25 = 0.25a

a = –5

Recall that h = 0.5 and k = 11.25. Therefore, the quadratic model is f(f(f x(x( ) = –5(x) = –5(x) = –5( – 0.5)2 + 11.25.


2.1



g. After how many seconds did the sponge fall to a height of 10 meters?

Use symmetry to determine how many seconds it took after the sponge was thrown for it to fall to a height of 10 meters.

The starting point, (0, 10), is 0.5 unit to the left of the axis of symmetry, x = 0.5. Therefore, the point that is 0.5 unit to the right of the axis of symm etry is (1, 10). Check by substituting (1, 10) into the model f(f(f x(x( ) = –5(x) = –5(x) = –5( – 0.5)2 + 11.25. The result, 10 = 10, is a true statement.

The sponge fell to a height of 10 meters 1 second after being thrown.

h. What is a reasonable domain of the function? How long was the sponge in the air?

A reasonable domain is the interval of time that the sponge was in the air. This interval begins a t x = 0, which represents when the sponge was thrown, and ends at the greatest x-intercept, which represents when the sponge hit the ground. Use graphing technology or algebra to find the greatest x-intercept. The sponge hit the ground at x = 2, or 2 seconds after being tossed. Therefore, the sponge was in the air for 2 seconds. This span of time can be represented as an inequ ality: 0 < x < 2.


Select one or more of the essential questions for a class discussion or as a journal entry prompt.


2.1

Implementation Guide



Task OverviewFocus

How can a model be determined from the given information of window washers? What is a reasonable domain for the time a sponge is tossed? Students will take given information from a problem and create a quadratic model. Then they’ll take the model and determine the time the sponge reaches a certain height and a reasonable amount of time the sponge is in the air.

This activity will provide practice with:

• analyzing the known and unknown information

• interpreting information to create a quadratic model

• examining the equation to determine the domain of a quadratic model

Introduction

This task should be used to apply the concept of a creating a quadratic model from given information. The domain and the meaning of specific ordered pairs is also discussed in this task. It is best to implement this task after students are able to determine known and unknown information and create a quadratic model, as well as finding values from the model and the domain for a scenario.

Begin by reading the problem and clarifying what information is known and unknown as well as the model needed for the problem. Using the general model, the values can be determined for the model of the wet sponge thrown by the window washer. From there, the time when the object reaches 10 meters and a reasonable domain can be determined from the quadratic model. Review the following term:

window washer a person whose job is to wash windows

Problem-Based Task 2.1 Implementation Guide: Window WashersNorth Carolina Math 2 Standard



2.1




Facilitating the TaskStandards for Mathematical Practice

Many or all of the Standards for Mathematical Practice are addressed through this activity. As students work, reinforce the importance of the following standards:

• SMP 1: Make sense of problems and persevere in solving them.

Some students will have difficulty determining the known and unknown information and creating a quadratic model. Encourage the students to identify key statements given in the problem. If necessary, provide various models (linear, quadratic, etc.) and allow the student to choose the appropriate model.

• SMP 2: Reason abstractly and quantitatively.

Some students will be able to find the domain without setting up the quadratic model. Allow this method as long as students can justify their explanations. Help students with the process of taking the verbal information and developing a quadratic model. Some students may use a table to help determine the domain of the model.

• SMP 6: Attend to precision.

Students will explain in their own words on what the meaning of the known and unknown information is in the context of the problem. Students may need to be provided the vertex form of a quadratic equation to assist in solving the problem. The accuracy of the model is necessary to answer questions e, f, g, and h.

Addressing Common Errors/Misconceptions

Be aware of common student errors and misconceptions associated with this task:

• identifying domain with output values instead of input values

The input values of the equation are the x-values instead of the y-values.

• using a form of a quadratic model other than vertex form

Provide students with the vertex form of a quadratic equation or a list for students to choose from.

• incorrectly calculating the number of seconds for the sponge to be at a height of 10 meters

Advise students to find the vertex first or create a table of values from the model.

Monitoring and Coaching

Ask questions as you circulate to monitor student understanding. Suggestions:

• What is the given information in the problem? (Answer: the maximum height, the vertex, and the initial height of the sponge)


2.1




• What is the information that is not known? (Answer: other heights of the sponge besides the initial and the vertex, as well as the amount of time the sponge is in the air)

• What is the vertex form of a quadratic equation? (Answer: The equation is y = a(x(x( – h)2 + k, where (h, k) is the vertex.)

• How can the a-value be located for the model? (Answer: Substitute the vertex and the initial height ordered pairs into the equation and solve for a.)

• How can the height of 10 meters be determined when the sponge is falling? (Answer: Use a table or the axis of symmetry to find when the sponge is at a height of 10 feet.)

• How is the domain determined for the equation? (Answer: The domain is determined from the starting point of the sponge until the sponge lands on the ground.)

• Ask students if they have questions about areas of the problem that are not clearly understood, and allow students to clarify these points for each other.

Debriefing the TaskCompare the strategies and the explanations used by various groups. Ask students to share the strategies they used and ask why they used these methods.

Connecting to Key Concepts

Make explicit connections to key concepts:

• A quadratic model can be created given two points. In this scenario, the vertex form of an equation is used to help determine the model.

• Using the given information, the data is turned into ordered pairs and identify as an initial height and the vertex.

• The applicable domain for this task is the time when the object is thrown into the air until the object hits the ground.

Extending the Task

To extend the task, add the following information. For example:

• A graph of the model is used to demonstrate the path of the thrown sponge.

• Find the range for the height of the sponge.

• Determine other values of the height of the sponge at a specific time.

• Change the initial height and/or vertex to create a new quadratic model.


2.1




Connecting to Standards for Mathematical Practice

Make explicit connections to the Standards for Mathematical Practice described previously for this task.

• For SMP 1, ASK: “How did you demonstrate perseverance?” (Answer: I kept reviewing the problem until I determined the known and unknown information. I focused on one question at a time until the solution or explanation was completed.)

• For SMP 2, ASK: “How did you reason abstractly and quantitatively? Which of your strategies represent abstract reasoning?” (Answer: I wrote the quadratic model from the given information.) “Which of your strategies represent quantitative reasoning?” (Answer:I used the vertex to help determine when the sponge would be at 10 feet off the ground. By using the x-intercept and y-intercept, I was able to find the domain of the equation.)

• For SMP 6, ASK: “How did you make sure you attended to precision?” (Answer: I was able to explain the steps of the problem in order from determining the known information, to the type of model, to finding the correct model, and obtaining information from the model.)

Alternate Strategies or Solutions

A graph of the quadratic model can be created as shown:

0

y

x

1 20.5 1.5 2.5–0.5–1

10

–2

8

6

4

2

12

Using the graph, questions g and h can be answered visually.

Technology

Students can use scientific calculators and graphing utilities.


2.1



Name: Date:

Practice 2.1: Identifying the Domain and Range of a Quadratic FunctionPractice 2.1: Identifying the Domain and Range of a Quadratic FunctionU se graphing technology to determine the domain and range of each quadratic function.

1. y = –x = –x = – 2 + 7x + 1

2. y x x= − + −3

521 32

3. f(f(f x(x( ) = 4x2 + 5x – 12

4. g(x(x( ) = x2 + 12x – 8

Describe the domain and range of each of the following functions in words and as an inequality.

5.

x

y

–1.5 –1 –0.5 0 0.5 1 1.5

–1

1

2

3

4

5

continued

AA


2.1



Name: Date:

6.

2 4 6 8 10

–8

–10

–6

–4

–2

2

0

y

x

7.

x

y

–3 –2 –1 0 1

–2

2

4

continued


2.1



Name: Date:

Use the given information to solve the following problems.

8. A soccer ball is kicked from the ground and travels a parabolic path. The path can be modeled by the function h(t) = –5t) = –5t t2t2t + 19.5t, where h(t) is the height of the soccer ball in meters above the t) is the height of the soccer ball in meters above the tground t seconds after being kicked. Assuming the ball lands on level ground, about how long is t seconds after being kicked. Assuming the ball lands on level ground, about how long is tthe ball in the air?

9. A golf ball is shot from the ground using a practice cannon and travels a parabolic path. The path of the ball can be modeled by the function h(t) = –16t) = –16t t2t2t + 150t, where h(t) is the height of t) is the height of tthe golf ball in meters above the ground t seconds after being shot. Assuming the ball lands on t seconds after being shot. Assuming the ball lands on tlevel ground, about how long does it take the golf ball to hit the ground?

10. The senior class is putting on a talent show to raise money for their senior trip. In the past, the profit from the talent show could be modeled by the function P(x(x( ) = –16x2 + 600x – 4000, where x represents the ticket price in dollars. What are a reasonable domain and range for this function? For what domain value will the profits be maximized?


2.1



Name: Date:

continued

BPractice 2.1: Identifying the Domain and Range of a Quadratic FunctionUse graphin g technology to describe the domain and range of each quadratic function.

1. y = 3x2 – 4x + 2

2. y x x= − − −7

432 52

3. f(f(f x(x( ) = 6x2 + 9x – 1

4. g(x(x( ) = 2x2 – 12x – 9

Describe the domain and range of each of the following functions in words and as an inequality.

5.

x

y

–8 –6 –4 –2 0

2

4

6

8

10

12

14

16


2.1



Name: Date:

6.

x

y

2 4 6 8

–14

–16

–12

–10

–8

–6

–4

–2

2

0

7.

x

y

–4 –2 0 2 4

–10

–8

–6

–4

–2

2

continued


2.1



Name: Date:

Use the given information to solve the following problems.

8. A kickball is kicked from the ground and travels a parabolic path. The path can be modeled by the function h(t) = –2t) = –2t t2t2t + 20t, where h(t) is the height of the kickball in feet above the ground t) is the height of the kickball in feet above the ground tt seconds after being kicked. Assuming the ball lands on level ground, about how long is the ball t seconds after being kicked. Assuming the ball lands on level ground, about how long is the ball tin the air?

9. The height of baseballs thrown by an automatic baseball-pitching machine can be modeled by the function h(t) = –16t) = –16t t2t2t + 48t + 3.5, where h(t) is the height of the ballt) is the height of the ballt t seconds after being released. If the batter misses the ball, how long does it take the ball to hit the ground? Assume there is no net or catcher behind the plate to stop the ball.

10. A movie theater manager believes that the theater loses money as ticket prices go up. The theater’s average weekly sales can be modeled by the quadratic function R(x(x( ) = –700x2 + 7700x + 245,000, where R(x(x( ) is the weekly revenue in dollars and x is the number of $0.50 increases in price. For what number of $0.50 increases will the theater continue to produce revenue? After how many $0.50 increases will the theater receive the greatest revenue?


2.1

U2-22

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS A–SSE.1a★

Lesson 2.2: Identifying Terms, Factors, and Coefficients

Name: Date:

North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource2.2

© Walch Education© Walch Education© Walch Education© Walch Education


Warm-Up 2.2

In order to pay for upkeep of a local highway, the transportation department has set up tollbooths at each of the highway’s exits. Drivers are charged a toll of $1.20 for use of the highway, and are then charged an additional $0.04 per mile driven.

1. Write an algebraic expression that can be used to represent the total toll charged if m represents the number of miles driven.

2. 2. What is the toll charged to drive 14 miles on this highway?What is the toll charged to drive 14 miles on this highway?

U2-23North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource

2.2© Walch Education© Walch Education

Instruction




Warm-Up 2.2 Debrief

1. Write an algebraic expression that can be used to represent the total toll charged if m represents the number of miles driven.

The constant is 1.20, the flat fee for the use of the highway.

The variable is m, the number of miles.

The coefficient is 0.04, the charge per mile driven.

Therefore, the toll can be written as the expression 1.20 + 0.04m.

2. What is the toll charged to drive 14 miles on this highway?

Substitute 14 for m and evaluate the expression.

1.20 + 0.04(14) = 1.20 + 0.56 = 1.76

The toll charged for driving 14 miles on the highway is $1.76.


• Students will extend their knowledge of linear models in context to interpreting quadratic models in context.

• Students will be asked to identify parts of expressions given in context.

• Students will translate verbal scenarios into quadratic expressions.




U2-24North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource2.2


Instruction



Introduction

Algebraic expressions are mathematical representatio ns that include numbers, operations, and variables to represent a number or quantity. We know that a variable is a letter used to represent a value or unknown quantity that can change or vary. We have seen several linear expressions, such as 2x + 1, in which the highest power of the variable x is 1. In this lesson, we will look at expressions in which the highest power of the variable is 2.

Key Concepts

• A qu adratic expression can be written in the form ax2 + bx + c, where x is the variable, a,b, and c are r eal numbers, and c are r eal numbers, and c a ≠ 0. In a quadratic expression, the highest power of the variable is 2.

• Both b and c can be any real number, but c can be any real number, but c a cannot be equal to 0 because quadratic expressions must contain a squared term.

• An example of a quadratic expression is 4x2 + 6x – 2. When a quadratic expression is set equal to 0, as in 4x2 + 6x – 2 = 0, the resulting equation is called a quadratic equation. A quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where c = 0, where c x is the variable, a, b, and c are real numbers, and c are real numbers, and c a ≠ 0. In a quadratic equation, the highest power of the variable is 2.

• The quadratic expression 4x2 + 6x – 2 is made up of many component parts: terms, factors, coefficients, and constant terms.

• A term is a number, a variable, or the product of a number and variable(s).

• There are 3 terms in the given expression: 4x2, 6x, and –2.

• A factor is one of two or more numbers or expressions that are multiplied to produce a product. factor is one of two or more numbers or expressions that are multiplied to produce a product. factorIn the given expression, the factors of 4xIn the given expression, the factors of 4xIn the given expression, the factors of 4 2 are 4 and x2, and the factors of 6x are 6 and x are 6 and x x.

Prerequisite Skills


• translating verbal expressions into algebraic expressions (6.EE.2a)

• evaluating expression s using the order of operations (5.OA.1)

• adding and subtracting polynomials (A–APR.2)



Instruction



• The number multiplied by a variable in an algebraic expression is called a coefficient. In the given expression, the coefficient o f 4x2 is 4 and the coefficient of 6x is 6.

• The coefficient of the term x is 1 and the coefficient of the term –x is 1 and the coefficient of the term –x x is 1 and the coefficient of the term –x is 1 and the coefficient of the term – is –1; that is, x is –1; that is, x x = 1x an d x an d x–x–x– = –1x = –1x x.

• A term that does not contain a variable is called a constant term because the value of the term does not change. In the given expression, –2 is the constant term.

• Two or more terms that contain the same variables raised to the same power are called like terms. Like terms can be combined by adding. Be sure to follow the order of operations when combining like terms.

• Th e given quadratic expression, 4x2 + 6x – 2, has no like terms.

• The quadratic expression 9x2 – 8x2 + 2x has two like terms: 9x2 and –8x2. After simplifying the expression by combining like terms (adding 9x2 and –8x2), the result is x2 + 2x.

• A monomial is a number, a variable, or the product of a number and variable(s). We can also think of a monomial as an expression containing only one term. 5x2 is an example of a monomial.

• A binomial is a polynomial with two terms. 6x + 9 is an example of a binomial.

• A trinom ial is a polynomial with three terms. 4x2 + 6x – 2 is an example of a trinomial.

• A polynomial is a mono mial or the sum of monomials. A polynomial can have any number of terms. 5x2, 6x + 9, and 4x2 + 6x – 2 are all polynomials.


• not following the order of operations

• incorrectly identifying like terms

• inaccurately combining terms involving subtraction

• incorrectly combining terms by changing exponents

U2-26



Name: Date:



Use the given information to complete problems 1–3.

1. Identify the terms, coefficients, and the constant term of –6x2 + 2x – 2.

2. Simplify the expression 6x3 – 2x(x(x( 2 – 4) + 4x2(2 – x) and classify it as a monomial, binomial, or trinomial.

3. Write a quadratic expression that contains two terms, a coefficient of 4, and a constant of 10.

For problems 4–6, determine whether each expression is a quadratic expression.

4. (x (x ( + 1)(5x – 2)

5. (x (x ( + 2)(4x + 2)

6. (x(x( 2 + 2x) – x(x (x ( + 4)

continued

Scaffolded Practice 2.2: Identifying Terms, Factors, and Coefficients

U2-27

Date:Name:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS A–SSE.1a★




For problems 7–10, write an algebraic expression.

7. the product of 4 and the cube of x, decreased by the difference of x minus 4x2

8. half of the difference of 3x minus 5, increased by 2x

9. the volume of a box with the dimensions of x, x – 3, and x2

10. one-third of the perimeter of a triangle with sides equal a, b, and c



Instruction



E xample 1

Identify each term, coefficient, and constant term of 6(xIdentify each term, coefficient, and constant term of 6(xIdentify each term, coefficient, and constant term of 6( – 1) – x(3 – 2x) + 12. Classify the expression as a monomial, binomial, or trinomial. Determine whether it is a quadratic expression.

1. Simplify the expression.

The expression can be simplified by following the order of operations and combining like terms.

6(x6(x6( – 1) – x(3 – 2x) + 12 Original expression

= 6x – 6 – x(3 – 2x) + 12 Distribute 6 over (x) + 12 Distribute 6 over (x) + 12 Distribute 6 over ( – 1).

= 6x – 6 – 3x + 2x2 + 12 Distribute –x + 12 Distribute –x + 12 Distribute – over (3 – 2x).

= 3x + 6 + 2x2 Combine like terms: 6x and –3x; –6 and 12.

= 2x2 + 3x + 6 Rearrange terms so the powers are in descending order.

2. Identify all terms.

There are three terms in the expression: 2x2, 3x, and 6.

3. Identify the coefficients of the variable terms.

The numerical factor in the term 2x2 is 2, and the numerical factor in the term 3x is 3. Therefore, the coefficients are 2 and 3.

4. Identify any constant terms.

The number that is not a factor of a variable in the expression is 6; therefore, 6 is the constant term.

Guided Practice 2.2



Instruction



5. Classify the expression as a monomial, binomial, or trinomial.

The polynomial is a trinomial because it has three terms.

6. Determine whether the expression is a quadratic expression.

The expression is quadratic because the highest power of x in the expression is 2.

Example 2

Translate the verbal expression “take triple the difference of 12 and the square of x, then increase the result by the sum of 3 and x” into an algebraic expression. Identify the terms, coefficients, and constant term(s) of the given expression. Is the expression quadratic?

1. Translate the expression by breaking it down into pieces.

“The difference of 12 and the square of x” translates to (12 – x2).

Triple this expression is 3(12 – x2).

“The sum of 3 and x” translates to (3 + x).

Increasing the original expression by this sum translates to 3(12 – x2) + (3 + x).

2. Simplify the expression.

3(12 – x2) + (3 + x) Expression from the previous step

= 36 – 3x2 + (3 + x) Distribute 3 over (12 – x2).

= –3x2 + x + 39 Combine like terms.

The simplified expression is –3x2 + x + 39.


There are three terms in the expression: –3x2, x, and 39.



Instruction



4. Identify all coefficients.

The numerical factor in the term –3x2 is –3, and the numerical factor in the term x is 1. Therefore, –3 and 1 are coefficients.


The number that is not a factor of a variable in the expression is 39; therefore, 39 is the constant term.


The expression is quadr atic because the highest power of x is 2. x is 2. x

E xample 3

A fence surrounds a park in the shape of a pentagon. The side lengths of the park in feet are given by the expressions 2x2, 3x + 1, 3x + 2, 4x, and 5x – 3. Find an expression for the perimeter of the park. Identify the terms, coefficients, and constant term(s) in your expression. Is the expression quadratic?

1. Find an expression for the perimeter of the park.

Add like terms to find the perimeter, P.

P = 2P = 2P x2 + (3x + 1) + (3x + 1) + (3x x + 2) + 4x + 2) + 4x x + (5x + (5x x – 3) Set up the equation using x – 3) Set up the equation using xthe given expressions.

= 2x2 + 3x + 3x + 4x + 5x + 1 + 2 – 3 Reorder like terms.

= 2x2 + 15x Combine like terms.

The expression for the park’s perimeter is 2x2 + 15x.


There are two terms in this expression: 2x2 and 15x.



Instruction



3. Identify all coefficients.

The numerical factor in the term 2x2 is 2, and the numerical factor in the term 15x is 15. Therefore, 2 and 15 are coefficients.


Every number in the expression is multiplied by a variable; therefore, there is no constant term.


The expression is quadratic because the highest power of x is 2. x is 2. x

U2-32

What are the terms, factors, and coefficients of the quadratic expression that

represents the area of the deck to be

decorated?



Name: Date:



Problem-Based Task 2.2: Deck the Dec k

Shanna wants to decorate the triangular deck behind her house. The base of the triangle is 10 meters less than the altitude. What are the terms, factors, and coefficients of the quadratic expression that represents the area of the deck to be decorated?

SMP1 ✓ 2 3 4 ✓5 6 7 ✓ 8

U2-33





Problem-Based Task 2.2: Deck the Deck

Coaching

a. How can you represent the altitude of the triangle?

b. What algebraic expression represents the base of the triangle in terms of the altitude?

c. What algebraic expression represents the area of the triangle?

d. What units will be used to express the area of the deck?

e. Simplify your expression for the area using the Distributive Property. Is this a quadratic expression?

f. How many terms does the expression from part e include?

g. What are the factors that make up each of the terms?

h. h. What are the coefficients of each term?What are the coefficients of each term?



Instruction



Problem-Based Task 2.2: Deck the Deck

Coaching Sample Responses

a. How can you represent the altitude of the triangle?

Let a represent the altitude of the triangle.

b. What algebraic expression represents the base of the triangle in terms of the altitude?

Since the base of the triangle is 10 meters less than the altitude, it can be represented by the expression a – 10.

c. What algebraic expression represents the area of the triangle?

An exp ression for the area of a triangle is 1

2bh , where b represents the base of the triangle and

h represents the altitude (or height) of the triangle.

Substitute the expressions from parts a and b to write the area of this triangle as 1

210a a( )− .

d. What units will be used to express the area of the deck?

The lengths of the altitude and base are in meters, so the area of the deck will be expressed in square meters.

e. Simplify your expression for the area using the Distributive Property. Is this a quadratic expression?

After simplifying, the expression becomes 1

252a a− . This is a quadratic expression because the

highest power of a is 2.

f. How many terms does the expression from part e include?

There are two terms in the expression 1

252a a− . The terms are

1

22a and –5a.

g. What are the factors that make up each of the terms?

The factors are the quantities that are being multiplied together. The product of 1

22a is made

up of the factors 1

2 and a2. The product of –5a is made up of the factors –5 and a.

h. What are the coefficients of each term?

The coefficient of the term 1

22a is

1

2. The coefficient of the term –5a is –5.

Recommended Closur e Activity







Task Overview

Focus

How can the area of a figure be represented with an algebraic expression? How can the parts of the expression be defined? In this task, students will write an algebraic expression to represent the area of a triangular deck. Then they will identify the terms, factors, and coefficients of the expression.


• writing an algebraic expression for the area of a triangle

• substituting expressions into parts of a formula

• labeling units in an area problem

• simplifying an expression using the Distributive Property

• identifying terms, factors, and coefficients in an algebraic expression

Introduction

This task should be used to explore or apply the skill of interpreting parts of an expression such as terms, factors, and coefficients. Students should already be familiar with translating verbal expressions to algebraic expressions, and simplifying expressions using order of operations and the Distributive Property.

Begin by reading the problem and clarifying the meaning of altitude, coefficient, factor, quadratic expression, and term.

altitude the perpendicular line from a vertex of a figure to its opposite side; the height of a figure

coefficient the number multiplied by a variable in an algebraic expression

factor one of two or more numbers or expressions that when multiplied produce a given product

quadratic expression an equation that can be written in the form ax2 + bx + c = 0, c = 0, cwhere x is the variable; a, b, and c are constants; and c are constants; and c a ≠ 0

term a number, a variable, or the product of a number and variable(s)

Problem-Based Task 2.2 Implementation Guide: Deck the DeckNorth Carolina Math 2 Standard








Facilitating the Task


Many or all of the Standards for Mathematical Practice are addressed in this activity. As students work, reinforce the importance of the following standards:


Students will analyze the given information and construct an algebraic expression that will represent the area of the deck. They will use their knowledge about order of operations and the Distributive Property to simplify the expression, then they will identify the parts of the expression.

• SMP 4: Model with mathematics.

Students will recognize that the scenario can be modeled by an algebraic expression. They will also recognize that a diagram representing the verbal description can be drawn.

• SMP 7: Look for and make use of structure.

Students will recognize the structure of the formula for the area of a triangle as bh1

2 as well

as the structure for a quadratic expression. They will use these structures to substitute the

expressions for the base and the height of the deck, and they will create a simplified quadratic

expression for the area once these expressions are substituted into the formula.



• not following the order of operations

Have students write the order of operations (parentheses, exponents, multiplication/division, addition/subtraction) on top of their papers to use as a reference when working through the task.

• incorrectly identifying like terms

Remind students that like terms are two or more terms that contain the same variables raised to the same power.

• inaccurately combining terms involving subtraction

Remind students to carefully check the signs when combining like terms and distribute the negative sign appropriately.

• incorrectly combining terms by changing exponents

Remind students that if two or more terms contain the same variable(s), the exponents on the variable(s) must be exactly the same before the terms can be combined.








• If students have difficulty beginning the task, encourage them to draw a diagram of a triangle and label the base and the altitude. Ask them, “What are the definitions of the base and the altitude of a triangle?” (Answer: The base of a triangle is the side that is perpendicular to the altitude of the triangle. The altitude is the triangle’s height.)

• As students are writing the algebraic expression to represent the base of the triangle, ask them, “How do you know that subtraction is the operation to use when representing the base of the triangle in terms of the altitude?” (Answer: The given information states that the base of the triangle is 10 meters shorter than the altitude. The word “shorter” means the triangle is 10 meters less than, or smaller than, the altitude, so subtraction is used to represent this.)

• Ask students, “What are the units for the area of the deck, and how do you know this?” (Answer: The area of a figure is represented in square units, since two components are multiplied together. In the task, the dimensions of the deck are measured in meters, so the units for the area are square meters.)

• If students are having difficulty simplifying the expression for the area using the Distributive

Property, ask them to write it out in parts. Ask them, “What are the steps for applying the

Distributive Property to a a1

2( 10)− ?” (Answer: To use the Distributive Property, multiply

a1

2 to both the a term and the –10 term. The result is a a a a a

1

2•

1

2( 10)

1

252+ − = − .)

• If students have difficulty understanding why a a1

252 − is a quadratic expression, ask,

“What is the general form of a quadratic expression? How can a a1

252 − be written in

this form?” (Answer: The general form of a quadratic expression is ax2 + bx + bx + bx c. The expression

a a1

252 − can be written in this form as a a

1

25 02 − + , so the value of c in the expression is 0.)c in the expression is 0.)c

• If students are having difficulty identifying the factors that make up each of the terms in

the expression for the area, encourage them to think of examples of numerical factors, such

as 2 • 3 as factors of 6. Ask them, “What factors are multiplied together to get a product

of a1

22? What factors are multiplied together to get a product of –5a?” (Answer: To get

a product of a1

22 , the factors of

1

2, a, and a are multiplied together. Another combination of

factors is 1

2 and a. To get a product of –5a, the factors of –5 and a are multiplied together.)






• Ask students, “What different components/items can a ‘term’ be made of?” (Answer: Terms can be composed of a number, a variable, or a product of a number and variable(s).)

• Ask students, “Suppose you were given the monomials –3x3y3y3 4z4z4 4 and 6x3y3y3 3z4. Are these like terms? Explain why or why not.” (Answer: No, these are not like terms. Although they have the same three variables, the exponents are different on the y variables, as the first monomial has a power of 4 on the y-variable, and the second monomial has a power of 3 on the y-variable.)


Debriefing the Task

• Ask for volunteers to discuss their plans for constructing the algebraic expression to represent the area of the deck. Discuss the process of identifying the terms, factors, and coefficients of the expression. Encourage students to discuss any difficulties or confusion they experienced when working through the various parts of the task.

• Compare students’ strategies and ways of justifying responses. Focus on the use of precise mathematical language and clarity, specifically when referring to the altitude, terms, factors, and coefficients of the expression.



• A quadratic expression can be written in the form ax2 + bx + c, where x is the variable, and a, b, and c are constants.c are constants.c

In this task, students will observe that the expression they create to represent the area of the

deck, a a1

252 − , can be written in the form of a quadratic expression as a a

1

25 02 − + .

• A term is a number, a variable, or the product of a number and variable(s).

Students will identify that there are two terms in their expression.

• A factor is one of two or more numbers or expressions that, when multiplied, produce a given product.

Students will identify the factors of each term of the expression.

• The number multiplied by a variable in an algebraic expression is called a coefficient.

In this task, students will identify the coefficient of 1

2 in the first term of the expression, and

a coefficient of –5 in the second term.






Extending the Task

• To extend the task, provide students with a different scenario in which they are given binomial expressions to represent the length and width of a rectangle. Ask them to find the expression that represents the area of the rectangle, then ask them to identify the terms, factors, and coefficients of the expression.

• Another option for extending the task is to provide students with a variety of polynomial expressions that have not been simplified. Ask students to work with a partner to simplify the expressions and identify the terms, factors, and coefficients of the expression. Then ask students to create their own polynomial expression, one with at least 5 or 6 terms, and ask their partners to simplify it.



• For SMP 1, ASK: “How did you make sense of the problem or demonstrate perseverance?” (Answer: I made sense of the problem by analyzing the given information and constructing an algebraic expression that represented the area of the deck.)

• For SMP 4, ASK: “How did you use mathematics to model this particular scenario?” (Answer: I modeled the scenario by creating an algebraic expression to represent the given verbal description.)

• For SMP 7, ASK: “How did you look for, and make use of, structure when solving this problem?” (Answer: I used the structure of a quadratic expression to write an expression that represented the area of the deck.)


• Students may choose to draw a diagram representing the triangular deck before they begin the process of writing the algebraic expression. Encourage them to carefully label the base and altitude of the triangle.

• Once the algebraic expression is written, students may choose to substitute values into the expression for the altitude, a. Discuss how this can help them understand the validity of the expression, as they will ensure the value of the base is 10 meters less that the value of the altitude.

Technology

No technology is needed for this task.

U2-40



Name: Date:



continued

Use what you know about the components of expressions to complete problems 1–3.

1. Identify the terms, coefficients, constant term, and factors of 16x2 – 12x + 20.

2. Simplify the expression 3x2 + 2(5 – x2) – 8(x) – 8(x) – 8( 2 + 9) and classify it as a monomial, binomial, or trinomial.

3. Write a quadratic expression that contains two terms, a coefficient of 7, and a constant of 10.

For problems 4 and 5, determine whether each expression is a quadratic expression. Explain your reasoning.

4. –3x(x(x( + 14)

5. 8x2 – 2x(1 + 4x) + 2

Practice 2.2: Identifying Terms, Factors, and Coefficients AA

U2-41





For problems 6–10, write an algebraic expression. Identify the terms, coefficients, and constant terms of the alge braic expression. Determine whether the expression is quadratic and explain your reasoning.

6. the product of 7 and the square of x, increased by the difference of 5 and x2

7. half the sum of 12 and x2 decreased by one-third x

8. t he perimeter of a square, which is the product of 4 and the length of its side, s

9. the surface area of a cube, which is the product of 6 and the square of the side length, s

10. the volume of a sphere with radius r, which is four-thirds times the product of π and the cube π and the cube πof the radius

U2-42



Name: Date:



Use what you know about the components of expressions to complete problems 1–3.

1. Identify the terms, coefficients, constant term, and factors of 30x2 – 18x + 72.

2. Simplify the expression 5x + 4(5x – x) – 2x(6) and classify it as a monomial, binomial, or trinomial.

3. Write a quadratic expression that contains three terms, coefficients of –1 and 1, and a constant of 52.


4. 20x(4 – 5x) + 3(x) + 3(x) + 3( – 8)

5. 12x(x(x( 2 – 4x) – 2(3 + x)

Practice 2.2: Identifying Terms, Factors, and Coefficients B

continued

U2-43





For problems 6–10, write an algebraic expression. Identify the terms, coefficients, and constant terms of the alge braic expression. Determine whether the expression is quadratic and explain your reasoning.

6. the product of 9 and x, decreased by the sum of 8 ad the square of x

7. double the sum of 2 and x increased by one-half x2

8. t he area of a square, which is the square of its side, s

9. the volume of a cube, which is the cube of the length of its side, s

10. the surface area of a sphere with radius r, which is four times the product of π and the square π and the square πof the radius

U2-44North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource2.32.3


UNIT 2 • GRAPHING QUADRATIC FUNCTIONS A–SSE.1b★

Lesson 2.3: Interpreting Complicated Expressions

Name: Date:

Warm-Up 2.3Warm-Up 2.3

A fence surrounds a park in the shape of a triangle. The side lengths of the park in feet are given by the expressions 8x, 10x, and 11x – 2, where x is the length in feet of one fence panel.

1. Write an expression in terms of x for the perimeter of the park.

2. What is the perimeter of the park if each fence panel is 4 feet long?

3. How does the perimeter change if the length of each fence panel increases to 5 feet?How does the perimeter change if the length of each fence panel increases to 5 feet?




Instruction




Warm-Up 2.3 Debrief

1. Write an expression in terms of x for the perimeter of the park.

Set up an equation using the given expressions. Then combine like terms to find an expression to represent the perimeter, P.

P = 8P = 8P x + 10x + (11x – 2)

= 8x + 10x + 11x – 2

= 29x – 2

2. What is the perimeter of the park if each fence panel is 4 feet long?

Substitute 4 for x in the equation for perimeter.

P = 29(4) – 2 = 116 – 2 = 114P = 29(4) – 2 = 116 – 2 = 114P

If a fence panel is 4 feet long, the perimeter of the park is 114 feet.

3. How does the perimeter change if the length of each fence panel increases to 5 feet?

Substitute 5 for x in the equation for perimeter and compare the result to the previous question’s answer.

P = 29(5) – 2 = 145 – 2 = 143P = 29(5) – 2 = 145 – 2 = 143P

If the length of each fence panel increases to 5 feet, the perimeter of the park increases to 143 feet. Subtracting 114 from 143 indicates this is an increase of 29 feet.


• Students will interpret the parts of given expressions and determine how changes to the parts affect the expressions.

• Students will evaluate quadratic expressions for given values of the variables.






Instruction



Introduction

Variables change, but constants remain the same. We need to understand how changing the value of a variable in an expression will affect the value of the expression. In this lesson, we will explore these relationships.

Key Concepts

• We can translate verbal expressions into algebraic expressions to analyze how changing the values of coefficients, constants, or variables will affect the value of the expression.

• Changing the value of a variable in an expression will not change the value of a constant, just as changing the value of a constant will not change the value of terms containing variables.

• It is important to always follow the order of operations.

• Simplifying expressions lets us more easily see how different terms interact with one another.

• The Distributive Property is applied when multiplying binomials.

• When presented with a polynomial in factored form, multiply the factors to see if the polynomial is quadratic. The high est power of the variable of a quadratic expression is 2, and quadratic expressions are of the form ax2 + bx + bx + bx c, where a ≠ 0.

• Sometimes it is easier to work with the factored form of an expression. For instance, since we know that the product of two positive quantities is always a positive number, it is easier to determine values of a variable that make an expression positive by finding the values that make all o f the variable’s factors positive.

• We can use similar logic to determine the values that make an expression negative or equal to 0.

Prerequisite Skills


• evaluating expressions using the order of operations (5.OA.1)

• evaluating expressions for a given value (6.EE.2c)

• identifying parts of an expression (6.EE.2b)


• forgetting to distribute when multiplying binomials

• incorrectly following the order of operations

• incorrectly translating verbal expressions



Date:Name:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS A–SSE.1b★


For problems 1 and 2, use what you know about expressions to complete the problem.

1. What values of x make the (x make the (x make the ( – 3)(x – 3)(x – 3)( + 5) expression negative?

2. What values of x make the (5x – 25)(4 – 2x) expression positive?

For problems 3 and 4, show that the expression is quadratic by writing it in the form ax2 + bx + c. Identify a, b, and c.

3. (7x – 4)(x – 4)(x – 4)( + 3)

4. (x (x ( – 1)(x – 1)(x – 1)( + 1) – (2x – 3)(x – 3)(x – 3)( + 2)

For problems 5 and 6, determine whether each expression is a quadratic expression.

5. (x (x ( – 3)(x – 3)(x – 3)( + 4) – 15

6. (x (x ( – 2)(x – 2)(x – 2)( – 1) + (x – 1) + (x – 1) + ( – 4)(6 – x)

continued

Scaffolded Practice 2.3: Interpreting Complicated Expressions





Name: Date:


7. How does the area of a square change when the side length is tripled?

8. How does the volume of a cube change when the side length is halved?

9. The surface area of a cube is the product of 6 and the square of the side length. How does the surface area of a cube change when the side length of a cube is halved?

10. The surface area of a sphere is the product of 4π and the square of the radius. How does the π and the square of the radius. How does the πsurface area change when the diameter is tripled?



Instruction



Example 1

Show that (xShow that (xShow that ( + 2)(2x + 2)(2x x – 1) is a quadratic expression by writing it in the form x – 1) is a quadratic expression by writing it in the form x ax2 + bx + bx + bx c. Identify a, b, and c.

1. Mult iply the factors using the Distributive Property.

(x(x( + 2)(2x – 1) Original expression

= x(2x) + x(–1) + 2(2x) + 2(–1) Distrib ute (x Distrib ute (x Distrib ute ( + 2) over (2x – 1).

= 2x2 – x + 4x – 2 Mult iply.

= 2x2 + 3x – 2 Combine like terms.

2. Compare the resulting polynomial to ax2 + bx + c.

Since 2x2 + 3x – 2 has the form ax2 + bx + c, where a ≠ 0, (x ≠ 0, (x ≠ 0, ( + 2)(2x – 1) is a quadratic expression.

3. Identify a, b, and c.

In the polynomial, a = 2, b = 3, and c = –2. c = –2. c

Guided Practice 2.3



Instruction



Example 2

What values of x make the expression (xpression (xpression ( + 2)(x + 2)(x + 2)( – 3) posi tive?

1. Determine the sign possibilities for each factor.

The expression will be positive if both factors are positive or if both factors are negative.

2. Determine the values of the variable that make both factors positive.

Solve the inequalities x + 2 > 0 and x – 3 > 0 for x.

x + 2 > 0 when x > –2

x – 3 > 0 when x > 3

Both factors are positive when x > –2 and x > 3, and the intersection of these two sets i s x > 3. Therefore, any values of x greater than 3 will make both factors positive.

3. Determine the values of the variable that make both factors negative.

Solve the inequalities x + 2 < 0 and x – 3 < 0 for x.

x + 2 < 0 when x < –2

x – 3 < 0 when x < 3

Both factors are negative when x < –2 and x < 3, and the intersection of these two sets is x < –2. Therefore, any values of x less than –2 will make both factors negative.

4. Determine the values of x for which the expression is always positive.

The value of the expression is positive when x > 3 or x < –2.



Instruction



E xample 3

The length of each side of a square is increased by 2 centimeters. How does the perimeter change? How does the area change?

1. Find an expression for the perimeter of the original square.

Let x be the length of one side of the square in centimeters.

Multiply the side length by 4 since a square has four sides of equal length.

The perimeter of the square is 4x centimeters.

2. Find an expression for the perimeter of the new square and compare the new perimeter to the original perimeter.

One side of the new square is x + 2 centimeters.

The new perimeter is 4(x The new perimeter is 4(x The new perimeter is 4( + 2) or 4x + 8 centimeters.

Next, subtract the old perimeter from the new perimeter to determine the difference.

4x + 8 – 4x = 8

The new perimeter is 8 centimeters more than the original perimeter.

3. Find an expression for the area of the original square.

The area in square centimeters of the original square is x2.

4. Find an expression for the area of the new square and compare the new area to the original area.

The new area is (xThe new area is (xThe new area is ( + 2)2 or x2 + 4x + 4 square centimeters.

Subtract the old area from the new area to determine the difference.

(x(x( 2 + 4x + 4) – x2 = 4x + 4

The new area is 4x + 4 square centimeters greater than the old area.



Instruction



E xample 4

A car’s total stopping distance in feet depends on many factors, but can be approximated by the

expression 11

10

1

192x x+ , where x is the speed of the car in miles per hour. Is this expression

quadratic? What effect does doubling the car’s speed from 10 mph to 20 mph have on the total

stopping distance?

1. Determine whether the expression is quadratic.

This expression is quadratic because it can be written in the form

ax2 + bx + c, with a = 1

19, b =

11

10, and c = 0.c = 0.c

2. Determine the approximate stopping distance of a car traveling 10 mph.

Substitute 10 for x to find the total stopping distance of a car traveling 10 mph.

11

1010

1

1910 2( ) ( )+ = 11 +

100

19≈ 16.26 feet

The stopping distance of a car traveling 10 mph is approximately 16.26 feet.

3. Determine the approximate stopping distance of a car traveling 20 mph.

Substitute 20 for x to find the total stopping distance of a car traveling 20 mph.

+11

10(20)

1

19(20)2 = 22 +

400

19≈ 43.05 feet

The stopping distance of a car traveling 20 mph is approximately 43.05 feet.

4. Compare the stopping distances for a car traveling 10 mph and a car traveling 20 mph.

When the speed of the car doubles from 10 mph to 20 mph, the stopping distance almost triples from about 16 feet to about 43 feet.

U2-53

If 2 feet of fencing are

damaged and cannot be used, how does this affect the

pen’s area?

North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource2.3




Problem-Based Task 2.3: Puppy Pen

Oscar has 20 feet of fencing. He wants to build a rectangular pen for his new puppy. Write an expression for the area of the puppy’s pen and show that the expression is quadratic. If 2 feet of fencing are damaged and cannot be used, how does this affect the pen’s area?

SMP1 ✓ 2 3 4 ✓5 6 7 ✓ 8





Name: Date:


Coaching

a. Let l be the length of the pen and w be the width of the pen. Write an expression for the perimeter of the rectangular pen.

b. Use what you know about the perimeter of the pen to write an expression for the pen’s width in terms of the pen’s length, l.

c. How can the area of the pen be expressed in terms of l ?

d. Show that the area formula is quadratic.

e. If 2 feet of fencing are damaged, what is the pen’s perimeter?

f. How does the change in perimeter affect the expression for width?

g. How will the area change if 2 feet of fencing are damaged?



Instruction





a. Let l be the length of the pen and w be the width of the pen. Write an expression for the perimeter of the rectangular pen.

The perimeter of the pen can be expressed as 2l + 2l + 2l w feet.

b. Use what you know about the perimeter of the pen to write an expression for the pen’s width in terms of the pen’s length, l.

We know that the perimeter is 20 feet. Set the perimeter equal to 20 and solve for w.

2l + 2l + 2l w = 20

2w = 20 – 2l

w = 10 – l feetl feetl

c. How can the area of the pen be expressed in terms of l ?

The area of a rectangle is found by multiplying the length by the width.

A = l • w = l(10 – l ) square feet

d. Show that the area formula is quadratic.

Multiply the factors of the expression.

A = l(10 – l ) = 10l – l – l l 2 square feet

This is a quadratic equation, with a = –1, b = 10, and c = 0.c = 0.c

e. If 2 feet of fencing are damaged, what is the pen’s perimeter?

If 2 feet of fencing are damaged, the pen’s perimeter will decrease from 20 feet to 18 feet.

f. How does the change in perimeter affect the expression for width?

Substitute 18 for the perimeter to find the new expression for width.

2l + 2l + 2l w = 18

2w = 18 – 2l

w = 9 – l feetl feetl

If 2 feet of fencing are damaged, the width will change from 10 – If 2 feet of fencing are damaged, the width will change from 10 – ll feet to 9 – feet to 9 – l feet to 9 – ll feet to 9 – l ll feet. feet.l feet.ll feet.l



Instruction



g. How will the area change if 2 feet of fencing are damaged?

Substitute the new expression for width into the area formula and multiply the factors.

A = l(9 – l) = 9l) = 9l l – l – l l 2 square feet

If 2 feet of fencing are damaged, the area will change from 10l – l – l l 2 square feet to 9l – l – l l 2 square feet. The decrease in area is equal to (10l – l – l l 2) – (9l – l – l l 2), which is equal to l, the length of the pen.








Task Overview

Focus

How does changing the perimeter of a rectangle affect its area? Students will use the perimeter of a rectangular pen to model its area with a quadratic expression in one variable. Then they will analyze how changing the perimeter of the rectangular pen affects the area.


• interpreting verbal expressions and translating them into algebraic expressions

• analyzing the meaning and impact of each part of an expression

• analyzing how a change to a constant of one expression impacts a related expression

• examining the terms of an expression to determine if it is quadratic

• comparing quadratic expressions to determine how they differ in terms of the context of the problem

Introduction

This task should be used to explore how changing the value of a constant in an expression affects the value of a related expression. It is best to implement this task after students have learned how to translate expressions from verbal statements into symbolic representations, how to solve for a variable in an expression, and how to identify a quadric expression.

Begin by reading the problem and reviewing the formulas for the area and perimeter of a rectangle. Explore the relationship between the two formulas and how the formulas are related because they both can be expressed in terms of the length and the width of the rectangle. Review the following terms: area and perimeter.

area the number of square units that are contained within a plane figure

perimeter the distance around a plane figure

Problem-Based Task 2.3 Implementation Guide: Puppy Pen













Some students will not know how to translate the scenario into symbolic representations. Encourage them to break the problem into parts by first connecting the given information with the unknowns of the perimeter formula and then using the resulting perimeter expression to model the area of the rectangle.


Students will recognize that the perimeter of the pen can be modeled with a linear equation, and that the area of the pen can be modeled with a quadratic equation.


When comparing the two area expressions, students can make use of the fact that the expressions differ only in the linear term, because the quadratic terms are the same. Ask students to compare the expressions by analyzing the coefficients of each term.



• forgetting to simplify expressions

Remind students that if each term of the numerator of an expression is divisible by the denominator, the expression can be simplified.

• not completely isolating a variable

Advise students that to express the width of the pen in terms of its length, the w term must have a coefficient of l. Since the coefficient of w is 2, the students must divide both sides of the equation by 2.

• forgetting to distribute to the second term of a binomial

When multiplying l(10 – l), remind students that thel), remind students that thel l must by multiplied by both the 10 and l must by multiplied by both the 10 and lthe –l.

• forgetting to distribute the negative when subtracting the area expressions

Remind students that when subtracting a binomial from an expression, the negative sign must be distributed to both terms of the binomial.






• difficulty with determining why the expression is quadratic

Review that the terms of a quadratic can be in any order. The leading term does not have to have a degree of 2.

• incorrectly comparing the two area expressions

Remind students how to compare the area of two figures with known constant areas by giving them an example of two areas and asking them to compare them. Emphasize that the subtraction strategy can be used with the area expressions.



• If students are having trouble getting started, ask them to review the information they are given and make connections using the formulas they know. Ask students, “What is the problem asking?” (Answer: How does the area of the pen change when the perimeter is decreased by 2?) “What formula represents the area of the pen?” (Answer: A = lw.) “What does the 20 feet of fencing represent?” (Answer: the perimeter of the pen.) “What formula represents the perimeter of the pen?” (Answer: P = 2P = 2P l + 2l + 2l w.)

• If students are having trouble comparing the areas, remind them that expressing the area in terms of one variable will allow a comparison of the areas when the perimeter changes. Ask students, “How can the given information be used to express the area in terms of one variable?” (Answer: Use the perimeter to solve for one variable in terms of the other, then substitute the expression into the area formula.)

• If students are unsure how to write an expression for the pen’s width in terms of its length, ask the following questions: “Which variable in the expression do you want to isolate?” (Answer: w.) “Which operation is needed to get the 2l to the other side of the equation?” l to the other side of the equation?” l(Answer: Subtraction.) “Which operation is needed to get the coefficient of w to the other side of the equation?” (Answer: Division.)

• If students are struggling with how to express the area of the pen in terms of l, ask the following questions: “What formula represents the area of the pen?” (Answer: A = lw.) “What does it mean to express the area of the pen in terms of l?” (l?” (l Answer: Write an area expression using only the l variable.) “What expression did you find to represent l variable.) “What expression did you find to represent l w?” (Answer: 10 – l.) “How can you use that expression to write the area of the pen in terms of l?” l?” l(Answer: Substitute 10 – l for l for l w in A = lw.)

• Once students have the two area expressions, ask them how to compare the areas. (Answer:Find the difference of the two expressions.) Ask students to analyze the terms of the expressions and determine the differences. (Answer: The expressions only differ in their linear terms.) Ask the students if it is necessary to subtract the entire area expressions to find the difference of the areas. (Answer: No; only the linear terms must be subtracted.)






• When students have determined that the difference of the areas is l, ask them how the solution relates to the context of the problem. (Answer: l is the length of the pen, so the areas l is the length of the pen, so the areas ldiffer by l square units.) Remind students that the side length labeled l square units.) Remind students that the side length labeled l l is arbitrary, so the l is arbitrary, so the ldifference of the areas in square units is equal to one of the dimensions of the pen.


Debriefing the Task

Review how the given information was used with the known perimeter and area formulas to create equations to model the given scenario. Highlight that the area formula is the product of two lengths, and that using the perimeter to solve for one variable in terms of the other allows the area to be expressed in one variable. Discuss how changing the value of the perimeter of the pen affected the area expression. Emphasize that simplifying the perimeter expression allowed for easier manipulation of the area formula. Review the relationship between the structure of the final two area expressions, and explore the relationship between the structure and the difference in the areas.



• Changing the value of a variable will not change the value of a constant, just as changing the value of a constant will not change terms containing variables.

When the perimeter of the pen is decreased by 2 units, the number of square units by which the area is decreased is equal to one of the lengths of the sides of the pen.

• Simplifying expressions lets us more easily see how different terms interact with one another.

Simplifying the perimeter expression allows the area expression to be more easily manipulated, and therefore easier to differentiate how the area changes when the perimeter is altered.

• When presented with an expression in factored form, multiply the factors to see if the polynomial is a quadratic. Quadratic expressions are of the form ax2 + bx + c, where a ≠ 0.

The Distributive Property was applied to expand l(10 – l) = 10l) = 10l l – l – l l 2 and l(9 – l) = 9l) = 9l l – l – l l 2. Both of these expressions are quadratic, with a = –1 and c = 0. c = 0. c

Extending the Task

• To extend the task, change either of the parameters. For example, change the initial amount of fencing to another even number, and write a conjecture about how the initial amount of fencing is related to the final answer. Or change the amount of fencing damaged to another even number, and write a conjecture about how the amount of fencing damaged is related to the final answer.

• Another option is to increase the amount of fencing instead of decreasing it. How does the change affect the difference in areas?








• For SMP 1, ASK: “How did you demonstrate perseverance?” (Answer: I reread the problem to find the information I needed. I broke the problem into smaller parts by connecting the perimeter and area formulas to the given information.)

• For SMP 4, ASK: “How did you use mathematics to model this particular scenario?” (Answer:I wrote expressions to model the width of the pen in terms of the length (10 – l and 9 – l and 9 – l l), and l), and lused these expressions to model the area of the pen with expressions l(10 – 1) and l(10 – 1) and l l(9 – l(9 – l l).l).l

• For SMP 7, ASK: “How did you make use of structure when solving the problem?” (Answer:When the perimeter of the pen is 20 units, I modeled the situation with the expression 10l – l – l l 2. When the perimeter of the pen is decreased by 2 units, I modeled the situation with the expression 9l – l – l l 2. I was able to use structure to see that these expressions are the same with the exception of the l terms and determine that the number of square units that the area l terms and determine that the number of square units that the area ldiffers by is equal to the difference of the l terms: 10l terms: 10l l – 9l – 9l l = l = l l.


• Students may try to create various dimensions of a rectangle of perimeter 20 and hypothesize how the area is affected by analyzing the results of various dimensions. For example:

Length Width Perimeter Area

Original area 5 5 20 25

Decrease perimeter by 2 5 4 18 10

Difference in area: 25 – 20 = 5














• After viewing the results of various sets of lengths and widths that fit the parameters of the problem, students may realize the difference of the areas in square units is always equal to one of the dimensions of the pen.

• Another common strategy is to solve for l in terms of l in terms of l w instead of solving for w in terms of l.In this case, the difference of the areas is w. Since the variables l and l and l w are arbitrarily assigned to the side lengths, this solution will help reinforce that the difference of the areas in square units is equal to one of the dimensions of the pen.

Technology

If using the alternative strategy, student can use spreadsheets to calculate the area of each rectangle, and to find the difference of the area for each of the related dimensions.





continued

For problems 1 and 2, use what you know about expressions to answer the questions.

1. What values of x make the expression (x make the expression (x make the expression ( + 7)(x + 7)(x + 7)( – 10) negative?

2. What values of x make the expression (2x + 1)(x + 1)(x + 1)( – 3) positive?

For problems 3 and 4, show that the expression is a quadratic expression by writing it in the formax2 + bx + c. Identify a, b, and c.

3. (3x + 5)(4x – 1)

4. 3x(2x + 8) + (x + 8) + (x + 8) + ( – 3)(x – 3)(x – 3)( + 10)


5. (x(x( + 4)(5x – 11)

6. (2x2 + 9)(x + 9)(x + 9)( – 2)

Practice 2.3: Interpreting Complicated ExpressionsPractice 2.3: Interpreting Complicated Expressions AA





Name: Date:

For problems 7–10, translate any verbal expressions into quadratic expressions, and then answer the questions.

7. The property tax in dollars for p acres of property can be found by using the quadratic expression 225pexpression 225pexpression 225 2. How does doubling the acreage of property from 4 acres to 8 acres affect the property taxes?

8. The area of an equilateral triangle can be found by multiplying the square of the side length

by 3

4. How does changing the side length from 2 inches to 10 inches affect the area of the

triangle?

9. How does the area of a square change when the side length is halved?

10. The surface area of a cube is the product of 6 and the square of the side length. How does the surface area of a cube change when the side of a cube doubles in length?





For problems 1 and 2, use what you know about expressions to answer the questions.

1. What values of x make the expression (x make the expression (x make the expression ( + 4)(x + 4)(x + 4)( – 6) negative?

2. What values of x make the expression (5x + 7)(2x – 8) positive?

For problems 3 and 4, show that the expression is a quadratic expression by writing it in the formax2 + bx + c. Identify a, b, and c.

3. (9x – 15)(2x + 1)

4. (x(x( – 2)(x – 2)(x – 2)( + 3) – (x + 3) – (x + 3) – ( – 4)(2x + 5)


5. (x(x( – 1)2 + 10

6. (x(x( + 4)(x + 4)(x + 4)( + 1)(x + 1)(x + 1)( – 1)

Practice 2.3: Interpreting Complicated Expressions B

continued





Name: Date:

For problems 7–10, translate any verbal expressions into quadratic expressions, and then answer the questions.

7. The population of a city t years after 2000 is represented by the quadratic expression t years after 2000 is represented by the quadratic expression t100t2t2t + 300t + 50,000. How did the city’s population change from 2000 to 2012?t + 50,000. How did the city’s population change from 2000 to 2012?t

8. The surface area in feet of a cylinder with a radius of 1 foot can be found by adding the product of 2π and the square of the radius to the product of 2π and the square of the radius to the product of 2π π and the radius. How does the surface π and the radius. How does the surface πarea change when the radius is tripled from 1 foot to 3 feet?

9. How does the area of a square change when the side length is tripled?

10. The surface area of a sphere is the product of 4π and the square of the radius. How does the π and the square of the radius. How does the πsurface area change when the radius is halved?surface area change when the radius is halved?

U2-67© Walch Education

Warm-Up 2.4

Fireworks displays are often set to music by pyrotechnicians. The function h(t) = –16t2 + 200t + 4 is used to represent the height, h, in feet, of the first firework set to go off in, t, seconds. The firework will go off when it reaches its highest point.

1. What is the maximum height the firework will reach before it explodes in the air?

2. How long after the music starts will the first firework explode?

3. Rewrite the function h(t) = –16t2 + 200t + 4 in a different form. Then explain how this form could better help a pyrotechnician do his or her job.

Lesson 2.4: Quadratic Modeling

Date:Name:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS A–CED.2•, F–BF.1•


North Carolina Math 2 Custom Teacher Resource 2.4

© Walch Education


Warm-Up 2.4 Debrief

1. What is the maximum height the firework will reach when it explodes in the air?

Use the axis of symmetry formula, xb

a=−2

, to find the x-value of the maximum point for a

quadratic function in the form y = ax2 + bx + c. Substitute values for a and b from the given

function, h(t) = –16t2 + 200t + 4, then solve for x.

xb

a=−2

Axis of symmetry

x =−−

(200)

2( 16)Substitute 200 for b and –16 for a.

x =−−

=200

326.25 Simplify.

The x-value of the maximum is 6.25, representing t = 6.25 seconds.

Substitute this value for t in the given function h(t), and solve.

h(t) = –16t2 + 200t + 4 Given function

h(6.25) = –16(6.25)2 + 200(6.25) + 4 Substitute 6.25 for t.

h(6.25) = –16(39.0625) + 200(6.25) + 4 Evaluate 6.252.

h(6.25) = –625 + 1,250 + 4 Multiply.

h(6.25) = 629 Simplify.

The maximum height of the firework when it explodes would be 629 feet.

North Carolina Math 2 StandardA–CED.2 Create and graph equations in two variables to represent quadratic, square root and

inverse variation relationships between quantities.★

F–BF.1 Write a function that describes a relationship between two quantities by building quadratic functions with real solution(s) and inverse variation functions given a graph, a description of a relationship, or ordered pairs (include reading these from a table).★


Instruction

© Walch Education

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS A–CED.2•, F–BF.1•




Instruction

© Walch Education




2. How long after the music starts will the first firework explode?

Using the maximum point that we found in problem 1, we know the firework will explode 6.25 seconds after the music starts.

3. Rewrite the function h(t) = –16t2 + 200t + 4 in a different form. Then explain how this form of the function could better help a pyrotechnician do his or her job.

Answers will vary depending on the chosen form for the rewritten function. Sample answer: The function rewritten in vertex form is h(x) = –16(x – 6.25)2 + 629. The vertex form of the quadratic function will enable the pyrotechnician to easily see the number of seconds before each firework will go off. This will allow him or her to more easily set the fireworks to the timing of the music.


• Students will solve quadratic equations.

• Students will analyze and interpret key features of quadratic functions.


Introduction

Even though we might not realize it, quadratic relationships are all around us. When a basketball player shoots a three-pointer to win a game, the ball’s arc forms a parabola with a quadratic relationship relating the height of the ball to the time of its flight. When an architect is designing a room to use the least amount of materials to build a wall for a given area, the minimum perimeter is sometimes based on a quadratic relationship relating the area to the length and width of the room. Being able to model, solve, and evaluate these equations can help us make predictions about the behavior of many real-world situations.

Key Concepts

• The standard form of a quadratic equation is y = ax2 + bx + c, with (x, y) representing the points on the parabola that satisfy the equation. The key features of the quadratic equation can be found using the following methods:

Key feature Method to determine the feature’s value

x-intercept Solve the equation by factoring or graphing, or by using the

quadratic formula xb b ac

a=− ± − 4

2

2

.

y-intercept Substitute 0 for x in the equation and evaluate. The y-intercept will be the c value in the equation.

vertex (minimum or maximum point)

Evaluate b

a

−2

to determine the x-value of the vertex. Then, substitute

the x-value into the equation and evaluate to determine the y-value

of the vertex.

Prerequisite Skills


• recognizing key points of a parabola on a graph (F–IF.7a★)

• interpreting word problems to determine the question and key facts (6.EE.2a)

Instruction

© Walch Education





Instruction

© Walch Education




• The key features can also be determined using a graphing calculator. The y-intercept is shown as the c value in the equation, so it is not necessary to use the calculator to find it.

• To find the x-intercepts:

On a TI-83/84:

Step 1: Press Y=, and type the quadratic equation in Y1 and y = 0 in Y2.

Step 2: Adjust the calculator window as necessary so that the x-intercepts are visible in the graph window.

Step 3: Press [2ND][TRACE][CALC], and scroll to 5: Intersect. Press ENTER to select the quadratic graph, then press ENTER again to select the horizontal line y = 0. Use the arrows to move the cursor near one of the x-intercepts. Press ENTER to find the intercept at (x, 0). The x-value represents the x-intercept.

Step 4: If there is another x-intercept, press [2ND][TRACE][CALC], and scroll to 5: Intersect. Press ENTER to select the quadratic graph, then press ENTER again to select the horizontal line y = 0. Use the arrows to move the cursor near the other x-intercept. Press ENTER to find the intercept at (x, 0). The x-value represents the second x-intercept.

On a TI-Nspire:

Step 1: Press the HOME key. Choose the Graphs and Geometry Application, then type the quadratic equation in f1(x) and type f2(x) = 0.

Step 2: Adjust the viewing window so that all x-intercepts are visible on the graph.

Step 3: Press MENU, select 6: Points & Lines, then select 3: Intersection Points.

Step 4: When the pointing hand appears, click on the quadratic graph and the line y = 0 (the x-axis). All x-intercepts in the viewing window will be displayed as (x, 0). The x-values represent the x-intercepts.


Instruction

© Walch Education




• To find the vertex:

On a TI-83/84:

Step 1: Press Y=, and type the quadratic equation into Y1.

Step 2: Press WINDOW. Adjust the window range so that the vertex is visible on the graph.

Step 3: Press GRAPH to see the graph. Then press [2ND][TRACE][CALC] and scroll to #3 if the vertex is a minimum and #4 if the vertex is a maximum. Set the left bound anywhere to the left of the vertex, and set the right bound anywhere to the right of the vertex. Press ENTER again to find the vertex.

On a TI-Nspire:

Step 1: Press the HOME key. Choose the Graphs and Geometry Application, then enter the quadratic equation as f1(x).

Step 2: Press MENU, then select 4: Window/Zoom and 1: Window Setting to adjust the window range so that the vertex is visible.

Step 3: If the vertex is a maximum, press MENU. Select 5: Trace, then 1: Graph Trace. Trace the graph using the navigation pad arrows until an uppercase “M” appears. The point where you see the “M” is the maximum, or the vertex.

Step 4: If the vertex is a minimum, press MENU. Select 5: Trace, then 1: Graph Trace. Trace the graph using the navigation pad arrows until a lowercase “m” appears. The point where you see the “m” is the minimum, or the vertex.

• When reading word problems in which you must use a quadratic equation to answer a question, certain keywords and phrases can help determine what key feature will help you answer the question. The context of the problem is also extremely important, but the following words and phrases can be a guide:

Key feature Keywords and phrases

x-intercept when will an object hit the ground; when will a value be 0

y-intercept initial value; starting value

vertex maximum, minimum, highest value, lowest value, apex, peak


• Generally, the domain of a quadratic equation is comprised of real numbers. In many contexts, this domain is limited by the real-world aspects of the problem. Often, especially when x represents time, the domain can only be positive numbers. In other cases, when x represents people or another discrete quantity, the domain could be limited to positive integers. It is important to consider the context of the problem when determining the potential answer to a word problem.

• One common application of quadratic equations involves projectile motion, when objects are launched from a given height at a given velocity. The rate at which they accelerate toward the ground due to gravity is represented by a quadratic equation. For the standard form equation y = ax2 + bx + c, a will usually be –4.9 meters/second2 or –16 feet/second2, representing the proven rate for gravity; b will represent the initial velocity at which the object was launched; and c will represent the initial height of the object.


• misinterpreting the question in a word problem and solving for the wrong feature

• giving answers that are outside the domain of a given context, such as negative x-intercepts

Instruction

© Walch Education





For each quadratic function in problems 1–4, algebraically determine the axis of symmetry and vertex. Identify whether the vertex represents a maximum or minimum.

1. f(x) = x2 + 6x + 9

2. f(x) = –3x2 – 4

3. f(x) = –x2 + 6x – 4

4. f(x) = 5x2 – 10x + 3

For each quadratic function in problems 5–7, use a graphing calculator to determine the location of the vertex. Identify whether the vertex represents a maximum or minimum.

5. f(x) = x2 + 4x – 7

continued

Scaffolded Practice 2.4: Quadratic Modeling



Name: Date:


© Walch Education


6. f(x) = –2x2 – 8x – 3

7. f(x) = –x2 + 6x + 1

For each quadratic function in problems 8–10, use a graphing calculator to determine the x-intercept(s).

8. f(x) = x2 – 2x – 8

9. f(x) = x2 + 4x – 5

10. f(x) = –x2 + 12x – 36




© Walch Education


Example 1

Given the quadratic function f(x) = –2x2 + 8x + 6, algebraically determine the axis of symmetry and the vertex. Identify whether the vertex represents a maximum or minimum.

1. Determine the axis of symmetry algebraically.

xb

a=−2

Formula for the axis of symmetry

x =−−8

2( 2)Substitute –2 for a and 8 for b.

x =−−

=8

42 Simplify.

The axis of the symmetry of f(x) is x = 2.

2. Determine the location of the vertex algebraically.

The axis of symmetry gives the x-value of the vertex. To find the y-value, substitute the value for the axis of symmetry into the quadratic and simplify.

b

af

b

a

− −

2

,2

Formula for location of vertex

(2, f(2)) Substitute the axis of symmetry value.

f(2)= –2(2)2 + 8(2) + 6 Substitute 2 for the x-value of f(x).

f(2)= –2(4) + 16 + 6 = 14 Simplify.

f(2)=14 Simplify.

(continued)

Guided Practice 2.4

Instruction

© Walch Education





Instruction

© Walch Education




The vertex of f(x) is located at (2, 14). Because the a-value is negative, the parabola is concave down. This means that the vertex is a maximum. The graph of f(x) below confirms this.

2 4 6 8 10

8

6

4

2

1

7

5

3

9

10

14

12

15(2, 14)

13

11

0 1 3 5 7 9

y

x

–1–2


Example 2

If a current, I, is flowing through a certain electrical circuit, the power, W, of the circuit can be determined by the formula W = 120I – 12I2. Using a graphing calculator, determine the amount of current it will take in amperes to supply the maximum amount of power in watts.

1. Graph the function W in a graphing calculator and find the maximum.

Use the directions appropriate to your calculator model.

On a TI-83/84:

Step 1: Press Y= and type 120x – 12x2 in Y1.

Step 2: Press GRAPH to graph the expression.

Step 3: Adjust the window ZOOM-ZoomFit until the graph is visible.

Step 4: Press [2ND][TRACE][CALC] and scroll to 4: Maximum. Set the left bound anywhere to the left of the vertex in the graph, and set the right bound anywhere to the right of the vertex in the graph to calculate the highest point. Press ENTER again to get point (4.9999971300).

On a TI-Nspire:



Step 3: Press MENU. Select 5: Trace, then 1: Graph Trace. Trace the graph using the navigation pad arrows until an uppercase “M” appears. The point where you see the “M” is the maximum, or the vertex.

The maximum of the quadratic is located at approximately (5, 300). This means it will take 5 amperes of current to supply the maximum amount of power of 300 watts.

Instruction

© Walch Education





Example 3

An apple falls from the branch of a tree to the ground 25 feet below. The distance, D, the apple is from the ground as it falls is represented by the equation D(t) = –16t2 + 25, where t represents time in seconds. Sketch a graph of the situation and calculate, to the nearest hundredth of a second, the time the apple will take to reach the ground.

1. Graph D(t) on the calculator.

5

25

20

15

10

5

0

y

x

–5

2. Algebraically determine the time it will take the apple to reach the ground.

The time it takes the apple to reach the ground represents the x-intercept of D(t). When the apple is on the ground, D(t), or the distance the apple is from the ground, equals 0. Solve for the x-intercept by replacing D(t) with 0.

0 = –16t2 + 25 Replace D(t) with 0.

–25 = –16t2 Subtract by 25.

t=25

162 Divide both sides by –16.

t=25

162 Square root both sides.

t= =5

41.25 Simplify.

Instruction

© Walch Education





Instruction

© Walch Education




3. Confirm the x-intercept found in the previous step using the graphing calculator.

On a TI-83/84:



Step 3: Press [2ND][TRACE][CALC] and scroll to 5: Intersect. Press [ENTER] to select the quadratic graph, then press [ENTER] again to select the horizontal line y = 0. Use the arrows to move the cursor near the right x-intercept. Press [ENTER] to find the intercept at (x, 0).

On a TI-Nspire:

Step 1: From the Graphs and Geometry Application, type the quadratic equation in f1(x) and type f2(x) = 0.


Step 3: Press [menu] and select 6: Points & Lines, then 3: Intersection Points.

Step 4: When the pointing hand appears, click on the quadratic graph. The line y = 0 (the x-axis) and all x-intercepts in the viewing window will be displayed as (x, 0).

The x-value of (1.25, 0) represents the positive x-intercept.

It will take the apple 1.25 seconds to reach the ground.


Example 4

The path of a rocket fired into the air follows a parabolic path. The formula to determine the height of the rocket in feet can be found by combining three terms that represent different forces acting on the rocket. The three terms affecting the rocket are identified in the formula:

height (t) = 1

2(effects of gravity) • t2 + (initial speed of rocket) • t + original height

The rocket is launched from the ground with an initial velocity of 128 feet per second. If the effect of gravity on any object on Earth is approximately –32 ft/sec2, write an equation that represents the height of the rocket as a function of time and answer the following questions.

a. What is the height of the rocket after 2 seconds?

b. How long will it take the rocket to reach its maximum height? What is its maximum height?

c. When will the rocket reach 100 feet?

d. What is a realistic domain for this function?

1. Write an equation that represents the height of the rocket as a function of time.

height (t) = 1

2(–32) • t2 + (128) • t + 0

Substitute known values into the formula.

h(t) = –16t2 + 128t Simplify.

2. Determine the height of the rocket after 2 seconds.

h(2) = –16(2)2 + 128(2) Substitute 2 for t.

h(2) = –16(4) + 256 Simplify.

h(2) = 192 Simplify.

After 2 seconds, the rocket is 192 feet high.

Instruction

© Walch Education





Instruction

© Walch Education




3. Determine the location of the maximum using a graphing calculator.

On a TI-83/84:

Step 1: Press Y= and type –16x2 + 128x in Y1.

Step 2: Press GRAPH to graph the expression.

Step 3: Adjust the window ZOOM-ZoomFit until the graph is visible.

Step 4: Press [2ND][TRACE][CALC] and scroll to 4: Maximum. Set the left bound anywhere to the left of the vertex in the graph, and set the right bound anywhere to the right of the vertex in the graph to calculate the highest point. Press [ENTER] again to get point (3.9999981256).

On a TI-Nspire:



Step 3: Press MENU. Select 5: Trace, then 1: Graph Trace. Trace the graph using the navigation pad arrows until an uppercase “M” appears. The point where you see the “M” is the maximum, or the vertex.

The maximum of the quadratic function is approximately (4, 256). It will take the rocket 4 seconds to reach its maximum height of 256 feet in the air.


Instruction

© Walch Education




4. Use the graphing calculator to determine when the rocket will reach 100 feet.

On a TI-83/84:


Step 2: Press GRAPH to graph both expressions on the same coordinate plane.

Step 3: Adjust the window ZOOM-ZoomFit until the point of intersection is visible.

Step 4: Press [2ND][TRACE][CALC] and scroll to 5: Intersect. Press [ENTER] to select the first graph, then press [ENTER] again to select the second graph. Press [ENTER] to find the left point of intersection. The left point of intersection is at x = 0.8775, y = 100.

Step 5: If there is another point of intersection, press [2ND][TRACE][CALC] and scroll to 5: Intersect. Press [ENTER] to select the quadratic graph. Then use the arrows to move the cursor near the right point of intersection and press [ENTER]. Press [ENTER] again to find the right point of intersection. The right point of intersection is at x = 7.1224, y = 100.

On a TI-Nspire:


Step 2: Press MENU, then select 4 Window to adjust the window until all points of intersection are visible, if necessary.

Step 3: Press MENU, then select 6: Points & Lines and 3: Intersection Points.

Step 4: When the pointing hand appears, click on each graph. All points of intersection in the viewing window will be displayed. The points of intersection are at x = 0.8775, y = 90 and x = 7.1224, y = 90.

The rocket will reach 100 feet on its way up at 0.88 seconds, and on its way down at 7.12 seconds.


Instruction

© Walch Education




5. Determine a realistic domain for the function by finding the x-intercept.

The domain of this function represents the time it takes the rocket to complete its flight path through the air. Since time cannot be negative, it would make sense to begin the time at 0. Height, or h(t), also cannot be negative, so to figure out the right-sided restriction on the domain, find the positive x-intercept of the function using the graphing calculator.

On a TI-83/84:



Step 3: Press [2ND][TRACE][CALC] and scroll to 5: Intersect. Press [ENTER] to select the quadratic graph, then press [ENTER] again to select the horizontal line y = 0. Use the arrows to move the cursor near the right x-intercept. Press [ENTER] to find the intercept at (x, 0). The x-value of (8, 0) represents the positive x-intercept.

On a TI-Nspire:



Step 3: Press MENU, then select 6: Points & Lines, and 3: Intersection Points.

Step 4: When the pointing hand appears, click on the quadratic graph. The line y = 0 (the x-axis) and all x-intercepts in the viewing window will be displayed as (x, 0). The x-value of (8, 0) represents the positive x-intercept.

The rocket will be in the air for 8 seconds. This means a realistic domain of the function would be 0 ≤ x ≤ 8.


You are nervous. Your high school basketball team is down 2 points in the last seconds of

a game against your biggest rivals. With 5 seconds left, your star player jumps to take a

30-foot shot worth 3 points, and it goes in! The height of the ball in the air is represented

by the equation 1676

382y x x= − + + , where x represents the time in seconds since the

basketball left his hand.

Your team is now ahead by 1 point. However, you know that the other team has some amazing shooters. If there are more than 2 seconds left on the clock, they could make a basket to win. Knowing that the shot went in and the basketball hoop is 10 feet off the ground, how much time is left on the clock when the shot goes in? Can the other team still win?

SMP 1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

Problem-Based Task 2.4: Three-Pointer for the Win!

How much time is left on the

clock when the shot goes in? Can the other team still win?




© Walch Education



Coaching

a. What question is the problem asking you to solve?

b. What key features of the quadratic equation will help answer the question?

c. What y-value will represent when the basketball passes through the hoop?

d. How long after the shot is taken does the basketball pass through the hoop?

e. How much time is left on the clock when the basketball passes through the hoop? Can the other team still win the game?



Name: Date:


© Walch Education




a. What question is the problem asking you to solve?

The problem is asking how much time is left on the clock when the basketball goes in. Therefore, the amount of time it takes for the ball to reach the hoop must be found.

b. What key features of the quadratic equation will help answer the question?

To solve the problem, determine the time, or x-value, for the basketball when its height, or y-value, is the same height as the basketball hoop. Also, you will need to find the second time when the ball is at the same height as the hoop, because you need to know when the ball is descending to the hoop.

c. What y-value will represent when the basketball passes through the hoop?

The y-value will be 10 to represent the basketball hoop being 10 feet off the ground.

d. How long after the shot is taken does the basketball pass through the hoop?

Given the equation 1676

382y x x= − + + , substitute 10 for y to represent the height of the

basketball hoop. Then solve the equation for x by either setting it equal to 0 and using the

quadratic formula or finding the intersection using the trace function of a graphing calculator.

The second x-value when y = 10 is at approximately (1.5, 10). The x-value represents the time

when the height of the ball is 10 feet, or about 1.5 seconds.

Instruction

© Walch Education





The graph should resemble the following:

3

20

15

10

(0, 8)

(1.5, 10)

0 21

y

x

5

–1

The shot takes 1.5 seconds to go in the basket.

e. How much time is left on the clock when the basketball passes through the hoop? Can the other team still win the game?

Since there were 5 seconds left when the player shot the ball, there are 5 – 1.5 = 3.5 seconds left after the shot. Therefore, the other team still has a chance to win the game.



Instruction

© Walch Education





1. Given the function f(x) = 3x2 – x + 3:

a. Does the function have a maximum or minimum?

b. What is the axis of symmetry?

c. What are the coordinates of the vertex?

d. What is the y-intercept?

e. What are the solutions?

Use the following information to complete problems 2–6:

A model rocket is launched from the roof of a building. It can be represented by the function h(t) = –5t2 + 30t + 10, where h(t) is the height of the rocket above the ground in meters and t is the time after the launch in seconds.

2. What was the initial height of the rocket?

3. In how many seconds did the rocket reach its maximum height? What was the maximum height?

4. What is the height of the rocket after 5 seconds?

Practice 2.4: Quadratic Modeling A

continued




© Walch Education


5. When will the rocket hit the ground?

6. What is the reasonable domain?


When something is thrown into the air, gravity starts to slow it down as soon as it leaves a person’s hand. At some point, it stops going up and starts coming back down. This type of situation can be modeled by the function h t t V t h= − + +( ) 16 2

0 0 , where h(t) is the height at the given time, t, in seconds; V0 is the initial velocity at the time of throwing the object; and h0 is the original height.

7. A ball is thrown in the air at a speed of 50 feet per second. It is 6 feet in the air when it leaves the person’s hand. Create an equation to model this situation.

8. How high above the person’s hand will it be when it reaches its maximum height?

9. When will the ball reach the ground?

10. What would be a reasonable domain?



Name: Date:


© Walch Education


1. Given the function f(x) = 3x2 + 8x – 3:

a. Does the function have a maximum or minimum?

b. What is the axis of symmetry?

c. What are the coordinates of the vertex?

d. What is the y-intercept?

e. What are the solutions?


A ball is thrown up from a rooftop. It will reach a maximum vertical height, then fall to the ground. The height of the ball from the ground at time t is modeled by the function h t t t= − + +( ) 16 64 802 .

2. What was the initial height of the ball?

3. In how many seconds did the ball reach its maximum height? What was the maximum height?

4. What is the height of the ball after 3 seconds?

Practice 2.4: Quadratic Modeling B

continued




© Walch Education


5. When will the ball hit the ground?

6. What is the reasonable domain?

Use the following information to complete problems 7–10.

When something is thrown into the air, gravity starts to slow it down as soon as it leaves a person’s hand. At some point, it stops going up and starts coming back down. This type of situation can be modeled by the function h t t V t h= − + +( ) 16 2

0 0 , where h(t) = height at the given time, t, in seconds; V0 = initial velocity at the time of throwing the object; and h0 = the original height.

7. A water balloon is thrown in the air at a speed of 16 feet per second. It is 2 feet in the air when it leaves the person’s hand. Create an equation to model this situation.

8. How high above the person’s hand will it be when it reaches its maximum height?

9. When will the balloon reach the ground?

10. What would be a reasonable domain?



Name: Date:


© Walch Education

U2-93

Date:Name:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–IF.4★

Lesson 2.5: Interpreting Key Features of Quadratic Functions

North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher ResourceCustom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource2.5



Warm-Up 2.5The object of a popular video game is to launch a boulder to knock over boxes, buildings, and other items. The graph shows an obstacle on the left that the boulder must clear in order to knock over the stack of boxes on the right. The boulder will follow a parabolic path and will launch from (0, 0) and end at (8, 0).

14

12

10

6

4

2

0

2

4

6

8

10

12

5 5 10Launch point

(3.5, 2.5) (6, 2.5)

1. What are the x-intercepts for the parabola formed by the path of the boulder?

2. What is the axis of symmetry for the parabola formed by the path of the boulder? How do you know?

3. One possible path for the boulder is y x x= − +3

832 . What is the vertex of the parabola created

by this equation?

4. Will the boulder clear the obstacle? How do you know?

5. Will the boulder knock down the boxes? How do you know?

U2-94North Carolina Math 2North Carolina Math 2 Custom Teacher ResourceCustom Teacher Resource2.52.5

Instruction





Warm-Up 2.5 Debrief1. W hat are the x-intercepts for the parabola formed by the path of the boulder?

The x-intercepts occur where the graph intersects the x-axis when y = 0. The x-intercepts are 0 and 8.

2. What is the axis of symmetry for the parabola formed by the path of the boulder? How do you know?

The axis of symmetry is a vertical line through the middle of the parabola. You can use the midpoint between the two x-intercepts to find the axis of symmetry. The midpoint between 0 and 8 is 4, so the axis of symmetry is x = 4.

3. One possible path for the boulder is y x x= − +3

832 . What is the vertex of the parabola created

by this equation?

The axis of symmetry goes through the vertex. The x-coordinate of the vertex is 4. To find the

y-coordinate, substitute 4 into the equation y x x= − +3

832 .

y x x= − +3

832 Original equation

y= − +3

84 3 42( ) ( ) Substitute 4 for x.

y= − +3

816 12( ) Simplify.

y = 6

The vertex is (4, 6).



U2-95North Carolina Math 2North Carolina Math 2 Custom Teacher ResourceCustom Teacher Resource

2.5

Instruction




4. Will the boulder clear the obstacle? How do you know?

Yes, it will. Sketch the boulder’s path using (0, 0), (4, 6), and (8, 0), and you can see that the path is above the obstacle.

14

12

10

6

4

2

0

2

4

6

8

10

12

5 5 10Launch point

(3.5, 2.5) (6, 2.5)

5. Will the boulder knock down the boxes? How do you know?

No, it will not. The top right corner of the boxes is at the point (7, 2.5). If you substitute 7 for xin the equation of the parabola, you find that the parabola contains the point (7, 2.625), which is slightly above the boxes. The boulder will not come in contact with the boxes, though it is close.


• Students will identify key features of quadratic functions such as the x-intercepts and vertex from the equation of the function.

• Students will use key features to graph the quadratic function.


Instruction




IntroductionThe tourism industry thrives on being able to provide travelers with an amazing travel experience. At the same time, tourism professionals need to know exactly how successful they are in their businesses. Tour planners might use quadratic models to determine when profits are increasing or decreasing, when they maximized, and how profits change month to month by looking at the key features of the quadratic functions. In this lesson, you will review the definitions of key features of a quadratic function and how to use graphs, tables, and verbal descriptions to identify and apply the key features.

Key Concepts

• The key features of a quadratic function are characteristics used to describe, draw, and compare quadratic functions. These characteristics are: the x-intercept(s); the y-intercept; values of x for which the function is increasing and decreasing and for which the graph of the function is positive and negative; the function’s minimum or maximum; the axis of symmetry; and the end behavior of the graph of the function.

• Recall each of the forms of quadratic functions.

Standard Form

• The standard form, or general form, of a quadratic function is written as f(f(f x(x( ) = ax2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. c is the constant term. c

• The y-intercept is the value of c.

• The vertex of the function can be found by first determining the x-coordinate, and then substituting this in the function to find the corresponding y-coordinate.

Vertex Form

• The vertex form of a quadratic function is written as f(f(f x(x( ) = a(x(x( – h)2 + k.

• The vertex of the parabola is (h, k).

• The axis of symmetry is identified from vertex form as x = h.

Prerequisite Skills


• knowing the standard form of quadratic functions (F–IF.8a)

• using grap hing technology to model quadratic functions (F–IF.4★)


2.5

Instruction




Factored Form

• The factored form, or intercept form, of a quadratic function is written as f(f(f x(x( ) = a(x(x( – p)(x(x( – q).

• The x-intercepts of the function are p and q.

• The x-intercepts of a quadratic function are where the parabola intersects the x-axis. In the graph that follows, the x-intercepts are 2 and –2.

x

y

–3 –2 –1 0 1 2 3

–4

–2

2

4

6

8

x-intercepts

• All points on the x-axis have a y-coordinate of 0; therefore, the x-intercepts can also be found by identifying the value(s) of x when y = 0.

• The following table of values corresponds to the parabola shown. Notice that the x-intercepts,2 and –2, can be found where the table shows that y is equal to 0.

x y–4 12–2 00 –42 04 12

• The x-intercepts are the x-coordinates of the points where the curve intersects the x-axis. The ordered pair that corresponds to an x-intercept is of the form (x-intercept is of the form (x-intercept is of the form ( , 0). The x-intercepts are also the solutions of the related quadratic equation, ax2 + bx + c = 0.c = 0.c


Instruction




• The y-intercept of a quadratic function is where the parabola intersects the y-axis. In the following graph, the y-intercept is –4.

x

y

–3 –2 –1 0 1 2 3

–4

–2

2

4

6

8

y-intercept

• All points on the y-axis have an x-coordinate of 0; therefore, the y-intercept can also be found by identifying the value of yby identifying the value of yby identifying the value of when x = 0.

• The following table of values corresponds to the parabola shown. Notice that the y-intercept, –4, can be found where the table shows that x is equal to 0.

x y–4 12–2 00 –42 04 12

• The y-intercept is the y-coordinate of the point where the curve intersects the y-axis. The ordered pair that corresponds to a y-intercept is of the form (0, y).


2.5

Instruction




• Recall that the vertex is the maximum or minimum of the function.

• The vertex is also the point where a parabola changes from increasing to decreasing or from decreasing to increasing.

• Increasing refers to the interval of a function for which the output values are becoming larger as the input values are becoming larger.

• Decreasing refers to the interval of a function for which the output values are becoming smaller as the input values are becoming larger.

• Recall that parabolas are symmetric to a vertical line that passes through the vertex, called the axis of symmetry.

• The reflection of every point on a parabola across the axis of symmetry is another point on the parabola. A point and its reflection are equidistant from the axis of symmetry.

• Read the graph from left to right to determine when the function is increasing or decreasing. Trace the path of the graph with a pencil tip going from left to right. If your pencil tip goes down as you move toward increasing values of x, then the function is decreasing. If your pencil tip goes up as you move toward increasing values of x, then the function is increasing.

• If the graph of a quadratic function has a minimum value, it will start by decreasing toward the vertex; after the vertex, it will increase.

• If the graph of a quadratic function has a maximum value, it will start by increasing toward the vertex; after the vertex, it will decrease.

• The vertex is called an extremum. Extrema are the maxima or minima of a function.

• The concavity of a parabola refers to the arch of the curve, either upward or downward.

• A quadratic function that has a minimum value has positive concavity and is concave upbecause the graph of the function is bent upward.

• A quadratic function that has a maximum value has negative concavity and is concave downbecause the graph of the function is bent downward.

• The graphs that follow demonstrate examples of parabolas as they decrease and then increase, and vice versa. Trace the path of each parabola from left to right with your pencil to see the difference.


Instruction




Decreasing then Increasing Increasing then Decreasing

Vertex: (0, –4); minimum

x < 0: decreasing

x > 0: increasing

Direction: concave up

Vertex: (0, 4); maximum

x < 0: increasing

x > 0: decreasing

Direction: concave down

x

y

–4 –2 0 2 4

–4

–2

2

4

x

y

–4 –2 0 2 4

–4

–2

2

4

• End behavior is the behavior of the graph as x approaches positive or negative infinity.

• If the highest exponent of a function is even, and the coefficient of the same term is positive, then the function is approaching positive infinity as x approaches both positive and negative infinity.

• If the highest exponent of a function is even, but the coefficient of the same term is negative, then the function is approaching negative infinity as x approaches both positive and negative infinity.


2.5

Instruction




Even and Positive Even and Negativef(f(f x(x( ) = x2 – 4

Highest exponent: 2

Coefficient of x2: positive

As x approaches positive infinity, f(f(f x(x( ) approaches positive infinity.

As x approaches negative infinity, f(f(f x(x( ) approaches positive infinity.

x

y

–4 –2 0 2 4

–4

–2

2

4

f(f(f x(x( ) = –x) = –x) = – 2 + 4

Highest exponent: 2

Coefficient of x2: negative

As x approaches positive infinity, f(f(f x(x( ) approaches negative infinity.

As x approaches negative infinity, f(f(f x(x( ) approaches negative infinity.

x

y

–4 –2 0 2 4

–4

–2

2

4


• incorrectly identifying when a function is increasing or decreasing

U2-102



Name: Date:

North Carolina Math 2North Carolina Math 2North Carolina Math 2North Carolina Math 2 Custom Teacher ResourceCustom Teacher Resource Custom Teacher ResourceCustom Teacher Resource Custom Teacher Resource2.52.52.52.5


For problems 1–4, use graphing technology to determine the intervals where each function is increasing or decreasing. Then, identify the x-value that maximizes or minimizes each function and any x-intercept(s). Finally, determine whether the function is even, odd, or neither.

1. f(f(f x(x( ) = 2x2 – 4x – 3

2. g(x(x( ) = –4x2 + 6x – 3

3. y = 3x2 + 2x – 2

4. h(x(x( ) = –2x2

For problems 5–8, determine what x-value maximizes or minimizes each function. Then, determine the intervals where each function is increasing or decreasing.

5. A function has a minimum value of –13.5 and x-intercepts of –4.5 and 2.3.

6. A function has a maximum value of 16.25 and x-intercepts of 5 and 2.


Scaffolded Practice 2.5: Interpreting Key Features of Quadratic Functions

continued

U2-103





8. A function has a minimum value of –9.5 and x-intercepts of –3.5 and –1.8.

For problems 9 and 10, use the tables and scenarios to complete each problem.

9. Oliver is on a fishing trip. He casts the rod so that the lure follows a parabolic path. The table shows the lure’s horizontal distance from Oliver and the ball’s height as it travels from a starting position of (0,0). Use a quadratic model to determine for what distances the height of the lure is increasing and decreasing.

Distance from Oliver (feet) Height of lure (feet)0 02 84 126 128 8

10 0

10. The table shows the number of visitors at an art gallery throughout the day. Use a quadratic model to determine when the maximum number of visitors to the gallery likely occurred.

Time Number of visitors9 A.M. 0

11 A.M. 1282 P.M. 1924 P.M. 1926 P.M. 128


Instruction




E xample 1

A local store’s monthly revenue from T-shirt sales, f(f(f x(x( ), as a function of price, x, is modeled by the function f(f(f x(x( ) = –5x2 + 150x – 7. Use the equation and graph to answer the following questions: At what prices is the revenue increasing? Decreasing? What is the maximum revenue? What prices yield no revenue?

x

y

0 5 10 15 20 25 30

200

400

600

800

1000

1200

Mon

thly

T-s

hirt

reve

nue

(dol

lars

)

T-shirt price (dollars)

1. Determine when the function is increasing and decreasing.

Use your pencil to determine for which x-values the function is increasing and decreasing.

Moving from left to right, trace your pencil along the function.

The function increases until it reaches the vertex, then decreases.

The vertex of this function has an x-value of 15.

The revenue is increasing when the price per shirt is less than $15, or when x < 15.

The revenue is decreasing when the price per shirt is more than $15, or when x > 15.

Guided Practice 2.5


2.5

Instruction




2. Determine the maximum revenue.

Use the vertex of the function to determine the maximum revenue.

The T-shirt price that maximizes revenue is $15, or x = 15.

The maximum revenue is the corresponding y-value.

Since it is difficult to estimate accurately from this graph, substitute the value of x into the function and solve.

f(f(f x(x( ) = –5x2 + 150x – 7 Original function

f(15) = –5(15)f(15) = –5(15)f 2 + 150(15) – 7 Substitute 15 for x.

f(15) = 1118 f(15) = 1118 f Simplify.

The maximum revenue is $1,118.

3. Determine the prices that yield no revenue.

Identify the x-intercepts.

The x-intercepts are 0 and 30, so the store has no revenue when the shirts cost $0 and when the shirts cost $30 or more.


Instruction




E xample 2

A quadratic function has a minimum value of –5 and x-intercepts of –8 and 4. What is the value of xthat minimizes the function? For what values of x is the function increasing? Decreasing?

1. Determine the x-value that minimizes the function.

The graphs of quadratic functions are symmetric about the axis of symmetry, the line that divides the parabola in half and passes through the vertex.

The x-value that minimizes the function is the midpoint of the two x-intercepts.

Find the midpoint of the x-intercepts by taking the average of the two values.

x =− +

= −8 4

22

The value of x that minimizes the function is –2.

2. Determine when the function is increasing and decreasing.

Use the vertex to determine for which x-values the function is increasing and for which x-values it is decreasing.

The minimum value is –5 and the vertex of the function is (–2, –5).

From left to right, the function decreases as it approaches the minimum and then increases.

The function is decreasing when x < –2 and increasing when x > –2.


2.5

Instruction




Example 3

The table shows the predicted temperatures for a summer day in Woodland, California. At what times is the temperature increasing? Decreasing?

Time Temperature (°F)

8 A.M. 52

10 A.M. 64

12 P.M. 72

2 P.M. 78

4 P.M. 81

6 P.M. 76

1. Use the table to determine approximate intervals of increasing and decreasing temperatures.

Examine what is happening to the temperatures as the day progresses from morning to evening.

The values are increasing when they get progressively larger in the table, and they are decreasing when they get progressively smaller in the table.

At 4 P.M. the temperature is at its highest, 81°. Before this time, the temperatures are increasing, and after this time, the temperatures are decreasing.


Instruction




2. Use graphing technology to verify the information that is assumed from the table.

On a TI-83/84:

Step 1: Press [STAT].

Step 2: Press [ENTER] to select Edit.

Step 3: Enter x-values into L1. Enter times based on a 24-hour clock for times after 12 P.M. For example, 1 P.M. should be entered as hour 13.

Step 4: Enter y-values into L2.

Step 5: Press [2ND][Y=].

Step 6: Press [ENTER] twice to turn on the Stat Plot.

Step 7: Press [ZOOM][9] to select ZoomStat and show the scatter plot.

Step 8: Press [STAT].

Step 9: Arrow to the right to select Calc.

Step 10: Press [5] to select QuadReg.

Step 11: Enter [L1][,][L2], Y1. To enter Y1, press [VARS] and arrow over to the right to “Y-VARS.” Select 1: Function. Select 1: Y1.

Step 12: Press [ENTER] to see the graph of the Data & the quadratic equation.

On a TI-Nspire:

Step 1: Press the [home] key and select the Lists & Spreadsheet icon.

Step 2: Name Column A “time” and Column B “temperature.”

Step 3: Enter x-values under Column A. Enter times based on a 24-hour clock for times after 12 P.M. For example, 1 P.M. should be entered as hour 13.

Step 4: Enter y-values under Column B.

Step 5: Select Menu, then 3: Data, and then 6: Quick Graph.

Step 6: Press [enter].

Step 7: Move the cursor to the x-axis and choose “time.”(continued)continued)continued


2.5

Instruction




Step 8: Move the mouse to the y-axis and choose “temperature.”

Step 9: Select Menu, then 4: Analyze, then 6: Regression, and then 4: Show Quadratic.

Step 10: Move the cursor over the equation and press the center key in the navigation pad to dra g the equation for viewing, if necessary.

5 10 15

20

0

40

60

80

Time (24-hour clock)

Tem

pera

ture

(ºF)

y

x

3. State your conclusion.

The highest temperature in the table occurs at the maximum y-value on the curve, or in this case, the time at which the temperature goes from increasing to decreasing.

The highest temperature is 81°, and this occurs at 4 P.M.

The maximum temperature appears to happen at hour 16, according to the quadratic model, or at around 4 P.M.

The hig h temperature in the graphed quadratic model is approximately 79°, which is slightly less than 81°, the predicted temperature for that hour.

U2-110

What is the maximum height that the ball reaches as it

continues toward the hoop?



Name: Date:



Problem-Based Task 2.5: One-on-One BasketballYou and a friend are playing one-on-one basketball at the park. You aim at the hoop an d release the ball, which follows a parabolic path. The ta ble represents the ball’s horizontal distance from you and the ball’s height as it travels toward the center of the hoop, which is represented by the point (14, 10). Use a quadratic model to determine for what horizontal distances the height of the ball is increasing and decreasing. What is the maximum height that the ball reaches as it continues toward the hoop?

Distance from shooter (feet) Heigh t of basketball (feet)

4 10

6 12

12 12

14 10

SMP1 ✓ 2 3 4 ✓5 6 ✓7 8

U2-111





Problem-Based Task 2.5: One-on-One Basketball

Coachinga. Is there enough information in the table to estimate where the ball is increasing and decreasing,

and at what distance it reaches its maximum height?

b. What’s the best method to develop a quadratic mo del given a table of values?

c. How do you determine the x-values for which the function is increasing and decreasing?

d. How do you determine the maximum value of the parabola?

e. What is the maximum height that the ball reaches as it continues toward the hoop?


Instruction




Problem-Based Task 2.5: One-on-One BasketballProblem-Based Task 2.5: One-on-One Basketball

Coaching Sample Responsesa. Is there enough information in the table to estimate where the ball is increasing and decreasing,

and at what distance it reaches its maximum height?

The heights, or output values, are the symmetric coordinates about some vertical line, the axis of symmetry.

It is difficult to estimate where the function is increasing and decreasing since this depends on the vertex.

b. What’s the best method to develop a quadratic model given a table of values?

A graphing calculator is the best method because it will fill in all of the blanks not included in the table. The table only gives a snapshot of the basketball’s travels along the way .

Enter ing the table of values into a graphing calculator yields the quadratic model f(x) = –0.125x2 + 2.25x + 3. The quadratic model and plotted points from the table are shown.

20 4 6 8 10 12 14

2

4

6

8

y

x

10

12

14

Distance from shooter (feet)

Hei

ght o

f bas

ketb

all (

feet

)


2.5

Instruction




c. How do you determine the x-values for which the function is increasing and decreasing?

This function has a maximum and therefore is increasing first, then decreasing.

The function will increase until reaching the vertex, which appears have an x-coordinate of 9.

This can be checked by using the fact that the axis of symmetry is halfway between symmetric points, and the vertex lies on the axis of symmetry.

If we use the symmetric points (6, 12) and (12, 12), the x-coordinates are 6 and 12. Calculate the midpoint between 6 an d 12:

6 12

2

18

29x =

+= =

The basketball’s height is increasing when its horizontal distance from the shooter is less than 9 feet and decreasing when its horizontal distance from the shooter is greater than 9 feet.

d. How do you determine the maximum value of the parabola?

The maximum of the parabola occurs at the vertex.

We’ve determined that the parabola’s axis of symmetry is x = 9, so we know that the x-coordinate of the vertex is 9.

From the graph, we can estimate the maximum to be about 13.1 feet.

However, substituting the x-value into the quadratic model will yield a more accurate estimate.

f(x) = –0.125(9)2 + 2.25(9) + 3 = 13.125

e. What is the maximum height that the ball reaches as it continues toward the hoop?

The maximum height the ball reaches is about 13.1 feet.



U2-114© Walch Education© Walch EducationNorth Carolina Math 2North Carolina Math 2 Custom Teacher ResourceCustom Teacher Resource

2.52.5




Task OverviewFocus

Have you ever wondered the path a basketball travels when shooting it into a basketball hoop? Students will create a quadratic model to determine the path the ball travels. They will also determine when the ball is increasing and decreasing as well as reaching its maximum height.


• graphing a quadratic model

• analyzing when the model is increasing and decreasing

• interpreting the maximum height of the object

Introduction

This task should be used to explore or to apply key features of quadratic functions. It is best to implement this task after students learn the various forms of a quadratic function, intercepts, maximum or minimum, increasing or decreasing, concave up or down, and other terms associated with these functions. Graphing may be done using technology, including graphing calculators or online graphing programs.

Begin by reading the problem and clarifying the meaning of path of the basketball. Explain that when shooting a basketball the ball often travels on a parabolic (quadratic) path.




Some students will have difficulty translating the problem and/or interpreting from a graph when a function is increasing/decreasing and the maximum height. Encourage them to look at the ball height to determine when the object is increasing and decreasing. Encourage students to look at the pattern of the model to help determine when the ball is at its highest point.

Problem-Based Task 2.5 Implementation Guide: One-on-One BasketballNorth Carolina Math 2 Standard


U2-115© Walch Education© Walch Education North Carolina Math 2North Carolina Math 2 Custom Teacher ResourceCustom Teacher Resource

2.5





Students can provide a model for the scenario by first plotting the given points for the graph and from there drawing a parabola. Some students will prefer using a model because it is easier for them to see the information in a visual context. Ask the students how the quadratic model can be modified for different distances.

• SMP 6: Attend to precision.

Some students may struggle with finding the exact values of the maximum height as well as when the graph transitions from increasing to decreasing. Work with students on the exact calculation of the values as well as using a graph to provide accurate estimations.



• misinterpreting identifying when a function is increasing or decreasing

Remind students that when the graph is going up, the values for y are increasing, and when the graph is going down, the values for y are decreasing.

• incorrectly identifying the model

Point out to students the model is quadratic since the basketball is shot in the air and must come down.

• incorrectly identifying the axis of symmetry

Suggest that students review the symmetric coordinates and recalculate the value.

• miscalculating the maximum height

Remind students that the value can be approximate from the maximum value on the graph or plug in the value from the axis of symmetry into the quadratic model.



• If students are struggling with determining if there is enough information to draw conclusions, remind them to look at the given table. (Answer: Yes, there is enough information since the ball height in feet is increasing, then decreasing.)

• If students are struggling with the best method, remind them of what tool they can use to help them graph. (Answer: graphing calculator)

• If students are struggling to determine when the function is increasing and decreasing, remind them of how they can find the axis of symmetry. (Answer: find the average of the distance of the shooter at the same height)

U2-116© Walch Education© Walch EducationNorth Carolina Math 2North Carolina Math 2 Custom Teacher ResourceCustom Teacher Resource

2.52.5




• If students are struggling to determine the maximum height, remind students that there are two options. (Answer: use a quadratic model or approximate the maximum height from a graph)


Debriefing the TaskCompare the strategies and explanations used by different students. Ask students to share their information and ask how they came up with these approaches.



• A quadratic model displays an object that increases in height, then decreases in height.

Using symmetric coordinates, the axis of symmetry can be calculated and tell when the graph changes from increasing to decreasing.

• In this task, the maximum value represents the highest point the basketball is when it is shot into the air.

Extending the Task

To extend the task, ask the following questions:

• What is the height of the ball when the ball is 0 feet from the shooter?

• What happens at the ordered pair (14, 9)? (14, 11)

• What is the distance from the shooter when the ball hits the ground?



• For SMP 1, ASK: “How did you make sense of the problem or demonstrate perseverance?” (Answer: I kept reading the problem to determine the information that I needed. I broke down the problem using the key terms (increasing, decreasing, maximum).)

• For SMP 4, ASK: “How did you use mathematics to model this particular scenario?” (Answer: I created a model to help demonstrate information needed to determine increasing and decreasing.)

• For SMP 6, ASK: “How did you make sure you attended to precision?” (Answer: I created an algebraic model for the graph. I extended the table to help create a problem first by finding the maximum height, and then working backward.)

U2-117© Walch Education© Walch Education North Carolina Math 2North Carolina Math 2 Custom Teacher ResourceCustom Teacher Resource

2.5





• Students can use a graphing utility to find the exact model and find the exact value of the maximum height.

• Students could find additional points for the table after finding the graph for the model.

Technology

Students can use graphing utilities to help create a quadratic model.

U2-118



Name: Date:



continued

For each of the given quadratic functions, use graphing technology to answer the following questions: What are the x-values for which the function is increasing? Decreasing? What is the maximum or minimum value of the function? What are the intercepts?

1. f(f(f x(x( ) = 3x2 – 2x – 5

2. g(x(x( ) = –3x2 + 10x + 1

3. y = 5x2 + 10x + 11

4. h(x(x( ) = 2x2 – 4x – 11

Given the following descriptions of quadratic functions, answer the questions: What is the value of x that minimizes or maximizes the function? For what values of x is the function increasing? Decreasing?

5. A function has a minimum value of –20 and x-intercepts of –1.72 and 0.38.

6. A function has a maximum value of 12.375 and x-intercepts of 0.41 and 1.84.

7. A function has a minimum value of –8.675 and x-intercepts of 1.23 and –0.48.

8. 8. A function has a minimum value of –8.167 and A function has a minimum value of –8.167 and xx-intercepts of 1.3 and –1.-intercepts of 1.3 and –1.

Practice 2.5: Interpreting Key Features of Quadratic Functions A

U2-119





Use the tables and scenarios that follow to complete the remaining problems.

9. You and a friend are playing softball. You throw the ball toward your friend’s mitt so that the ball follows a parabolic path. The table represents the ball’s horizontal distance from you and the ball’s height as it travels from a starting position of (0, 4). Use a quadratic model to determine for what distances the height of the ball is increasing and decreasing.

Distance from you (feet) Height of softball (feet)

0 4

20 10

30 10

40 8

10. The table shows the predicted temperatures for an autumn day in Annapolis, Maryland. Use a quadratic model to determine the maximum temperature that Annapolis reaches on this day.

Time Temperature (°F)

11 A.M. 59

2 P.M. 63

5 P.M. 63

8 P.M. 58

11 P.M. 56

U2-120



Name: Date:



Practice 2.5: Interpreting Key Features of Quadratic Functions BFor each of the given quadratic functions, use graphing technology to answer the following questions: What are the x-values for which the function is increasing? Decreasing? What is the maximum or minimum value of the function? What are the x-intercepts?

1. f(f(f x(x( ) = x2 – 3x – 6

2. g(x(x( ) = –x) = –x) = – 2 – 4x + 7

3. y = –4x2 + 8x + 12

4. h(x(x( ) = 5x2

Given the f ollowing descriptions of quadratic functions, answer the questions: What is the value of x that minimizes or maximizes the function? For what values of x is the function increasing? Decreasing?

5. A function has a minimum value of –16.3 and x-intercepts of 9.3 and –41.9.

6. A function has a minimum value of –8.125 and x-intercepts of 4.27 and 0.23.

7. A function has a maximum value of 0.417 and x-intercepts of –1.618 and 0.618.


continued

U2-121





Use the tables and scenarios that follow to complete the remaining problems.

9. You are practicing punting the football before football tryouts. You kick the ball from the ground represented by the point (0, 0), and the path of the ball is parabolic. The table represents the height of the ball seconds after being kicked. Use a quadratic model to determine at what times the height of the ball is increasing and decreasing.

Time (seconds) Height of football (feet)

0 0

0.5 12

1 16

1.5 12

2 0

10. The table shows the height of a signal flare seconds after it is shot from the deck of a ship. Signal flares explode when they reach their highest point. Use a quadratic model to determine how high the flare will be when it explodes.

Time (seconds) Height of flare (feet)

0 112

1 192

2 240

3 256

4 240

5 192

North Carolina Math 2 Custom Teacher Resource2.6

© Walch EducationU2-122

Name: Date:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–IF.8Lesson 2.6: Interpreting Various Forms of Quadratic Functions

Lesson 2.6: Interpreting Various Forms of Quadratic Functions

Warm-Up 2.6Joe is trying to use a hose to spray water over a stack of moving boxes and into a bird bath. The graph represents a stack of boxes on the left that the water must clear and a birdbath on the right that the water must fill. The water will follow a parabolic path.

16

14

12

10

8

6

4

2

2

4

6

8

10

12

14

10 5 5 10

(4, 5) (6, 5)

0

1. One possible path the water could travel is given by y x x= − + +1

5

8

5

9

52 , where y represents the

height in feet and x represents the horizontal distance traveled in feet. What is the vertex of

this parabola?

2. Determine the second x-intercept if one x-intercept of the path of the water is –1.

3. What is the maximum value of the quadratic function?

4. Sketch the graph of the path of the water.

5. Based on the graph, will the water clear the boxes? If it clears the boxes, will the water fill the birdbath?



Instruction

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–IF.8Lesson 2.6: Interpreting Various Forms of Quadratic Functions

Lesson 2.6: Interpreting Various Forms of Quadratic Functions

Warm-Up 2.6 Debrief

1. One possible path the water could travel is given by y x x= − + +1

5

8

5

9

52 , where y represents the

height in feet and x represents the horizontal distance traveled in feet. What is the vertex of

this parabola?

The vertex is of the form 2

,2

b

af

b

a

− −

.

Use the original equation, 1

5

8

5

9

52y x x=− + + , to find the values of a and b in order to find the

x-coo rdinate of the vertex.

2x

b

a=− Formula to find the x-coordinate of the

vertex of a parabola

8

5

21

5

x =−

−

Substitute 1

5− for a and

8

5 for b.

x = 4 Simplify.

The x-coordinate of the vertex is 4.

Substitute 4 into the original equation to find the y-coordinate of the vertex.

1

5

8

5

9

52y x x=− + + Original equation

1

5(4)

8

5(4)

9

52y=− + + Substitute 4 for x.

y = 5 Simplify.

The y-coordinate of the vertex is 5.

The vertex is located at (4, 5).





Instruction


2. Determine the second x-intercept if one x-intercept of the path of the water is –1.

Use symmetry to identify the second x-intercept.

The axis of symmetry goes through the vertex, so the axis of symmetry is x = 4.

For each point to the left of the axis of symmetry, there is another point the same distance on the right side of the axis.

The point (–1, 0) is on the graph; x = –1 is 5 units to the left of the axis of symmetry.

The point that is 5 units to the right of the axis is 9, so the point (9, 0) is also on the graph. The other x-intercept is 9.

3. What is the maximum value of the quadratic function?

The maximum value is the y-coordinate of the vertex, 5.

4. Sketch the graph of the path of the w ater.

16

14

12

10

8

6

4

2

0

2

4

6

8

10

12

14

10 5 5 10

(4, 5) (6, 5)

y

x

5. Based on the graph, will the water clear the boxes? If it clears the boxes, will the water fill the birdbath?

No, the water will not clear the boxes. If the boxes were not in the way, the water would still not fill the birdbath because the birdbath is too tall.


• Students will extend their knowledge of identifying key features of a quadratic function in standard form in order to identify key features of a quadratic function written in alternative forms.



Instruction


I ntroductionQuadratic equations can be written in several forms, including standard form, vertex form, and factored form. While each form is equivalent, certain forms easily reveal different features of the graph of the quadratic function. In this lesson, you will learn to use the various forms of quadratic functions to show the key features of the graph and determine how these key features relate to the characteristics of a real-world situation.

Key Concepts

Standard Form

• Recall that the standard form, or general form, of a quadratic function is written as f(f(f x(x( ) = ax2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. c is the constant term. c

• When a function is written in standard form, the y-intercept is the value of c.

• The vertex of the function can be found by first determining the value of x, 2

xb

a=−

, and

then finding the corresponding y-value, 2

y fb

a=

−

.

• The vertex is often written as 2

,2

b

af

b

a

− −

.

• If a > 0, the function has a minimum and the graph opens up.

• If a < 0, the function has a maximum and the graph opens down.

Vertex Form

• The vertex form of a quadratic function is written as f(f(f x(x( ) = a(x(x( – h)2 + k.

• In vertex form, the maximum or minimum of the function is identified using the vertex of the parabola, the point (h, k).

Prerequisite Skills


• applying the order of operations (5.OA.1)

• identifying key features of a quadratic function in standard form (F–IF.4★)



Instruction


• If a > 0, the function has a minimum. The minimum is the value of k, the y-coordinate of the vertex.

• If a < 0, the function has a maximum. The maximum is the value of k, the y-coordinate of the vertex.

• Because the axis of symmetry goes through the vertex, the axis of symmetry can be identified from vertex form as x = h.

• The graph of a quadratic function is symmetric about the axis of symmetry.

Factored Form

• The factored form, or intercept form, of a quadratic function is written as f(f(f x(x( ) = a(x(x( – p)(x)(x)( – q).

• Recall that the x-intercepts of a function are the x-values when y = 0.

• In factored form, the x-intercepts of the function are identified as p and q.

• Recall that the y-intercept of a function is the point at which the function intersects the y-axis.

• To determine the y-intercept, substitute 0 for x and simplify.

• The axis of symmetry can be identified from the factored form since it passes through the

midpoint between the x-intercepts. Therefore, the axis of symmetry is 2

xp q

=+

.

• To determine the vertex of the parabola, calculate the y-value that corresponds to the x-value of the axis of symmetry.

• If a > 0, the function has a minimum and the graph opens up.

• If a < 0, the function has a maximum and the graph opens down.


• confusing the attributes of different forms

• incorrectly identifying x-intercepts of the factored form

• incorrectly identifying the vertex as a maximum or minimum




Use the given functions to complete all parts of problems 1–3.

1.

a. Identify the y-intercept.

b. Identify the vertex.

c. Identify whether the function has a maximum or minimum.

2.

a. Identify the x-intercepts.

b. Identify the y-intercepts.

c. Identify the axis of symmetry.

d. Identify the vertex.

3.

a. Identify the vertex.

b. Identify whether the function has a maximum or minimum.

continued

Scaffolded Practice 2.6: Interpreting Various Forms of Quadratic Functions




For problems 4–10, use the given information to complete the problems.

4. Aerobatic pilots can maneuver jet planes with impressive control. A popular stunt performed by pilots involves decreasing altitude to a height near the ground and then re-ascending. A particular pilot’s path during this stunt is modeled by the function , where h denotes the pilot’s height above the ground (in meters) at time t (in seconds). Graph the t (in seconds). Graph the tfunction, determine its vertex, and interpret the minimum value in this context.

5. Find the maximum or minimum of the function . What are the coordinates of this point?

6. Find the x-intercepts of the function .

continued




7. The curveball is an essential pitch in baseball. It follows a parabolic motion in which it “breaks,”

or descends sharply, after reaching its maximum height, fooling the batter. Suppose a pitcher’s

curveball follows the path , where r denotes the ball’s height r denotes the ball’s height r

(in feet) at a given distance x (in feet) away from the pitcher and toward home plate. Graph the

function. Then, determine the vertex and interpret it in this context.

8. Determine whether has a maximum or a minimum. Find the coordinates of that extremum.

continued




9. A volleyball server must have exceptional precision when serving. Suppose a particular

server wants to launch the ball at the farthest corner on the opposing side. Let the function

describe the height B (in meters) of the ball at a given distance x

(in meters) from the server. Graph the function. Then, determine the x-intercepts and interpret

them in this context.

10. Find the x-intercepts of the function .



Instruction


E xample 1

Suppose that the flight of a launched bottle rocket can be modeled by the function fSuppose that the flight of a launched bottle rocket can be modeled by the function fSuppose that the flight of a launched bottle rocket can be modeled by the function (f(f x(x( ) = –(x(x( – 1)x – 1)x (x(x( – 6), x – 6), xwhere fwhere fwhere (f(f x(x( ) measures the height above the ground in meters and x represents the horizontal distance x represents the horizontal distance xin meters from the launching spot at the point (1, 0). How far does the bottle rocket travel in the horizontal direction from launch to landing? What is the maximum height the bottle rocket reaches? How far has the bottle rocket traveled horizontally when it reaches its maximum height? Graph the function.

1. Identify the x-intercepts of the function.

In the function, f(f(f x(x( ) represents the height of the bottle rocket. At launch and landing, the height of the bottle rocket is 0.

The function f(f(f x(x( ) = –(x) = –(x) = –( – 1)(x – 1)(x – 1)( – 6) is of the form f(f(f x(x( ) = a(x(x( – p)(x)(x)( – q), where p and q are the x-intercepts.

The x-intercepts of the function are 1 and 6.

Find the distance between the two points to determine how far the bottle rocket traveled in the horizontal direction.

6 – 1 = 5

The bottle rocket traveled 5 meters in the horizontal direction from launch to landing.

2. Determine the maximum height of the bottle rocket.

The maximum height occurs at the vertex.

Find the x-coordinate of the vertex using the formula 2

xp q

=+

.

2x

p q=

+ Formula to determine the x-coordinate of the vertex of a parabola

6 1

2x =

+Substitute 6 for p and 1 for q.

x = 3.5 Simplify.

The x-coordinate of the vertex is 3.5.(continued)continued)continued

Guided Practice 2.6



Instruction


Use this value to determine the vertex of the function.

f(f(f x(x( ) = –(x) = –(x) = –( – 1)(x – 1)(x – 1)( – 6) Original function

f(3.5) = –[(3.5) – 1][(3.5) – 6]f(3.5) = –[(3.5) – 1][(3.5) – 6]f Substitute 3.5 for x.

f(3.5) = –(2.5)(–2.5)f(3.5) = –(2.5)(–2.5)f Simplify.

f(3.5) = 6.25f(3.5) = 6.25f Multiply.

The y-coordinate of the vertex is 6.25.

The maximum height reached by the bottle rocket is 6.25 meters.

3. Determine the horizontal distance from the launch point to the maximum height of the bottle rocket.

We know that the bottle rocket is launched from the point (1, 0) and reaches a maximum height at (3.5, 6.25). Subtract the x-value of the two points to find the distance traveled horizontally.

3.5 – 1 = 2.5

Another method is to take the total distance traveled horizontally from launch to landing and divide by 2 to find the same answer. This is because the maximum value occurs halfway between the x-intercepts of the function.

5

22.5=

The bottle rocket travels 2.5 meters horizontally when it reaches its maximum.



Instruction


4. Graph the function.

Use a graphing calculator or complete a table of values. Use the x-intercepts and vertex as three of the known points. Choose x-values on either side of the vertex for two additional x-values.

x y1 02

3.5 6.2556 0

To determine the y-coordinates of the additional points, substitute each x-value into the original function and evaluate.


f(2) = –[(2) – 1][(2) – 6]f(2) = –[(2) – 1][(2) – 6]f Substitute 2 for x.

f(2) = –(1)(–4)f(2) = –(1)(–4)f Simplify.

f(2) = 4f(2) = 4f Mul tiply.


f(5) = –[(5) – 1][(5) – 6]f(5) = –[(5) – 1][(5) – 6]f Substitute 5 for x.

f(5) = –(4)(–1)f(5) = –(4)(–1)f Simplify.

f(5) = 4f(5) = 4f Multiply.

Fill in the missing table values.

x y1 02 4

3.5 6.255 46 0

(continued)continued)continued



Instruction


P lot the points on a coordinate plane and connect them using a smooth curve.

Since the function models the flight of a bottle rocket, it is important to on ly show the portion of the graph whe re both horizontal distance and height are positive.

-1 0 1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

(1, 0) (6, 0)

(5, 4)

(3.5, 6.25)

(2, 4)

Horizontal distance (m)

Hei

ght (

m)



Instruction


Example 2

Reducing the cost of an item can result in a greater number of sales. The revenue function that predicts the revenue in dollars, R(x(x( ), for each $1 change in price, x, for a particular item is R(x(x( ) = –100(x) = –100(x) = –100( – 7)2 + 28,900. What is the maximum value of the function? What does the maximum value mean in the context of the problem? What price increase maximizes the revenue and what does it mean in the context of the problem? Graph the function.

1. Determine the maximum value of the function.

The function R(x(x( ) = –100(x) = –100(x) = –100( – 7)2 + 28,900 is written in vertex form, f(x) = a(x – h)2 + k, where (h, k) is the vertex.

The vertex of the function is (7, 28,900); therefore, the maximum value is 28,900.

2. Determine what the maximum value means in the context of the problem.

The maximum value of 28,900 means that the maximum revenue resulting from increasing the price by x dollars is $28,900.

3. Determine the price increase that will maximize the revenue and what it means in the context of the problem.

The maximum value occurs at the vertex, (7, 28,900).

This means an increase in price of $7 will result in the maximum revenue.



Instruction



Use a graphing calculator or complete a table of coordinates. Use the vertex as one known point. Choose x-values on either side of the vertex to have four additional x-values.

x y

0

5

7 28,900

9

14

To determine the y-coordinates of the additional points, substitute each x-value into the original function and solve.

R(x(x( ) = –100(x) = –100(x) = –100( – 7)2 + 28,900 Original function

R(0) = –100[(0) – 7]2 + 28,900 Substitute 0 for x.

R(0) = 24,000 Simplify.



R(5) = 28,500 Simplify.



R(9) = 28,500 Simplify.



R(14) = 24,000 Simplify.




Instruction



x y

0 24,000

5 28,500

7 28,900

9 28,500

14 24,000

Notice that the points (0, 24,000) and (14, 24,000) are the same horizontal distance fr om the vertex on either side. The same is true for (5, 28,500) and (9, 28,500).

Plot the points on a coordinate plane and connect using a smooth cu rve .

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1,0002,0003,0004,0005,0006,0007,0008,0009,000

10,00011,00012,00013,00014,00015,00016,00017,00018,00019,00020,00021,00022,00023,00024,00025,00026,00027,00028,00029,000

(0, 24,000)

(5, 28,500)

(7, 28,900)

(9, 28,500)

(14, 24,000)

Change in price ($)

Reve

nue

($)



Instruction


E xample 3

A football is kicked and follows a path given by f(f(f x(x( ) = –0.03x2 + 1.8x, where f(f(f x(x( ) represents the height of the ball in feet and x represents the horizontal distance in feet. What is the maximum height the ball reaches? What horizontal distance maximizes the height? Graph the function.

1. Determine the maximum height of the ball.

The function f(f(f x(x( ) = –0.03x2 + 1.8x is written in standard form, f(f(f x(x( ) = ax2 + bx + c, where a = –0.03, b = 1.8, and c = 0.

The maximum occurs at the vertex, 2

,2

b

af

b

a

− −

.

Determine the x-coordinate of the vertex.

2x

b

a=− Formula to determine the x-coordinate

of the vertex of a parabola(1.8)

2( 0.03)x =

−−

Substitute values for a and b.

x = 30 Simplify.

Determine the y-coordinate of the vertex.

f(f(f x(x( ) = –0.03x2 + 1.8x Original function

f(30) = –0.03(30)f(30) = –0.03(30)f 2 + 1.8(30) Substitute 30 for x.

f(30) = 27f(30) = 27f Simplify.

The vertex is (30, 27), so the maximum value is 27 feet.

The maximum height the ball reaches is 27 feet.

2. Determine the horizontal distance of the ball when it reaches its maximum height.

This horizontal distance is determined by the x-coordinate of the vertex.

The vertex is (30, 27).

The ball will have traveled 30 feet in the horizontal direction when it reaches its maximum height.



Instruction



Use a graphing calculator or complete a table of coordinates. Use the vertex as one known point. Choose x-values on either side of the vertex to have four additional x-values.

x y

5

20

30 27

40

55

To determine the y-coordinates of the additional points, substitute each x-value into the original function and solve.



f(5) = 8.25f(5) = 8.25f Simplify.



f(20) = 24f(20) = 24f Simplify.



f(40) = 24f(40) = 24f Simplify.



f(55) = 8.25f(55) = 8.25f Simplify.




Instruction



x y

5 8.25

20 24

30 27

40 24

55 8.25

Notice that the points (5, 8.25) and (55, 8.25) are the same horizontal distance fr om the vertex on either side. The same is true for (20, 24) and (40, 24).

Plot the points on a coordinate plane and connect them using a smooth curve.

Since the function models the path of a kicked football, it is important to only show the portion of the graph where both height and horizontal distance are positive.

0 5 10 15 20 25 30 35 40 45 50 55 60 65

2

4

6

8

10

12

14

16

18

20

22

24

26

28

(5, 8.25)

(20, 24)

Horizontal distance

Hei

ght

(30, 27)

(40, 24)

(55, 8.25)




Problem-Based Task 2.6: Is the Glider Safe?Shin is a beginner hang glider. He’s practicing jumping from a certain height, dipping initially, and then rising. Shin should dip to a height no lower than 6 feet above the ground, which is considered a safe height, before changing direction and beginning to rise. The position of Shin’s hang glider is given by y = (x = (x = ( – 4)(x – 4)(x – 4)( – 6), with x representing the time in seconds since Shin starts the initial jump and y representing the distance in feet from the safe height. Will Shin stay above the safe height? How long will it take for Shin to reach the initial height of the jump?

Will Shin stay above the safe

height?How long will it take for Shin to How long will it take for Shin to How long will it

reach the initial height of the

jump?

SMP1 ✓ 2 3 ✓ 4 ✓5 6 7 8




Problem-Based Task 2.6: Is the Glider Safe?Problem-Based Task 2.6: Is the Glider Safe?

Coachinga. What is the minimum height Shin reaches?

b. Does a negative answer make sense in the context of the problem? Explain.

c. Find the time at which the minimum height occurs.

d. What are the x-intercepts and what do they represent?

e. Will Shin stay above the safe height? Explain.

f. f. How long will it take for Shin to reach his initial height?How long will it take for Shin to reach his initial height?



Instruction


Problem-Based Task 2.6: Is the Glider Safe?

Coaching Sample Responsesa. What is the minimum height Shin reaches?

The minimum occurs at the vertex. Write the equation in standard form.

y = (x = (x = ( – 4)(x – 4)(x – 4)( – 6) Original equation

y = x2 – 6x – 4x + 24 Distribute.

y = x2 – 10x + 24 Simplify.


,2

b

af

b

a

− −

.

Use the equation y = x2 – 10x + 24 to find the values of a and b in order to find the x-coordinate of the vertex.

2x

b

a=− Formula to determine the x-coordinate of the

vertex of a parabola

( 10)

2(1)x =

− −Substitute 1 for a and –10 for b.

x = 5 Simplify.


Substitute 5 into the original equation to find the y-coordinate.

y = x2 – 10x + 24 Original equation

y = (5)2 – 10(5) + 24 Substitute 5 for x.

y = –1 Simplify.

The y-coordinate of the vertex is –1.

The vertex is located at (5, –1). T he minimum height that Shin reaches is –1 foot.

b. Does a negative answer make sense in the context of the problem? Explain.

Yes. The value of –1 means 1 foot below the safe height.

c. Find the tim e at which the minimum height occurs.


Shin reaches the minimum 5 seconds after the jump.



Instruction


d. What are the x-intercepts and what do they represent?

The x-intercepts can be identified from the original equation written in factored form.

y = (x = (x = ( – 4)(x – 4)(x – 4)( – 6)

The x-intercepts are 4 and 6.

The x-intercepts represent times when the glider goes from a safe height to an unsafe height or vice versa.

The glider is at the safe height for the first 4 seconds, and then again after 6 seconds.

e. Does Shin stay above the safe height? Explain.

No. Shin is above the safe height for the first 4 seconds after the jump, and again after 6 seconds.

He dips below the safe height between 4 seconds and 6 seconds.

For 2 seconds, the glider is below the safe height.

f. How long will it take for Shin to reach his initial height?

The initial height occurs when x = 0.

The axis of symmetry passes through the vertex, (5, –1).

The axis of symmetry is x = 5.

Zero is 5 units to the left of the axis of symmetry.

The y-value is the same for the point 5 units to the right of the axis of symmetry.

At x = 10, the height will be the same as the initial height.

Substitute 0 and 10 for x to check.

y = (x = (x = ( – 4)(x – 4)(x – 4)( – 6) y = (x = (x = ( – 4)(x – 4)(x – 4)( – 6)

y = [(0) – 4][(0) – 6] y = [(10) – 4][(10) – 6]

y = 24 y = 24

Shin will reach his initial heigh t of 24 feet above the safe height (or 30 feet above the ground) after 10 seconds.







Task OverviewFocus

Students will use a quadratic function to analyze the path of a hang glider and determine whether the ride is safe. They will use the key features of a quadratic function to make the decision.


• writing a quadratic equation in standard form

• determining the vertex of a parabola from an equation

• interpreting the attributes of a quadratic function in terms of the context of the problem

Introduction

This task should be used to explore a quadratic function which models a real-world situation by analyzing the vertex and intercepts of the function. It is best to implement this task after students have analyzed several quadratic functions and their key features. Graphing may be done using technology, including graphing calculators or online graphing programs.

Begin by reading the problem and clarifying how a hang glider moves through the air. Review the following term:

hang glider an unpowered vehicle used to glide through the air




Encourage students to create a graph of the function to help make sense of the problem. Discuss various points on the graph and what those points mean in the context of the problem. For example, the point (7, 3) means that after 7 seconds Shin is 3 feet above the safe height.

Problem-Based Task 2.6 Implementation Guide: Is the Glider Safe?North Carolina Math 2 Standard





Implementation Guide• SMP 3: Construct viable arguments and critique the reasoning of others.

Students more often see negative numbers that do not apply to the problem context, than they do ones that do apply. Use a graph of the function to help students make a valid argument as to why a y-coordinate of –1 makes sense in the context of this problem.


The path of the hang glider is modeled by the quadratic equation given in the problem statement. Some students may have difficulty understanding how and why a quadratic function models the path of the glider. Allow students to graph the equation and use the graph to aid in understanding the various steps in the solution.



• using 10 instead of –10 for the value of b

Have students write the equation in the form y = x2 + (–10x) + 24.

• assuming a negative value for the y-coordinate does not make sense in the context of the problem

Tell students that a safe height occurs when y ≥ 0, not when y ≥ 6 because y represents the distance from the safe height, not the distance from the ground.

• assuming the initial height occurs when y = 0

Explain that the initial height occurs when the time is 0, so it occurs when x = 0. Remind students that the initial height is not 0. Nor is it 6. The initial height is the height from which Shin jumps.



• If students say that the minimum height is 0, ask them to describe the path of a hang glider. (Answer: A hang glider starts from a height well above 6 feet. Then the glider dips to no more than 6 feet and rises again. The only time the height is 0 is when the glider lands. The landing is not discussed in this model.)

• If students are unsure of where the minimum value occurs with respect to the key features of the function, have them graph the function and describe where they think the minimum value appears graphically. (Answer: at the vertex)

• If students say that the value of –1 does not make sense, ask them to describe what a y-value of 0 means. (Answer: The glider is at the safe height.)




Implementation Guide• If students say that x-intercepts represent when the glider is at 0 feet, review what x and y

represent in the context of the problem. (Answer: x represents the number of seconds since Shin jumps, and y represents the distance from the safe height. So when y is 0, the x-values are the times when the glider is at the safe height.)


Debriefing the Task Compare solving the problem algebraically and graphically. Ask students to decide which method they prefer and why. Discuss how one method can be used to solve the problem and the other method can be used to check your work.



• Recall that the standard form, or general form, of a quadratic function is written as f(f(f x(x( ) = ax2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term.c is the constant term.c

• The vertex of a function can be found by first determining the value of x, =−

xb

a2, and then

finding the corresponding y-value, =−

y fb

a2.

Students use the standard form of the function to identify the vertex.

• Recall that the x-intercepts of a function are the x-values where the function is 0.

Students use the x-intercepts to identify when the glider is at the safe height.

Extending the Task

To extend the task, have students analyze the jump in more detail. For example:

• Have students draw a graph of the height of the glider with respect to the time. Ask, “How does that graph compare to the graph of the function for the safe height?” Discuss whether the graph accurately represents all phases of the jump.

• Have students write a function that models a safe hang gliding ride, one that does not go below the safe height.

• Show one or more videos of hang gliding from the following website:

Vimeo.com. “Videos Tagged ‘Hang Gliding’.”





Implementation GuideConnecting to Standards for Mathematical Practice


• For SMP 1, ASK: “How did you make sense of the problem or demonstrate perseverance?” (Answer: I reread the problem, listing what I knew. I made a graph to help me understand the problem.)

• For SMP 3, ASK: “Did you construct viable arguments and did you critique the reasoning of others?” (Answer: I determined whether a negative value made sense for the problem and justified my answer.)

• For SMP 4, ASK: “How did you use mathematics to model this particular scenario?” (Answer: I graphed the function to help me better understand the problem and what I was asked to find.)


This task could first be explored graphically instead of algebraically with students using the graph to answer the questions.

Technology

Students can use a graphing calculator or online software to graph the function.





1. f(f(f x(x( ) = x2 – 6x + 8




2. f(f(f x(x( ) = –0.5(x) = –0.5(x) = –0.5( + 2)(x + 2)(x + 2)( – 4)


b. Identify the y-intercept.



3. f(f(f x(x( ) = –16(x) = –16(x) = –16( – 1)2 + 10



Practice 2.6: Interpreting Various Forms of Quadratic Functions AA

continued




Use the given information in each scenario that follows to complete the remaining problems.

4. A bird is descending toward a lake to catch a fish. The bird’s flight can be modeled by the equation h(t) = t) = t t2t2t – 14t + 40, where t + 40, where t h(t) is the bird’s height above the water in feet and t) is the bird’s height above the water in feet and t t is the t is the ttime in seconds since you saw the bird. Graph the function. What is the vertex? What does the minimum value mean in the context of the problem?

5. A military pilot fires a test missile whose path can be modeled by the equation f A military pilot fires a test missile whose path can be modeled by the equation f A military pilot fires a test missile whose path can be modeled by the equation (f(f x(x( ) = –(x) = –(x) = –( – 40)(x – 40)(x x – 40)(x – 40)( + 2), x + 2), xwhere fwhere fwhere (f(f x(x( ) is the height of the missile in miles and x is the number of seconds since the missile was x is the number of seconds since the missile was xfired. Graph this function. What are the x-intercepts and what do they mean in the context of the problem? After how many seconds is the height of the missile the same as the initial height?

continued




6. The path of a snowboarder performing stunts is given by the equation f(f(f x(x( ) = –16(x) = –16(x) = –16( – 2)(x – 2)(x – 2)( + 1), where x is time in seconds and f(f(f x(x( ) is the height of the snowboarder above the ground. Graph the function. What are the x-intercepts? Explain the meaning of the x-intercepts in the context of the problem. How long does the stunt last?

7. The flight of a paper airplane follows the quadratic equation H(H(H x(x( ) = –(x) = –(x) = –( – 3)2 + 25, where H(H(H x(x( ) represents the height of the paper airplane in feet, and x is the horizontal distance in feet the airplane travels after it is thrown. Graph the function. What is the vertex? Explain the meaning of the vertex in the context of the problem.

continued




8. The height of a golfer’s ball is given by the equation y = –16x2 + 32x, where y represents the height in feet and x represents the time in seconds. Graph the function. What is the vertex and what does it mean in the context of the problem?

9. The revenue, R(x(x( ), generated by an increase in price of x dollars for an item is represented by the equation R(x(x( ) = –5(x) = –5(x) = –5( – 15)(x – 15)(x – 15)( + 5). Graph the function. What are the x-intercepts and what do they represent in the context of the problem? What value of x maximizes the revenue?

continued




10. Reducing the cost of an item can result in a greater number of sales. The revenue function that predicts the revenue in dollars, R(x(x( ), for each $1 decrease in price, x, is R(x(x( ) = – (x) = – (x) = – ( – 7)2 + 289. Graph the function. What is the vertex and what does it represent in the context of the problem?




B

continued

Practice 2.6: Interpreting Various Forms of Quadratic Functions


1. f(f(f x(x( ) = x2 – 8x + 12




2. f(f(f x(x( ) = –2(x) = –2(x) = –2( – 3)(x – 3)(x – 3)( + 5)


b. Identify the y-intercept.



3. f(f(f x(x( ) = –16(x= –16(x= –16( – 3)2






Use the given information in each scenario that follows to complete the remaining problems.

4. A butterfly descends toward the ground and then flies back up. The butterfly’s descent can be modeled by the equation h(t) = t) = t t2t2t – 10t + 26, where t + 26, where t h(t) is the butterfly’s height above the ground t) is the butterfly’s height above the ground tin feet and t is the time in seconds since you saw the butterfly. Graph the function and identify t is the time in seconds since you saw the butterfly. Graph the function and identify tthe vertex. What is the meaning of the vertex in the context of the problem?

5. A cliff diver jumps upward from the edge of a cliff then begins to descend, so that his path follows a parabola. The diver’s height, h(t), above the water in feet is given by t), above the water in feet is given by th(t) = –2(t) = –2(t t – 1)t – 1)t 2 + 52, where t represents the time in seconds. Graph the function. What is the t represents the time in seconds. Graph the function. What is the tvertex and what does it represent in the context of the problem? How many seconds after the start of the dive does the diver reach the initial height?

continued




6. The revenue of producing and selling widgets is given by the function R(x(x( ) = –8(x) = –8(x) = –8( – 50)(x – 50)(x – 50)( – 2), where x is the number of widgets produced and R(x(x( ) is the amount of revenue in dollars. Graph the function. What are the x-intercepts and what do they represent in the context of the problem? What number of widgets maximizes the revenue?

7. A football is kicked and follows a path given by y = –0.03x2 + 1.2x, where y represents the height of the ball in feet and x represents the horizontal distance in feet. Graph the function. What is the vertex and what does it mean in the context of the problem? How far does the ball travel in the horizontal direction?

continued




8. A frog hops from the bank of a creek onto a lily pad. The path of the jump can be modeled

by the equation ( )1

2( 2) 42h x x=− − + , where h(x(x( ) is the frog’s height in feet above the water

and x is the number of seconds since the frog jumped. Graph the function. What does the

vertex represent in the context of the problem? What is the axis of symmetry? After how many

seconds does the height of the frog reach the initial height?

9. The revenue, R(x(x( ), generated by an increase in price of x dollars for an item is represented by the equation R(x(x( ) = –2x2 + 20x + 150. Graph the function and identify the vertex. What does the vertex represent in the context of the problem? What is the axis of symmetry? What increase in price results in the same revenue as not increasing the price at all?

continued




10. Reducing the cost of an item can result in a greater number of sales. The revenue function that predicts the revenue in dollars, R(x(x( ), for each $1 decrease in price, x, for a certain item is R(x(x( ) = –(x) = –(x) = –( – 26)(x – 26)(x – 26)( + 10). Graph the function. Identify the x-intercepts. What do the x-intercepts represent in the context of the problem? What is the axis of symmetry? What increase in price results in the same revenue as not increasing the price at all?


2.7

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–IF.9Lesson 2.7: Comparing Properties of Quadratic Functions Given in Different Forms

Name: Date:

Warm-Up 2.7

Two kingfishers are perched on a tree branch at the edge of a lake, when they dive into the lake to snatch fish. The given quadratic functions represent the height of each kingfisher relative to the surface of the lake, with f(f(f x(x( ) representing the first bird and g(x(x( ) representing the second bird. For each set of position functions, determine which bird dives deeper into the lake.

1. f x x1

5– 10 – 22( ) ( )= or g(x(x( ) = {(0, 15), (10, –1), (20, 15)}

2. f x x x1

102 9.52( ) = − + or the quadratic function that passes through the following points:

x 0 2 4 6 8

g(g(g x)x)x 17 7 1 –1 1

3. the quadratic function that passes through the points on the following table or the function described by the following graph:

x f(f(f x)x)x

0 85 3

10 015 –120 0

4 8 12 16 182 6 10 14 20

y

x22 24

6

4

2

0

10

8

12

14

–2

g(x)

Lesson 2.7: Comparing Properties of Quadratic Functions Given in Different Forms


2.7


Instruction

Warm-U p 2.7 Debrief

1. f x x1

5– 10 – 22( ) ( )= or g(x(x( ) = {(0, 15), (10, –1), (20, 15)}

The function f(f(f x(x( ) is written in vertex form; its vertex is (10, –2). The y-values of g(x(x( ) are symmetric around –1, so the point (10, –1) must be the vertex of g(x(x( ).

The y-coordinate of the vertex represents the deepest point of the dive, so the first bird dove deeper.

2. f x x x1

102 9.52( ) = − + or the quadratic function that passes through the following points:

x 0 2 4 6 8

g(g(g x)x)x 17 7 1 –1 1

The y-values of g(x(x( ) are symmetric around –1; this means the vertex of g(x(x( ) must be (6, –1). We

can find the vertex of f x x x1

102 9.52( ) = − + by using the formula

b

af

b

a2,

2

− −

. In this

case, a = 1

10 and b = –2.

For the x-coordinate, substitute the values of a and b into the formula, then simplify:

b

a2

2

21

10

2 • 10

210

( )−=

− −

= =

For the y-coordinate:

f x x x1

102 9.52( ) = − + Original function

f 101

1010 2 10 9.52( ) ( ) ( )= − + Substitute 10 for x.

f 10 10 20 9.5( ) = − + Simplify.

f 10 0.5( ) = −

The vertex of f(f(f x(x( ) is (10, –0.5), while the vertex of g(x(x( ) is (6, –1). The y-coordinate of the vertex represents the deepest point of the dive, so the second bird dove deeper.


F–IF.9 Compare key features of two functions (linear, quadratic, square root, or inverse variation functions) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

Lesson 2.7: Comparing Properties of Quadratic Functions Given in Different Forms


2.7


Instruction3. the quadratic function that passes through the points on the following table or the function

described by the following graph:

x f(f(f x)x)x

0 85 3

10 015 –120 0

4 8 12 16 182 6 10 14 20

y

x22 24

6

4

2

0

10

8

12

14

–2

g(x)

The y-values of f(f(f x(x( ) are symmetric around –1; this means the vertex of f(f(f x(x( ) must be (15, –1). To find the vertex of g(x(x( ), find the lowest point of the graph. This occurs at (16, –2). The y-coordinate of the vertex represents the deepest point of the dive, so the second bird dove deeper.


• Students will compare quadratic functions presented in different forms.


2.7


Instruction

Introduction

Relationships between two quantities can be presented in a variety of different ways. Sometimes a relationship may be given in tables or graphs; other times, the relationship may be given in terms of an explicit equation. Relationships can also be described in a context. In this lesson, we will learn how to compare quadratic functions given in different forms.

Key Con cepts

• Recall that the standard form of a quadratic function is f(f(f x(x( ) = ax2 + bx + c, where a ≠ 0.

• Recall that the intercept form of a quadratic function is f(f(f x(x( ) = a(x(x( – p)(x)(x)( – q), where a ≠ 0 and p and q are the zeros of the function.

• Recall that the vertex form of a quadratic function is f(f(f x(x( ) = a(x(x( –h)2 + k, where a ≠ 0.

• Recall that the graph of a quadratic function is a U-shaped curve, known as a parabola.

• Recall that the value of a in any of the general forms determines whether the quadratic has a local maximum or a local minimum.

• If a is negative, the quadratic function will have a local maximum.

• If a is positive, the quadratic function will have a local minimum.

• The local maximum or minimum occurs at the vertex of the function.

• A quadratic function can only have both a maximum and a minimum if its domain is restricted.

• Recall that the concavity of a parabola refers to the direction it faces. The parabola will be concave up if a is positive, and it will be concave down if a is negative.

• A quadratic function is symmetric around its axis of symmetry.

Prerequisite Skills


• finding the vertex of a parabola (F–IF.8a)

• creating quadratic functions from verbal descriptions (A–CED.2)

• creating quadratic functions using vertex form (F–IF.8a)


2.7


Instruction


• thinking that an equation with a higher y-intercept has a higher maximum value

• thinking that a positive leading coefficient means the function has a maximum

• thinking that a negative leading coefficient means the function has a minimum

• The vertex lies on the axis of symmetry.

• To compare functions given in different forms, analyze each function for key features (intercepts, maximum/minimum, concavity, symmetry, etc.). Then, compare the key features.


2.7


Instruction

Example 1

Which function has the greatest x-intercept, f(f(f x(x( ) = 5x2 – 35x + 30 or g(x(x( ) = (x) = (x) = ( – 3)(x – 3)(x – 3)( – 4)?

1. Determine the x-intercept of each function.

To find the x-intercepts, put each function into intercept form.

For f(f(f x(x( ):

f(f(f x(x( ) = 5x2 – 35x + 30 Original function

f(f(f x(x( ) = 5(x) = 5(x) = 5( 2 – 7x + 6) Factor out the 5.

f(f(f x(x( ) = 5(x) = 5(x) = 5( – 6)(x – 6)(x – 6)( – 1) Factor the quadratic expression.

The x-intercepts of f(f(f x(x( ) are x = 1 and x = 6.

Because g(x(x( ) = (x) = (x) = ( – 3)(x – 3)(x – 3)( – 4) is already in intercept form, we can identify the x-intercepts as x = 3 and x = 4.

2. Compare the x-intercepts.

The greatest x-intercept of f(f(f x(x( ) occurs at x = 6. The greatest x-intercept of g(x(x( ) occurs at x = 4. Therefore, the function f(f(f x(x( ) has the greatest x-intercept.

Guided Practice 2.7


2.7


InstructionExample 2

Which of the following quadratic functions has the vertex with the larger x-value? The graph of f(f(f x(x( ) is shown, and values for g(x(x( ) are given in the table.

2 4 6 8

10

–1

8

6

4

2

7

5

3

1

0

9

1 3 5 7

y

x

–1–2–3–4–5–6–7–8

11

12

13

14

15f(x)

x g(g(g x)x)x

–3 29–2 24–1 210 201 212 243 294 36

1. Determine the vertex of each function.

The vertex corresponds to the local minimum or maximum of each function. On the graph of f(f(f x(x( ), we can see that the vertex is (3, 15).

In the table of values for the quadratic function g(x(x( ), there appears to be a minimum value of 20 at x = 0. Notice that the function values are symmetric around this point; this means that the point (0, 20) is indeed the vertex of g(x(x( ). This is because the vertex lies on the axis of symmetry.

2. Determine which x-coordinate is larger.

The x-coordinate of the vertex of f(f(f x(x( ) is 3. The x-coordinate of the vertex of g(x(x( ) is 0. Therefore, the x-coordinate of the vertex is larger in f(f(f x(x( ) than in g(x(x( ).


2.7


Instruction Example 3

A small manufacturing company has designed a new product, but the company has to upgrade its equipment to begin production. Before paying for the upgrade, the company wants to be sure it will make enough profit. Two different consultants made profit predictions for the company. Consultant A thinks the profit will follow the function P(x(x( ) = –50x2 + 42,500x – 7,500,000, where x is the sale price of the product. Consultant B also thinks the profit is dependent on the sale price of the product, but came up with a different prediction model, F(F(F x(x( ). The predicted profit from Consultant B is shown in the following graph. Which consultant predicts a higher maximum profit ?

2,500,000

2,000,000

1,500,000

1,000,000

500,000Pred

icte

d pr

o�t (

$)

Sale price ($)0

y

x

100 550150 200 250 300 350 400 450 500

F (x)

1. Translate the information into a mathematical question.

If we want to know the maximum predicted profit for both functions, we need to find the maximum y-value that lies on each function. Both functions are quadratics, so the maximum will be either at the vertex or at the end of a restricted domain. From the function equation P(x(x( ) = –50x2 + 42,500x – 7,500,000, we see that the leading coefficient is negative. This means that the maximum will occur at the vertex. The graph of F(F(F x(x( ) indicates that the maximum will occur at the vertex for that function as well.

Therefore, the maximum profit predicted by each function corresponds to the y-coordinate of the vertex of each function. We wish to know which vertex has the greater y-coordinate.


2.7


Instruction

2. Determine the vertex for each function.

For P(x(x( ) = –50x2 + 42,500x – 7,500,000, we will use the formula b

af

b

a2,

2

− −

to find the vertex. In this case, a = –50 and

b = 42,500.

To find the x-coordinate, substitute these values of a and b into the first part of the formula :

−=−

−=−−

=b

a2

(42,500)

2( 50)

42,500

100425

The x-coordinate of the vertex of P(x(x( ) is 425.

To find the y-coordinate, evaluate the function at x = 425.

P(x(x( ) = –50x2 + 42,500x – 7,500,000 Original function

P(425) = –50(425)2 + 42,500(425) – 7,500,000

Substitute 425 for x.

P(425) = –50(180,625) + 18,062,500 – 7,500,000

Simplify.

P(425) = –9,031,250 + 10,562,500 Simplify.

P(425) = 1,531,250

The vertex of P(x(x( ) is (425, 1,531,250).

To find the vertex of F(F(F x(x( ), examine the graph. The vertex is at the maximum of the graph, (350, 2,250,000).

3. Determine which function predicts a greater maximum profit.

The maximum profit predicted by P(x(x( ) is $1,531,250, or about $1.5 million. The maximum profit predicted by F(F(F x(x( ) is $2,250,000, or about $2.3 million. The analysis by Consultant B predicts a greater maximum profit.


2.7


Name: Date:

Three students throw basketballs from the center of the court toward one hoop. The height of each student’s ball is given as a quadratic function of the horizontal distance it has traveled over the floor. Assume the hoop is located at the point (37, 10). Which students, if any, make the shot?

• The path of Andrea’s basketball is modeled by the function

f x x( )1

100( 30) 142= − − + .

• Saul’s basketball is estimated to pass through the following points:

x 0 10 20 30y 3 10.5 13 10.5

• Ichigo’s basketball leaves her hand from a height of about 5 feet. When the ball is 12 feet from the basket, it achieves a maximum height of 11.5 feet.

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

Problem-Base d Task 2.7: Half-Court Shot

Which students, if any, make the shot?


2.7


Name: Date:

Problem-Base d Task 2.7: Half-Court Shot

Coaching

a. How can we determine whether a student makes the shot?

b. What is the function for the path of Andrea’s ball?

c. Does Andrea make her shot?

d. What information do we need to build a function g(x(x( ) for the path of Saul’s ball?

e. What is the vertex of g(x(x( )?

f. What is the equation of g(x(x( )?

g. Does Saul make his shot?

h. What information do we need to build a function h(x(x( ) for the path of Ichigo’s ball?

i. What is the vertex of h(x(x( )?

j. What is the equation of h(x(x( )?

k. Does Ichigo make her shot?

l. Summarize your findings.


2.7


Instruction

Problem-Bas ed Task 2.7: Half-Court Shot


a. How can we determine whether a student makes the shot?

Given the three different functions, we want to determine which students make their shots. The coordinates of the hoop are given as (37, 10), so we need to figure out which functions, if any, pass through the point (37, 10). We can do this by building an explicit equation for each function, then evaluating the functions at x = 37.

b. What is the function for the path of Andrea’s ball?

The path of Andrea’s ball already has an explicit function, f x x( )1

100( 30) 142= − − + .

c. Does Andrea make her shot?

Evaluate the function f x x( )1

100( 30) 142= − − + at x = 37 to see whether the function passes

through (37, 10).

f x x( )1

100( 30) 142= − − +

f (37)1

100(37 30) 142= − − +

f (37)1

100(7) 142= − +

f (37)1

100(49) 14= − +

f (37)49

10014 13.51= − + =

The function f(f(f x(x( ) does not pass through (37,10). Andrea does not make her shot.

d. What information do we need to build a function g(x(x( ) for the path of Saul’s ball?

We need the vertex of the function and also 1 other point on the curve of the function. Look at the table to determine these points.

e. What is the vertex of g(x(x( )?

For the path of Saul’s ball, notice that the function values are symmetric around x = 20. This means that the vertex of the function is (20, 13).


2.7


Instructionf. What is the equation of g(x(x( )?

The vertex of the function is (20, 13). Use the vertex form for quadratic functions to start building the equation:

g(x(x( ) = a(x(x( – h)2 + k

g(x(x( ) = a(x(x( – 20)2 + 13

Now we need to find a value for a. We can do this by substituting another (x. We can do this by substituting another (x. We can do this by substituting another ( , y) coordinate into the function, then solving. Let’s use (0, 3), because this makes calculation easier.

g(x(x( ) = a(x(x( – 20)2 + 13

3 = a(0 – 20)2 + 13

–10 = a(– 20)2

–10 = a(400)

a10

400

−=

a1

40= −

Substituting this value of a back into the equation, we get g x x1

40– 20 132( ) ( )= − + for the

path of Saul’s ball.

g. Does Saul make his shot?

Evaluate the function g x x1

40– 20 132( ) ( )= − + at x = 37 to see whether the function passes

through (37, 10).

g x x1

40– 20 132( ) ( )= − +

g 371

4037 – 20 132( ) ( )= − +

g 371

4017 132( ) ( )= − +

g 371

40289 13( ) ( )= − +

g 37289

4013 5.78( ) = − + ≈

The function g(x(x( ) does not pass through (37, 10). Saul does not make his shot.


2.7


Instructionh. What information do we need to build a function h(x(x( ) for the path of Ichigo’s ball?

We need the vertex of the function and also 1 other point on the curve of the function. Analyze the problem context to find this information.

i. What is the vertex of h(x(x( )?

For the path of Ichigo’s ball, we are given a starting height, 5 feet, and a maximum height, 11.5 feet. The maximum must occur at the vertex. Because the maximum occurs 12 feet from the basket, it must have traveled 37 –12 = 25 feet after leaving Ichigo’s hand. Therefore, the coordinates for the vertex are (25, 11.5).

j. What is the equation of h(x(x( )?

Use the vertex form for quadratic functions to start building the equation:

h(x(x( ) = a(x(x( – h)2 + k

h(x(x( ) = a(x(x( – 25)2 + 11.5

To find the value of a, we need one other point on the parabola. Recall that the ball leaves Ichigo’s hand from a height of 5 feet. This means that the coordinates for the starting height must be (0, 5) because x = 0 corresponds to the point where the ball leaves Ichigo’s hand. Substitute this point into the equation and solve for a:

h(x(x( ) = a(x(x( – 25)2 + 11.5

5 = a(0 – 25)2 + 11.5

–6.5 = a(–25)2

–6.5 = a(625)

a6.5

625

−=

a65

6250

−=

a13

1250= −

Substituting this value of a back into the equation, we get h x x13

1250– 25 11.52( ) ( )= − + for

the path of Ichigo’s ball.


2.7


Instructionk. Does Ichigo make her shot?

Evaluate the function h x x13

1250– 25 11.52( ) ( )= − + at x = 37 to see whether the function

passes through (37, 10).

h x x13

1250– 25 11.52( ) ( )= − +

h 3713

125037 – 25 11.52( ) ( )= − +

h 3713

125012 11.52( ) ( )= − +

h 3713

1250144 11.5( ) ( )= − +

h 371872

125011.5 10.00( ) = − + ≈

The function h(x(x( ) passes through (37, 10). Ichigo makes her shot.

l. Summarize your findings.

Of the three students, only Ichigo makes the shot from half-court.




2.7


Name: Date:

Use the given information to solve problems 1–3.

1. Which function has a lower vertex: a parabola with two x-intercepts and a > 0, or a parabola with two x-intercepts and a < 0?

2. Which function has a higher y-intercept: a parabola with a > 0 and x-intercepts p and –p and –p and – , or a parabola with a > 0 and no x-intercepts?

3. Which function’s vertex is farther to the right: a parabola with x-intercepts p and –p and –p and – , or a parabola with two x-intercepts that are both less than 0?


Shelly is testing the braking distance for three different car models. The

distance traveled by Model A after she applies the brakes follows the function

A x xx

( ) 2.2 177

= +

, where A(x(x( ) represents the distance traveled in feet and x

represents her speed in miles per hour. The distance traveled (in feet) by Model B is

given in the following table. The distance traveled (in feet) by Model C is given by the

function C x xx

( ) 2.226

2

= + .

x 25 35 45 55

B(x)x)x 83.4 132.7 191.0 258.5

4. Is there a point after x = 0 where two or more models have the same stopping distance?

continued

Practice 2.7: Comparing Properties of Quadratic Functions Given in Different Forms AA AA


2.7


Name: Date:

5. Which model has the shortest stopping distance at x = 25? Will it always have the shortest stopping distance?

6. Which model has the longest stopping distance at x = 55? Will it always have the longest stopping distance?


A new fountain has been installed in a city park. The fountain has three different types of water jets that stream out of nozzles set in the basin of the fountain. The streams from the water jets all have a quadratic shape. Type A shoots water 12 feet across the basin, and the water stream reaches a maximum height of 12 feet. Type B sprays water along the path shown in the following graph. The path of water shot out of Type C is given in the table.

2 4 12 14 1610 13 156 8 111 3 5 7 9

8

6

4

2

7

5

3

1

9

0

Distance from nozzle (feet)

Hei

ght a

bove

bas

in (f

eet)

y

x

B(x)

x (ft) C(C(C x) (ft)x) (ft)x

0 02 124 166 128 0

7. Which type reaches the highest point?

continued


2.7


Name: Date:

8. Which type crosses the longest span?

9. The three nozzle types are arranged in a horizontal line so that the vertexes of the water arcs line up vertically. Suppose the widest arc has its nozzle at (0, 0). Where are the nozzles of each type located?

10. What are the endpoints of each arc?


2.7


Name: Date:

Use the given information to solve problems 1–3.

1. Which function has a higher vertex: a parabola with two x-intercepts and a > 0, or a parabola with no x-intercepts and a > 0?

2. Which function has a higher y-intercept: a parabola with a < 0 and no x-intercepts, or a parabola with a > 0 and no x-intercepts?

3. Which function’s vertex is farther to the left: a parabola with x-intercepts p and –3p and –3p and –3 , or a parabola with two x-intercepts that are both greater than 0?


Sabine is testing the braking distance for three different car models. The distance traveled

by Model A after Sabine applies the brakes follows the function A x xx

( ) 2.2 166

= +

,

where A(x(x( ) represents the distance traveled in feet and x represents her speed in miles per

hour. The distance traveled by Model B is given in the following table. The distance traveled

by Model C is given by the function C x xx

( ) 2.224

2

= + .

x 25 35 45 55

B(x)x)x 73.94 114.12 160.36 212.67

4. Is there a point after x = 0 where two or more models have the same stopping distance?

continued

Practice 2.7: Comparing Properties of Quadratic Functions Given in Different Forms BB BB


2.7


Name: Date:

5. Which model has the shortest stopping distance at x = 25? Will it always have the shortest stopping distance?

6. Which model has the longest stopping distance at x = 55? Will it always have the longest stopping distance?


A new fountain has been installed in a city park. The fountain has three different types of water jets that stream out of nozzles set in the basin of the fountain. The streams from the water jets all have a quadratic shape. Type A shoots water 10 feet across the basin, and the water stream reaches a maximum height of 10 feet. Type B sprays water along the path shown in the following graph. The path of water shot out of Type C is given in the following table.

2 4 12 14 1610 13 156 8 111 3 5 7 9

8

6

4

2

7

5

3

1

9

0

Distance from nozzle (feet)

Hei

ght a

bove

bas

in (f

eet)

x

B(x)

y

16

14

12

10

15

13

11

17

1918

20 x (ft)x (ft)x C(C(C x) (ft)x) (ft)x

0 04 38 4

12 316 0

continued


2.7


Name: Date:

7. Which type reaches the highest point?

8. Which type crosses the longest span?

9. The three nozzle types are arranged in a horizontal line so that the vertexes of the water arcs line up vertically. Suppose the widest arc has its nozzle at (0, 0). Where are the other two nozzles located?

10. What are the endpoints of each arc?



Name: Date:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–BF.3Lesson 2.8: Replacing f (f (f x ) with f (f (f x ) + k and k and k f (f (f x + x + x k)k)k

Lesson 2.8: Replacing f (f (f x ) with f (f (f x ) + k and k and k f (f (f x + x + x k)k)kWarm-Up 2.8As a fund-raiser for the senior prom, the student council has decided to charge a small fee for parking in the school parking lot. Students can purchase yearly parking passes for $5 each. The school spent a total of $10 to buy a year’s supply of the tags the students hang in their windows to verify that they paid for the parking spot.

1. Build a function that models the profit the school will make selling parking passes.


3. What would be the effect on the graph if the school were able to find a company that only charges $8 for the tags?



Instruction

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–BF.3Lesson 2.8: Replacing f (f (f x ) with f (f (f x ) + k and k and k f (f (f x + x + x k)k)k

Lesson 2.8: Replacing f (f (f x ) with f (f (f x ) + k and k and k f (f (f x + x + x k)k)k

Warm-Up 2.8 Debrief1. Build a function that models the profit the school will make selling parking passes.

Each parking pass sells for $5, but the school spent $10 up front to purchase the window tags. The function that models this situation is f(f(f x(x( ) = 5x – 10.


0 20 40 60 80 100 120 140

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

Number of tags sold

y

x

Pro�

t in

dolla

rs ($

)

3. What would be the effect on the graph if the school were able to find a company that only charges $8 for the tags?

The profit would increase by $2. This would affect the y-intercept by changing it from –10 to –8. The graph would move up 2 units.


• Students will extend their study of the effects of transformations on linear and exponential functions to quadratic functions.

• Students will graph functions and analyze those graphs.





Instruction


IntroductionYou can change a function’s position or shape by adding a constant to that function or to the independent variable in the function, or b y multiplying the function or independent variable by a constant. This is called a transformation. When adding a constant, you can transform a function in two distinct ways. The first is a transformation on the independent variable of the function; that is, given a function f(f(f x(x( ), we add some constant k to x: f(f(f x(x( ) becomes f(f(f x(x( + k). The second is a transformation on the dependent variable; given a function f(f(f x(x( ), we add some constant k to f(f(f x(x( ):

f(f(f x(x( ) becomes f(f(f x(x( ) + k.

In this lesson, we consider the transformation of a quadratic function by a constant k, either when k is added to the independent variable, x, or when k is added to the dependent variable, f(f(f x). Given f(f(f x) and a constant k, we will observe the transformations f(f(f x) + k and f(f(f x + k), and examine how transformations affect the vertex of the graph of a quadratic function.

Key Concepts

• To determine the effects of the constant on a graph, compare the vertex of the original function to the vertex of the transformed function.

• Neither f(f(f x(x( + k) nor f(f(f x(x( ) + k will change the shape, size, or orientation of the function so long as k is a constant.

• Transformations that do not change the shape, size, or orientation of the function but do move it horizontally and/or vertically are called translations.

• Translations are performed by adding a constant to the independent or dependent variable.

Prerequisite Skills


• graphing quadratic functions (A–REI.10)

• evaluating quadratic functions (F–IF.2)



Instruction


Vertical Translations—Adding a Constant to the Dependent Variable, f Vertical Translations—Adding a Constant to the Dependent Variable, f Vertical Translations—Adding a Constant to the Dependent Variable, (f(f x) +x) +x k

• f( f( f x(x( ) + k moves the graph of the function k units up or down depending on whether k is greater than or less than 0.

• If k is positive in f(f(f x(x( ) + k, the graph of the function will be moved up.

• If k is negative in f(f(f x(x( ) + k, the graph of the function will be moved down.

Vertical translations: fVertical translations: fVertical translations: (f(f x) + x) + x k

When k is positive, k > 0, the graph moves up:

f(x) + k

f(x)

When k is negative, k < 0, the graph moves down:

f(x) + k

f(x)

Horizontal Translations—Adding a Constant to the Independent Variable, fHorizontal Translations—Adding a Constant to the Independent Variable, fHorizontal Translations—Adding a Constant to the Independent Variable, (f(f x(x( + k)

• f(f(f x(x( + k) moves the graph of the function k units to the right or left depending on whether k is greater than or less than 0.

• If k is positive in f(f(f x(x( + k), the function w ill be moved to the left.

• If k is negative in f(f(f x(x( + k), the function will be moved to the right.



Instruction


Common Errors/Mis conceptions

• incorrectly moving the graph in the direction opposite that indicated by k, especially in horizontal shifts; for example, moving the graph left when it should be moved right

• incorrectly moving the graph left and right versus up and down (and vice versa) when operating with f(f(f x(x( + k) and f(f(f x(x( ) + k

Horizontal translation s: fHorizontal translation s: fHorizontal translation s: (f(f x + x + x k)

When k is positive, k > 0, the graph moves left:

f(x + k) f(x)

When k is negative, k < 0, the graph moves right:

f(x + k)f(x)




For problems 1–5, let f(f(f x(x( ) = x2. Write a function that translates f(f(f x(x( ) as described.

1. 3 units to the left

2. 2 units to the right

3. 3 units up

4. 6 units to the right and 2 units down

5. 4 units to the left and 3 units up

For problem 6–8, let f(f(f x(x( ) = x2. Graph g(x(x( ) by translating the graph of f) by translating the graph of f) by translating the graph of (f(f x(x( ).

6. f(f(f x(x( ) = (x= (x= ( – 3)2

continued

Scaffolded Practice 2.8: Replacing f(f(f x) with x) with x f(f(f x) + x) + x k and k and k f(f(f x + x + x k)k)k




7. f(f(f x(x( ) = (x= (x= ( )2 – 5

8. f(f(f x(x( ) = (x= (x= ( – 6)2 + 2

continued




Use the given information to complete problems 9 and 10.

9. The following problem is a translation of f The following problem is a translation of f The following problem is a translation of (f(f x(x( ) = –x= –x= – 2. Write an equation for the graph.

2 4 6 8 10

10

1 3 5 7 9

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

y

x

–10

–1–2–3–4–5–6–7–8–9–10

10. A football is thrown from a height of 6 feet so that its path is modeled by the function f(f(f x(x( ) = –0.03x2 + 1.2x + 6. If the exact same pass is taken from a balcony that is 13 feet above where the original passer was standing, how far away will the ball hit the ground? What is the equation that models this pass?



Instruction


E xample 1

Consider the function f(f(f x(x( ) = x2 and the constant k = 2. What is f(f(f x(x( ) + k? How are the graphs of fgraphs of fgraphs of (f(f x(x( ) and f(f(f x(x( ) + k different?

1. Substitute the value of k into the function.

If f(f(f x(x( ) = x2 and k = 2, then f(f(f x(x( ) + k = x2 + 2.

2. Use a table of values to graph the functions on the same coordinate plane.

x f (f (f x)x)x f (f (f x) + 2x) + 2x–2 4 6–1 1 30 0 21 1 32 4 6

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10 f(x) + 2

f(x)

Guided Practice 2.8



Instruction


3. Compare the graphs of the functions.

Notice the shape and horizontal position of the two graphs are the same. The only difference between the two graphs is that the value of f(f(f x(x( ) + 2 is 2 more than f(f(f x(x( ) for all values of x. In other words, the transformed graph is 2 units up from the original graph.

Example 2

Consider the function f(f(f x(x( ) = x2 and the constant k = –3. What is f(f(f x(x( ) + k? How are the graphs of f? How are the graphs of f? How are the graphs of (f(f x(x( ) and f(f(f x(x( ) + k different?


If f(f(f x(x( ) = x2 and k = –3, then f(f(f x(x( ) + k = x2 – 3.

2. Use a table of values to graph the two functions on the same coordinate plane.

x f (f (f x)x)x f (f (f x) – 3x) – 3x–4 16 13–2 4 10 0 –32 4 14 16 13




Instruction


-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

f(x) – 3

f(x)


Notice both the shape and horizontal position of the two graphs are the same. The only difference between the two graphs is that the value of f(f(f x(x( ) – 3 is 3 less than f(f(f x(x( ) for all values of x.

The easiest point to analyze on the graphs is the vertex. For f(f(f x(x( ), the vertex occurs at (0, 0). For the graph of f(f(f x(x( ) – 3, the vertex occurs at (0, –3). The transformed graph has moved down 3 units.



Instruction


E xample 3

Consider the function f(f(f x(x( ) = x2, its graph, and the constant k = 4. What is f(f(f x(x( + k)? How are the graphs of fof fof (f(f x(x( ) and f(f(f x(x( + k) different?


If f(f(f x(x( ) = x2 and k = 4, then f(f(f x(x( + k) = f(f(f x(x( + 4) = (x + 4) = (x + 4) = ( + 4)2.

2. Use a table of values to graph the two functions on the same coordinate plane.

x f (f (f x)x)x f (f (f x + 4)x + 4)x–6 36 4–4 16 0–2 4 40 0 162 4 364 16 64

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10f(x + 4) f(x)



Instruction



Notice the shape and vertical position of the two graphs are the same. The only difference between the two graphs is that every point on the curve of f(f(f x(x( ) has been shifted 4 units to the left in the graph of f(f(f x(x( + 4).

Example 4

Consider the function f(f(f x(x( ) = x2 and the constant k = –1. What is f(f(f x(x( + k)? How are the graphs of f)? How are the graphs of f)? How are the graphs of (f(f x(x( ) and f(f(f x(x( + k) different?


If f(f(f x(x( ) = x2 and k = –1, then f(f(f x(x( + k) = f(f(f x(x( – 1) = (x – 1) = (x – 1) = ( – 1)2.

2. Using vertex or standard form, graph the two functions on the same coordinate plane.

Recall from Example 2 that the vertex of the parent function, f(f(f x(x( ) = x2, is (0, 0).

The vertex of the transformed function, f(f(f x(x( – 1) = (x – 1) = (x – 1) = ( – 1)2, is (1, 0). This

can be verified by using − −

b

af

b

a2 2, .

To use this formula, expand the function so that it’s in the form of a quadratic, then find a, b, and c.

f(f(f x(x( – 1) = (x – 1) = (x – 1) = ( – 1)2 = x2 – 2x + 1

a = 1, b = –2, and c = 1c = 1c

The x-coordinate of the vertex is given by 2

( 2)

2(1)1=

−=− −

=xb

a.

Substitute the x-value of the vertex into the function.

f(f(f x(x( – 1) = (x – 1) = (x – 1) = ( – 1)2

f(1) = [(1) – 1]f(1) = [(1) – 1]f 2

f(1) = 0f(1) = 0f

Therefore, the vertex is (1, 0).(continued)continued)continued



Instruction


Graph the functions.

-5 -4 -3 -2 -1 0 1 2 3 4 5

-10-9-8-7-6-5-4-3-2-1

123456789

10

f(x – 1)f(x)


Notice both the shape and vertical position of the two graphs are the same. The only difference between the two graphs is that every point on the curve of f(f(f x(x( ) has been shifted 1 unit to the right in the graph of f(f(f x(x( – 1).



Instruction


Example 5

The revenue function for a model helicopter company is modeled by the curve f(f(f x(x( ) = –5x2 + 400x, where x is the number of helicopters built per month and f(f(f x(x( ) is the revenue. The owner wants to include rent in the revenue equation to determine the company’s profit per month. The company pays $2,250 per month to rent its warehouse. In terms of fcompany pays $2,250 per month to rent its warehouse. In terms of fcompany pays $2,250 per month to rent its warehouse. In terms of (f(f x(x( ), what equation now describes the company’s profit per month? Compare the vertices of the original function and the transformed function.

1. Build the new function.

The rent is being subtracted from the entire function. Therefore, the function follows the format f(f(f x(x( ) + k, where k is negative.

The company makes f(f(f x(x( ) = –5x2 + 400x dollars per month. The company also pays $2,250 dollars per month in rent. Therefore, the new function that describes the company’s profit, P(x(x( ), is as follows:

P(x(x( ) = f(f(f x(x( ) – rent = f(f(f x(x( ) – 2250 = –5x2 + 400x – 2250

2. Determine the vertex of fDetermine the vertex of fDetermine the vertex of ( f( f x(x( ).

The vertex of f(f(f x(x( ) has the x-coordinate −

=−

−=−−

=b

a2

400

2 5

400

1040

( )

( ).

Evaluate f( f( f x(x( ) at the x-coordinate of the vertex to find the y-coordinate.

f(f(f x(x( ) = –5x2 + 400x

f(40) = –5(40)f(40) = –5(40)f 2 + 400(40)

f(40) = –8000 + 16,000f(40) = –8000 + 16,000f

f(40) = 8000f(40) = 8000f

The vertex of f(f(f x(x( ) is (40, 8000).



Instruction


3. Determine the vertex of P(x(x( ).

The vertex of P(x(x( ) has the same x-coordinate as f(f(f x(x( ) since the values of a and b are the same. The x-coordinate of the vertex of P(x(x( ) is 40.

Evaluate P(x(x( ) at the x-coordinate of the vertex to find the y-coordinate.

P(x(x( ) = –5x2 + 400x – 2250

P(40) = –5(40)2 + 400(40) – 2250

P(40) = 8000 – 2250

P(40) = 5750

The vertex of P(x(x( ) is (40, 5750).

4. Compare the vertices.

The vertex of P(x(x( ) is 2,250 units lower than the vertex of f(f(f x(x( ). The model was shifted down 2,250 units.




Problem-Based Task 2.8: The CatchOn the last play of a football game, the offense is on the opposing team’s 35-yard line. The offense is losing by 4 points, but can win by making a touchdown. The quarterback backs away 5 yards from behind the line of scrimmage and throws the ball to his receiver, who makes the catch at the goal line for the touchdown and the win. The quarterback’s release point is 6 feet above the ground, the same height at which the receiver caught the ball. Also, the ball was thrown such that its maximum height was 15 feet above the ground.

Given this information, build and graph the equation of the football’s path, with the x-axis representing the distance from the line of scrimmage and the y-axis as the height of the football above the ground, with distances measured in yards.

Build and graph the equation of the football’s path.equation of the football’s path.equation of the

SMP1 2 ✓3 4 ✓5 6 7 ✓ 8




Problem-Based Task 2.8: The Catch

Coachinga. What is the total horizontal distance the ball was thrown?

b. What is the height diff erence between the release of the ball and its maximum height?

c. What are the x-intercepts of the equation describing the path of the ball, disregarding the point of release and allowing the parabola to begin at the origin, (0, 0)?

d. What value of a gives the parabola the correct arch?

e. Using the value of a and the vertex from part d, write the equation of the path of the ball in vertex form.

f. Translate the equation to account for the line of scrimmage.

g. Translate the equation to account for the height of the point of release and catch.

h. Graph the equation.



Instruction


Problem-Based Task 2.8: The CatchProblem-Based Task 2.8: The Catch

Coaching Sample Responsesa. What is the total horizontal distance the ball was thrown?

The ball was thrown from 5 yards behind the 35-yard line to the goal line. The ball traveled 40 yards.

b. What is the height difference between the release of the ball and its maximum height?

The ball reached a height of 15 feet, or 5 yards, and was thrown from a height of 6 feet, or 2 yards. Therefore, the height difference between the release of the ball and the maximum height is 3 yards.

c. What are the x-intercepts of the equation describing the path of the ball, disregarding the point of release and allowing the parabola to begin at the origin, (0, 0)?

If the path of the ball started at the origin (0, 0), then x = 0 is one of the x-intercepts. If the ball was caught 40 yards later, then it was caught at the coordinates (40, 0), making x = 40 the other x-intercept of the equation.

d. What value of a gives the parabola the correct arch?

The vertex of the path of the ball must happen at midway between the x-intercepts 40 0

220

−=

, and from part b we know the height difference was 3 yards. Therefore, the vertex

of the equation is (20, 3).

Using vertex form, one of the known coordinates (0, 0), and the vertex, we can establish the value of a for the path of the ball.

y a x

a

a

a

20 3

0 0 20 3

3 400

3

400

2

2

( )( )( )

= − +

= − +− =

− =

e. Using the value of a and the vertex from part d, write the equation of the path of the ball in vertex form.

f x x3

40020 32( ) ( )= − − +



Instruction


f. Translate the equation to account for the release point.

f x x x53

4005 20 3

3

40015 32 2( ) ( ) ( )+ = − + − + = − − +

g. Translate the equation to account for the height of the point of release and catch.

f x x x5 23

40015 3 2

3

40015 52 2( ) ( ) ( )+ + = − − +

+ = − − +

h. Graph the function.

-15 -10 -5 0 5 10 15 20 25 30 35 40

-10-9-8-7-6-5-4-3-2-1

123456789

10

Distance in yards

Hei

ght i

n ya

rds







Task OverviewFocus

How can you model the path of a football using a quadratic function? Students will write and graph a function that does this using transformations of functions.


• writing quadratic equations to model a real-world problem

• writing quadratic equations in vertex form

• identifying key features of a quadratic equation

• graphing a quadratic function

• comparing a quadratic function and a transformation of the function

• interpreting key features of a quadratic equation

Introduction

This task should be used to explore or to apply transformations of quadratic functions using a real-world model. It is best to implement this task after students have graphed horizontal and vertical transformations of quadratic function. Graphing may be done using technology, including graphing calculators or online graphing programs.

Begin by reading the problem and clarifying the meaning of various football terms:

line of scrimmage the place where two teams meet to start the next play

offense the team with the ball

quarterback the player who throws the ball to the receiver

receiver the player who receives the ball thrown by the quarterback

The following website provides a discussion of how football is played:

• NFL.com. “NFL Beginner’s Guide to Football.”


Problem-Based Task 2.8 Implementation Guide: The CatchNorth Carolina Math 2 Standard








• SMP 2: Reason abstractly and quantitatively.

Some students may have difficulty reasoning that (40, 0) is the second x-intercept. Have students reread the problem and review what the x- and y-axes represent. The x-axis represents the distance from the line of scrimmage and the y-axis represents the height of the football. There are two times when the height of the football is 0: when the quarterback releases the ball and when the receiver catches it. Remind students that the quarterback releases the ball and the receiver catches the ball at the same height.


Students should understand the reason they are being asked to model the problem using a quadratic function. Discuss the path of the ball and why a parabola models its path.


Some students may not understand why they are writing the equation of a function that intersects (0, 0) when the ball is thrown from behind the line of scrimmage. Discuss why it is easier to model this path and then transform the function than it is to write the function of the correct path to begin with.



• not accounting for the 5 yards when the quarterback backs away from the line of scrimmage

Have students draw a diagram that is a number line showing the line of scrimmage (0), the position of the quarterback as he starts to throw (–5) and the place where the receiver catches the ball (35). Have them use the number line to find the total distance.

• measuring the horizontal distance in yards but the height differential in feet

Remind students that all measures need to be the same unit of measure for the function to be correct. Students can convert all measures to feet or to yards, but because the lines on a football field are measured in yards, encourage students to use that unit of measure.

• placing the vertex at (20, 0) instead of (20, 3)

Have students again draw a diagram to show the height differential. They can use a vertical number line to label the distances.

• finding f(f(f x(x( ) + 5 instead of f) + 5 instead of f) + 5 instead of (f(f x(x( + 5)

Explain that the release point is a horizontal transformation representing the distance the quarterback moves back from the line of scrimmage to throw the ball.







• If students struggle to find the second x-intercept for part c, ask then what the point (0, 0) represents. (Answer: the position of the ball after the quarterback steps back 5 yards and the height of the football relative to 6 feet at the time the ball is thrown)

Then ask when the ball will have a height of 6 feet again. (Answer: when the ball is caught)

Then ask what distance the ball traveled in that time. (Answer: 40 yards)

• If students have difficulty finding a for the equation, ask what a represents. (Answer: It tells you which way the parabola opens and how wide the parabola is.)

Then ask, “How can you find a when you know the vertex of the parabola and one point on the parabola?” (Answer: Substitute for x, y, h, and k and then solve for a.)

• If students graph the function incorrectly, ask how you would enter the function to ensure the

operations are performed in the correct order? (Answer:3

400( 15) 52= − − +y x )


Debriefing the TaskDiscuss why we solved the problem by first finding the function that intersected the origin and why this function could not be used to model the problem. Ask students to share the strategies they used and discuss other possible ways to solve the problem. Discuss the advantages and disadvantages of solving the problem graphically as opposed to algebraically.



• Neither f(f(f x(x( + k) nor f(f(f x(x( ) + k will change the shape of the function as long as k is constant.

Students graph the function f(f(f x(x( + 5) + 2 and see that it has the same shape as the function f(f(f x(x( ).

• Translations are performed by adding a constant to the independent or dependent variable.

Students graph a horizontal and vertical translation of a function f(f(f x(x( ).

• f( f( f x(x( + k) moves the graph of a function k units to the left if k is positive and k units to the right if k is negative.

Students graph f(f(f x(x( + 5), which is a horizontal translation of the function f(f(f x(x( ).

• f( f( f x(x( + x + x k) moves the graph of a function k units up if k units up if k k is positive and k is positive and k k units down if k units down if k k is negative.k is negative.k

Students graph f(f(f x(x( + 5) + 2, which is a vertical translation of the function f(f(f x(x( + 5).





Extending the Task

To extend the task, have students analyze the graph of the functions fTo extend the task, have students analyze the graph of the functions fTo extend the task, have students analyze the graph of the functions (f(f x(x( ), f(f(f x(x( + 5), and f(f(f x(x( + 5) + 2. For example:

• Graph the function f(f(f x(x( ). Ensure that it intersects the origin and (40,0).

• Graph the function f(f(f x(x( + 5). Confirm that it is a horizontal translation of f + 5). Confirm that it is a horizontal translation of f + 5). Confirm that it is a horizontal translation of (f(f x(x( ).

• Graph the function f(f(f x(x( + 5) + 2. Confirm that it is a vertical translation of f + 5) + 2. Confirm that it is a vertical translation of f + 5) + 2. Confirm that it is a vertical translation of (f(f x(x( + 5).

Confirm that the shape of the parabola does not change. If students are using graphing software they can move the original function to the transformed function to see that this is true.



• For SMP 2, ASK: “How did you reason abstractly and quantitatively? Which of your strategies represent abstract reasoning?” (Answer: I had to figure out what the problem was asking me to find.) “Which of your strategies represent quantitative reasoning?” (Answer: I had to figure out how to use the number in the problem to calculate the distances I needed to write the function.)

• For SMP 4, ASK: “How did you use mathematics to model this particular scenario?” (Answer: I modeled the path of the football with a quadratic function.)

• For SMP 7, ASK: “How did you look for and make use of structure when solving this problem?” (Answer: I had to use what I knew about vertex form to write transformations of a quadratic function.)


• Perform the vertical translation first.

• Have students graph the function ( )3

400( 20) 32= − − +f x x and then move the parabola to

translate it 5 units to the left and 2 units up. They can use the graph to find the equation of

the function.

Technology

Students can use a graphing calculator or online software to graph the function.




Practice 2.8: Replacing f (f (f x ) with f (f (f x ) + k and k and k f (f (f x + x + x k)k)kFor problems 1–3, let f(f(f x(x( ) = x2. Write a function that translat es f(f(f x(x( ) as described.

1. 2 units to the left

2. 3 units up

3. 5 units to the right and 2 units down

For problems 4–6, let f(f(f x(x( ) = x2. Graph g(x(x( ) by translating the graph of f) by translating the graph of f) by translating the graph of (f(f x(x( ). State the vertex of the translated function.

4. g(x(x( ) = (x) = (x) = ( – 2)2

AA

continued




5. g(x(x( ) = x2 – 4 6. g(x(x( ) = (x) = (x) = ( – 5)2 – 2

Use what you know about translations of functions to solve each problem.

7. The following graph is a translation of f The following graph is a translation of f The following graph is a translation of (f(f x(x( ) = –x) = –x) = – 2. Write an equation for the graph and state the value of k that was used to transform the function.

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

-10-9-8-7-6-5-4-3-2-1

123456789

10

continued




8. A mother and her daughter went golfing. The mother hit first. Her ball followed the path modeled by the equation f(f(f x(x( ) = –0.0009x2 + 0.2088x in the direction of the hole, and landed 18 yards short of the hole. The daughter teed off 18 yards closer to the hole because she is a beginner. She realized that if she could hit the ball on the same trajectory as her mother, her ball would land right by the hole. What is the equation that describes the path that the daughter’s ball should follow?

9. A basketball is thrown from a height of 4 feet so that its path is modeled by the function f(f(f x(x( ) = –0.03x2 + 1.3x + 4. If the exact same shot is taken from a balcony that is 12 feet above where the original shooter was standing, how far away will the ball hit the ground? What is the equation that models this shot?

10. Simon has a toy that launches hollow plastic balls. The launched balls always follow a path

modeled by the function f x x( )= − −( ) +1

88 82 when the launcher is at the “origin.” If the

launcher is lifted up 2 feet and moved forward 5 feet, will a launched ball land in a basket that

is on a 4-foot high stool 20 feet from the origin? What is the function that models this new

launcher position?




BPractice 2.8: Replacing f (f (f x ) with f (f (f x ) + k and k and k f (f (f x + x + x k)k)kFor probl ems 1–3, let f(f(f x(x( ) = x2. Write a function that tra nslates f(f(f x(x( ) as described.

1. 3 units to the right

2. 4 units down

3. 6 units to the left and 1 unit down

For problems 4–6, let f(f(f x(x( ) = x2. Graph g(x(x( ) by translating the graph of f) by translating the graph of f) by translating the graph of (f(f x(x( ).

4. g(x(x( ) = (x) = (x) = ( + 4)2

continued




5. g(x(x( ) = x2 + 1 6. g(x(x( ) = (x) = (x) = ( – 2)2 – 5

Use what you know about translations of functions to solve each problem.

7. The following graph is a translation of f The following graph is a translation of f The following graph is a translation of (f(f x(x( ) = –x) = –x) = – 2. Write an equation for the graph. State the value of k that was used to transform the function horizontally and the value of k used to transform the function vertically.

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

-10-9-8-7-6-5-4-3-2-1

123456789

10

continued




8. A mother and her son went golfing. The mother hit first. Her ball followed the path modeled by the equation f(f(f x(x( ) = –0.0008x2 + 0.24x in the direction of the hole, and landed 12 yards short of the hole. The son teed off 12 yards closer to the hole because he is a beginner. He realized that if he could hit the ball on the same trajectory as his mother, his ball would land right by the hole. What is the equation that describes the path that the son’s ball should follow?

9. A paper wad is thrown from a height of 4 feet so that its path is modeled by the function f(f(f x(x( ) = –0.05x2 + x + 4. If the exact same shot is taken from a balcony that is 15 feet above where the original shooter was standing, how far away will the paper wad hit the ground? What is the equation that models this shot?

10. Suzanne has a toy that launches rubber bands. The rubber bands always follow a path modeled by the function f(f(f x(x( ) = –0.4(x) = –0.4(x) = –0.4( – 5)2 + 10 when the launcher is at the “origin.” If the launcher is lifted up 4 feet and moved forward 4 feet, will a launched rubber band hit a painted target on a 10-foot-tall tree branch that is 12 feet from the origin? What is the function that models this new launcher position?



Name: Date:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–BF.3Lesson 2.9: Replacing f (x ) with x ) with x k •k •k f (x ) and x ) and x f (k •k •k x ) x ) x

Lesson 2.9: Replacing f (x ) with x ) with x k •k •k f (x ) and x ) and x f (k •k •k x ) x ) xWarm-Up 2.9An architectural firm is designing a one-story, square office building that is required to have a 4-foot space in every exterior wall for an exit. The area of the office building is given by the equationf(f(f x(x( ) = (x) = (x) = ( + 4)(x + 4)(x + 4)( + 4) = (x + 4) = (x + 4) = ( + 4)2.

1. Given f(f(f x(x( ) = (x ) = (x ) = ( + 4)2, what would be the area of the building if x = 20 feet?

2. If the firm used the same blueprint with the area found in problem 1 to design a building with 3 identical floors, what would be the total area of the new building’s office space?

3. If the firm were to double the length of each wall in the original design, what would be the new area of the one-story building?



Instruction

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–BF.3Lesson 2.9: Replacing f (x ) with x ) with x k •k •k f (x ) and x ) and x f (k •k •k x ) x ) x

Lesson 2.9: Replacing f (x ) with x ) with x k •k •k f (x ) and x ) and x f (k •k •k x ) x ) x

Warm-Up 2.9 Debrief1. Given f(f(f x(x( ) = (x ) = (x ) = ( + 4)2, what would be the area of the building if x = 20 feet?

Evaluate f(f(f x(x( ) = (x ) = (x ) = ( + 4)2 for x = 20.

f(20) = (20f(20) = (20f + 4)2

f(20) = (24)f(20) = (24)f 2

f(20) = 576 f(20) = 576 f

The area of the building would be 576 ft2.

2. If the firm used the same blueprint with the area found in problem 1 to design a building with 3 identical floors, what would be the total area of the new building’s office space?

If the area is 576 ft2 when the office building is only 1 floor, then the area of 3 identical floors would be 3 • 576 = 1728 ft2. Written in function notation, 3 • f(20) = 3 • 576 = 1728 ftf(20) = 3 • 576 = 1728 ftf 2.

The area of the new building woul d be 1,728 ft2.

3. If the firm were to double the length of each wall in the original design, what would be the new area of the one-story building?

Build the new function.

The original function is f(f(f x(x( ) = (x ) = (x ) = ( + 4)2.

If the firm doubled the length of each office wall, then x would be twice as long, or 2x.

Therefore, the function that gives the area for the newly designed office building isf(2f(2f x) = (2x + 4)2.





Instruction


Evaluate f(2f(2f x) = (2x + 4)2 for x = 20.

f(2 • 20) = (2 • 20f(2 • 20) = (2 • 20f + 4)2

f(40) = (40f(40) = (40f + 4)2

f(40) = (44)f(40) = (44)f 2

f(40) = (44)f(40) = (44)f 2 = 1936

The new area of the one-story building would be 1,936 ft2.


• Students will extend their understanding of multiplication of numbers to the effects of multiplication in functions.

• Students will explore the effects on a graph when f(f(f x(x( ) is replaced with k • f(f(f x(x( ).

• Students will explore the effects on a graph when f(f(f x(x( ) is replaced with f(f(f k • x).

• Students will learn how these function transformations differ when applied to quadratics.



Instruction


IntroductionFunctions can be transformed in two distinct ways: by transforming the independent variable of the function (su ch as by multiplying the independent variable by k), and by transforming the function as a whole (such as by multiplying the dependent variable by k). Previously, we saw how adding some constant k to the variable of a function or to the entire function affected the graph of the function.

In this lesson, we will see how multiplying by a constant affects the graph of a quadratic function. Given f(f(f x(x( ) and a constant k, we will observe the transformations f(f(f k • x) and k • f(f(f x(x( ).

Key Concepts

Graphing and Points of Interest

• In the graph of a function, there are key points of interest that define the graph and represent the characteristics of the function.

• When a function is transformed, the key points of the graph define the transformation.

• Key points that can be used to graph the translation of a quadratic function are the vertex and the x- and y-intercepts.

Multiplying the Dependent Variable by a Constant, k: k • f • f • (f(f x)x)x

• In general, multiplying a function by a constant will stretch or shrink (compress) the graph of f(f(f x(x( ) vertically.

• If k > 1, the graph of f > 1, the graph of f > 1, the graph of (f(f x(x( ) will stretch vertically by a factor of k (so the parabola will appear narrower).

• A vertical stretch pulls the par abola and stretches it away from the x-axis.

• If 0 < k < 1, the graph of f < 1, the graph of f < 1, the graph of (f(f x(x( ) will shrink or compress vertically by a factor of k (so the parabola will appear wider).

• A vertical compression squeezes the parabola towards the x-axis.

Prerequisite Skills


• graphing quadratic functions (A–REI.10)

• finding intercepts and vertices of quadratic functions (F–IF.7a★)



Instruction


• If k < 0, the parabola will be first stretched or compressed and then reflected over the x-axis.

• When the graph of a function is stretched, compressed, or reflected over the x-axis, the x-intercept s (zeros) remain the same, as does the x-coordinate of the vertex (the axis of sy mmetry).

• While k • f(k • f(k • f x(x( ) = f(f(f k • x) can be true, generally k • f(k • f(k • f x(x( ) ≠ f(f(f k • x).

Vertical stretches: when k > 1 in k • f • f • (f(f x)x)x

k • f(x)f(x)

• The graph is stretched vertically by a factor of k.

• The x-coordinate of the vertex remains the same.

• The y-coordinate of the vertex changes.

• The x-intercepts remain the same.

Vertical compressions: when 0 < k < 1 in k • f • f • (f(f x)x)x

k • f(x) f(x)

• The graph is compressed vertically by a factor of k.






Instruction


Reflections over the x-axis: when x-axis: when x k < 0 in k • f • f • (f(f x)x)x

k • f(x)

f(x)

• The parabola is reflected over the x-axis.




• When k < 0, first perform the vertical stretch or compression, and then reflect the function over the x-axis.

Multiplying the Independent Variable by a Constant, k: f: f: (f(f k • x)x)x

• In general, multiplying the independent variable in a function by a constant will stretch or shrink the graph of the function horizontally.

• If k > 1, the graph of f > 1, the graph of f > 1, the graph of (f(f x(x( ) will shrink or compress horizontally by a factor of 1

k (so the parabola

will appear narrower).

• A horizontal compression squeezes the parabola toward the y-axis.

• If 0 < k < 1, the graph of f < 1, the graph of f < 1, the graph of (f(f x(x( ) will stretch horizontally by a factor of 1

k (so the parabola will

appear wider).

• A horizontal stretch pulls the parabola and stretches it away from the y-axis.

• If k < 0, the graph is first horizontally stretched or compressed and then reflected over the y-axis.

• The y-intercept remains the same, as does the y-coordinate of the vertex.

• When a constant k is multiplied by the variable x of a function f(f(f x(x( ), the interval of the intercepts of the function is increased or decreased depending on the value of k.



Instruction


Horizontal compressions: when k > 1 in f> 1 in f> 1 in (f(f k • x)x)x

f(k • x)f(x)

• The graph is compressed

horizontally by a factor of 1

k.

• The y-coordinate of the vertex remains the same.

• The x-coordinate of the vertex changes.

• The y-intercept remains the same.

Horizontal stretches: when 0 < k < 1 in f< 1 in f< 1 in (f(f k • x) x) x

f(k • x)

f(x)

• The graph is stretched

horizontally by a factor of 1

k.


• The x-coordinate of the vertex chan ges.



Instruction


Reflections over the y-axis: when y-axis: when y k < 0 in f< 0 in f< 0 in (f(f k • x)x)x

f(k • x) f(x)

• The parabola is reflected over the y-axis.


• The x-coordinate of the vertex changes.

• The y-intercept remains the same.

• When k < 0, first perform the horizontal compression or stretch, and then reflect the function over the y-axis.

Common Errors/Misc onceptions

• thinking that multiplying the dependent variable by a constant k yields the same equation as multiplying the independent variable by the same constant k in all cases (i.e., that k • f(f(f x(x( ) is always equal to f(f(f k • x), but this is not always true)

• forgetting to substitute all values of x withx withx k • x when working with the transformation k • x when working with the transformation k • x f when working with the transformation f when working with the transformation (f(f k • x)

• forgetting to square the constant k when substituting f(f(f k • x) into ax2 + bx + c

• confusing horizontal with vertical transformations and vice versa

• confusing stretches and compressions




Use what you have learned about transformations of functions to solve problems 1 and 2.

1. For the function f(f(f x(x( ) = x2 + x + 1, find 3 • f(f(f x(x( ) and describe the changes that occur to the graph of fof fof (f(f x(x( ) as a result of multiplying the function by 3. Check your answers by comparing the two functions on your graphing calculator.

2. For the function f(f(f x(x( ) = x2 – x, find f(5f(5f x) and describe the changes that occur to the graph of f) and describe the changes that occur to the graph of f) and describe the changes that occur to the graph of (f(f x(x( ) as a result of multiplying the variable x by 5. Check your answers by comparing the two functions on your graphing calculator.

Use the graphs and the given information to complete problems 3 and 4.

3. Consider the graphs of the functions f(f(f x(x( ) and g(x(x( ). The equation for f(f(f x(x( ) is f(f(f x(x( ) = x2 – 2x + 1. What could be the equation for g(x(x( )?

8

6

7

1 2 3 4 5

5

–1

–2

4

3

2

1

0

y

x

–1–2–3–4–5

g(x)

f(x)

continued

Scaffolded Practice 2.9: Replacing f (x ) with x ) with x k •k •k f (x ) and x ) and x f (k •k •k x ) x ) x




4. Consider the graphs of the functions f(f(f x(x( ) and g(x(x( ). The equation for f(f(f x(x( ) is f(f(f x(x( ) = 2x2 + x – 3. What could be the equation for g(x(x( )?

1 2 3 4 5

5

–1

–2

–3

–4

4

3

2

1

0

y

x

–5

–1–2–3–4–5

For problems 5–7, graph both functions on your graphing calculator, and determine the scale factor and the transformation(s): horizontal stretch, horizontal compression, vertical stretch, vertical compression, reflection over the x-axis, or reflection over the y-axis.

5. ;

continued




6. ;

7. ;

continued




Read each scenario and use the given information to solve problems 8–10.

8. An architect wants to add more square footage to a house. She could either double the lengths of each wall of one room, or add another room of equal size to the first. Which option would give her more square footage? Explain using functions and transformations.

9. A company’s profit is given by f(f(f x(x( ), where x is the number of products they sell. What function would represent tripling the company’s profits?

10. A thrown object follows the path given by h(x(x( ) = –x) = –x) = – 2 + 10. Would the ball go farther if it

followed the path or ? Which of these options would preserve the maximum

height of the ball?



Instruction


E xample 1

Consider the function f(f(f x(x( ) = x2, its graph, and the constant k = 2. What is k • f(f(f x(x( )? How are the graphs of fare the graphs of fare the graphs of (f(f x(x( ) and k • f(f(f x(x( ) different? How are they the same?


If f(f(f x(x( ) = x2 and k = 2, then k • f(f(f x(x( ) = 2 • f(f(f x(x( ) = 2x2.

2. Use a table of values to graph the functions.

x f (f (f x)x)x k • f • f • (f (f x)x)x–2 4 8–1 1 20 0 01 1 22 4 8

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10f(x) k • f(x)

Guided Practice 2.9



Instruction


3. Compare the graphs.

Notice the position of the vertex has not changed in the transformation of f(f(f x(x( ). Therefore, both equations have same x- and y-intercepts. However, notice the inner graph, 2x2, is narro wer than the graph of x2 because the y-coordinate of 2 • f(f(f x(x( ) is twice as much as the y-coordinate of f(f(f x(x( ). Since k > 1, and is being multiplied by the entire function as a whole, the graph of f(f(f x(x( ) will stretch vertically by a factor of 2. The parabola appears narrower.

E xample 2

Consider the function f(f(f x(x( ) = x2 – 81, its graph, and the constant k = 3. What is f(f(f k • x)? How do the vertices and intercepts of f of f of (f(f x(x( ) and f(f(f k • x) compare?


If f(f(f x(x( ) = x2 – 81 and k = 3, then f(f(f k • x) = f(3f(3f x) = (3x)2 – 81 = 9x2 – 81.

2. Using a graphing calculator, graph the two functions on the same coordinate plane.

From step 1, the equations are as follows: f(f(f x(x( ) = x2 – 81 and f(f(f k • x) = 9x2 – 81.

On a TI-83/84:

Step 1: Press [Y=].

Step 2: Type in the equation for f(f(f x(x( ) at Y1: [X,T, θ, n][x2][–][81]. Press [ENTER].

Step 3: Type in the equation for f(f(f k • x) at Y2: [9][X,T, θ, n][x2][–][81].

Step 4: Press [GRAPH].

Step 5: If necessary, set the viewing window by pressing [WINDOW]. A list of suggested parameters follows. Use the arrow keys to navigate between fields. Xmin = –10, Xmax = 10, Xscale = 1, Ymin = –90, Ymax = 50, Yscale = 10, Xres = 1.





Instruction


On a TI-Nspire:

Step 1: From the home screen, choose a graphing window.

Step 2: Type in the equation for f(f(f x(x( ) at f1(x): [x][x2][–][81]. Press [enter].

Step 3: Use the navigation pad to move the cursor over to the double arrows in the lower left corner and press the “click” key.

Step 4: Type in the equation for f(f(f k • x) at f2(x): [9][x][x2][–][81]. Press [enter].

Step 5: Set the viewing window by pressing [menu] and selecting 4: Window/Zoom and then 1: Window Settings.

Step 6: Fill in the fields as follows using the [tab] key to navigate between fields: Xmin = –10, Xmax = 10, Xscale = 1, Ymin = –90, Ymax = 50, Yscale = 10, Xres = 1.

Step 7: Press [tab] to navigate to “OK” and press [enter].

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-80

-70

-60

-50

-40

-30

-20

-10

10

20

30

40

f(x)

f(3x)



Instruction



Notice the position of the vertex has not changed in the transformation

of f(f(f x(x( ), so the y-intercept remains the same. However, notice that the

inner graph, f(3f(3f x) = 9x2 – 81, is narrower than f(f(f x(x( ). The x-intercepts

change. Specifically, the interval between the x-intercepts becomes

smaller. Since k > 1, and is being multiplied by the independent

variable of the function, x, the graph of f(f(f x(x( ) will shrink horizontally

by a factor of 1

3, so the parabola appears narrower.

Example 3

Consider the function f(f(f x(x( ) = x2 – 6x + 8, its graph, and the constant k = –1. What is k • f(f(f x(x( )? How do the graphs of fthe graphs of fthe graphs of (f(f x(x( ) and k • f(f(f x(x( ) compare?


If f(f(f x(x( ) = x2 – 6x + 8 and k = –1, then k • f(f(f x(x( ) = –f) = –f) = – (f(f x(x( ) = –(x) = –(x) = –( 2 – 6x + 8) = –x–x– 2 + 6x – 8.


From step 1, the equations are as follows: f(f(f x(x( ) = x2 – 6x + 8 and k • f(f(f x(x( ) = –x) = –x) = – 2 + 6x – 8. Refer to Example 1 for steps for graphing. Use a viewing window of –10 to 10 on the x-axis and from –10 to 10 on the y-axis. Each axis has a scale of 1.




Instruction


-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

f(x)

–f(x)


Notice that f(f(f x(x( ) is reflected over the x-axis for k • f(f(f x(x( ) because k is negative. Also notice that the x-intercepts of the two parabolas are the same. This is because k is being multiplied by the entire function. Therefore, the x-intercepts stay the same.



Instruction


Example 4

Consider the function f(f(f x(x( ) = x2 – 6x + 8, its graph, and the constant k = –1. What is f(f(f k • x)? How do the graphs of fthe graphs of fthe graphs of (f(f x(x( ) and f(f(f k • x) compare?


If f(f(f x(x( ) = x2 – 6x + 8 and k = –1, then f(f(f k • x) = f(–f(–f x(–x(– ) = (–x) = (–x) = (– )2 – 6(–x – 6(–x – 6(– ) + 8 = x2 + 6x + 8.


From step 1, the equations are as follows: f(f(f x(x( ) = x2 – 6x + 8 and k • f(f(f x(x( ) = x2 + 6x + 8. Refer to Example 1 for steps for graphing. Use a viewing window of –10 to 10 on the x-axis and –10 to 10 on the y-axis. Each axis has a scale of 1.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

f(x)f(–x)



Instruction



Notice that f(f(f x(x( ) is reflected over the y-axis for f(f(f k • x) because k is negative. Also notice that the x-intercepts of the equations have changed, but the y-intercept remains the same.

Exam ple 5

The dimensions of a rectangular garden edged with wood are such that the longer sides are 3 times the length of the shorter sides. Keeping the same ratio of side lengths, which would result in having a larger garden area: making the existing sides 5 times lo nger, or building 4 more gardens that are equal in size to the existing garden?

1. Write an equa tion that models the area of the existing garden.

Area is length times width, and the length of the garden is 3 times longer than its width. Therefore, the area can be modeled by the equation a(x(x( ) = x • 3x = 3x2, where x is the width (the garden’s shorter side).

2. Use the equation to determine the area of the garden when the sides of the existing garden are made 5 times longer.

The garden’s width would now be 5x, and the area of the garden would be a(5x) = 3(5x)2 = 3(25x2) = 75x2.

3. Use the equation from step 1 to determine the total garden area when 4 more gardens equal in size to the existing garden are built, for a total of 5 gardens.

If the area of the existing garden is a(x(x( ) = 3x2, then the area of 5 gardens that are the same size is 5a(x(x( ) = 5(3x2) = 15x2.

4. Compare the results of the two scenarios.

The area of the garden in the first scenario is 75x2 square units.

The area of the 5 gardens in the second scenario is 15x2 square units.

The area of the garden in the first scenario is 5 times the area of the gardens in the second scenario; therefore, making the sides of the existing garden 5 times longer would result in a larger area.




Problem-Based Task 2.9: Fewer Parabolas, PleaseA city hired a civil engineering firm to draw up plans for a 420-foot bridge that would use 7 downward-facing parabolic arches to support the span. The resulting plans called for the arches to be 75 feet high, with a distance of 60 feet between the bases of each arch, as shown in the following diagram.

After seeing the drawings, city councilors asked the civil engineering firm to make a second set of drawings using only 5 evenly spaced parabolic curves but covering the same 420-foot spa n. What are the eq uations of the 5 parabolas in the new plan? Let the y-axis represent the axis of symmetry for the left-most parabola, and let the x-axis represent the bottom of the arches.

What are the equations of the 5 parabolas in the new plan?

What are the equations of the 5 parabolas in the new plan?

What are the equations of the

SMP1 ✓ 2 3 4 ✓5 6 7 8 ✓




Problem-Based Task 2.9: Fewer Parabolas, Please

Coachinga. Identify the vertex and x-intercepts for the first of the original 7 parabolas.

b. Build the equation for the left-most parabola in vertex form us ing the coordinates of the vertex.

c. Solve for a using a point on the parabola.

d. Write the equation that describes the first of the 7 parabolas.

e. Determine whether keeping the same height and span of the bridge while reducing the number of arches is an k • f(k • f(k • f x(x( ) problem or an f(f(f k • x) problem.

f. What is the value of k?

g. What is the transformed equation for the left-most parabola?

h. What is the distance between the bases of each arch if the span must be 420 feet?

i. What type of transformation problem must be used to create the remaining four arches?

j. Use the information from parts h and i to create the eq uations for the four remaining arches.



Instruction


Problem-Based Task 2.9: Fewer Parabolas, Please

Coaching Sample Responsesa. Identify the vertex and x-intercepts for the first of the original 7 parabolas.

Let the y-axis be the line of symmetry for the parabola. The x-value of the vertex is 0. The y-value of the vertex is the maximum value of the arch. The arches are to be 75 feet tall, so the vertex is (0, 75).

Th e x-intercepts are the points on the graph where y = 0. The arches are 60 feet apart at their bases. Divide by 2 to determine the distance each base is from the axis of symmetry.

60 ÷ 2 = 30

Since the y-axis is the axis of symmetry, the x-intercepts are –30 and 30.

b. Build the equation for the left-most parabola in vertex form using the coordinates of the vertex.

Substitute the known information to build the vertex form of the equation, then simplify.

f(f(f x(x( ) = a(x(x( – h)2 + k

f(f(f x(x( ) = a(x(x( – 0)2 + 75

f(f(f x(x( ) = ax2 + 75

c. Solve for a using a point on the parabola.

Substitute one of the x-intercepts into f(f(f x(x( ) = ax2 + 75 to solve for a.

f(f(f x(x( ) = ax2 + 75

0 = a(30)2 + 75

–75 = a • 900

− = = −75

900

1

12a

d. Write the equation that describes the first of the 7 parabolas.

f x x( )= − +1

12752



Instruction


e. Determine whether keeping the same height and span of the bridge while reducing the number of arches is an k • f(k • f(k • f x(x( ) problem or an f(f(f k • x) problem.

The problem is asking for a horizontal stretch, so that each arch has the same height as in the previous plan, but a greater width. This is an f(f(f k • x) problem.

f. What is the value of k?

Starting with the current function, consider the ratio of the number of arches in the old and

new plans. We now want 5 parabolas instead of the 7 we had before, so our ratio is 5 : 7.

Remember that a value of k between 0 and 1 will increase the interval between the x-intercepts,

so we want our value to be 5

7=k .

g. What is the transformed equation for the left-most parabola?

Substitute the value of k into f(f(f k • x).

( • )5

7

5

7

1

12

5

775

5

7

1

12

25

4975

5

7

25

58875

2

2

2

=

= −

+

= −

+

= − +

f k x f x

f x x

f x x

f x x

h. What is the distance between the bases of each arch if the span must be 420 feet?

The problem states the span of the 5 parabolas must stay at 420 feet. There are 5 parabolas. Therefore, divide 420 feet by 5.

420

584=

The distance between each pair of bases for each arch is 84 feet.



Instruction


i. What type of transformation problem must be used to create the remaining four arches?

The first arch needs to be translated to the right 4 times. This is an f(f(f x(x( + k) problem.

j. Use the information from parts h and i to create the equations for the four remaining arches.

For each of the remaining 4 parabolas, translate the function to the right by 84 units each, the distance between the arches.

• Equation 2: f x x( )= − − +25

58884 752( )

• Equation 3 (translating Equation 2 by 84 units to the right): f x x( )= − − +25

588168 752( )

• Equation 4 (translating Equation 3): f x x( )= − − +25

588252 752( )

• Equation 5 (translating Equation 4): f x x( )= − − +25

588336 752( )

-42 0 42 84 126 168 210 252 294 336 378-5

5101520253035404550556065707580859095







Task OverviewFocus

This task provides students with a real-world example using transformations of functions of the form f(f(f kx). Students will apply transformations to redesign the structure of an arched bridge.


• writing quadratic equations to model a real-world problem

• writing equations of transformed quadratic functions

• identifying key features of a quadratic equation

• graphing a quadratic function

• comparing a quadratic function and a transformation of the function

• interpreting key features of a quadratic equation

• analyzing the type of transformation described in a real-world scenario

Introduction

This task should be used to explore or to apply transformations of quadratic functions that change the shape of the parabola. It is best to implement this task after students have graphed transformations that compress or stretch a function. Graphing may be done using technology, including graphing calculators or online graphing programs.

Begin by reading the problem and clarifying which part of the bridge the parabolas form. If possible, show a diagram of a bridge formed this way. Here is one example.

The Living New Deal. “Mapping the New Deal: Stone Arches in Texas.”


Problem-Based Task 2.9 Implementation Guide: Fewer Parabolas, PleaseNorth Carolina Math 2 Standard






Review the following terms:

civil engineering a branch of engineering that deals with the design and implementation of bridges, roads, buildings, and other structures in the physical world

critical points the key features of a function




Students may struggle with the many steps involved in solving this problem. Even though the problem is scaffolded, students must still complete, interpret, and remember each step along the way to the final solution. Suggest that students highlight or underline the parts of each step that they feel will be important as they make their way through each subsequent step.


Some students process information visually. It may help to have students graph the quadratic functions as they move through the steps to help in understand the problem and solution.

• SMP 8: Look for and express regularity in repeated reasoning.

Ensure that students understand they will use the same information to write the five transformed functions. Steps h and i are repeatedly used to do this.



• using the height of the bridge instead of the height of the arches to find the vertex

Remind students to use the diagram provided to understand that the vertices of the parabolas represent the height of the arches.

• thinking that this is a k • f(f(f x(x( ) problem

Review the key concepts for vertical and horizontal compressions and stretches. Discuss how changing from seven parabolas to five will change the shape of the parabolas.





• writing the ratio as 7 : 5 instead of 5 : 7

Remind students that if 0 < k < 1, the graph of f(f(f x(x( ) will stretch horizontally. We are changing from seven arches to five, so each of the five arches will need to be stretched horizontally to cover the same region as the seven arches. The ratio needs to be between 0 and 1.

• not squaring 5

7 when finding

f x5

7 Discuss the difference between f(f(f kx) and kf(kf(kf x(x( ). Remind students that in general, f(f(f kx) ≠ kf(kf(kf x(x( ).



• If students struggle to find the critical points of the first parabola ask what the vertex of the parabola would be if the diagram were placed in the first quadrant of the coordinate plane. (Answer: (0, 75)) Use this same strategy to help students who struggle to find the x-intercept.

• If students try to solve for a before substituting for x in part c, remind them that doing the substitution first will make the equation easier to solve.

• If students confuse vertical compressions with horizontal stretches, review the graphs of each. Ask whether the height of a parabola changes in a vertical compression (Answer: yes) in a horizontal stretch (Answer: no). This will help students understand that the problem is a horizontal stretch.


Debriefing the TaskCompare the graphs of the original and transformed parabolas. Discuss the key features of each. Compare strategies for solving the problem and discuss whether students thought a diagram (graph) was helpful in understanding and solving the problem.



• In the graph of a function, there are key points of interest that define the graph and represent the characteristics of the function. When a function is transformed, the key points of the graph define the transformation. The key points in the graph of a quadratic function are the vertex and the x-intercepts.

Students find and analyze the key points of the original functions as well as the transformed functions.





• In general, multiplying the independent variable in a function by a constant will stretch or

shrink the graph of fph of fph of horizontally. If 0 < f horizontally. If 0 < f k < 1, the graph of f < 1, the graph of f < 1, the graph of (f(f x(x( ) will stretch horizontally by a

factor of k

1. A horizontal stretch pulls the parabola and stretches it away from the y-axis.

The number of arches will change from seven to five, using the same amount of space. This means that each of the transformed functions will be wider than the original functions. Students will calculate k to determine the stretch.

Extending the Task

To extend the task, perform other types of transformations. For example:

• The height of the arch changes.

• The height of the bridge changes.

• The design of the bridge changes so that the parabolas are reflected.



• For SMP 1, ASK: “How did you make sense of the problem or demonstrate perseverance?” (Answer: I worked through each step making sure I understood why the step was necessary.)

• For SMP 4, ASK: “How did you use mathematics to model this particular scenario?” (Answer: The transformed equations modeled the arches of the bridge.)

• For SMP 8, ASK: “How did you look for and express regularity in repeated reasoning?” (Answer: I had to perform the same type of transformation on the equations for all of the five arches.)


Students can explore and solve this problem graphically using an online graphing tool.

Technology

Students can use a graphing calculator or online software to graph any or all of the functions representing the arches.




Practice 2.9: Replacing f (x ) with x ) with x k •k •k f (x ) and x ) and x f (k •k •k x ) x ) xUse what you have learned about transformations of functions to solve problems 1 and 2.

1. For the function f(f(f x(x( ) = x2 + x – 6, find 2 • f(f(f x(x( ), and describe the changes that occur to the graph o f fo f fo f (f(f x(x( ) as a result of multiplying the function by 2. Check your answers by comparing the two functions on your graphing calculator.

2. For the function f(f(f x(x( ) = x2 + x, find f(3f(3f x), and describe the changes that occur to the graph of f(f(f x(x( ) as a result of multiplying the variable x by 3. Check your answers by comparing the two functions on your graphing calculator.


3. Consider the graphs of the functions f(f(f x(x( ) and g(x(x( ). The equation for f(f(f x(x( ) is f(f(f x(x( ) = x2 – x – 2. What could be the equation for g(x(x( )?

-5 -4 -3 -2 -1 0 1 2 3 4 5

-10-9-8-7-6-5-4-3-2-1

123456789

10

f(x)g(x)

continued

AA




4. Consider the graphs of the functions f(f(f x(x( ) and g(x(x( ). The equation for f(f(f x(x( ) is f(f(f x(x( ) = x2 + 3x – 4. What could be the equation for g(x(x( )?

-5 -4 -3 -2 -1 0 1 2 3 4 5

-10-9-8-7-6-5-4-3-2-1

123456789

10

f(x) g(x)

Complete each of the following tasks for the functions in problems 5–7.

• Graph f(f(f x(x( ) and g(x(x( ) on your graphing calculator.

• Determine the scale factor and the transformation(s): horizontal stretch, horizontal compression, vertical stretch, vertical compression, reflection over the x-axis, or reflection over the y-axis.

• Describe the similarities and differences of the graphs.

5. f(f(f x(x( ) = x2 – x – 2; g(x(x( ) = –2f) = –2f) = –2 (f(f x(x( )

6. f(f(f x(x( ) = x2 – x – 2; g(x(x( ) = f(–2f(–2f x)

7. f(f(f x(x( ) = x2 – 1; g(x(x( ) = −1

2 • f(f(f x(x( )

continued





8. A farmer has a rectangular goat pen such that one side is 2 times as long as the other side. He would like to have more space for his goats, and he is deciding between two options. He could either double the lengths of the sides of the existing pen, or he could build a second pen of the same size as the first. Which option would give him the most area for his goats? Explain your answer in terms of k • f( f( f x(x( ) and f(f(f k • x).

9. A company that produces skateboards knows the equation that models profit per month is

f(f(f x(x( ) = 3x2 + 300x, where x is the price charged per skateboard. If the company plans to expand

with the hopes of doubling its profits, should the new model for the company’s profit be f(2f(2f x),

f x1

2

, 2 • f(f(f x(x( ), or 1

2• ( )f x ? Explain.

10. Jada and Jayla are twins on the same softball team. They can each hit the ball so that it follows a path modeled by the equation f(f(f x(x( ) = –0.01x2 + 0.98x + 2. Jada says that the ball would go farther if it followed the path g(x(x( ) = f(2f(2f x). Jayla says the ball would go farther if it followed the path g(x(x( ) = 2 • f(f(f x(x( ). Who is correct? Which equation for g(x(x( ) would allow the ball to achieve the same height as the ball in the original equation?




B

continued

Practice 2.9: Replacing f (x ) with x ) with x k •k •k f (x ) and x ) and x f (k •k •k x ) x ) x

Use what you have learned about transformations of functions to solve problems 1 and 2.

1. For the function f(f(f x(x( ) = x2 – 3x – 4, find 3 • f(f(f x(x( ), and describe the changes that occur to the graph of f of f of (f(f x(x( ) as a result of multiplying the function by 3. Check your answers by comparing the two functions on your graphing calculator.

2. For the function f(f(f x(x( ) = x2 – 3x + 2, find f(2f(2f x), and describe the changes that occur to the graph of fof fof (f(f x(x( ) as a result of multiplying the variable x by 2. Check your answers by comparing the two functions on your graphing calculator.


3. Consider the graphs of the functions f(f(f x(x( ) and g(x(x( ). The equation for f(f(f x(x( ) is f(f(f x(x( ) = x2 – 4x + 3. What could be the equation for g(x(x( )?

-5 -4 -3 -2 -1 0 1 2 3 4 5

-10-9-8-7-6-5-4-3-2-1

123456789

10

f(x) g(x)




4. Consider the graphs of the functions f(f(f x(x( ) and g(x(x( ). The equation for f(f(f x(x( ) is f(f(f x(x( ) = x2 – x – 1. What could be the equation for g(x(x( )?

-5 -4 -3 -2 -1 0 1 2 3 4 5

-10-9-8-7-6-5-4-3-2-1

123456789

10

f(x) g(x)

Complete each of the following tasks for the functions in problems 5–7.

• Graph f(f(f x(x( ) and g(x(x( ) on your graphing calculator.

• Determine the scale factor and the transformation(s): horizontal stretch, horizontal compression, vertical stretch, vertical compression, reflection over the x-axis, or reflection over the y-axis.

• Describe the similarities and differences of the graphs.

5. f(f(f x(x( ) = x2 + 2x – 3; g(x(x( ) = –2f) = –2f) = –2 (f(f x(x( )

6. f(f(f x(x( ) = x2 + 2x – 3; g(x(x( ) = f(–2f(–2f x)

7. f(f(f x(x( ) = x2 – 4; g(x(x( ) = 1

2− • f(f(f x(x( )

continued





8. A celebrity has a rectangular closet such that one side is 3 times as long as the other side. He would like to have more space for his coats, and he is deciding between two options. He could either triple the lengths of the sides of the existing closet, or he could build 2 more closets of the same size. Which option would give him the most area for his coats? Explain.

9. Dina manages a swimwear store in a beach town. She knows the equation that models the

store’s profit per month in the summer is f(f(f x(x( ) = 3x2 + 300x, where x is the average price charged

per swimsuit. If swimsuit sales drop by half in the winter, is the new model for the store’s profit

f(2f(2f x), f x1

2

, 2 • f(f(f x(x( ), or 1

2f x( )? Explain.

10. Zion and Zavier built a small catapult that launches beanbags for physics class.

The catapult can launch a beanbag so that the bag follows a path modeled by the equation

f(f(f x(x( ) = –0.004x2 + 0.792x + 1.6. Zion says that the beanbag would go farther if it followed

the path g x f x( )=

1

2. Zavier says the beanbag would go farther if it followed the path

g(x(x( ) = 2 • f(f(f x(x( ). Who is correct? Which equation for g(x(x( ) would allow a launched beanbag to

achieve the same height as a beanbag in the original equation? Which equation for g(x(x( ) would

allow a launched beanbag to go the same distance as a beanbag in the original equation?

U2-244


Lesson 2.10: Interpreting Quadratic Functions

Name: Date:

North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource2.102.102.102.10



Warm-Up 2.10

The table represents the amount of money a car owner spends on repairs as a function of the number of oil changes the owner gets each year.

Yearly number of oil changes Amount spent on car repairs ($)

0 500

1 420

2 340

3 260

4 180

5 100

1. Write a linear model of the amount the owner will spend on car repairs as a function of the number of oil changes in 1 year.

2. What is the y-intercept? What does it represent?

3. What is the rate of change? What does it mean for the owner?


2.10

Instruction





Warm-Up 2.10 Debrief

1. Write a linear model of the amount the owner will spend on car repairs as a function of the number of oil changes in 1 year.

y = –80x + 500, where x is the number of oil changes in 1 year and y is the total amount spent on car repairs in 1 year.

2. What is the y-intercept? What does it represent?

The y-intercept is 500. This is the amount spent on car repairs if the owner does not get any oil changes over the course of a year.

3. What is the rate of change? What does it mean for the owner?

The rate of change is the slope of the function, –80. For the owner, it means that for every oil change performed, he will save $80 on car repairs.


• Students will extend their knowledge of interpreting linear functions to identify and interpret the key features of a quadratic function.




Instruction




Introduction

You may recall that a line is the graph of a linear function and that all linear functions can be written in the form f(f(f x(x( ) = mx + b, where m is the slope and b is the y-intercept. The solutions to a linear function are the infinite set of points on the line. In this lesson, you will learn about a second type of function known as a quadratic function.

Key Concepts

• A quadratic function is a function that can be written in the form f(f(f x(x( ) = ax2 + bx + c, where x is the variable, a, b, and c are real numbers, and c are real numbers, and c a ≠ 0. This form is also known as the standard form of a quadratic function, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. c is the constant term. c

• Quadratic functions can be graphed on a coordinate plane.

• One method of graphing a quadratic function is to create a table of at least five x-values and calculate the corresponding y-values.

• Once graphed, all quadratic functions will have a U-shape called a parabola.

• Distinguishing characteristics can be used to describe, draw, and compare quadratic functions. These characteristics include the y-intercept, x-intercepts, the maximum or minimum of the function, and the axis of symmetry.

• The intercept of a graph is the point where a line or a curve intersects the x- or y-axis.

• The x-interceptx-interceptx is the x-coordinate of the point where a line or a curve intersects the x-axis. It is written as (xis written as (xis written as ( , 0).

• The x-intercepts of a quadratic function occur when the parabola intersects the x-axis at (x-axis at (x-axis at ( , 0).

Prerequisite Skills


• graphing functions by creating a table of values (A–CED.2★)

• graphing usi ng technology (F–IF.7a★)

• identifying key features of linear functions (F–IF.4★)


2.10

Instruction




• The following graph of a quadratic function, f(f(f x(x( ) = x2 – 2x – 3, shows the location of the parabola’s x-intercepts.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

(–1, 0) (3, 0)

x-intercepts

• Note that the x-intercepts of this function are (–1, 0) and (3, 0).

• The equation of the x-axis is y = 0; therefore, the x-intercepts can also be found in a table by identifying which values of x have a corresponding x have a corresponding x y-value that is 0.

• The following table of values corresponds to the function f(f(f x(x( ) = x2 – 2x – 3. Notice that the same x-intercepts noted in the graph can be found where the table shows y is equal to 0.

x y

–2 5

–1 0

0 –3

1 –4

2 –3

3 0


Instruction




• The y-intercepty-intercepty of a quadratic function is the y-coordinate of the point where a line or a curve intersects the y-axis. It is written as (0, y).

• The y-intercept of a quadratic is the c value of the quadratic function when written in c value of the quadratic function when written in cstandard form.

• The following graph of a quadratic function, f(f(f x(x( ) = x2 – 2x – 3, shows the location of the parabola’s y-intercept.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

(0, –3)y-intercept

• Note that the y-intercept of this function is (0, –3). The c-value of the function is –3.

• The axis of symmetry of a parabola is the line through the vertex of a parabola about which the parabola is symmetric.

• For a qua dratic function in the form f(f(f x(x( ) = ax2 + bx + c, the equation of the axis of sym metry is

2x

b

a=−

.


2.10

Instruction




• The equation of the axis of symmetry for the function f(f(f x(x( ) = x2 – 2x – 3 is x = 1 because the vertical line through 1 is the line that cuts the parabola in half.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

Axis of symmetry

• The vertex of a parabola is the point on a parabola that is the maximum or minimum of the function.

• The maximum is the greatest value of y in the graph of a function, and the minimum is the least value of y.

• The extrema of a graph are the minima or maxima of a function. In other words, an extremum is the function value that achieves either a minimum or maximum.

• The vertex of a parabola lies on the axis of symmetry.

• The vertex is often written as (h, k).

• The formula 2

xb

a=−

is also used to find the x-coordinate of the vertex.

• To find the y-coordinate, sub stitute the value of x into the original function,

( , )2

,2

h kb

af

b

a=

− −

.


Instruction




• The graph that follows shows the relationship between the vertex and the axis of symmetry of a par abola.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

Axis of symmetry

Vertex (1, –4)

• Notice that the vertex of the function f(f(f x(x( ) = x2 – 2x – 3 is (1, –4).

• If you know the x-intercepts of the graph, or any two points on the graph with the same y-value, the x-coordinate of the vertex is the point halfway between the values of the x-coordinates.

• For x-intercepts (r, 0) and (s, 0), the x-coordinate of the vertex is 2

r s+.

• From the equation of a function in standard form, you can determine if the function has a maximum or a minimum based on the sign of the coefficient of the quadratic term, a.

• If a > 0, then the parabola opens up and therefore has a minimum value. If a < 0, the parabola opens down and therefore has a maximum value.

• The value of a in the function f(f(f x(x( ) = x2 – 2x – 3 is 1; therefore, the vertex is a minimum.

• To graph a funct ion using a graphing calculator, follow these general steps for your calculator model.


2.10

Instruction




On a TI-83/84:

Step 1: Press the [Y=] button.

Step 2: Type the function into Y1, or any available equation. Use the [X, T, θ, n] button for the variable x. Use the [x2] button for a square.

Step 3: Press [WINDOW]. Enter values for Xmin, Xmax, Ymin, and Ymax. The Xscl and Yscl are arbitrary. Leave Xres = 1.


On a TI-Nspire:

Step 1: Press the [home] key.

Step 2: Arrow over to the graphing icon and press [enter].

Step 3: Type the function next to f1(x(x( ), or any available equation, and press [enter]. Use the [X] button for the variable x. Use the [x2] button for a square.

Step 4: To change the viewing window, press [menu]. Select 4: Window/Zoom and select A: Zoom – Fit.

Common Errors/Misc onceptions

• incorrectly replacing x with 0 instead of x with 0 instead of x y when determining the x-intercept (and vice versa)

• using the incorrect sign when calculating the x-coordinate of the vertex

U2-252



Name: Date:



For problems 1–4, sketch the graph for each of the quadratic functions.

1. f( f( f x(x( ) = (x ) = (x ) = ( – 3)(x – 3)(x – 3)( – 2)

2. g(x(x( ) = x2+ 6x – 7

3. n(m) = m2+ 14m + 49

4. l(s) = –s2 – s + 12

For problems 5–8, find the y-intercept and vertex of each function. State whether the vertex is a minimum or maximum point on the graph and explain your reasoning.

5. l(k) = k2+ 4k

continued

Scaffolded Practice 2.10: Interpreting Quadratic Functions

U2-253





6. s(r) = r) = r r2 – 2r + 3

7. y(e) = –e2 – 4e – 4

8. p(w) = –w2 – 1

Complete problems 9 and 10 as directed.

9. Does the following graph represent the given function? Explain your reasoning.

y(x(x( ) = 21x2+ 6x – 37

0

y

x

–2–1

21

2 4 6 8 101 3 5 7 9–1–2–3–4–5–6–7–8–9–10

10

8

6

4

7

5

3

9

1112131415

–4–3

–5

10. Create a quadratic equation with a vertex of (–2, 16) and a y-intercept of 12.


Instruction




Example 1

Given the function f(f(f x(x( ) = x2, identify the key features of the graph: the extremum, vertex, and y-intercept. Then sketch the graph.

1. Determine the extremum of the graph.

The extreme value is a minimum when a > 0. It is a maximum when a < 0.

Because a = 1, the graph opens upward and the quadratic has a minimum.

2. Determine the vertex of the graph.

The minimum value occurs at the vertex.


,2

b

af

b

a

− −

.

Use the original function f(x) = x2 to find the values of a and b in order to find the x-coordinate of the vertex.

2x

b

a=− Formula to find the x-coordinate of

the vertex of a parabola(0)

2(1)x =

−Substitute 1 for a and 0 for b.

x = 0 Simplify.



f(f(f x(x( ) = x2 Original equation

f(0) = (0)f(0) = (0)f 2 Substitute 0 for x.

f(0) = 0f(0) = 0f Simplify.



Guided Practice 2.10


2.10

Instruction




3. Determine the y-intercept of the graph.

The y-intercept occurs when x = 0.

The y-intercept of the function f(f(f x(x( ) = x2 is the same as the ver tex, (0, 0).

When the equation is written in standard form, the y-intercept is c.


Create a table of values and axis of symmetry to identify points on the graph.


For each point to the left of the axis of symmetry, there is another point the same distance on the right side of the axis and vice versa.

Choose at least two values of x that are to the right and left of 0.

Let’s start with x = 2.




An additional point is (2, 4).

(2, 4) is 2 units to the right of the vertex. The point (–2, 4) is 2 units to the left of the vertex, so (–2, 4) is also on the graph.

To find another set of points on the graph, let’s evaluate the original equation for x = 3.




An additional point is (3, 9).

(3, 9) is 3 units to the right of the vertex. The point (–3, 9) is 3 units to the left of the vertex, so (–3, 9) is also on the graph.



Instruction




Plot the points and join them with a smooth curve.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

(0, 0)

(2, 4)(–2, 4)

(3, 9)(–3, 9)

f(x) = x2

Example 2

Given the function f(f(f x(x( ) = –2x2 + 16x – 30, identify the key features of the graph: the extremum, vertex, and y-intercept. Then sketch the graph.



Because a = –2, the graph opens downward and the quadratic has a maximum.


2.10

Instruction





The maximum value occurs at the vertex.

The vertex is of the form b

af

b

a2,

2

− −

.

Use the original equation f (x) = –2x2 + 16x – 30 to find the values of aand b in order to find the x-coordinate of the vertex.

xb

a2=− Formula t o find the x-coordinate of

the vertex of a parabola

x16

2( 2)

( )=−−

Substitute –2 for a and 16 for b.

x = 4 Simplify.



f(f(f x(x( ) = –2x2 + 16x – 30 Original equation







Substitute 0 for x in the original equation.



f(0) = –30f(0) = –30f Simplify.

The y-intercept is (0, –30).

When the quadratic equation is written in standard form, the y-intercept is c.


Instruction





Use symmetry to identify additional points on the graph.



The point (0, –30) is on the graph, and 0 is 4 units to the left of the axis of symmetry.

The point that is 4 units to the right of the axis is 8, so the point (8, –30) is also on the graph.

Determine two additional points on the graph.

Choose an x-value to the left or right of the vertex and find the corresponding y-value.



f(1) = –16f(1) = –16f Simplify.

An additional point is (1, –16).

(1, –16) is 3 units to the left of the axis of symmetry.

The point that is 3 units to the right of the axis is 7, so the point (7, –16) is also on the graph.


35

30

25

20

15

5

5

10

15

20

25

30

35

40

45

50

55

60

2 2 4 6 8 1

–

(0, –30) (8, –30)

(3, 0)

f(x) = –2x2 + 16x – 30

(4, 2)

(5, 0)

(1, –16) (7, –16)

0


2.10

Instruction




Example 3

Given the function f(f(f x(x( ) = x2 + 6x + 9, identify the key features of its graph: the extremum, vertex, and y-intercept. Then sketch the graph.



Because a = 1, the graph opens upward and the quadratic has a minimum.


The minimum value occurs at the vertex.


,2

b

af

b

a

− −

.

Use the original function f(f(f x(x( ) = x2 + 6x + 9 to find the values of a and bin order to find the x-coordinate of the vertex.

2x

b



(6)

2(1)x =

−Substitute 1 for a and 6 for b.

x = –3 Simplify.

The x-coordinate of the vertex is –3.

Substitute –3 into the original equation to find the y-coordinate.

f(f(f x(x( ) = x2 + 6x + 9 Original equation

f(–3) f(–3) f = (–3)2 + 6(–3) + 9 Substitute –3 for x.

f(–3) = 0f(–3) = 0f Simplify.


The vertex is located at (–3, 0).


Instruction








f(0) = (0)f(0) = (0)f 2 + 6(0) + 9 Substitute 0 for x.


The y-intercept is (0, 9).


Use symmetry to identify an additional point on the graph.

The axis of symmetry goes through the vertex, so the axis of symmetry is x = –3.


The point (0, 9) is on the graph, and 0 is 3 units to the right of the axis of symmetry.

The point that is 3 units to the left of the axis is –6, so the point (–6, 9) is also on the graph.




f(–1) = (–1)f(–1) = (–1)f 2 + 6(–1) + 9 Substitute –1 for x.

f(–1) = 4f(–1) = 4f Simplify.

An additional point is (–1, 4).

(–1, 4) is 2 units to right of the axis of symmetry.

The point that is 2 units to the left of the axis is –5, so the point (–5, 4) is also on the graph.



2.10

Instruction





14

13

12

10

9

8

7

6

5

4

3

2

1

1

3

4

5

6

7

12 10 8 6 4 2 2 4

(–6, 9) (0, 9)

f(x) = x2 + 6x + 9

(–3, 0)

(–5, 4) (–1, 4)

0

Example 4

Given the function f(f(f x(x( ) = –2x2x2x – 12x – 10, identify the key features of its graph: the extremum, vertex, and y-intercept. Then sketch the graph.


The extreme value is either a minimum, when a > 0, or a maximum, when a < 0.

Because a = –2, the graph opens down and the quadratic has a maximum.


Instruction






,2

b

af

b

a

− −

.

Use the original function f(x) = –2x2 – 12x – 10 to find the values of a and b in order to find the x-coordinate of the vertex.

2x

b



( 12)

2( 2)x =

− −−

Substitute –2 for a and –12 for b.

x = –3 Simplify.

The x-coordinate of the vertex is –3.

Substitute –3 into the original equation to find the y-coordinate.

f(f(f x(x( ) = –2x2 – 12x – 10 Original equation

f(–3) f(–3) f = –2(–3)2 – 12(–3) – 10 Substitute –3 for x.

f(–3) = 8f(–3) = 8f Simplify.


The vertex is (–3, 8).





f(0) = –2(0)f(0) = –2(0)f 2 – 12(0) – 10 Substitute 0 for x.

f(0) = –10f(0) = –10f Simplify.

The y-intercept is (0, –10).


2.10

Instruction





Use symmetry to identify another point on the graph.

Because 0 is 3 units to the right of the axis of symmetry, the point 3 units to the left of the axis will have the same value, so (–6, –10) is also on the graph.




f(0) = –2(–2)f(0) = –2(–2)f 2 – 12(–2) – 10 Substitute –2 for x.

f(–2) = 6f(–2) = 6f Simplify.

An additional point is (–2, 6).

(–2, 6) is 1 unit to right of the axis of symmetry.

The point that is 1 unit to the left of the axis is –4, so the point (–4, 6) is also on the graph.


24

22

20

18

16

14

10

8

6

4

2

2

4

6

8

10

14

16

18

20

22

24

26

28

2 10 8 6 4 2 2 4

(–3, 8)

(–5, 0)

(–4, 6) (–2, 6)

(–1, 0)

(0, –10)(–6, –10)

0

f(x) = –2x2 – 12x – 10


Instruction




E xample 5

h(x(x( ) = 2x2 – 11x + 5 is a quadratic function. Determine the direction in which the function opens, the coordinates of the vertex, the axis of symmetry, the x-intercept(s), if any, and the y-intercept. Use this information to sketch the graph.

1. Determine whether the graph opens up or down.

h(x(x( ) = 2x2 – 11x + 5 is in standard form; therefore, a = 2.

Since a > 0, the parabola opens up.

2. Find the vertex and the equation of the axis of symmetry.

h(x(x( ) = 2x2 – 11x + 5 is in standard form; therefore, a = 2 and b = –11.

=−

xb

a2The x-coordinate of the vertex

( )( )=

− −x

11

2 2Substitute 2 for a and –11 for b.

x = 2.75 Simplify.

The vertex has an x-coordinate of 2.75.

Since the input value is 2.75, find the output value by evaluating the function for x = 2.75.

h(x(x( ) = 2x2 – 11x + 5 Original equation

h(2.75) = 2(2.75)2 – 11(2.75) + 5 Substitute 2.75 for x.

h(2.75) = –10.125 Simplify.

The y-coordinate of the vertex is –10.125.

The vertex is the point (2.75, –10.125).

Since the axis of symmetry is the vertical line through the vertex, the equation of the axis of symmetry is x = 2.75.


2.10

Instruction




3. Find the y-intercept.

h(x(x( ) = 2x2 – 11x + 5 is in standard form, so the y-intercept is the con stant term c, which is 5.

The y-intercept is 5.

4. Find the x-intercepts, if any exist.

The x-intercepts occur when y = 0.

Substitute 0 for the output, h(x(x( ), and solve.

This equation is factorable, but if we cannot easily identify the factors, the quadratic formula always works.

Note both methods.

Solved by factoring: Solved using the quadratic formula:

h(x(x( ) = 2x2 – 11x + 5

0 = 2x2 – 11x + 5

0 = (2x – 1)(x – 1)(x – 1)( – 5)

0 = 2x – 1 or 0 = x – 5

x = 0.5 or x = 5

h(x(x( ) = 2x2 – 11x + 5

0 = 2x2 – 11x + 5

a = 2, b = –11, and c = 5c = 5c

xb b ac

a=− ± −2 4

2

x =− −( )± −( ) − ( )( )

( )11 11 4 2 5

2 2

2

x =±11 81

4

x =±11 9

4

=−

=+

x x11 9

4or

11 9

4

= =x x2

4or

20

4x = 0.5 or x = 5

The x-intercepts are 0.5 and 5.


Instruction




5. Plot the points from steps 2–4 and their symmetric points over the axis of symmetry.

Connect the points with a smooth curve.

x

y

0 2 4 6 8

–10

–8

–6

–4

–2

2

4

6

Axis of symmetry

2.75 units 2.75 units(0, 5) (5.5, 5)

(0.5, 0) (5, 0)

(2.75, –10.125)


2.10

Instruction




Example 6

g(x(x( ) = –x) = –x) = – 2 + 8x – 17 is a qu adratic function. Determine the direction in which the function opens, the vertex, the equation of the axis of symmetry, the x-intercept(s), if any, and the y-intercept. Use this information to sketch the graph.

1. Determine whether the graph opens up or down.

g(x(x( ) = –x) = –x) = – 2 + 8x – 17 is written in standard form; therefore, a = –1.

Since a < 0, the parabola opens down.

2. Find the vertex and the equation of the axis of symmetry.

g(g(g x(x( ) = –x) = –x x) = –x) = – 2x2x + 8x + 8x + 8 – 17 is written in standard form; therefore, x – 17 is written in standard form; therefore, x a = –1 and b = 8.

=−

xb

a2The x-coordinate of the vertex

( )( )=−−

x8

2 1Substitute –1 for a and 8 for b.

x = 4 Simplify.

The vertex has an x-coordinate of 4.

Since the input value is 4, find the output value by evaluating the function for x = 4.

g(x(x( ) = –x) = –x) = – 2 + 8x – 17 Original equation

g(4) = –(4)2 + 8(4) – 17 Substitute 4 for x.

g(4) = –1 Simplify.

The y-coordinate of the vertex is –1.

The vertex is the point (4, –1).

Since the axis of symmetry is the vertical line through the vertex, the equation of the axis of symmetry is x = 4.


Instruction




3. Find the y-intercept.

The function g(x(x( ) = –x) = –x) = – 2 + 8x – 17 is in standard form, so the y-intercept is the constant term c, which equals –17.

The y-intercept is –17.

4. Find the x-intercepts, if any exist.

The x-intercepts occur when y = 0.

Substitute 0 for the output, g(x(x( ), and solve using the quadratic formula since the function is not factorable over the rational numbers.


–x–x– 2 + 8x – 17 = 0 Set the equation equal to 0.

Determine the values of a, b, and c.

a = –1, b = 8, and c = –17c = –17c

xb b ac

a=− ± −2 4

2Quadratic formula

x =−( )± ( ) − −( ) −( )

−( )8 8 4 1 17

2 1

2Substitute –1 for a, 8 for b, and –17 for c.

x =− ± −

−8 4

2Simplify.

In this case, the discriminant, –4, is negative, which means there are no real solutions.

This also means that there are no x-intercepts.


2.10

Instruction




5. Plot the points from steps 2–4 and their symmetric points over the axis of symmetry .

xy

0–1 1 2 3 5 7 94 6 8

–20

–14

–16

–18

–10

–2

–4

–6

–8

–12

Axis of symmetry

(0, –17) (8, –17)

(4, –1)

4 units 4 units

For a more accurate graph, determine an additional pair of symmetric points.

Choose any x-value on the left or right of the axis of symmetry.

Evaluate the function for the chosen value of x to determine the output value.

Let’s choose x = 1.


g(1) = –(1)2 + 8(1) – 17 Substitute 1 for x.

g(1) = –10 Simplify.

(1, –10) is an additional point on the parabola.

Plot (1, –10) on the same graph.

(1, –10) is 3 units from the axis of symmetry.

Locate the point that is symmetric to the point (1, –10) with respect to the axis of symmetry.



Instruction




(7, –10) is also 3 units from the axis of symmetry and is symmetrical to the original point (1, –10) with respect to the axis of symmetry.

You can verify that (1, –10) and (7, –10) are the same distance from the axis of symmetry and are also symmetrical by referring to the graph.

xy

0–1 1 2 3 5 7 94 6 8

–20

–14

–16

–18

–10

–2

–4

–6

–8

–12

Axis of symmetry

(0, –17) (8, –17)

(4, –1)

4 units 4 units

3 units 3 units(1, –10) (7, –10)

U2-271

After how many seconds and at

what height was the streamer at its maximum distance from the ground?





Problem-Based Task 2.10: Parabolic Party Streamers

Students in a physics class are testing the pull of gravity at varying heights. With permission of the building manager, each student went to a different floor of a tall office building and tossed a roll of paper streamer up into the air from the window. Another student videotaped the streamers’ paths downward so that the class could determine the approximate equations of the parabolas the streamers created as they unraveled. When a streamer was thrown upward from the highest story of the building, the students determined that the distance, in feet, between the streamer and the ground t seconds after the t seconds after the tstreamer was thrown could be expressed by h(t) = –16t) = –16t t2t2t + 32t + 5 6. After how many seconds and at t + 5 6. After how many seconds and at twhat height was the streamer at its maximum distance from the ground?

SMP1 ✓ 2 3 4 ✓5 6 7 ✓ 8

U2-272



Name: Date:




Coaching

a. What is the formula stated in the problem?

b. Is the formula written in standard form?

c. What are the values of a, b, and c?

d. How do you determine when the streamer reached its highest point?

e. After how many seconds was the streamer at its maximum height?

f. What height did the streamer reach at its maximum?


2.10

Instruction






a. What is the formula stated in the problem?

h(t) = –16t) = –16t t2t2t + 32t + 56t + 56t

b. Is the formula written in standard form?

Yes, the equation h(t) = –16t) = –16t t2t2t + 32t + 56 is written in the form t + 56 is written in the form t f(f(f x(x( ) = ax2 + bx + c, which is the standard form of a quadratic equation.

c. What are the values of a, b, and c?

The value of a is –16.

The value of b is 32.

The value of c is 56.c is 56.c

d. How do you determine when the streamer reached its highest point?

Identify the y-coordinate of the vertex by using the formula for axis of symmetry to find the input value, t. Then substitute that into the function to find the output value, h(t).t).t

e. After how many seconds was the streamer at its maximum heigh t?

tb

a=−2

t( )( )=−−32

2 16

t =−−

32

32t = 1t = 1t

The streamer was at its maximum height after 1 second.

f. What height did the streamer reach at its maximum?

h(t) = –16t) = –16t t2t2t + 32t + 56t + 56t

h(1) = –16(1)2 + 32(1) + 56

h(1) = 72

The streamer reached a maximum height of 72 feet.



U2-274© Walch Education© Walch EducationNorth Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource

2.102.10




Task Overview

Focus

When an object is thrown into the air from a window, how can its path be described? Students will use a quadratic model to find the time and height of a streamer when it is at its maximum distance from the ground.


• modeling the situation with a quadratic equation

• identifying the vertex of a parabola

• applying the model to find the maximum distance from the ground

Introduction

This task should be used to explore or to apply the concept of solving quadratic equations. It is best to implement this task after students have learned to graph quadratic equations in standard form. Graphing may be done using technology, including graphing calculators or online graphing programs.

Begin by reading the problem and clarifying the meaning of the students’ experiment and how the streamer was thrown. Explain that the streamer went out the window and up before it fell to the ground. The distance between the streamer and the ground is measured in inches and the time it takes to fall to the ground is measured in seconds. Review the following terms: parabola, physics, gravity, streamers, and unraveled.

gravity a force that pulls things down towards the center of Earth

parabola the U-shaped graph of a quadratic equation

physics the study of the nature and properties of matter and energy

streamers long narrow paper strips that are used to decorate for parties

unraveled to come undone, or unrolled

Problem-Based Task 2.10 Implementation Guide: Parabolic Party StreamersNorth Carolina Math 2 Standard


U2-275© Walch Education© Walch Education North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher Resource

2.10








Students may struggle to visualize the path of the streamers. If possible, have students use streamers or toilet paper to model the path. Then, have them sketch the path.


In order to understand the equation as a mathematical model, students may need to graph the equation so that they can see the path of the streamer. Then, have them identify the point at which the streamer would be the farthest from the ground.


The vertex of a quadratic function that opens down represents the maximum point, or in

this case, the maximum height from the ground. Students continue to use the same process

for finding the vertex. That is they use the equation =−

xb

a2 to find its x-coordinate and then

substitute that value into the equation to find its y-coordinate.



• not finding the important points on a parabola

Have students sketch the path of the streamers and identify where the streamer is at its maximum height. Ask them to identify which important point this is.

• finding the x-coordinate of the vertex, but not the y-coordinate

Remind students that the vertex is a point. The formula −b

a2 gives the x-coordinate, but it still

needs to be substituted into the equation to find the y-coordinate of the vertex.



• If students do not understand the given model, ask “What does the given equation model?” (Answer: how the height of the streamers changes over time after it was thrown) “What kind of equation is it?” (Answer: quadratic) “What is the shape of the graph of this equation?” (Answer: a parabola) “What is the task for?” (Answer: when the streamers were at the highest point) “What part of a parabola would represent the highest point?” (Answer: the vertex)


2.102.10




• If students cannot find the vertex, ask “What formula can be used to find the x-coordinate?”

(Answer: =−

xb

a2) “How will you find the y-coordinate?” (Answer: Substitute the

x-coordinate into the equation and calculate y.)

• If students struggle to identify the answer, ask “What represents the time the streamer was at the highest point?” (Answer: The x-coordinate of the vertex.)

• “What represents the height at the maximum distance from the ground?” (Answer: They-coordinate of the vertex.)

• Challenge students to think about how they would use the model to find the time the streamer is at another height, such as 675 inches above the ground.

• “How would you find the time?” (Answer: Substitute 675 for y and then solve the equation forx.) “Why would there be two answers?” (Answer: Because the streamer is at that height on the way up and then again on the way down.)


Debriefing the Task

• Ask for volunteers to discuss their plan for determining the number of seconds. Discuss the process of identifying the coefficients of the quadratic equation in standard form. Encourage students to discuss any difficulties or confusion they experienced when working through the task.

• Compare students’ strategies and ways of justifying responses. Focus on the use of precise mathematical language and clarity, specifically when referring to the maximum height and vertex.

• Ask students whether they need to graph the equation completely to solve this problem. Discuss the importance of the important points in the graph, and ask them to interpret what the other important points would represent in this situation.



• The standard form of a quadratic equation is f(f(f x(x( ) = ax2 + bx + c.

The formula for the path of the streamers is given in standard form. Students need not rearrange the formula to any different form.

• The formula =−

xb

a2 is not only used to find the equation of the axis of symmetry, but also

the x-coordinate of the vertex.

To find the highest point, the vertex, of the path, students must use this formula to find the time, x, that the distance between the streamers and the ground is the greatest.


2.10




• The vertex gives information about the maximum and minimum value of a quadratic. The maximum is the largest y-value of a quadratic.

The task asks students to find the point when the distance between the streamers and the ground is maximized. Students should recognize that this point is the vertex of the graph.

Extending the Task

To extend the task, further analyze the model. For example, have students identify when the streamers are going up and when they are going down, identify the time that the streamers hit the ground, identify the height of the streamers at other times.



• For SMP 1, ASK: “How did you demonstrate perseverance?” (Answer: I didn’t just state that the vertex was the maximum point, I used the equation model to find the exact coordinates so that I could answer the question.)

• For SMP 4, ASK: “How did you use mathematics to model this particular scenario?” (Answer: I used the equation to model the path of the streamers.)

• For SMP 7, ASK: “How did you look for and make use of structure when solving this problem?” (Answer: I know that the vertex represents the maximum value for a quadratic function with a < 0, so I used the formula for the x-coordinate to find the time, and then substituted into the equation to find the maximum height.)


Students could use a graphing calculator to graph the function f(f(f x(x( ) = –16x2 + 32x + 673 and then find the maximum value to identify the vertex.

Technology

Students can use a graphing calculator to graph the function. Students can also use a table on the graphing calculator or a spreadsheet to generate a list of heights at different times and identify the maximum height.

U2-278



Name: Date:



Sketch the graph for each of the following quadratic functions.

1. q(x(x( ) = –x) = –x) = – 2 – 6x – 8

2. f(f(f x(x( ) = –3x2 + 24x – 48

3. m(b) = b2 – 6b + 10

Find the y-intercept and vertex of the following functions. State whether the vertex is a minimum or maximum point on the graph and explain your reasoning.

4. k(h) = h2 – 4h + 3

5. l(d) = d) = d d2d2d – 6d

6. f(f(f x(x( ) = –7x2 – 14x – 6

Does the following graph represent the given function? Explain your reasoning.

7. y(x(x( ) = x2 + 12x – 28

xy

–10–12–14 –4–6–8 0 2–2

–60

–50

–40

–30

–20

–10

continued

Practice 2.10: Interpreting Quadratic Functions A

U2-279





Use your knowledge of quadratic functions to complete the problems that follow.

8. Create the equation of a quadratic function with a vertex of (–3, 7) and a y-intercept of –2.

9. The path of an arrow shot in the air can be modeled by the function h(t) = –16t) = –16t t2t2t + 144t + 4, where h is the height, in feet, of the arrow above ground t seconds after it is released. What was t seconds after it is released. What was tthe height of the arrow when it was released from the bow?

10. The demand, d, for plastic storage containers depends on their price. A retail manager determines that the number of containers she can sell at a price of x dollars each is given by the formula d(x(x( ) = –3x2 + 220x – 200. What price produces a maximum value in the demand function?

U2-280



Name: Date:



Sketch the graph for each of the following quadratic functions.

1. a(x(x( ) = 2x2 – 6x + 4

2. e(x(x( ) = x2

3. f(f(f x(x( ) = x2 + 2

Find the y-intercept and vertex of the following functions. State whether the vertex is a minimum or maximum point on the graph and explain your reasoning.

4. n(h) = –2h2 – 7h

5. l(r) = 4r) = 4r r 2 + 40r + 7 r + 7 r

6. f(f(f x(x( ) = –2x2 + 4x + 3

Does the following graph represent the given function? Explain your reasoning.

7. d(t) = t) = t t2t2t – 3t – 5 t – 5 t

x

y

–2 0 2 4 6

–15

–10

–5

5

continued

Practice 2.10: Interpreting Quadratic Functions B

U2-281





Use your knowledge of quadratic functions to complete the problems that follow.

8. Create the equation of a quadratic function with a vertex of (5, 6) and a y-intercept of –69.

9. The path of a ball shot up in the air from a slingshot can be modeled by the function h t t t( )= − + +16 150 42 , where h is the height, in feet, of the ball above ground t seconds after it t seconds after it tis released. At what height was the ball released into the air?

10. A sock manufacturing company’s profit p (in hundreds of dollars) after selling x thousand pairs of socks can be modeled by the function p(x(x( ) = –4x2 + 40x – 2. How many pairs of socks must be sold in order to maximize profits?

U2-282

Warm-Up 2.11

As the population in North Carolina continues to grow, the solid waste generated grows with it. Governments must plan for disposal and recycling of the ever-growing amounts of waste by using data to predict the amount of recycling centers and landfills they will need in the future. The table below shows the tons (in thousands) of solid waste generated in North Carolina between 2010 and 2014.

YearTons of solid waste

generated (in thousands)

2010 18,358

2011 19,484

2012 20,293

2013 21,499

2014 23,561

1. Use a graphing calculator to fit a linear function to the data. Let x represent the number of years since 2010.

2. Use the best fit line to predict the amount of waste that will be produced in 2030.

3. How might your prediction change if the amount of waste generated in 2010 was 20,358 thousand tons?

Lesson 2.11: Quadratic Regression

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS A–CED.2★


Name: Date:



U2-283

Warm-Up 2.11 Debrief

1. Use a graphing calculator to fit a linear function to the data. Let x represent the number of years since 2010.

On a TI-83/84:

Step 1: Press [STAT], then [ENTER]. Enter the x values in the L1 column of the table and the y values in the L2 column. Since x represents the number of years since 2010, the values entered into L1 should be 0, 1, 2, 3, and 4.

Step 2: Press [STAT], right arrow key over to [CALC], select 4: LinReg, then continue to press [ENTER] until the function is “calculated.”

On a TI-Nspire:

Step 1: Enter data on List & Spreadsheet App. Be sure to name the list.

Step 2: From [home], choose 5: Data & Statistics, then press [enter].

Step 3: Using NavPad, move to the bottom and choose an x-variable list name, then move left and choose a y-variable list name. Be sure to use 0, 1, 2, 3, and 4 to represent the years for the x-values.

Step 4: Press [menu], then select 3: Actions, then 5: Regression, then 1: Show Linear (mx+b).

The linear regression line is y = 1,242.1x + 18,154.8.


A–CED.2 Create and graph equations in two variables to represent quadratic, square root and inverse variation relationships between quantities.★


Instruction





U2-284

Instruction





2. Use the best fit line to predict the amount of waste that will be produced in 2030.

Let x represent the number of years since 2010; therefore, x = 20.

y = 1,242.1(20) + 18,154.8 Substitute x into the linear regression equation.

y = 42,996.8 Simplify the equation.

The predicted amount of solid waste that will be produced in 2030 is 42,996.8 thousand tons.

3. How might your prediction change if the amount of waste generated in 2010 was 20,358 thousand tons?

Answers may vary. Graphing the new point shows that the linear function is no longer a good fit; a different function model is needed.


• Students will write quadratic equations.

• Students will analyze quadratic functions for key features.

U2-285

Instruction





Prerequisite Skills


• recognizing coordinate points from a table, graph, or written description of a situation (F-IF.9)

• evaluating quadratic equations for a given input (F–IF.2)

• solving quadratic equations using technology (A-REI.4)

• finding the vertex of a quadratic equation (A-SSE.3b)

Introduction

When a golfer hits a ball, a basketball player takes a shot, or a firefighter aims her hose at a fire, they don’t often consider the quadratic equation that models the path of the ball or water stream. However, by tracking the height of the ball or water at different times throughout its flight, you can predict where the ball will land or if the water will reach the flames. You have written equations for linear models in previous lessons, so we will extend this concept to quadratics now.

Key Concepts

• The standard form of a quadratic equation is y = ax2 + bx + c, with (x, with (x, with ( , y) representing the points that satisfy the equation. The graph is in the shape of a parabola, or U. When the a value is positive, the parabola opens up. When the a value is negative, the parabola opens down.

• You have already learned to use linear regression to write the equation representing a linear relationship, y = mx + b, when you know a minimum of 2 points on the line.

• Similarly, you can use quadratic regression to write an equation representing a quadratic relationship in the form y = ax2 + bx + c when you know a minimum of 3 points on the c when you know a minimum of 3 points on the cparabola. The equation will be more accurate if you know more points, but 3 points is the minimum.

• The points used to determine a quadratic equation can be represented either in a table, on a graph, or as part of a written description.

U2-286


• mixing up x- and y-values in the lists in the calculator

• only using two points instead of three to calculate the quadratic regression

• misinterpreting the question in the word problem to solve for the wrong characteristic

Instruction





• It is possible to use algebraic substitution or matrices to write a quadratic equation through 3 points, but the most efficient method is using graphing calculator technology to perform the regression. Here are the steps:

On a TI-83/84:

Step 1: Determine 3 or more points on the quadratic equation from a given graph, table, or written explanation.

Step 2: Press [STAT], then [EDIT], then select 1: Edit . . ., and enter x-values in L1

of the table and y-values in L2.

Step 3: Press [STAT], then [CALC], and select 5: QuadReg. Press Calculate. The calculator will calculate the coefficients of the terms in the quadratic equation.

Step 4: Substitute the coefficients before the variables in the standard form of the quadratic equation.

On a TI-Nspire:

Step 1: Determine three or more points on the quadratic equation from a given graph, table, or written explanation.

Step 2: Enter data on the List and Spreadsheet App, with the x-coordinates in List A and the y-coordinates in List B. Be sure to name your lists as x for List A and y for List B.

Step 3: From the HOME menu, choose 5: Data & Statistics. Press [enter].

Step 4: Using the NavPad, move to the bottom and choose the x-variable list name. Then move to the left and choose the y-variable list name to create the scatter plot.

Step 5: To determine the best-fit regression equation, press [menu], then select 3: Actions, then 5: Regression. Then select 4: Show Quadratic. The curve and equation will appear on the screen.

U2-287

For problems 1–5, calculate the quadratic regression equation of the ordered pairs.

1. (0, 5), (2, 1), and (3, 2)

2.

x f (f (f x)x)x–4 39–2 30 –9

3. (0, –3), (2, 1), (3, 6), and (4, 13)

4.

2 4 6 8 10

10

1 3 5 7 9

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

y

x

–10

–1–2–3–4–5–6–7–8–9–10

continued

Scaffolded Practice 2.11: Quadratic Regression

Date:Name:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS A–CED.2★




U2-288

5.

x h(x)x)x3 –84 –55 –47 –8

For problems 6–10, calculate the quadratic regression equation of the ordered pairs. Then use your graphing calculator to find the vertex of each equation.

6. (1, 4), (2, 1), and (4, 1)

7.

x –2 0 2g(g(g x)x)x 0 4 4

continued



Name: Date:



U2-289





8. (–6, 0), (–2, 0), and (0, 3)

9. (–3, 0), (–2, 3), (0, 3), and (2, –5)

10.

x f (f (f x)x)x–2 00 –83 –5

U2-290

Example 1

The table of values below represents the height in feet of a rocket as a function of time in seconds. Use regression to write a quadratic equation representing the table of values, and identify when the rocket will return to the ground both algebraically and graphically.

x h(x)x)x

1 199

3 441

10 280

1. Determine a quadratic equation using regression.

On a TI-83/84:

Step 1: Press [STAT], then [EDIT], then select 1: Edit … . Enter x-values 1, 3, and 10 in L1 of the table and y-values 199, 441, and 280 in L2.

Step 2: Press [STAT], then [CALC], and select 5: QuadReg. Press Calculate. The calculator will calculate the coefficients of the terms in the quadratic equation.

Step 3: Substitute the coefficients before the variables in the standard form of the quadratic equation.

On a TI-Nspire:

Step 1: Enter data on the List and Spreadsheet App with the x-coordinates 1, 3, and 10 in List A and the y-coordinates 199, 441, and 280 in List B. Be sure to name your lists as xfor List A and y for List B.



Guided Practice 2.11

Instruction





U2-291

Instruction






Step 4: To determine the best-fit regression equation, press [menu], then select 3: Actions, then 5: Regression. Then select 4: Show Quadratic. The curve and equation will appear on the screen.

The quadratic equation that represents the table of values is h(x(x( ) = –16x2 + 185x + 30.

2. Determine when the rocket will return to the ground algebraically.

In the context of this problem, time is represented by the variable x and cannot be negative. The rocket will reach the ground when the height, or h(x(x( ), equals 0. Therefore, you are finding the positive x-intercept.

0 = –16x2 + 185x + 30 Replace h(x(x( ) with 0.

x185 (185) 4( 16)(30)

32

2

=− ± − −

−

Substitute the values of a, b, and c into the quadratic c into the quadratic cformula.

x185 36,145

32=− ±

−Simplify.

x185 36,145

320.1599=

− +−

= − Simplify.

x185 36,145

3211.7224=

− −−

= Simplify.

The positive x-intercept is approximately (11.72, 0). Therefore, the rocket will return to the ground after approximately 11.72 seconds.

U2-292

Instruction





3. Confirm the positive x-intercept found in the previous step using a graphing calculator.

On a TI-83/84:



Step 3: Press [2ND][TRACE][CALC] and scroll to 5: Intersect. Press [ENTER] to select the quadratic graph, then press [ENTER] again to select the horizontal line y = 0. Use the arrows to move the cursor near the right x-intercept. Press [ENTER] to find the intercept at (xto find the intercept at (xto find the intercept at ( , 0). The x-value of (11.72, 0) represents the positive x-intercept.

On a TI-Nspire:

Step 1: From the Graphs and Geometry Application, type the quadratic equation in f1(x(x( ) and type f2(x(x( ) = 0.


Step 3: Press [menu], then select 6: Points & Lines, then 3: Intersection Points.


U2-293

Instruction





Step 4: When the pointing hand appears, click on the quadratic graph. The line y = 0 (the x-axis) and all x-intercepts in the viewing window will be displayed as (xviewing window will be displayed as (xviewing window will be displayed as ( , 0). The x-value of (11.72, 0) represents the positive x-intercept.

400400

600600

y

x

200200

0 (–0.16, 0)(–0.16, 0) (11.722, 0)(11.722, 0)(11.722, 0)

0 20 2–2–2 4 64 6 8 18 10 10 10 122 1414

It will take the rocket approximately 11.72 seconds to reach the ground.

Example 2

The height of a ball that has been thrown is tracked at different times as it flies through the air. Three points, (1, 58), (3, 66) and (4, 22), are recorded. If the height of the ball is recorded in feet and the time is recorded in seconds, determine the height of the ball after 2 seconds from when it is thrown.

1. Determine a quadratic equation using regression.

Using the regression feature in the calculator, the quadratic equation that models the flight of the ball is h(x(x( ) = –16x2 + 68x + 6.

2. Determine the height of the ball after 2 seconds.

h(2) = –16(2)2 + 68(2) + 6 Substitute 2 for x.h(2) = –16(4) + 136 + 6 Simplify.

h(2) = 78 Simplify.

The height of the ball after 2 seconds is 78 feet.

U2-294

Example 3

You and a friend are vacationing in Acapulco, Mexico, and decide to jump off a cliff and into the ocean. Your height as a function of time can be represented by the following graph of f(f(f t), and your t), and your tfriend’s height can be represented in the table of values, g(t). Determine the approximate difference in t). Determine the approximate difference in ttime that it takes for both of you to reach the surface of the water.

404040

5050

y

x

2020

1010

3030

0

(0, 48)(0, 48)

(0.5, 52)(0.5, 52)

0 10 1 22–2–2 –1–1

(2, 16)(2, 16)

ff((f(ff(f tt))t)tt)t

t g(g(g t)t)t

0 48

1 53

2 26

1. Determine a quadratic equation for f( f( f t) using regression.t) using regression.t

Using the three points given on the graph of f(f(f t)—(0, 48), t)—(0, 48), t(0.5, 52), and (2, 16)—and the quadratic regression feature in the calculator, the quadratic equation that represents the graph of f(f(f t) is t) is tf(f(f x(x( ) = –16x2 + 16x + 48.

Instruction





U2-295

Instruction





2. Determine the x-intercept of f(f(f t) using the graphing calculator.t) using the graphing calculator.t

Graphing the equation found in the previous problem, the positive x-intercept on the calculator shows that it will take approximately 2.3 seconds to reach the surface of the water.

3. Determine a quadratic equation for g(t) using regression.t) using regression.t

Using the quadratic regression feature in the calculator, the quadratic equation that represents the graph of g(t) is t) is t g(x(x( ) = –16x2 + 21x + 48.

4. Determine the x-intercept of g(t) using the graphing calculator.t) using the graphing calculator.t

Graphing the equation found in problem 3, the positive x-intercept on the calculator shows it will take your friend approximately 2.51 seconds to reach the surface of the water.

5. Subtract the time it takes you and the time it takes your friend to reach the surface of the water.

The approximate difference in time that it takes for both you and your friend to reach the surface of the water is 0.21 seconds.

U2-296

There is a fire on the 57th floor of a skyscraper. When the fire truck pulls up to the blaze, the truck’s hose is 5 feet off the ground. The firefighters must figure out where to aim the hose to make sure it reaches the window of the 57th floor, which is about 550 feet off the ground. They know that after 1 second, the water is 250 feet high, and after 2 seconds, the water stream has reached 425 feet. Eventually, gravity will start to pull the water back down, so they know a quadratic equation will model the height of the water after xseconds. Calculate the quadratic equation that models the water’s height after x seconds. Will the water reach the 57th floor?

Problem-Based Task 2.11: Put Out That Fire!SMP

1 ✓✓2 ✓3 ✓✓4 ✓5 ✓✓6 ✓7 ✓✓8✓

Will the water reach the 57th

floor?



Name: Date:



U2-297

Problem-Based Task 2.11: Put Out That Fire!

Coaching

a. What question does the problem ask you to answer?

b. What key feature of the equation will help you answer the question?

c. What points can we use to write the quadratic equation?

d. What quadratic equation represents this situation?

e. Based on the equation and its vertex, will the water reach high enough to put out the fire?





U2-298

Problem-Based Task 2.11: Put Out That Fire!


a. What question does the problem ask you to answer?

The problem asks if the water’s height, represented by the y-value in the quadratic equation, will reach 550 feet up the building to put out the fire.

b. What key feature of the equation will help you answer the question?

If the y-value of the vertex of the parabola is greater than 550, the water will reach high enough to put out the fire.

c. What points can we use to write the quadratic equation?

Based on the word problem, we know that the hose is 5 feet off the ground when it starts shooting water. We also know that the water is 250 feet off the ground after 1 second and 425 feet off the ground after 2 seconds. Since the time in seconds is represented by the x-coordinate and the height in feet is represented by the y-coordinate, the points are (0, 5), (1, 250), and (2, 425).

d. What quadratic equation represents this situation?

To find the equation, use the regression feature of a graphing calculator.

On a TI-83/84:

Step 1: Use the three points determined from the written explanation of the fire hose: (0, 5), (1, 250), and (2, 425).

Step 2: Press [STAT], then [EDIT], then select 1: Edit … . Enter x-values 0, 1, and 2 in L1 of the table and y-values 5, 250, and 425 in L2.

Step 3: Press [STAT], then [CALC], and select 5: QuadReg. Press Calculate. The calculator will yield the coefficients of the terms in the quadratic equation: a = –35, b = 280, and c = 5.c = 5.c

Step 4: Substitute the coefficients before the variables in the standard form of the quadratic equation to determine the equation y = –35x2 + 280x + 5.

Instruction





U2-299

On a TI-Nspire:

Step 1: Use the three points determined from the written explanation of the fire hose: (0, 5), (1, 250), and (2, 425).

Step 2: Enter data on the List and Spreadsheet App, with the x-coordinates 0, 1, and 2 in List A and the y-coordinates 5, 250, and 425 in List B. Be sure to name your lists as x for List A and y for List B.



Step 5: To determine the best-fit regression equation, press [menu], then select 3: Actions, then 5: Regression. Then select 4: Show Quadratic. The curve and equation, y = –35x2 + 280x + 5, will appear on the screen.

The quadratic equation that represents the water’s height is y = –35x2 + 280x + 5.

e. Based on the equation and its vertex, will the water reach high enough to put out the fire?

To find the vertex of the equation, we can either use algebraic steps or graph the equation using a graphing calculator. Remember, the quadratic equation that models the water’s height is

y = –35x2 + 280x + 5. Therefore, use the formula xb

a2=−

to find the x-coordinate of the vertex.

Let a = –35 and b = –280.

xb

a2=−

x280

2( 35)

280

704=

−−

=−−

=

Substitute this x-value into the quadratic equation and solve for y.

y = –35x2 + 280x + 5

y = –35(4)2 + 280(4) + 5

y = 565

The vertex of the equation is (4, 565).

565 feet is higher than 550 feet, or the 57th floor, so the water will reach the fire.

Instruction





U2-300

Instruction





Using a graphing calculator’s trace function also shows that the vertex is at x = 4, y = 565. The resulting graph should resemble the following:

400400

600600

x

y

(4, 565)(4, 565)

200200

0

–2–2 0 20 2 4 64 6 888



U2-301

For problem 1, calculate the quadratic regression equation of the ordered pairs.

1. (2, 37), (5, 10), and (9, 2)

Use the following information to complete problems 2– 4.

A punter kicks a football straight up in the air, and the following points are on the quadratic model representing the football’s height based on the time: (1, 48), (0, 4), (3, 40), (2, 60).

2. Use the ordered pairs to create the quadratic regression equation.

3. What was the initial height of the football?

4. When will the football hit the ground?

A basketball player shoots a ball toward the basket. Use the given ordered pairs to complete problems 5–7, where x represents time and y represents the height of the ball.

5. Use the following ordered pairs to create the quadratic regression equation: (14, 7.26), (17, 1.155), (0, 6), (10, 11.9)

6. When does the basketball reach its maximum height, and what is that height? Round your answer to the nearest tenth.

7. Suppose the basketball does not touch the basketball net or the rim. When will the basketball hit the floor? Round your answer to the nearest second.

Practice 2.11: Quadratic Regression A

continued





U2-302


Your physics teacher conducts an experiment. She throws a ball off the top of a building. She records the time in seconds and the distance in feet at 1, 2, and 3 seconds. The results are in the following table:

Time 0 1 2 3

Distance 46 63 48 1

8. Calculate the quadratic regression equation from the data in the table.

9. What was the height of the building?

10. When did the ball hit the ground?



Name: Date:



U2-303

For problem 1, calculate the quadratic regression equation of the ordered pairs.

1. (10, 3), (9, –3), (11, 13), (4, 27)


A soccer ball is kicked, and the height of the ball is tracked after x seconds. The quadratic path that the ball models includes the following points: (0, 0), (1, 1.57), (2, 3.08), (5, 7.25), (7, 9.73)

2. Use the ordered pairs to create the quadratic regression equation, with y representing the height of the ball and x representing the seconds since it was kicked.

3. What was the initial height of the soccer ball?

4. What is the maximum height that the soccer ball reached?


Kiara tosses a coin off a bridge into the stream below. The quadratic regression equation can be found using the following ordered pairs, where y is the height (in feet) and x is the time (in seconds).

5. Use the following ordered pairs to create the quadratic regression equation: (0, 112), (2, 240), (5, 192), (7, 0)

6. What is the maximum height of the coin?

7. When will the coin hit the water?

Practice 2.11: Quadratic Regression B

continued





U2-304


A tennis player hits balls across the court and records his maximum height of each hit.

Time 0 1 2 3

Height 5.5 6.0 5.5 4.0

8. Calculate the quadratic regression equation from the data in the table.

9. What was the maximum height of the ball?

10. When did the ball hit the ground?



Name: Date:




Teacher Resource/Student Workbook Answer KeyTeacher Resource/Student Workbook Answer Key© Walch Education© Walch Education

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS

Answer KeyLesson 2.1: Identifying the Domain and Range of a Quadratic Functi on (F–IF.4★Lesson 2.1: Identifying the Domain and Range of

★Lesson 2.1: Identifying the Domain and Range of

)

Warm-Up 2.1, p. U2-11. y = 3x + 642. all wh ole numbers3. The domain repres ents the possible number of games that

Gina could sell in one 8-hour workday.

Practice 2.1 A: Identifying the Domain and Range of a Quadratic Functi on, p. U2-16

1. Domain: all real numbers or −∞< <∞x ; range: all real numbers greater than or equal to 13.25, or –∞ < y ≤ 13.25

2. Domain: all real numbers or −∞< <∞x ; range: all real numbers up to 180.75, or –∞ < y ≤ 180.75

3. Domain: all real numbers or −∞< <∞x ; range: all real numbers up to about 13.6, or –∞ < f(f(f x(x( ) ≤ 13.6

4. Domain: all real numbers or −∞< <∞x ; range: all real numbers greater than or equal to –44, or –44 ≤ g(x(x( ) < ∞

5. Domain: all real numbers; −∞< <∞x ; range: all real numbers up to about 5.3, or –∞ < y ≤ 5.3

6. Domain: all real numbers; −∞< <∞x ; range: all real numbers greater than or equal to –9.6, or –9.6 ≤ y < ∞


8. less than 3.9 seconds9. 9.4 seconds

10. Domain: 8.67 ≤ x ≤ 28.83; range: 0 ≤ P(x(x( ) ≤ 1625; x = $18.75

Practice 2.1 B: Identifying the Domain and Range of a Quadratic Function, p. U2-19

1. Domain: all real numbers or −∞< <∞x ; range: all real numbers greater than or equal to about 0.7, or 0.7 ≤ y < ∞

2. Domain: all real numbers or −∞< <∞x ; range: all real numbers up to about 141.3, or –∞ < y ≤ 141.3

3. Domain: all real numbers or −∞< <∞x ; range: all real numbers greater than or equal to about –4.4, or–4.4 ≤ f(f(f x(x( ) < ∞

4. Domain: all real numbers or −∞< <∞x ; range: all real numbers greater than or equal to –27, or –27 ≤ g(x(x( ) < ∞


6. Domain: all real numbers; −∞< <∞x ; range: all real numbers greater than or equal to –15, or –15 ≤ y < ∞

7. Domain: all real numbers; −∞< <∞x ; range: all real numbers greater than or equal to –10, or –10 ≤ y < ∞

8. less than 10 seconds9. about 3 seconds

10. 25 in creases; 5.5 increases

Lesson 2.2: Identifying Terms, Factors, and Coefficients (A–SSE.1a★Lesson 2.2: Identifying Terms, Factors, and

★Lesson 2.2: Identifying Terms, Factors, and

)

Warm-Up 2.2, p. U2-221. The toll can be written as the expression 1.20 + 0.04m.2. The toll charged for driving 14 miles on the highway is $1.76.

Scaffolded Practice 2.2: Identifying Terms, Factors, and Coefficients, p. U2-26

1. terms: –6x2; 2x; –2; coefficients: –6, 2; constant: –2 2. 8x2+ 8x; binomial3. 4x2+ 104. 5x2 + 3x – 2; quadratic5. 4x2 + 10x + 4; quadratic6. –2x; linear7. 4x3– (x – (x – ( – 4x2)8. 0.5(3x – 5) + 2x9. x(x (x ( – 3)x2

10.

Practice 2.2 A: Identifying Terms, Factors, and Coefficients, p. U2-40

1. terms: 16x2, –12x, 20; coefficients: 16, –12; constant term: 20; factors: 16 and x2; –12 and x

2. –7x2 – 62; binomial3. 7x2 + 104. The expression is quadratic because it simplifies to –3x2x2x – 42x

and can be written in the form ax2ax2ax + bx + c, with c = 0.c = 0.c5. The expression is not quadratic because it simplifies to

–2x + 2. This is a linear expression and cannot be written in the form ax2 + bx + c.

6. Expression: 7x2 + (5 – x2) = 6x2 + 5; terms: 6x2, 5; coefficient: 6; constant term: 5. The expression is quadratic because it can be written in the form ax2 + bx + c.

7. Expression: (1/2)(12 + x2) – (1/3)x = (1/2)x = (1/2)x x2 – (1/3)x + 6; x + 6; xterms: (1/2)x2, (–1/3)x, 6; coefficients: 1/2, (–1/3); constant term: 6. The expression is quadratic because it can be written in the form ax2 + bx + c.

8. Expression: 4s; term: 4s; coefficient: 4; constant term: none. The expression is not quadratic because it cannot be written in the form ax2 + bx + c. It is a linear expression.

9. Expression: 6s2; term: 6s2 ; coefficient: 6; constant term: none. The expression is quadratic because it can be written in the form ax2 + bx + c.

10. Expression: (4/3)πrπrπ 3r3r ; term: (4/3)πrπrπ 3r3r ; coefficient: (4/3)π ; constant term: none. The expression is not quadratic because it cannot be written in the form ax2 + bx + c.

U2-306North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher ResourceTeacher Resource/Student Workbook Answer KeyTeacher Resource/Student Workbook Answer Key


Practice 2.3 A: Interpreting Complicated Expressions, p. U2-63

1. –7 < x < 102. x > 3 or x < –1/23. 12x2 + 17x – 5; a = 12; b = 17; c = –5c = –5c4. 7x2 + 31x – 30; a = 7; b = 31; c = –30c = –30c5. The expression is quadratic because it can be written in

the form ax2 + bx + c.6. The expression is not quadratic because it cannot be

written in the form ax2 + bx + c.7. The taxes increase from $3,600 to $14,400.8. The area increases from 3 square inches to

25 3 square inches.9. The area decreases by a factor of 4.

10. The area increases by a factor of 4.

Practice 2.3 B: Interpreting Complicated Expressions, p. U2-65

1. –4 < x < 62. x > 4 or x <−

7

53. 18x2 – 21x – 15; a = 18; b = –21; c = –15c = –15c4. –x–x– 2 + 4x + 14; a = –1; b = 4; c = 14c = 14c5. The expression is quadratic because it can be written in the

form ax2 + bx + c.6. The expression is not quadratic because it cannot be

written in the form ax2 + bx + c.7. The population increases from 50,000 to 68,000.8. The area increases from 4π square feet to 24π square feet to 24π π square feet.π square feet.π9. The area increases by a factor of 9.

10. The area decreases by a factor of 4.

Lesson 2.4: Quadratic Modeling (A–CED.1★)

Warm-Up 2.4, p. U2-671. 629 feet2. 6.25 seconds3. Answers may vary. One possible answer is h(x(x( ) = –16(x) = –16(x) = –16( –

6.25)2 + 629. The vertex form of the quadratic function will enable the pyrotechnician to easily see the seconds he has for each firework to go off. This will allow him to set the fireworks to the timing of the music faster.

Scaffolded Practice 2.4: Quadratic Modeling, p. U2-74

1. x = –3; (–3, 0); minimum 2. x = 0; (0, –4); maximum 3. x = 3; (3, 5); maximum 4. x = 1; (1, –2); minimum 5. (–2, –11); minimum 6. (–2, 5); maximum7. (3, 10); maximum 8. (–2, 0) and (4, 0)9. (–5, 0) and (1, 0)

10. (6, 0)

Practice 2.2 B: Identifying Terms, Factors, and Coefficients, p. U2-42

1. terms: 30x2, –18x, 72; coefficients: 30, –18; constant term: 72; factors: 30 and x2, –18 and x

2. 9x; monomial3. –x–x– 2 + x + 52 or x2 – x + 524. The expression is quadratic because it can be written in the

form ax2 + bx + c.5. The expression is not quadratic because it cannot be

written in the form ax2 + bx + c.6. Expression: 9x – (8 + x2) = –x) = –x) = – 2 + 9x – 8; terms: –x – 8; terms: –x – 8; terms: – 2, 9x, –8;

coefficients: –1, 9; constant term: –8. The expression is quadratic because it can be written in the form ax2 + bx + c.

7. Expression: 2(2 + x) + (1/2)x2 = (1/2)x2 + 2x + 4; terms: (1/2)x2, 2x, 4; coefficients: 1/2, 2; constant term: 4. The expression is quadratic because it can be written in the form ax2 + bx + c.

8. Expression: s2; term: s2; coefficient: 1; constant term: none. The expression is quadratic because it can be written in the form ax2 + bx + c.

9. Expression: s3; term: s3; coefficient: 1; constant term: none. The expression is not quadratic because it cannot be written in the form ax2 + bx + c.

10. Expression: 4πrπrπ 2r2r ; term: 4πrπrπ 2r2r ; coefficient: 4π; constant π; constant πterm: none. The expression is quadratic because it can be written in the form ax2 + bx + c.

Lesson 2.3: Interpreting Complicated Expressions (A–SSE.1b★Lesson 2.3: Interpreting Complicated

★Lesson 2.3: Interpreting Complicated

)

Warm-Up 2.3, p. U2-441. 29x – 22. If a fence panel is 4 feet long, the perimeter of the park is

114 feet.3. If the length of each fence panel increases to 5 feet, the

perimeter of the park increases to 143 feet. Subtracting 114 from 143 indicates this is an increase of 29 feet.

Scaffolded Practice 2.3: Interpreting Complicated Expressions, p. U2-47

1. x < 32. x > 23. 7x2 + 17x – 12; a = 7, b = 17, c = –12c = –12c4. –x –x – 2 – x + 5; a = –1, b = –1, c = 5c = 5c5. The expression is quadratic because it is equal to x2 + x – 27.6. The expression is not quadratic because it is equal to 7x The expression is not quadratic because it is equal to 7x The expression is not quadratic because it is equal to 7 – 22.7. It increases ninefold.8. It decreases eightfold.9. It decreases fourfold.

10. It increases ninefold.



Practice 2.4 A: Quadratic Modeling, p. U2-891. a. minimum

b. x =1

6

c. 1

6,

35

12

d. 3e. No real solutions

2. 10 meters3. At 3 seconds with the height of 55 meters4. 35 meters5. About 6.3 seconds6. 0 to 6.37. h t t t= − + +( ) 16 50 62

8. About 39 feet (45 – 6)9. About 3.2 seconds

10. 0 to 3.2

Practice 2.4 B: Quadratic Modeling, p. U2-911. a. minimum

b. x =4

3

c.

4

3,25

3d. –3e. No real solutions

2. 80 meters3. At 2 seconds with a height of 144 meters4. 128 meters5. 5 seconds6. 0 to 57. h t t t= − + +( ) 16 16 22

8. About 4 feet (6 – 2)9. About 1.1 seconds

10. 0 to 1.1

3. increasing: (–0.33 < x < ∞); decreasing: (–∞ < x < –0.33); minimum: x = –0.33; x-intercepts: –1.22, 0.55; neither even nor odd

4. increasing: (–∞ < x < 0); decreasing: (0 < x < ∞); maximum: x = 0; x-intercept: 0; even

5. minimum at x = –1.1; increasing: (–1.1 < x < ∞); decreasing: (–∞ < x < –1.1)

6. maximum at x = 3.5; increasing: (–∞ < x < 3.5); decreasing: (3.5 < x < ∞)

7. minimum at x = 1.75; increasing: (1.75 < x < ∞); decreasing: (–∞ < x < 1.75)

8. minimum at x = –2.65; increasing: (–2.65 < x < ∞); decreasing: (–∞ < x < –2.65)

9. increasing: (0 < x < 5); decreasing: (5 < x < 10)10. 3 P.M.

Practice 2.5 A: Interpreting Key Features of Quadratic Functions, p. U2-118

1. x > 0.3, x < 0.3; minimum = –5.3; x-intercepts: –1 and 1.67; y-intercept: –5

2. x < 1.667, x > 1.667; maximum = 9.333; x-intercepts: –0.097 and 3.431; y-intercept: 1

3. x > –1, x < –1; minimum = 6; no x-intercepts; y-intercept: 114. x > 1, x < 1; minimum = –13; x-intercepts: –1.55 and 3.55;

y-intercept: –115. x = –0.67; x > –0.67, x < –0.67 6. x = 1.125; x < 1.125, x > 1.125 7. x = 0.375; x > 0.375, x < 0.375 8. x = 0.15; x > 0.15, x < 0.159. The height of the softball increases until it reaches a

distance of 25 feet, and then it decreases.10. According to the model, the maximum temperature is

about 62.5°F. The actual temperature could be higher.

Practice 2.5 B: Interpreting Key Features of Quadratic Functions, p. U2-120

1. x > 1.5, x < 1.5; minimum = –8.25; x-intercepts: –1.37 and 4.37; y-intercept: –6

2. x < –2, x > –2; maximum = 11; x-intercepts: –5.32 and 1.32; y-intercept: 7

3. x < 1, x > 1; maximum = 16; x-intercepts: –1 and 3; y-intercept: 12

4. x > 0, x < 0; minimum = 0; x-intercept: 0; y-intercept: 05. x = –16.3; x > –16.3, x < –16.36. x = 2.25; x > 2.25, x < 2.25 7. x = –0.5; x < –0.5, x > –0.58. x = –0.02; x > –0.02, x < –0.029. The height of the football is increasing for the first second

and then it is decreasing.10. The flare will explode at 256 feet.

Lesson 2.5: Interpreting Key Features of Quadratic Functions (F–IF.4★Lesson 2.5: Interpreting Key Features of

★Lesson 2.5: Interpreting Key Features of

)

Warm-Up 2.5, p. U2-931. The x-intercepts are 0 and 8.2. The equation for the axis of symmetry is x = 4. It is midway

between the x-intercepts.3. The vertex is (4, 6).4. Yes, it will. Sketch the boulder’s path using (0, 0), (4, 6),

and (8, 0); the path is above the obstacle.5. No, it will not. If you substitute x = 7 into the equation of

the parabola, you find that the y-value of the boulder is slightly above the boxes.

Scaffolded Practice 2.5: Interpreting Key Features of Quadratic Functions, p. U2-102

1. increasing: (1 < x < ∞); decreasing: (–∞ < x < 1); minimum: x = 1; x-intercepts: –0.58, 2.58; neither even nor odd

2. increasing: (–∞ < x < 0.75); decreasing: (0.75 < x < ∞); maximum: x = 0.75; no x-intercepts; neither even nor odd



5. The function has a maximum at (–1, 18).6. x-intercepts: (–3, 0) and (3, 0)7. vertex: (40, 10); the point at which the baseball “breaks” or

descends sharply, which is when the ball is 40 feet from the pitcher (toward home plate) and 10 feet above the ground.

6

2

4

14

10

8

12

18

16

22

20

0

y

x

10 20 30 40 50 60 70 80 90 100 110 120

8. The function has a minimum at (–3, –6).9. x-intercepts: (–1, 0) and (18, 0); (–1, 0) is irrelevant to

the context, but (18, 0) represents the point at which the volleyball hits the farthest corner of the opposing court, horizontally 18 meters from the server.

12 14 16 18

4

2

1

3

8

6

5

7

10

9

14

12

11

13

18

16

15

17

19

0 11 13 15 17 19

y

x

9 1087654321

10. x-intercepts: (6, 0) and (8, 0)

Practice 2.6 A: Interpreting Various Forms of Quadratic Functions, p. U2-149

1. a. 8 b. (3, –1) c. minimum

2. a. –2 and 4 b. 4 c. x = 1 d. (1, 4.5)

Lesson 2.6: Interpreting Various Forms of Quadratic Functions (F–IF.8)

Warm-Up 2.6, p. U2-1221. (4, 5)2. x = 93. y = 54.

16

14

12

10

8

6

4

2

0

2

4

6

8

10

12

14

10 5 5 10

(4, 5) (6, 5)

y

x

5. No, the water will not clear the boxes. If the boxes were not in the way, the water would still not fill the birdbath because the birdbath is too tall.

Scaffolded Practice 2.6: Interpreting Various Forms of Quadratic Functions, p. U2-127

1. y-intercept: ; vertex: ; minimum

2. x-intercepts: and ; y-intercept: ;

axis of symmetry: ; vertex:

3. vertex: (–3, 5); maximum4. vertex: (15, 25); The minimum height at which the pilot

flies and the point transitioning to re-ascension, which is when the pilot flies 25 meters above the ground.

–150

–100

–50

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

0

y

x

5–5 10 15 20 25 30 35



6. –1 and 2; the x-intercepts represent the times when the snowboarder is on the ground. The x-intercept –1 does not make sense because time cannot be negative; 2 seconds.

0 1 2 3 4

5

10

15

20

25

30

35

7. (3, 25); the paper airplane will reach a maximum height of 25 feet after traveling 3 feet in the horizontal direction.

0 1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

3. a. (1, 10) b. maximum

4. (7, –9); after the bird travels for 7 seconds, it reaches its minimum height, which is 9 feet below the surface of the water.

-5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

5. –2 and 40; the x-intercepts represent the times when the height is 0 miles, which is when the missile is on the ground. The x-intercept of –2 does not make sense in the problem because time cannot be negative; 38 seconds.

0 5 10 15 20 25 30 35 40 45

50

100

150

200

250

300

350

400

450

500



Practice 2.6 B: Interpreting Various Forms of Quadratic Functions, p. U2-154

1. a. 12 b. (4, –4) c. minimum

2. a. 3 and –5 b. 30 c. x = –1 d. (–1, 32)

3. a. (3, 0) b. maximum

4. (5, 1); the butterfly reaches its minimum height of 1 foot above the ground 5 seconds after you see it.

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10-1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

5. (1, 52); the cliff diver reaches a maximum height of 52 feet 1 second after starting the dive; 2 seconds

0 1 2 3 4 5 6 7 8 9 10

-5

5

10

15

20

25

30

35

40

45

50

55

8. (1, 16); the golf ball will reach a maximum height of 16 feet 1 second after being hit.

0 1 2 3 4 5 6

2

4

6

8

10

12

14

16

18

20

9. –5 and 15; the x-intercepts represent the number of dollars in price increase/decrease that would result in no revenue; 5.

– 5 50 10 15

500

450

550

400

350

300

250

200

150

100

50

– 50

10. (7, 289): the maximum revenue of $289 will occur when the price is reduced by $7.

0 5 10 15 20 25

50

100

150

200

250

300

350



8. (2, 4); the frog reaches its maximum height of 4 feet after 2 seconds; x = 2; 4 seconds

-1 0 1 2 3 4 5 6

-1

1

2

3

4

5

9. (5, 200); the revenue reaches its maximum value of $200 when the price is increased by $5; x = 5; $10

0 2 4 6 8 10 12 14 16 18 20

-20

20

40

60

80

100

120

140

160

180

200

220

6. 2 and 50; the x-intercepts represent the number of widgets sold that result in no revenue; 26 widgets

-5 0 5 10 15 20 25 30 35 40 45 50 55

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

4,500

5,000

5,500

7. (20, 12); the football reaches a maximum height of 12 feet after it has traveled 20 feet in the horizontal direction; 40 feet

-5 0 5 10 15 20 25 30 35 40

-2

2

4

6

8

10

12



10. –10 and 26; the x-intercepts represent price decreases that result in no revenue. The x-intercept –10 does not make sense in the context of the problem because a price decrease should not be negative; x = 8; $16

0 5 10 15 20 25 30

-50

50

100

150

200

250

300

350

10. Type A: (14, 0)Type B: (16, 0)Type C: (12, 0)

Prac tice 2.7 B: Comparing Properties of Quadratic Functions Given in Different Forms, p. U2-177

1. the function with no x-intercepts2. the function with a > 03. the function with x-intercepts p and –3p and –3p and –34. No; the functions do not cross anywhere but the origin.5. Model B has the shortest stopping distance at x = 25. It

will always have the shortest stopping distance because the functions for the stopping distances do not cross anywhere but the origin.

6. Model C has the longest stopping distance at x = 55. It will always have the longest stopping distance because the functions for the stopping distances do not cross anywhere but the origin.

7. Type B8. Type C9. Type A: (3, 0)

Type B: (5, 0)Type C: (0, 0)

10. Type A: (13, 0)Type B: (11, 0)Type C: (16, 0)

Lesson 2.7: Comparing Properties of Quadratic Functions Given in Different Forms (F–IF.9)

Wa rm-Up 2.7, p. U2-1591. the first bird2. the second bird3. the second bird

Pra ctice 2.7 A: Comparing Properties of Quadratic Functions Given in Different Forms, p. U2-174

1. the function with a > 02. the function with no x-intercepts3. the function with x-intercepts p and –p and –p and –4. No; the functions do not cross anywhere but the origin.5. Model A has the shortest stopping distance at x = 25. It

will always have the shortest stopping distance because the functions for the stopping distances do not cross anywhere but the origin.

6. Model B has the longest stopping distance at x = 55. It will always have the longest stopping distance because the functions for the stopping distances do not cross anywhere but the origin.

7. Type C8. Type B9. Type A: (2, 0)

Type B: (0, 0)Type C: (4, 0)

Lesson 2.8: Replacing f (f (f x) with x) with x f) with f) with (f (f x) + x) + x k and f (f (f x + x + x k) (F–BF.3)

Warm-Up 2.8, p. U2-1801. f(f(f x(x( ) = 5x – 102.

0 20 40 60 80 100 120 140

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

Number of tags sold

y

x

Pro�

t in

dolla

rs ($

)

3. The profit would increase by $2. This would affect the y-intercept by changing it from –10 to –8. The graph would move up 2 units.



8.

2 4 6 8 10

10

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

1 3 5 7 9

y

x

–10

–1–2–3–4–5–6–7–8–9–10

9. f(f(f x(x( ) = –(x= –(x= –( + 4)2 + 4 10. The ball will hit the ground 48.87 feet away;

f(f(f x(x( ) = –0.03x2 + 1.2x + 13

Practice 2.8 A: Replacing f (f (f x) with x) with x f) with f) with (f (f x) + x) + x k and f (f (f x + x + x k), p. U2-204

1. y = (x = (x = ( + 2)2

2. y = x2 + 33. y = (x = (x = ( – 5)2 – 2 4. (2, 0)

0 1 2 3 4 5

1

2

3

4

5

Scaffolded Practice 2.8: Replacing f (f (f x) with x) with xf (f (f x) + x) + x k and f and f and (f (f x + x + x k), p. U2-185

1. f(f(f x(x( ) = (x= (x= ( + 3)2

2. f(f(f x(x( ) = (x= (x= ( – 2)2

3. f(f(f x(x( ) = x2 + 34. f(f(f x(x( ) = (x= (x= ( – 6)2 – 25. f(f(f x(x( ) = (x= (x= ( + 4)2 + 3 6.

2 4 6 8 10

10

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

1 3 5 7 9

y

x

–10

–1–2–3–4–5–6–7–8–9–10

7.

2 4 6 8 10

10

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

1 3 5 7 9

y

x

–10

–1–2–3–4–5–6–7–8–9–10



Practice 2.8 B: Replacing f (f (f x) with x) with x f) with f) with (f (f x) + x) + x k and f (f (f x + x + x k), p. U2-207

1. y = (x = (x = ( – 3)2

2. y = x2 – 4 3. y = (x = (x = ( + 6)2 – 14.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1

–3

–2

–1

1

2

3

4

5

6

7

8

9

10

5.

–5 –4 –3 –2 –1 0 1 2 3 4 5

–3

–2

–1

1

2

3

4

5

6

7

8

9

10

5. (0, –4)

–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

6. (5, –2)

0 1 2 3 4 5 6 7 8 9

–3

–2

–1

1

2

3

4

5

6

7

8

9

10

7. y = –(x = –(x = –( + 4)2; k = 48. f(f(f x(x( ) = –0.0009(x) = –0.0009(x) = –0.0009( – 18)2 + 0.2088(x + 0.2088(x + 0.2088( – 18)9. f(f(f x(x( ) = –0.03x2x2x + 1.3x + 16; the ball will land about 53 feet away.x + 16; the ball will land about 53 feet away.x

10. f x x( ) ( )= − − +1

813 102 ; yes, it will land in the basket.



9.

10. The ball would go farther following , which would

also preserve the maximum height attained by the ball.

Practice 2.9 A: Replacing f (f (f x) with x) with x k • f (k • f (k • f x)x)x andf (f (f k • x), p. U2-238k • x), p. U2-238k • x

1. 2 • f(f(f x(x( ) = 2x2 + 2x – 12. Since the entire function is being multiplied by 2, the x-intercepts of the two functions are the same. However, the second parabola, 2 • f(f(f x(x( ), is stretched vertically because each y-coordinate has been multiplied by 2. As a result, the new parabola is narrower than the original parabola.

–5 –4 –3 –2 –1 0 1 2 3 4 5

–14–13–12–11–10

–9–8–7–6–5–4–3–2–1

123456789

10

2 • f(x)

f(x)

2. f(3f(3f x) = 9x2x2x + 3x. Since the variable x is multiplied by 3, the x is multiplied by 3, the x

graph is compressed horizontally by a factor of 1

3. That is,

each x-value is multiplied by 1

3, so the width of the parabola

becomes narrower. The y-intercept remains the same, as do

the minimum values of the functions. The x-intercepts change

and the interval between them becomes narrower.

–5 –4 –3 –2 –1 0 1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

f(3x)f(x)

Lesson 2.9: Replacing f (f (f x) with x) with x k • f (k • f (k • f x)x)x andf (f (f k • x) (F–BF.3)k • x) (F–BF.3)k • x

Warm-Up 2.9, p. U2-2101. 576 ft2

2. 1,728 ft2

3. 1,936 ft2

Scaffolded Practice 2.9: Replacing f (f (f x) with x) with x k • f (k • f (k • f x)x)xand fand fand ( f ( f k • x), p. U2-218k • x), p. U2-218k • x

1. The function is vertically stretched by a factor of 3.2. The function is horizontally compressed by a factor of 5.

3.

4.

5. ; vertical compression and reflection over the x-axis

6. ; horizontal stretching and reflection over the y-axis

7. ; vertical compression

8. Increasing the size of one room would transform into while adding another room would transform into . This means that doubling the lengths of the original room will add more square footage.

6.

–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

7. y = –(x = –(x = –( + 2)2 + 5. For the horizontal translation, k = 2. For the vertical translation, k = 5.

8. f(f(f x(x( ) = –0.0008(x) = –0.0008(x) = –0.0008( – 12)2 + 0.24(x + 0.24(x + 0.24( – 12)9. f(f(f x(x( ) = –0.05x2 + x + 19; the paper wad will land about 32 feet

away.10. f(f(f x(x( ) = –0.4 (x) = –0.4 (x) = –0.4 ( – 9)2 + 14; yes, it will hit the target.



7. Graph of f(f(f x(x( ) and g(x(x( ):

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5

–4

–3

–2

–1

1

2

3

4

5

6

7

8

9

10

f(x)

g(x)

Since g(x(x( ) is a multiple of f(f(f x(x( ), it will have the same

x-intercepts as f(f(f x(x( ): –1 and 1. Since the value of k is −1

2, the

graph will be compressed vertically by a factor of 1

2, and it

will reflect over the x-axis because k < 0.

8. If the sides of the original pen were doubled, then the side lengths would be 2x and 4x. The area would be f(2f(2f x) = 2(2x)2 = 8x2. If the farmer were to build a second pen of the same size, the area of the two pens would be 2 • f(f(f x(x( ) = 2(2x2) = 4x2. Doubling the side length gives a larger area.

9. f(f(f x(x( ) represents the profit, so a doubling of the profit is modeled by 2 • f(f(f x(x( ).

10. Neither player is correct because neither of the equations leads to a ball going farther than the original ball. Jada’s equation, f(2f(2f x) = –0.01(2x)2 + 0.98(2x) + 2, leads to a ball going the same height as the original ball but only half as far; Jayla’s equation, 2 • f(f(f x(x( ) = 2(–0.01x2 + 0.98x + 2), leads to a ball going the same distance as the original ball but twice as high.

0 10 20 30 40 50 60 70 80 90 100

5

10

15

20

25

30

35

40

45

50

f(x)

2 • f(x)

f(2x)

3. Answers may vary. Apply the transformation k • f(f(f x(x( ), where 0 < k < 1. Sample answer: 1/2 • f(f(f x(x( ) = g(x(x( ) = 0.5x2 – 0.5x – 1

4. Answers may vary. Apply the transformation f(f(f k • x), where k > 1. Sample answer: f(2f(2f x) = g(x(x( ) = 4x2 + 6x – 4


–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

g(x)

f(x)

Since g(x(x( ) is a multiple of f(f(f x(x( ), it will have the same x-intercepts as f(f(f x(x( ): –1 and 2. Since the value of k is –2, the graph will be stretched vertically by a factor of 2, and it will reflect over the x-axis because k < 0.


–5 –4 –3 –2 –1 0 1 2 3 4 5

–5

–4

–3

–2

–1

1

2

3

4

5

6

7

8

9

10

f(x) g(x)

The function g(x(x( ) is obtained by multiplying the x-values in f(f(f x(x( ) by –2. This means that the y-intercept will be the same, the minimum y-value will be the same, but the x-intercept will change and the interval becomes narrower. The graph of g(x(x( ) will be compressed horizontally by a scale factor of 1/2, and the graph will be reflected over the y-axis.



3. Answers may vary. Apply the transformation k • f(f(f x(x( ), where k > 1. Sample answer: 2 • f(f(f x(x( ) = g(x(x( ) = 2x2 – 8x + 6

4. Answers may vary. Apply the transformation f(f(f k • x), where k > 1. Sample answer: f(2f(2f x) = g(x(x( ) = 4x2 – 2x – 1


–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

f(x)

g(x)

Since g(x(x( ) is a multiple of f(f(f x(x( ), it will have the same x-intercepts as f(f(f x(x( ): –3 and 1. Since the value of k is –2, the graph will be stretched vertically by a factor of 2, and it will reflect over the x-axis because k < 0.


–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

f(x) g(x)

The function g(x(x( ) is obtained by multiplying the x-values in f(f(f x(x( ) by –2. This means that the y-intercept will be the same, the minimum y-value will be the same, the x-intercepts will change and the interval between them will become narrower. The graph of g(x(x( ) will be compressed horizontally by a factor of 1/2, and the graph will be reflected over the y-axis.

Practice 2.9 B: Replacing f (f (f x) with x) with x k • f (k • f (k • f x)x)x andf (f (f k • x), p. U2-241k • x), p. U2-241k • x

1. 3 • f(f(f x(x( ) = 3x2 – 9x – 12. Since the entire function is being multiplied by 3, the x-intercepts of the two functions are the same (–1 and 4). However, the second parabola, 3 • f(f(f x(x( ), is stretched vertically because each y-coordinate is multiplied by 3. As a result, the new parabola is narrower than the original parabola.

–5 –4 –3 –2 –1 0 1 2 3 4 5

–20

–18

–16

–14

–12

–10

–8

–6

–4

–2

2

4

6

8

10

3 • f(x)

f(x)

2. f(2f(2f x) = 4x2 – 6x + 2. Since the variable x is multiplied by

2, the graph shrinks horizontally by a factor of 1

2. That

is, each x-value is multiplied by 1

2, so the width of the

parabola becomes narrower. The y-intercepts remain the

same, as do the minimum values of the functions. The

x-intercepts change and the interval becomes narrower.

–5 –4 –3 –2 –1 0 1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

f(x)f(2x)




–5 –4 –3 –2 –1 0 1 2 3 4 5

–10–9–8–7–6–5–4–3–2–1

123456789

10

f(x)

g(x)

Since g(x(x( ) is a multiple of f(f(f x(x( ), it will have the same

x-intercepts as f(f(f x(x( ): –2 and 2. Since the value of k is −1

2, the

graph will be condensed vertically by a factor of 1

2, and it

will reflect over the x-axis because k < 0.

8. The original closet has side lengths of x and 3x. Let the area of the original closet be f(f(f x(x( ) = 3x2. If the sides of the original closet were tripled, then the side lengths would be 3x and 9x. The area would be f(3f(3f x) = 3(3x)2 = 27x2. If the celebrity builds two additional closets of the same size, the area of the three closets would be 3 • f(f(f x(x( ) = 3(3x2) = 9x2. Tripling the side length gives a larger area.

9. f(f(f x(x( ) represents the profit, so earning half of the profit is

modeled by 1

2 • f(f(f x(x( ).

10. Zion is correct because his equation, f x1

2

, leads to a

beanbag going twice as far as the original beanbag. This

beanbag reaches the same height as the original beanbag.

Zavier’s equation, 2 • f(f(f x(x( ), leads to a beanbag going the

same distance as the original beanbag but twice as high.

0 40 80 120 160 200 240 280 320 360 400–10

10

20

30

40

50

60

70

80

90

100

110

2 • f(x)

f(x)f( x)1

2

Lesson 2.10: Interpreting Quadratic Functions (F–IF.4★)

Warm-Up 2.10, p. U2-2441. y = –80x + 500, where x is the number of oil changes in

1 year and y is the total amount spent on car repairs in 1 year.

2. The y-intercept is 500. This is the amount spent on car repairs if the owner does not get any oil changes over the course of a year.

3. The rate of change is the slope of the function, –80. For every oil change performed, he will save $80 on car repairs.

Scaffolded Practice 2.10: Interpreting Quadratic Functions, p. U2-252

1.

2 4 6 8 10

10

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

1 3 5 7 9

y

x

–10

–1–2–3–4–5–6–7–8–9–10

2.

2 4 6 8 10

3

–9

–11

–13

–15–14

–16–17

–10

–12

–8–7

1

–1

–3

–5

–2

–4

–6

0

2

1 3 5 7 9

y

x

–1–2–3–4–5–6–7–8–9–10



3.

–3 –1–5 3 5

70

10

20

–10

–20

–25

5

–5

–15

15

60

50

40

30

55

45

35

25

0

65

–4 –2 1 2 4

y

x

–30

–6–7–8–9–10–11–12–13–14–15

4.

2–1 8 10

15

3

5

–2

–4

–5

2

1

–1

–3

4

13

11

9

7

12

10

8

6

0

14

1 3 6 754 9

y

x

–2–3–4–5–6–7–8–9–10

5. y-intercept: 0; vertex: (–2, –4); minimum6. y-intercept: 3; vertex: (1, 2); minimum7. y-intercept: –4; vertex: (–2, 0); maximum8. y-intercept: –1; vertex: (0, –1); maximum9. No. Reasons may vary; the y-intercepts are different, they

have different x2 coefficients, etc.10. f(f(f x(x( ) = –x) = –x) = – 2 – 4x + 12

Practice 2.10 A: Interpreting Quadratic Functions, p. U2-278

1.

x

y

–7 –6 –5 –4 –3 –2 –1 0

–8

–6

–4

–2

2. xy

0 2 4 6 8

–40

–30

–20

–10



2.

x

y

–4 –2 0 2 4

–1

1

2

3

4

5

3.

x

y

–4 –2 0 2 4

2

4

6

4. y-intercept: 0; vertex: (–1.75, 6.125); the vertex is a maximum because a < 0.

5. y-intercept: 7; vertex: (–5, –93); the vertex is a minimum because a > 0.

6. y-intercept: 3; vertex: (1, 5); the vertex is a maximum because a < 0.

7. No. The y-intercept is –5 because the constant term is –5. However, the graph shows a y-intercept of –10.

8. y = –3x2 + 30x – 699. 4 feet

10. 5,000 pairs of socks

3.

x

y

0 1 2 3 4 5 6

2

4

6

8

10

4. y-intercept: 3; vertex: (2, –1); the vertex is a minimum because a > 0.

5. y-intercept: 0; vertex: (3, –9); it has a minimum because a > 0.

6. y-intercept: –6; vertex: (–1, 1); it has a maximum because a < 0.

7. Yes, the y-intercept is –28 and the graph opens up because a > 0.

8. y = –x = –x = – 2 – 6x – 2 9. 4 feet

10. $36.67

Practice 2.10 B: Interpreting Quadratic Functions, p. U2-280

1.

x

y

–4 –2 0 2 4

2

4

6



Lesson 2.11: Quadratic Regression (A–CED.2★)

Warm-Up 2.11, p. U2-2821. y = 1242.1x + 18154.82. 42,996.8 thousand tons3. Answers may vary. However, graphing the new point, the

idea is to see that the function becomes quadratic in nature and that a quadratic regression might best fit the data.

Scaffolded Practice 2.11: Quadratic Regression, p. U2-287

1. f(f(f x(x( ) = x2 – 4x + 52. f(f(f x(x( ) = 3x2 – 9 3. f(f(f x(x( ) = x2 – 3 4. f(f(f x(x( ) = 2x2 + 3x – 5 5. h(x(x( ) = –x) = –x) = – 2 + 10x – 29 6. f(f(f x(x( ) = x2 – 6x + 9; (3, 0)

7. ( )1

242= − + +g x x x ; (1, 4.5)

8. ( )1

42 32= + +f x x x ; (–4, –1)

9. f(f(f x(x( ) = –x) = –x) = – 2 – 2x + 3; (–1, 4)10. f(f(f x(x( ) = x2 – 2x – 8; (1, –9)

Practice 2.11 A: Quadratic Regression, p. U2-3011. y x x16 1652= − +2. h t t t( ) 16 60 42= − + +3. 4 feet4. About 3.8 seconds5. h t t t( ) 0.125 1.84 62= − + +6. About 7.4 seconds and 12.8 feet high7. About 17 seconds8. h t t t( ) 16 33 462= − + +9. 46 feet tall

10. About 3 seconds

Practice 2.11 B: Quadratic Regression, p. U2-3031. y x x2 32 1232= − +2. h t t t( ) 0.03 1.62= − +3. 0 meters4. About 21 meters5. y x x16 96 1122= − + +6. 256 feet7. 7 seconds8. y x x0.5 5.52= − + +9. 6 feet

10. About 4.5 seconds


Station Activities Set 1: Graphing Quadratic EquationsStation Activities Set 1: Graphing Quadratic Equations

Instruction


Station Activities Set 1: Graphing Quadratic Equations

Goal: To provide opportunities for students to develop concepts and skills related to graphing quadratic equations and functions

Station 1

Students will be given graph paper and a ruler. Students will derive how to find the vertex of the graph of a quadratic function. Then they will find the x-intercept and y-intercept of the function. They will graph the quadratic function and describe its shape.

Answers

1. f x x x( ) = + +2 6 9

2. a = 1; b = 6; and c = 9c = 9c

3. −b

a2

4. fba

−

2

5. − −

ba

fba2 2

, ; (–3, 0)

6. Use the vertex as the axis of symmetry and a table of values to determine the corresponding y-value when x = 0.

7. Set x = 0 and solve for y.

8. (–3, 0) and (0, 9)

Student Activities Overview and Answer Key





Instruction



9.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

8

7

6

5

0

10. Parabola; the square term gives twoParabola; the square term gives two x-values for each yy-value due to the nature of square roots.-value due to the nature of square roots.

Station 2

Students will be given a graphing calculator. Students will use the graphing calculator to graph two quadratic functions. They will describe and analyze the characteristics of both graphs, including the vertex.

Answers

1. parabola

2. upward

3. x2, because it has a positive coefficient

4. table of (xtable of (xtable of ( , y) values that satisfy the function

5. Find the y-value where x = 0. This point is the vertex; (0, 4)

6. parabola

7. upward

8. x2, because it has a positive coefficient

9. table of (xtable of (xtable of ( , y) values for both functions

10. Find the y-value where x = 0. This point is the vertex; (0, –4)

11. The value of the constant determines the vertex of the graph.



Instruction



Station 4

Students will be given graph paper and a ruler. Students will analyze and graph quadratic equations using vertices, x-intercepts, and a table of values. They will determine why certain parabolas open upward while others open downward.

Answers

1. f x x x

a

b

c

( ) = − −== −= −

2 6

1

1

6

and f x x x

a

b

c

( ) = − + −= −== −

2 6

1

1

6

2. For f(f(f x(x( ) = x2 – x – 6, the vertex is (1/2, –25/4).

For f(f(f x(x( ) = –x) = –x) = – 2 + x – 6, the vertex is (1/2, –23/4).

3. (3, 0), (–2, 0)

Station 3

Students will be given a graphing calculator. Students will use the graphing calculator to graph three quadratic functions. The quadratic functions have the same vertex, but varying widths. Students will describe the relationships between the widths of the parabolas.

Answers

1. vertex at (0, 0) because there is no constant

2. y = x2; because the coefficient is 1 versus 3. A smaller coefficient yields a wider parabola.

3. table of x- table of x- table of and y-values for both graphs

4. Y2 = 3Y1; it shows that the y-value for each x-value of Y2 is three times as large as Y1.

5. because it has a smaller coefficient

6. table of x- table of x- table of and y-values for all three graphs

7. Y3 = 1/2(Y1); it shows that the y-value for each x-value of Y3 is half the size of Y1.

8. Y3 = 1/6(Y2); it shows that the y-value for each x-value of Y3 is one-sixth the size of Y2.



Instruction



4.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x3

6

5. upward, because the coefficient of x2 is positive

6. Table:

x y = f= f= (f(f x)x)x–4 –260 –64 –18

Graph:

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

–11

–12

–13

–14

–15

y

x

–7

7. downward, because the coefficient of x2 is negative

8. no, because the parabola opens downward and is below the no, because the parabola opens downward and is below the x-axis



Instruction



Materials List/SetupStation 1 graph paper; ruler

Station 2 graphing calculator

Station 3 graphing calculator

Station 4 graph paper; ruler



Instruction



Discussion GuideTo support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to “debrief” the station activities.

Prompts/Questions

1. What does the graph of a quadratic equation look like?

2. Why does the graph have this shape?

3. How do you find the vertex of a quadratic equation?

4. Does a quadratic equation with a term of x2 open upward or downward? Why?

5. Does a quadratic equation with a term of –xDoes a quadratic equation with a term of –xDoes a quadratic equation with a term of – 2 open upward or downward? Why?

6. How can you find the x-intercepts of a quadratic equation?

Think, Pair, Share

Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class.

Suggested Appropriate Responses

1. The graph of a quadratic equation looks like a parabola.

2. There are two x-values for each y-value because of the x2 term.

3. vertex = − −

ba

fba2 2

,

4. upward, because the coefficient is positive

5. downward, because the coefficient is negative

6. Use the vertex as the axis of symmetry and a table of values to determine the corresponding y-value when x = 0.

Possible Misunderstandings/Mistakes

• Not realizing that graphs of quadratic equations are parabolas

• Not using enough data points to construct a parabola for the graph of the quadratic equation

• Ignoring the sign of the coefficient of the x2 term when constructing a parabola that opens upward or downward





Station 1At this station, you will find graph paper and a ruler. Work together to graph the following quadratic equation:

y x x= + +2 6 9

1. Write this quadratic equation as a quadratic function.

2. What are the values of a, b, and c in the quadratic function?c in the quadratic function?c

a = ______

b = ______

c = ______c = ______c

To graph the function, you need the vertex, x-intercept, and y-intercept.

3. If the x-value of the vertex is found by x =−

= −6

2 13

( ), then write this x calculation using the

general terms a, b, and/or c.

4. If the y-value of the vertex is found by y f f=−

= − =

62 1

3 0( )

( ) , then write this

y calculation using the general terms a, b, and/or c.

5. Based on problems 3 and 4, how can you find the vertex of the graph for f(f(f x(x( ) = ax2 + bx + c?

What is the vertex of the quadratic function x x2 6 9 0+ + = ? continued



Name: Date:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–IF.7★


6. How do you find the x-intercept of a function? (Hint-intercept of a function? (Hint-intercept of a function? ( : y = f(f(f x(x( ))

7. How do you find the y-intercept of a function?

8. What are the intercepts for y x x= + +2 6 9 ?

9. On your graph paper, graph the function using the vertex, x-intercept, and y-intercept.

10. What shape is the graph? Why do you think the graph has this shape?





Station 2At this station, you will find a graphing calculator. As a group, follow the steps according to your calculator model to graph y x= +2 4 and y x= −2 4 .

On a TI-83/84:

Step 1: Press [Y=]. At Y1, type [X,T,θ,n][x2][+][4].


On a TI-Nspire:

Step 1: Arrow over to the graphing icon and press [enter]. At f1(x(x( ), enter [x], hit the [x2] key, then type [+][4].


1. What shape is the graph?

2. Does the graph open upward or downward?

3. Which term do you think makes the graph open upward or downward? Explain your reasoning.

On a TI-83/84:

Step 3: Press [2ND], then [GRAPH].

On a TI-Nspire:

Step 3: Press [ctrl], then [T].

4. What information does your calculator show?

5. How can you use this information to find the vertex of the graph?

What is the vertex of the graph? continued





On a TI-83/84:

Step 4: Press [Y=]. At Y2, type [X,T,θ,n][x2][–][4].


On a TI-Nspire:

Step 4: Press [ctrl][tab] to go back to the graphing window. Use the touch pad to select “>>” on the bottom left of the screen. At f2(x(x( ), enter [x], hit the [x2] key, then type [–][4].


6. What shape is the graph?

7. Does the graph open upward or downward?

8. Which term do you think makes the graph open upward or downward? Explain your reasoning.

On a TI-83/84:


On a TI-Nspire:

Step 6: Press [ctrl], then [T]. Press [ctrl], then [T] a second time to refresh the screen.


10. How can you use this information to find the vertex of the graph of y x= −2 4 ?

What is the vertex of y x= −2 4 ?

11. Why do the graphs for Why do the graphs for y x= +2 4 and y x= −2 4 have different vertices?





Station 3At this station, you will find a graphing calculator. As a group, follow the steps according to your

calculator model to graph y x= 2 , y x= 3 2 , and y x=12

2 .

On a TI-83/84:

Step 1: Press [Y=]. At Y1, type [X,T,θ,n][x2]. At Y2, type [3][X,T,θ,n][x2].


On a TI-Nspire:

Step 1: Arrow over to the graphing icon and press [enter]. At f1(x(x( ), enter [x], then hit the [x2] key. Arrow down. At f2(x(x( ), enter [3][x], then hit the [x2] key.


1. Why do both graphs have the same vertex?

2. Which graph is wider, y x= 2 or y x= 3 2 ?

Why is one graph wider than the other?

On a TI-83/84:


On a TI-Nspire:

Step 3: Press [ctrl], then [T].


4. What is the relationship between Y1 and Y2 in the table?

How does this relationship relate to y x= 2 and y x= 3 2 ? continued





On a TI-83/84:

Step 4: Press [Y=]. At Y3, type [0][.][5][X,T,θ,n][x2].


On a TI-Nspire:

Step 4: Press [ctrl][tab] to go back to the graphing window. Use the touch pad to select “>>” on the bottom left of the screen. At f3(x(x( ), enter [0][.][5][x], then hit the [x2] key.


5. Why is the graph of yWhy is the graph of yWhy is the graph of = 0.5x2 wider than y x= 2 and y x= 3 2 ?

On a TI-83/84:


On a TI-Nspire:

Step 6: Press [ctrl], then [T]. Press [ctrl], then [T] a second time to refresh the screen.



How does this relationship relate to y x= 2 and y = 0.5x2?


How does this relationship relate to y x= 3 2 and y = 0.5x2?





Station 4At this station, you will find graph paper and a ruler. Work together to graph the following quadratic equations:

f(f(f x(x( ) = x2 – x – 6 and f(f(f x(x( ) = –x) = –x) = – 2 + x – 6

1. What are the values of a, b, and c in each quadratic function?c in each quadratic function?c

f(f(f x(x( ) = x2 – x – 6 f(f(f x(x( ) = –x) = –x) = – 2 + x – 6

a = ______ a = ______

b = ______ b = ______

c = ______ c = ______ c c = ______c = ______c

2. Use the information in problem 1 to find the vertex − −

ba

fba2 2

, for each function. Show your work.

3. Find the x-intercepts of f x x x( ) = − −2 6 using factoring. Show your work.

4. On your graph paper, graph f x x x( ) = − −2 6 using its vertex and x-intercepts.

5. Does the parabola open upward or downward? Explain your answer.

continued





6. Fill out the table below to help you graph f(f(f x(x( ) = –x) = –x) = – 2 + x – 6.

x y = y = y f = f = (f(f x)x)x

–4

0

4

Graph f(f(f x(x( ) = –x) = –x) = – 2 + x – 6 on your graph paper.

7. Does the graph open upward or downward? Explain your answer.

8. Will the graph of fWill the graph of fWill the graph of (f(f x(x( ) = –x) = –x) = – 2 + x – 6 have x-intercepts? Why or why not?


Station Activities Set 2: Quadratic Transformations in Vertex FormStation Activities Set 2: Quadratic Transformations in Vertex Form

Instruction

UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–IF.7★; F–IF.8★; F–BF.3Station Activities Set 2: Quadratic Transformations in Vertex Form

Goal: To provide opportunities for students to analyze the relationship between the equation of a parabola and its graph

Student Activities Overview and Answer KeyStation 1

Given equations in the form y = x2 + k and y = (x – h(x – h( )2, where h and k are integers, students graph a series of parabolas, finding the y-intercept and the axis of symmetry. They explore the relationship between the value of h and k and the position of the parabola with respect to the x- and y-axes. Students should also begin to understand the relationship between the equation of the parabola and the axis of symmetry.

North Carolina Math 2 Standards



F–BF.3 Understand the effects of the graphical and tabular representations of a linear, quadratic, square root, and inverse variation function fwith k • f(f(f x(x( ), f(f(f x(x( ) + k, f(f(f x(x( + k) for specific values of k (both positive and negative).



Instruction


Answers

1. x yyyy0 01 12 43 9

–1 1–2 4–3 9

2.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x0

3. (0, 0)

4. (0, 2)

5. y = x2 + 2

6. y = x2 – 5



Instruction


7.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x–2

–5

8.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

5

4

9. x = 2

10. x = –3



Instruction


Station 2

Given equations in the form y = ax2, students graph parabolas. Students compare graphs to explore the relationship between the coefficient of x and the width of the parabola.

Answers

1. x y0 01 32 123 27

–1 3–2 12–3 27

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x0

2. (0, 0)

3. x = 0



Instruction


4.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x0

yy = x2y = 3x2

The parabola 3x2 is narrower than the parabola x2.

5–6.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x–1

1

–1

y = –2x2

y = –2x2 + 2

2 3 4

7. (0, 2)

8. The parabola shifts vertically.

9. The parabola changes in width.

00



Instruction


Station 3

Given equations in the form y = (x = (x = ( – h)2 + k, students graph parabolas. Students find the y-intercept and the axis of symmetry from both the graph and the equation, and begin working towards an understanding of the vertex of a parabola.

Answers

1. x y0 51 32 53 114 21

–1 11–2 21–3 35

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

7

6

5

10

2. x = 1

3. (1, 3)

4. (0, 5)

5. The parabola would open downward but have the same vertex.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x0

1

–2

–



Instruction


Station 4

Students use two methods (completing the square and finding the midpoint of the x-intercepts) to convert the equations of parabolas from quadratic form to vertex form. They graph to check their work and to understand the correlation between the different forms and the graph. Students should recognize that a parabola’s axis of symmetry always runs through its vertex. They should also understand the relationship between the coordinates of the vertex and the equation in vertex form.

6. (0, 3)

7. (2, 1)

8. It will be wider, because the higher the coefficient of the x2 term, the narrower the parabola. The coefficient of the second x2 term is 1, which is higher than 1⁄1⁄1

2⁄2⁄ , the coefficient of the first x2 term.

y = (x = (x = ( – 2)2

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

3

2

66

5

y x

y

y

y

= − +

= − +

= +

=

12

2 1

12

0 2 1

12

4 1

3

2

2

( )

( )

( )

9. (0, 17); (2, 5)

10. y = 3(x = 3(x = 3( – 2)2 + 10

The new y-y-intercept is (0, 22), so solve for the value of intercept is (0, 22), so solve for the value of a.



Instruction


Answers

1.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x–4 –1

6

5

4

2. x = –3

3.

y x x

y x x

y x x

y x

= + +

= + + + −

= + + −

= + −

2

2

2

2

6 7

6 7 2 2

6 9 2

3 2

( )

( )

4. (–3, –2)

5. (4, 1)

6.

yx

x

yx

x

y x x

y x

= − +

= − + +

= − + +

= −

24 9

24 8 1

12

8 16 1

12

4

2

2

2( )

( )22 1+

7. (0, 9)

8. x = 4

9. Yes. The parabola opens out from the vertex. The vertex contains the only y-coordinate that is not repeated in the range.

10. Because the parabola is symmetrical, the axis of symmetry will intersect the midpoint of the line between the x-intercepts. The midpoint is at (0, 0). That means the x-coordinate at the vertex must be 0, because the axis of symmetry intersects the vertex. If x = 0, y = –4, so the coordinates of the vertex are (0, –4).

Materials List/SetupStation 1 graph paper

Station 2 colored pencils or pens; graph paper

Station 3 colored pencils or pens; graph paper

Station 4 graph paper



Instruction


Discussion GuideDiscussion GuideTo support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to “debrief” the station activities.

Prompts/Questions

1. What is a function’s axis of symmetry? Does every parabola have one?

2. What is a y-intercept?

3. How do you find the coordinates of a parabola’s y-intercept?

4. Compare the equations y a x h k= − +( )2 and y ax axh ah k= − + +2 22 . Do you think they express the same thing? How could you find out?

Think, Pair, Share

Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class.

Suggested Appropriate Responses

1. An axis of symmetry is the line that divides the graph of the function into two symmetrical halves. Every parabola has an axis of symmetry.

2. A y-intercept is the point at which a function crosses the y-axis.

3. Set x equal to 0 and solve the function for y.

4. Yes, they express the same thing. Students should multiply out the equation in vertex form to find the equation in quadratic form.

Possible Misunderstandings/Mistakes

• Incorrectly calculating the value of yIncorrectly calculating the value of yIncorrectly calculating the value of -coordinates from x-coordinates

• Incorrectly graphing parabolas, either from incorrect calculations or from a misunderstanding of graphing itself

• Not understanding the definition of the vertex

• Assuming that the vertex is unrelated to the axis of symmetry

• Incorrectly factoring quadratic equations



Instruction


• Making simple arithmetical errors in completing the square

• Not understanding the arithmetical manipulations involved in completing the square

• Confusing the y-intercept with the x-intercept

• Confusing the vertex coordinates h and k

• English language learners may struggle with the questions that ask for written explanations. Encourage these students to write out the numeric operations involved and then describe their work out loud.



Date:Name:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–IF.7★; F–IF.8★; F–BF.3Station Activities Set 2: Quadratic Transformations in Vertex Form

Station 1Station 1Work as a group to answer the questions. Construct graphs without the aid of a graphing calculator. Show all your work and label the axes of each graph.

1. Given the equation y = x2, complete the table below with the y coordinates for the following values of x.

x y0123

–1–2–3

2. Use the coordinates from your table to graph the parabola on graph paper.

3. What are the coordinates for the parabola’s y-intercept?

4. Look at the parabola below. What is its y-intercept?

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

2

continued



Name: Date:UNIT 2 • GRAPHING QUADRATIC FUNCTIONS F–IF.7★; F–IF.8★; F–BF.3Station Activities Set 2: Quadratic Transformations in Vertex Form

5. What is the equation for this parabola?

6. How would you write the equation for a similar parabola of yHow would you write the equation for a similar parabola of yHow would you write the equation for a similar parabola of -intercept (0, –5)?

7. Graph the parabola from problem 6.

8. Graph the parabola y = (x (x ( – 2)2.

9. What is the equation for the axis of symmetry of this parabola?

10. Without graphing, predict the equation for the axis of symmetry of the parabola Without graphing, predict the equation for the axis of symmetry of the parabola y = y = (x(x( + 3)2.




Station 2Station 2Work with your group to explore the relationship between a quadratic function and its graph.

1. Given the equation y = 3x2, complete the table with the values of y, complete the table with the values of y, complete the table with the values of and graph the parabola.

x y0123

–1–2–3

2. What are the coordinates of this parabola’s y-intercept?

3. What is the equation of its axis of symmetry?

4. On the graph from problem 1, draw the parabola y = x2 in a contrasting color. In words, compare the two parabolas.

5. Graph the parabola y = –2x2. Complete the table if you need a reference.

x y0123

–1–2–3 continued




6. On the same graph, in a contrasting color, graph the parabola y = –2x2 + 2. Label each parabola.

7. What are the coordinates of the y-intercept of y = -intercept of y = -intercept of –2x2 + 2?

8. What happens to the graph of a parabola when you add a constant to its equation, as in problem 6?

9. What happens to the graph of a parabola when the x2 expression is given a coefficient other than 1, as in problems 1 and 5? (than 1, as in problems 1 and 5? (Hintthan 1, as in problems 1 and 5? (Hintthan 1, as in problems 1 and 5? ( : Compare the parabola : Compare the parabola y = y = 3x2 to y = xy = x2.).)




Station 3Station 3Work with your group to answer the following questions.

1. Complete the table for the parabola y = 2(x2(x2( – 1)2 + 3. Graph the parabola on graph paper.

x y01234

–1–2–3

2. What is the equation for this parabola’s axis of symmetry?

3. What is the vertex?

4. What are the coordinates of this parabola’s y-intercept?

5. How would this graph change if the parabola’s equation changed to y = –2(x–2(x–2( – 1)2 + 3? Graph the new parabola to check your answer.

6. What are the coordinates of the y-intercept of the parabola y x= − +12

2 12( ) ?

continued




7. What is the vertex?

8. Do you think that the graph of y x= −12

2 2( ) will be wider or narrower than the graph

of y x= −( )2 2 ? Why? Graph both parabolas, in contrasting colors, to check your answer.

9. Look at the graph below. The equation for this parabola is y = 3(x 3(x 3( – 2)2 + 5. What is the vertex? What is the parabola’s y-intercept?

–5–5 –4–4 –3–3 –2–2 –1–1 11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515

353534343333323231313030292928282727272726262626252525252424242423232323222221212020191918181717161615151414131312121010

998877665544332211000

–1–1–2–2–3–3–4–4–5–5

y

x

302929282827272626252524242323222221212020

10. How would you write the equation for a similar parabola with a y-intercept 5 units higher? Show your work. Write out an explanation in words if necessary.Show your work. Write out an explanation in words if necessary.




Station 4Station 4Work with your group to answer the following questions.

1. Graph the parabola y x x= + +2 6 7 on graph paper.

2. Give the equation for its axis of symmetry.

3. Optional: Complete the square to give the equation for the parabola in vertex form. Show your work.

4. What are the coordinates of the vertex of this parabola?

5. Look at the graph below. What are the coordinates of the vertex of this parabola?

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

10

9

1010

continued




6. The equation for this parabola is yx

x= − +2

24 9 . Use the coordinates found in problem 5 to

convert the equation to vertex form. Show your work.

7. What are the coordinates of the y-intercept?

8. What is the equation for the axis of symmetry?

9. Does a parabola’s axis of symmetry always run through its vertex? Why or why not?

10. Look at the graph below, which shows the parabola y = x2 – 4. The coordinates of the parabola’s x-intercepts are (2, 0) and (–2, 0). How could you use this information to find the coordinates of the parabola’s vertex? Explain, showing your work.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

y

x

–4

2

North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher ResourceUnit 2 Mid-Unit Assessment

© Walch Education© Walch EducationMA-1

UNIT 2 • GRAPHING QUADRATIC FUNCTIONSMid-Unit Assessment

Assessment

Name: Date:

Unit 2 Mid-Unit Assessment

Circle the letter of the best answer.

1. The area of a circle is given by A = πr πr π 2, for which r is the radius of the circle. What is the value r is the radius of the circle. What is the value rof the coefficient?

a. 2

b. r

c. π

d. There is no coefficient.

2. Which of the following expressions is quadratic?

a. x2 – x2(x(x( + 2)

b. 5 – t

c. s(5s + 4)

d. 3 2 12x x− +

3. For the function f(f(f x(x( ) = –(x) = –(x) = –( – 2)2 + 0.25, what is the value of x that changes the function from increasing to decreasing?

a. 0.25

b. 2

c. –2

d. –0.25

4. A tourist jumps off a 16-foot cliff to swim with pink river dolphins in Bolivia. The tourist’s height in feet above the water is modeled by h(t) = –16t) = –16t t2t2t + 16, where t is the time in seconds after t is the time in seconds after tthe tourist jumps from the cliff. About how long will it take the tourist to reach the water?

a. 2 seconds

b. 0.5 second

c. 1.5 seconds

d. 1 second

5. A local company manufactures custom truck parts for big rigs. The company’s average profit can be modeled by a(p(p( ) = –3(p) = –3(p) = –3( – 20)(p – 20)(p – 20)( ), where a(p(p( ) is the average profit in dollars per p parts produced. What is a reasonable domain for this function?

a. 0 to 10 parts

b. 0 to 14 parts

c. 0 to 18 parts

d. 0 to 20 parts

continued




Assessment

Name: Date:

6. The area of a memorial park was determined by the height of a flagpole in the center of the memorial. The park is rectangular, and the length of the north side of the park is 9 times the height of the flagpole. The east side of the park is 11 times the height of the flagpole. If the height of the flagpole is h feet, what is the area of the park?

a. (h + 9)(h + 11) ft2

b. (9h + 11h) ft2

c. 99h2 ft2

d. (h2 – 99) ft2

7. A suspension bridge has two supporting towers and cables secured at either end of the span. The section of the cables between the two supporting towers forms a parabolic curve. At the lowest point, the cables are 12 feet from the surface of the bridge. The supporting towers are 600 feet apart and 412 feet tall. What is a quadratic equation that describes the curve of the cables between the towers? Use the vertex as the y-intercept.

a. f x x( )= +1

225122

b. f x x( ) ( )= −1

60012 2

c. f x x( )= +1

300122

d. f x x( ) ( )= − +1

41212 3002

8. The path of a water balloon can be described by the equation h(x(x( ) = –2x2 + 8x + 10, where x + 10, where x h(x(x( ) is the height of the water balloon in feet and x is the number of seconds that have passed since it was x is the number of seconds that have passed since it was xlaunched. Which of the following points does not make sense within the context of the problem?

a. (–1, 0)

b. (0, 10)

c. (2, 18)

d. (5, 0)

9. A missile is launched along the path determined by the equation f(f(f x(x( ) = –2x2 + 72x, where f(f(f x(x( ) is the height of the missile in feet x seconds after it has been launched. A plane is flying at a height of 650 feet. Is the plane in danger? Why or why not?

a. Yes. The value of a is negative.

b. Yes. The missile reaches a height of greater than 650 feet.

c. No. The missile does not reach a height of 650 feet.

d. No. The y-intercept is the origin.continued




Assessment

Name: Date:

Use the given information to answer the questions that follow.

10. The function f(f(f t) = –16t) = –16t t2t2t + 40t + 3 models the height t + 3 models the height t f(f(f t) in feet of a golf ball t) in feet of a golf ball t t seconds after t seconds after tbeing hit.

a. What is the height of the golf ball after 2 seconds?

b. After how many seconds does the golf ball reach its maximum height?

c. What is the maximum height of the golf ball?



Instruction


Unit 2 Mid-Unit Assessment Answer Key

Multiple Choice

Answer Standard(s)

1. c A–SSE.1a★

2. c A–SSE.1b★

3. b F–IF.4★

4. d F–IF.4★

5. d F–IF.4★

6. c F–BF.1★

7. a F–BF.1★

8. a F–IF.8

9. c F–IF.8

Extended Response

Answer Standard(s)

10. a. 19 feet

b. 1.25 seconds

c. 28 feet

F–IF.7★, F–IF.8

North Carolina Math 2 Custom Teacher ResourceNorth Carolina Math 2 Custom Teacher ResourceUnit 2 End-of-Unit Assessment

© Walch Education© Walch EducationEA-1

UNIT 2 • GRAPHING QUADRATIC FUNCTIONSEnd-of-Unit Assessment

Assessment

Name: Date:

Unit 2 End-of-Unit Assessment

Circle the letter of the best answer.

1. What is the value of b when (x when (x when ( + 14)(2x – 3) is written in the form ax2 + bx + c ?

a. –3

b. 2

c. 14

d. 25

2. The income in dollars from ticket sales for a concert is represented by 400p The income in dollars from ticket sales for a concert is represented by 400p The income in dollars from ticket sales for a concert is represented by 400 – 20p – 20p – 20 2, where p is the ticket price in dollars. How does an increase in the ticket price from $15 to $20 affect the income?

a. The income decreases by $1,500.

b. The income decreases by $2,000.

c. The income increases by $1,500.

d. The income increases by $2,000.

3. The table shows the predicted temperatures for a summer day in Salt Lake City, Utah. At what times is the temperature increasing?

Time Temperature (°F)6 A.M. 639 A.M. 76

12 P.M. 863 P.M. 896 P.M. 869 P.M. 76

a. from 6 A.M. until 6 P.M.

b. from 6 A.M. until 3 P.M.

c. from 6 A.M. until 5:30 P.M.

d. from 6 A.M. until 6:30 P.M

4. Kara tosses a volleyball across the gym to her friend, who is standing 50 feet away. The ball travels in a quadratic arc. Which function could represent the path of the ball?

a. f(f(f x(x( ) = –0.04(x) = –0.04(x) = –0.04( – 50)x

b. f(f(f x(x( ) = –0.04(x) = –0.04(x) = –0.04( 2 – 50)

c. f(f(f x(x( ) = –2(x) = –2(x) = –2( – 5)(x – 5)(x – 5)( + 5)

d. f(f(f x(x( ) = –0.04(x) = –0.04(x) = –0.04( – 50)(x – 50)(x – 50)( + 50)

continued




Assessment

Name: Date:

5. Which quadratic regression equation models the following data points?

(1, 6), (3, 8), (7, 0)

a. f(f(f x(x( ) = –0.5x2 + 3x + 3.5

b. f(f(f x(x( ) = –0.5(x) = –0.5(x) = –0.5( – 3)(x – 3)(x – 3)( – 3.5)

c. f(f(f x(x( ) = 0.5x2 + 3x+ 3.5

d. f(f(f x(x( ) = –0.5(x) = –0.5(x) = –0.5( + 3)(x + 3)(x + 3)( + 3.5)

6. What characteristic of a quadratic function is found in its vertex form?

a. a

b. vertex

c. x-intercept(s)

d. y-intercept

7. The path of a marshmallow launched from a slingshot can be described by the equation f(f(f x(x( ) = –x) = –x) = – 2 + 4x + 5, where f(f(f x(x( ) is the height of the marshmallow and x is the number of seconds that have passed since the slingshot’s band was released. Which of the following points shows the extremum for the function?

a. (0, 5)

b. (2, 9)

c. (0, –1)

d. (4, 5)

8. The dimensions of a community garden are such that the length is 6 feet shorter than 3 times its width. What expression describes the area of the community garden in terms of its width, w?

a. (w + 3)(w + 6) ft2

b. w(6w – 3) ft2

c. 3w(w – 6) ft2

d. w(3w – 6) ft2

9. If the vertex of f(f(f x(x( ) is (3, –5), what is the vertex of f(f(f x(x( + 3)?

a. (3, –8)

b. (6, –5)

c. (3, –2)

d. (0, –5)

continued




Assessment

Name: Date:

10. If the transformation of f(f(f k • x) is applied to f(f(f x(x( ), with 0 < k < 1, which is true about the graph of the function?

a. The graph of the original function is compressed horizontally by a factor of 1

k.

b. The graph of the original function is stretched horizontally by a factor of 1

k.

c. The graph of the original function is stretched vertically by a factor of k.

d. The graph of the original function is compressed vertically by a factor of k.

11. For the function f(f(f x(x( ) = –(x) = –(x) = –( – 2)2 + 0.25, what is the value of x that changes the function from increasing to decreasing?

a. 0.25

b. 2

c. –2

d. –0.25

12. The function h(t) = –16(t) = –16(t t – 1.5)(t – 1.5)(t t + 1.5) represents the height of a nut t + 1.5) represents the height of a nut t t seconds after falling t seconds after falling tfrom a tree. What is a reasonable domain for this function?

a. –1.5 to 1.5 seconds

b. 0 to 0.75 seconds

c. 0 to 1.5 seconds

d. 0 to 3 seconds

13. Given two quadratic functions, what characteristics do you look at to see which achieves a higher value?

a. a-values and vertices

b. a-values and y-intercepts

c. a-values and x-intercepts

d. x- and y-intercepts

14. Which of the following quadratic functions has a vertex with a lower y-value?

a. f(f(f x(x( ) = 3(x) = 3(x) = 3( + 1)(x + 1)(x + 1)( – 2)

b. g(x(x( ) = 3x2 + 2x + 1

c. h(x(x( ) = 3(x) = 3(x) = 3( + 1)2 + 2

d. It cannot be determined from this information. continued




Assessment

Name: Date:

15. Which of the following statements is true about the functions f(f(f x(x( ) and g(x(x( )?

x f(f(f x)x)x–1 –10 21 32 23 –1

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

–10

–9

–8

–7

–6

–5

–4

–3

–2

–1

1

2

3

4

5

6

7

8

9

10y

g(x)

x

a. The function f(f(f x(x( ) has a greater maximum than the function g(x(x( ).

b. The function g(x(x( ) has a greater maximum than the function f(f(f x(x( ).

c. The function f(f(f x(x( ) has a greater minimum than the function g(x(x( ).

d. The function g(x(x( ) has a greater minimum than the function f(f(f x(x( ).



Instruction


Unit 2 End-of-Unit Assessment Answer Key

Answer Standard(s)

1. d A–SSE.1a★

2. a A–SSE.1b★

3. b F–IF.4★

4. a A–SSE.3a★

5. a A–CED.2★

6. b F–IF.8a

7. b F–IF.4★

8. d F–BF.1★

9. d F–BF.3

10. b F–BF.3

11. b F–IF.4★

12. c F–IF.4★

13. a F–IF.9

14. a F–IF.9

15. b F–IF.9

North Carolina Math 2 - Walch

Documents

Transcript of North Carolina Math 2 - Walch