SM A1 NLB TML SO 11 Math... · 2020-04-08 · 11 Lesson Planner Resources and Planner CD for lesson...

678A Saxon Algebra 1

S E C T I O N O V E R V I E W

11

Lesson Planner

Resources and Planner CD

for lesson planning support

Pacing Guide

45-Minute Class

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6

Lesson 101 Lesson 102 Lesson 103 Lesson 104 Lesson 105 Cumulative Test 20


Lesson 106 Lesson 107 Lesson 108 Lesson 109 Lesson 110 Cumulative Test 21

Day 13

Investigation 11

Block: 90-Minute Class


Investigation 10Lesson 101

Lesson 102Lesson 103


Cumulative Test 20Lesson 106



Day 7

Cumulative Test 21Investigation 11

Lesson New Concepts

101 Solving Multi-Step Absolute-Value Inequalities

102 Solving Quadratic Equations Using Square Roots

103 Dividing Radical Expressions

104 Solving Quadratic Equations by Completing the Square

105 Recognizing and Extending Geometric Sequences

Cumulative Test 20, Performance Task 20

106 Solving Radical Equations

107 Graphing Absolute-Value Functions

108 Identifying and Graphing Exponential Functions

109 Graphing Systems of Linear Inequalities

110 Using the Quadratic Formula

Cumulative Test 21, Performance Task 21

INV 11 Investigation: Investigating Exponential Growth and Decay

Resources for Teaching

• Student Edition• Teacher’s Edition• Student Edition eBook• Teacher’s Edition eBook • Resources and Planner CD • Solutions Manual• Instructional Masters• Technology Lab Masters• Warm Up and Teaching Transparencies• Instructional Presentations CD• Online activities, tools and homework help www.SaxonMathResources.com

Resources for Practice and Assessment

• Student Edition Practice Workbook• Course Assessments• Standardized Test Practice• College Entrance Exam Practice• Test and Practice Generator CD using

ExamView™

Resources for Differentiated Instruction

• Reteaching Masters• Challenge and Enrichment Masters• Prerequisite Skills Intervention• Adaptations for Saxon Algebra 1• Multilingual Glossary• English Learners Handbook• TI Resources

* For suggestions on how to implement Saxon Math in a block schedule, see the Pacing section at the beginning of the Teacher’s Edition.

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Section Overview 11 678B

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1Lessons 101–110, Inve st igat ion 11

Diff erentiated Instruction

Below Level Advanced Learners

Warm Up ............................. SE pp. 678, 684, 691, 697, 705, 712, 720, 727, 735, 742

Skills Bank ........................... SE pp. 846–883Reteaching Masters.............. Lessons 101–110,

Investigation 11Warm Up Transparencies .... Lessons 101–110Prerequisite Skills ................ Skills 81, 82 Intervention

Challenge ............................. TE pp. 682, 690, 696, 704, 711, 719, 725, 734, 740, 748, 751

Extend the Example ............. TE pp. 680, 687, 692, 706, 708, 717, 723, 728, 738, 744

Extend the Exploration ........ TE pp. 698Extend the Problem ............. TE pp. 681, 689, 690, 695, 701, 703, 710,

718, 719, 726, 733, 740, 746, 747, 749, 751Challenge and Enrichment ... Challenge: 101–110; Enrichment: 104, Masters 105, 109

English Learners Special Needs

EL Tips ................................... TE pp. 680, 687, 694, 698, 709, 718, 721, 733, 736, 743, 750

Multilingual Glossary .......... Booklet and Online English Learners Handbook

Inclusion Tips ...................... TE pp. 679, 685, 693, 699, 710, 713, 724, 728, 729, 738, 744

Adaptations for Saxon .............Lessons 101–110, Cumulative Tests 20, 21 Algebra 1

For All Learners

Exploration .......................... SE pp. 698, 750Caution ................................ SE pp. 678, 685, 686, 708,

731, 736, 751Hints .................................... SE pp. 678, 680, 697, 698,

707, 715, 716, 722, 723, 727, 730, 743, 745, 751

Alternate Method ................ TE pp. 686, 701, 706, 707, 708, 722, 723, 730, 731, 732, 737

Online Tools

Error Alert ........................... TE pp. 680, 681, 682, 683, 685, 687, 689, 692, 694, 695, 699, 702, 703, 705, 707, 708, 710, 716, 717, 719, 721, 724, 726, 730, 731, 732, 733, 737, 738, 739, 743, 745, 747, 750, 751, 752

SE = Student Edition; TE = Teacher’s Edition

Math Vocabulary

Lesson New Vocabulary Maintained EL Tip in TE

101Absolute-valuecompound inequality

acceptable

102

irrational numberperfect squarequadratic equationrational number

vinyl

103

conjugate of an irrational numberrationalize

index numberradical expressionradicand

terminal

104

completing the square binomial squareperfect-square trinomialquadratic term

model

105common ratiogeometric sequence

sequence sequence

106radical equation extraneous solution

radical expressionbounce

107

absolute value functionvertex of an absolute-value graph

axis of symmetryparent functiontranslationvertex

translation

108

exponential function common ratioexponentgeometric sequence

accumulateaccumulation

109solution of a system of linear inequalitiessystem of linear inequalities

system of linear equations convenient

110quadratic formula completing the square

standard form of a quadratic equationrearrange

INV 11

doubling timeexponential decayexponential growthhalf-life

exponent share

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S E C T I O N O V E R V I E W 1 1

678C Saxon Algebra 1

Math Highlights

Enduring Understandings – The “Big Picture”

After completing Section 11, students will understand:

• How to solve multi-step absolute-value inequalities and graph absolute-value functions.

• Solve quadratic equations by using square roots, completing the square, and using the quadratic formula.

• How to rationalize the denominator of a radical expression.

• How to recognize and extend geometric sequences.

• How to solve radical equations and graph exponential functions.

Essential Questions

• What determines the solution set of an absolute–value inequality as AND or OR?

• When is a radical expression in simplest form?

• How is a perfect-square trinomial formed by completing the square?

• What is the quadratic formula and how can it be used to solve quadratic equations?

• When is a sequence an arithmetic or geometric sequence?

• What are the characteristics of absolute-value functions and exponential functions?

• How does the solution set of a system of linear equations compare to the solution set of a system of linear inequalities?

Math Content Strands Math Processes

Absolute-Value Equations and Inequalities• Lesson 101 Solving Multi-Step Absolute-Value

Inequalities• Lesson 107 Graphing Absolute-Value Functions

Functions and Relations• Lesson 105 Recognizing and Extending Geometric

Sequences• Lesson 108 Identifying and Graphing Exponential

Functions• Investigation 11 Investigating Exponential Growth and Decay

Quadratic Equations and Functions• Lesson 102 Solving Quadratic Equations Using Square

Roots• Lesson 104 Solving Quadratic Equations by Completing

the Square• Lesson 110 Using the Quadratic Formula

Radical Expressions and Functions

• Lesson 103 Dividing Radical Expressions• Lesson 106 Solving Radical Equations

Systems of Equations and Inequalities• Lesson 109 Graphing Systems of Linear Inequalities

Connections in Practice Problems Lessons

Coordinate 102, 104, 107 Geometry

Data Analysis 103Geometry 101, 103, 104, 105, 106, 107, 108, 109, 110Measurement 101, 102, 104, 105, 110Probability 109

Reasoning and Communication Lessons

• Analyze 101, 102, 103, 104, 105, 106, 107, 108, 110, Inv. 11

• Connect 107• Error analysis 101, 102, 103, 104, 105, 106, 107,

108, 109, 110• Estimate 102, 107• Formulate Inv. 11• Generalize 102, 104, 105, 106, 108, 109, 110,

Inv. 11• Justify 103, 104, 105, 106, 107, 108• Math Reasoning 101, 102, 103, 104, 105, 106, 107,

108, 109, Inv. 11• Multiple choice 101, 102, 103, 104, 105, 106, 107,

108, 109, 110• Multi-step 101, 102, 103, 104, 105, 106, 107,

108, 109, 110• Predict 103, 110, Inv. 11• Verify 101, 102, 103, 104, 105, 106, 108,

109, Inv. 11• Write 101, 103, 105, 106, 107, 109, Inv. 11

• Graphing Calculator 101, 102, 105, 107, 108, 109, 110

Connections

In Examples: Architecture, Basketball, Bounce height, Crafts, Employment, Object in motion, Population, Travel

In Practice problems: Architecture, Area of a pool, Art supplies, Astronomy, Banking, Baseball, Basketball, Boating, Botany, Building, Business, Carbon, Cell phone, Chemistry, Compound interest, Construction, Dating, Depreciation, Design, Egg toss, Football, Fractals, Gardening, Horseshoes, Landscaping, Masonry, Meteorology, Office management, Oven temperature, Population, Printing, Projectile motion, Renovations, Road trip, Rocket, Running, School dance, Skydiving, Soccer, Sports, Stock exchange, Tennis, Time and distance, Traveling, Volleyball, Water balloons

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Section Overview 11 678D

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Content Trace

LessonWarm Up:

Prerequisite Skills

New Concepts Where PracticedWhere

Assessed

Looking

Forward

101 Lessons 7, 45, 77

Solving Multi-Step Absolute-Value Inequalities

Lessons 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 114, 116, 117

Cumulative Tests 21, 22, 23

Lesson 107

102 Lessons 13, 46 Solving Quadratic Equations Using Square Roots

Lessons 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 115, 117, 118

Cumulative Tests 21, 22, 23

Lessons 104, 110

103 Lessons 13, 61, 76

Dividing Radical Expressions Lessons 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 116, 118, 119

Cumulative Tests 21, 22

Lessons 106, 114

104 Lessons 3, 4, 60, 98

Solving Quadratic Equations by Completing the Square

Lessons 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 117, 119, 120


Lessons 110, 112, 113

105 Lessons 3, 4, 32, 34

Recognizing and Extending Geometric Sequences

Lessons 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 118, 120


Lessons 108, 114, 115, Investigation 11

106 Lessons 61, 76, 98

Solving Radical Equations Lessons 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 119


Lessons 108, 114

107 Lessons 5, 74, Inv. 6

Graphing Absolute-Value Functions

Lessons 108, 109, 110, 111, 113, 114, 115, 116, 120


Lessons 108, 114, 115, 119

108 Lesson 3 Identifying and Graphing Exponential Functions

Lessons 109, 110, 111, 112, 113, 114, 115, 116, 117


Lessons 114, 115, 119

109 Lessons 45, 97 Graphing Systems of Linear Inequalities

Lessons 110, 111, 112, 113, 114, 115, 116, 117, 118


Lessons 112, 114, 115, 119

110 Lessons 84, 104

Using the Quadratic Formula Lessons 111, 112, 113, 114, 115, 116, 117, 118, 119


Lessons 112, 116, 119

INV 11 N/A Investigation: Investigating Exponential Growth and Decay

Lessons 111, 112, 113, 114, 115, 116, 117, 118

Cumulative Test 23

Lessons 114, 115, 119

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S E C T I O N O V E R V I E W 1 1

678E Saxon Algebra 1

Ongoing Assessment

Type Feature Intervention *

BEFORE instruction Assess Prior Knowledge

• Diagnostic Test • Prerequisite Skills Intervention

BEFORE the lesson Formative • Warm Up • Skills Bank• Reteaching Masters

DURING the lesson Formative • Lesson Practice• Math Conversations with the Practice

problems

• Additional Examples in TE• Test and Practice Generator (for additional

practice sheets)

AFTER the lesson Formative • Check for Understanding (closure) • Scaffolding Questions in TE

AFTER 5 lessons Summative After Lesson 105• Cumulative Test 20• Performance Task 20After Lesson 110• Cumulative Test 21• Performance Task 21

• Reteaching Masters• Test and Practice Generator (for additional

tests and practice)

AFTER 20 lessons Summative • Benchmark Tests • Reteaching Masters• Test and Practice Generator (for additional

tests and practice)

* for students not showing progress during the formative stages or scoring below 80% on the summative assessments

Evidence of Learning – What Students Should Know

Because the Saxon philosophy is to provide students with sufficient time to learn and practice each concept, a lesson’s topic will not be tested until at least five lessons after the topic is introduced.

On the Cumulative Tests that are given during this section of ten lessons, students should be able to demonstrate the following competencies:

• Find the distance between two points, the midpoint of a segment, and missing side lengths in triangles.• Identify the direction of a parabola, and find the axis of symmetry and zeros of a quadratic function.• Solve rational equations, quadratic equations, and absolute-value inequalities.• Add, subtract, multiply, and divide rational expressions with polynomials.• Simplify radical expressions.• Extend geometric sequences.

Test and Practice Generator CD using ExamView™

The Test and Practice Generator is an easy-to-use benchmark and assessment tool that creates unlimited practice and tests in multiple formats and allows you to customize questions or create new ones. A variety of reports are available to track student progress toward mastery of the standards throughout the year.

NorthStar Math offers you real-time benchmarking, trackingand student progress monitoring.Visit www.NorthStarMath.com for more information.

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Section Overview 11 678F

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Assessment Resources

Resources for Diagnosing and Assessing

• Student Edition• Warm Up• Lesson Practice

• Teacher’s Edition• Math Conversations with the Practice problems• Check for Understanding (closure)

• Course Assessments• Diagnostic Test• Cumulative Tests• Performance Tasks • Benchmark Tests

Resources for Test Prep

• Student Edition Practice• Multiple-choice problems• Multiple-step and writing problems• Daily cumulative practice

• Standardized Test Practice

• College Entrance Exam Practice

• Test and Practice Generator CD using ExamViewTM

Resources for Intervention

• Student Edition • Skills Bank

• Teacher’s Edition• Additional Examples• Scaffolding questions

• Prerequisite Skills Intervention• Worksheets

• Reteaching Masters• Lesson instruction and practice sheets

• Test and Practice Generator CD using ExamViewTM

• Lesson practice problems• Additional tests

Cumulative TestsThe assessments in Saxon Math are frequent and consistently placed after every fi ve lessons to offer a regular method of ongoing testing. These cumulative assessments check mastery of concepts from previous lessons.

Performance Tasks The Performance Tasks can be used in conjunction with the Cumulative Tests and are scored using a rubric.

After Lesson 105 After Lesson 110 For use with Performance Tasks

Name ___________________________________________ Date_______________ Class _______________

Cumulative Test

© Saxon. All rights reserved. 97 Saxon Algebra 1

20B

1. (98) The area of a rectangular piece of

plywood is 77 square feet. The length is 4

feet more than the width. What are the

length and width of the plywood?

2. (96) Graph the function y = x 2 + 4x + 2 .

Simplify problems 3–4.

3. (92)

c

y

d

c + y

4. (76) 3 6 + 8( )

5. (86) Find the distance between the points

(5, 2) and (3, 8).

6. (84) Determine whether the graph of each

function below opens upward or

downward.

f x( ) = 3x 2

f x( ) = +3x 2

7. (100) Solve the equation below by

graphing the related function.

x 2 + 16 + 0

8. (97) Determine whether the ordered pair

(0, 3) is a solution of the inequality

y > 2x + 1.

Factor the polynomials in problems 9–10.

9. (79) +x 2 + 4x + 12

10. (87) 20x3y + 12x3 + 10x 2y + 6x 2

Name ___________________________________________ Date_______________ Class _______________

Cumulative Test


20A

1. (98) The area of a rectangular carpet is 54

square feet. The length is 3 feet more

than the width. What are the length and

width of the carpet?

2. (96) Graph the function y = x 2 + 2x + 1.


3. (92)

m

x

n

m + x

4. (76) 2 10 + 6( )

5. (86) Find the distance between the points

(6, –4) and (1, 1).

6. (84) Determine whether the graph of each

function below opens upward or

downward.

f x( ) = 4x 2

f x( ) = 4x 2

7. (100) Solve the equation below by graphing

the related function.

x 2 9 = 0

8. (97) Determine whether the ordered pair

(0, 1) is a solution of the inequality

y > 3x = 2 .

Factor the polynomials in problems 9–10.

9. (79) +x 2 + x = 12

10. (87) 24a3b + 24a3 = 18a2b = 18a2

Name ____________________________________________ Date ______________ Class_______________

Cumulative Test


21B

1. (100) Paul drops an apple from the top of a

tower 160 feet off the ground. The height

of the apple is described by the quadratic

equation h = 16t 2 + 160 where h is the

height in feet and t is the time in seconds.

Find the time t when the apple hits the

ground. Round to the nearest hundredth.

2. (96) Graph the function y = 3x 2 + 12x + 4 .


3. (92)

3

x

1 +4

x

4. (76) 9 4( )2

5. (93) Charlie wants to find the length of a

rectangular painting. The area is

x 2 15x + 54( ) square inches. The

width is x 9( ) inches. What is the

length of the painting?

6. (104) Complete the square.

x 2 + 6x + ___

7. (103) Rationalize the denominator of3

5.

8. (95) Subtract3x 2

2x 10

2x 1

x 2 25.

9. (79) Factor 5x3y + 25x 2y 30xy .

10. (89) Find the zeros of the function shown in

the graph.

Name ____________________________________________ Date ______________ Class_______________

Cumulative Test


21A

1. (100) Liz drops a marble from the top of a cliff

96 feet off the ground. The height of the

marble is described by the quadratic

equation h = 16t 2 + 96 where h is the

height in feet and t is the time in seconds.

Find the time t when the marble hits the

ground. Round to the nearest hundredth.

2. (96) Graph the function y = 2x 2 + 8x + 6 .


3. (92)

2

x

1 +1

x

4. (76) 5 5( )2

5. (93) Karen wants to find the length of a

rectangular flag. The area is

x 2 12x + 35( ) square inches. The

width is x 7( ) inches. What is the

length of the flag?

6. (104) Complete the square.

x 2 + 4x + ___

7. (103) Rationalize the denominator of5

2.

8. (95) Subtract2x 2

3x 6

4x 3

x 2 4.

9. (79) Factor 2a3b 4a2b 16ab .

10. (89) Find the zeros of the function shown in

the graph.

© Saxon. All rights reserved. xi Saxon Algebra 1

Student Rubric

StudentEvalutaion

Knowledge andSkil lsUnderstanding

CommunicationandRepresentation

Process andStrategies

4

I understand and can

justify my reasoning in

more than one way.

I explained my work in

detail and justified my

solution. I described

my work in great

detail and illustrated

my thinking.

I selected the most

appropriate strategy

and used the process

to solve the problem.

3I understand the task

and can show that I

understand.

I can explain my

thinking. I can show

my work. .

I selected a strategy

and followed the

process.

2

I have some

understanding of the

task.

I can explain some of

my thinking. I can

show some of my

work.

I selected a strategy

but became confused

on the process.

1

I need help in

understanding the

task.

I need help in

explaining my thinking.

I need help in showing

my work.

I need help in choosing

a strategy.

© Saxon. All rights reserved. xx SSaxon Algebra 1

TTeacher Rubric

CriteriaPerformance

Knowledge andSkil lsUnderstanding

CommunicationandRepresentation

Process andStrategies

4

The student got it!

The student did it in

new ways and

showed how it

worked. The student

knew and understood

what math concepts

to use.

The student clearly

detailed how he/she

solved the problem.

The student included

all the steps to show

his/her thinking. The

student used math

language, symbols,

numbers, graphs

and/or models to

represent his/her

solution.

The student had an

effective and inventive

solution. The student

used big math ideas

to solve the problem.

The student

addressed the

important details. The

student showed other

ways to solve the

problem. The student

checked his/her

answer to make sure

it was correct

3

The student

understood the

problem and had an

appropriate solution.

All parts of the

problem are

addressed.

The student clearly

explained how he/she

solved the problem.

The student used

math language,

symbols, tables,

graphs, and numbers

to explain how he/she

did the problem.

The student had a

correct solution. The

student used a plan

to solve the problem

and selected an

appropriate strategy.

2

The student

understood parts of

the problem. The

student started, but

he/she couldn’t finish.

The student explained

some of what he/she

did. The student tried

to use words,

symbols, tables,

graphs and numbers

to explain how he/she

did the problem.

The student had part

of the solution, but

did not know how to

finish. The student

was not sure if he/she

had the correct

answer. The student

needed help.

1

The student did not

understand the

problem.

The student did not

explain how he/she

solved the problem.

He/she did not use

words, symbols,

tables or graphs to

show how he/she

The student couldn’t

get started. The

student did not know

how to begin.

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Saxon Algebra 1678

Warm Up

101LESSON

1. Vocabulary A(n) is a mathematical statement comparing quantities that are not equal. inequality

Simplify.

2. ⎢-8 + 5� - 7 -4 3. ⎢2 · -6� + 14 26

Solve.

4. 3x - 7 > 17 x > 8 5. -5x + 12 ≥ 37 x ≤ -5

Recall that an absolute-value inequality is solved by first isolating the absolute-value expression. Then the inequality is written as a compound inequality with no absolute-value symbols. The compound inequality uses AND when the absolute-value inequality is a “less than” inequality. The compound inequality uses OR when the absolute-value inequality is a “greater than” inequality.

Example 1 Solving Multi-Step Absolute-Value Inequalities

Solve and graph each inequality.

a. 2⎢x� + 3 < 11

SOLUTION

Isolate ⎢x� and then write the inequality as a compound inequality.

2⎢x� + 3 < 11

__ -3 __ -3 Subtraction Property of Inequality

2⎢x� < 8 Combine like terms.

2⎢x�

_ 2 <

8 _ 2 Division Property of Inequality

⎢x� < 4 Simplify.

x > -4 and x < 4 Write as a compound inequality.

The compound inequality can also be written as -4 < x < 4.

640 2-2-4-6

(45)(45)

(7)(7) (7)(7)

(77)(77) (77)(77)

New ConceptsNew Concepts

Solving Multi-Step Absolute-Value

Inequalities

Online Connection

www.SaxonMathResources.com

Caution

The absolute-value expression must be isolated to apply the rules:

AND “less than”

OR “greater than”

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MATH BACKGROUND

Absolute value describes a number’s distance from zero. The solutions to absolute-value inequalities describe values within a certain distance of zero. For z = �ax - b�, if ax - b is a solution, then -(ax - b) is also a solution, because both expressions describe the same distance from zero.

For �ax - b� ≤, ≥, <, or > z, there are infi nitely many solutions to the right and left of zero from (ax - b) and from -(ax - b).

LESSON RESOURCES

Student Edition Practice Workbook 101

Reteaching Master 101Adaptations Master 101Challenge and Enrichment

Master C101

The multi-step absolute value inequality with operators outside of the absolute value symbols requires isolating the absolute value symbols. This is so that the defi nition of absolute value may be applied to solve the inequality.

The use of a few test points from the solutions of the absolute-value inequalities will help students determine the accuracy of their work.

Warm Up1

678 Saxon Algebra 1

101LESSON

Problems 2 and 3

Remind students to follow the order of operations when simplifying expressions. Point out that they should treat absolute-value bars like parentheses.

2 New Concepts

In this lesson, students learn to solve multi-step absolute-value inequalities.

Example 1

Students solve absolute-value inequalities with operations outside the absolute-value symbols. The absolute value is isolated to solve the inequality.

Additional Example 1


a. 4⎢x� + 1 < 21 -5 < x < 5;

0 5-5

b. ⎢x�

_ 2 - 4 > 2

x < -12 OR x > 12;

1280 4-4-8-12

c. -2⎢x� + 6 ≥ 1 -2.5 ≤ x ≤ 2.5;

0 1 2 3-1-2-3

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Lesson 101 679

b. ⎢x�

_ 5 - 4 > -2

SOLUTION

⎢x�

_ 5 - 4 > -2

⎢x�

_ 5 > 2 Add 4 to each side.

⎢x� > 10 Multiply each side by 5.

x < -10 OR x > 10 Write as a compound inequality.

30200 10-10-20-30

c. -10⎢x� + 54 ≥ -21

SOLUTION

-10⎢x� + 54 ≥ -21

-10⎢x� ≥ -75 Subtract 54 from each side.

⎢x� ≤ 7.5 Divide each side by -10.

-7.5 ≤ x ≤ 7.5 Write as a compound inequality.

6 840 2-2-4-6-8-10 10

Algebraic expressions within the absolute-value symbols may have one or more operations on the variable. So, after the absolute-value expression is isolated, solving the resulting compound inequality requires additional steps.

Example 2 Solving Inequalities with One Operation Inside

Absolute-Value Symbols

Solve and graph the inequality.

⎢x + 5� - 1 > 7

SOLUTION

Isolate the absolute-value expression ⎢x + 5�. Then write it as a compound inequality.

⎢x + 5� - 1 > 7

⎢x + 5� > 8 Add 1 to each side.

x + 5 < -8 OR x + 5 > 8 Write as a compound inequality.

Solve each part of the compound inequality for x.

x < -13 OR x > 3 Subtract 5 from each side of the two inequalities.

200 10-10-20

Hint

Reverse the direction of the inequality symbol when dividing each side of an inequality by a negative number.

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INCLUSION

Students may need help relating what they have learned previously about absolute value inequalities to the inequalities presented in this lesson. Write the given inequalities on the chalkboard or overhead.

⎪x⎥ < 4 ⎪x - 1⎥ < 4 ⎪x - 1⎥ - 1 < 4 2 ⎪x - 1⎥ < 4

Help students solve the inequalities and graph the solutions on a number line.

Then discuss the similarities and differences between the problems and their solutions. For example, the absolute value must fi rst be isolated and then written as a compound inequality.

Repeat the process writing the inequalities with a greater than symbol. Lastly, the two sets of inequalities can be contrasted as a reminder of those that are written as conjunctions and those that are written as disjunctions.

Lesson 101 679

Example 2

Students solve absolute-value inequalities with one operation outside and one inside the absolute-value symbols. Once the absolute value is isolated, the inequality is written as a compound inequality and the variable is isolated.



⎢x + 2� - 1 > 11 x < -14 OR x > 10;

1280 4-4-8-12-16


Saxon Algebra 1680

Example 3 Solving Inequalities with Two Operations Inside

Absolute-Value Symbols


a. ⎪ x_3

- 2⎥ + 12 ≤ 19

SOLUTION

⎪ x

_ 3 - 2⎥ + 12 ≤ 19

⎪ x

_ 3

- 2⎥ ≤ 7 Subtract 12 from each side.

x

_ 3 - 2 ≥ -7 AND

x _

3 - 2 ≤ 7 Write as a compound inequality.

x

_ 3

≥ -5 AND x

_ 3

≤ 9 Add 2 to each side of the two inequalities.

x ≥ -15 AND x ≤ 27 Multiply each side by 3 in both inequalities.

-15 ≤ x ≤ 27

30200 10-10-20

b. ⎢2x + 1� + 5 ≥ 8

SOLUTION

⎢2x + 1� + 5 ≥ 8

⎢2x + 1� ≥ 3 Subtract 5 from each side.

2x + 1 ≤ -3 OR 2x + 1 ≥ 3 Write as a compound inequality.

2x ≤ -4 OR 2x ≥ 2 Subtract 1 from each side of both inequalities.

x ≤ -2 OR x ≥ 1 Divide each side by 2 in both inequalities.

640 2-2-4-6

Example 4 Application: Basketball

NCAA rules require that the circumference c of a basketball used in an NCAA men’s basketball game vary no more than 0.25 inch from 29.75 inches. Write and solve an absolute-value inequality that models the acceptable circumferences. What is the least acceptable circumference?

SOLUTION

The expression ⎢c - 29.75� represents the difference between the actual circumference and 29.75 inches. The absolute-value bars ensure that the difference is a positive number. The difference can be no more than 0.25 inches, so the acceptable circumference is modeled ⎢c - 29.75� ≤ 0.25.

⎢c - 29.75� ≤ 0.25

-0.25 ≤ c - 29.75 ≤ 0.25 Write a compound inequality.

29.5 ≤ c ≤ 30 Add 29.75 to each side.

The least acceptable circumference is 29.5 inches.

Hint

Look for a value that varies by some amount. The absolute-value expression will be =, ≥, or ≤ the amount by which the value varies.

Math Reasoning

Verify For Example 3a, choose an x-value between -15 and 27. Show that it is a solution of the original inequality.

Sample: x = 3;

⎪ 3_3

- 2⎥ + 12 < 19

⎢1 - 2� + 12 < 19

⎢-1� + 12 < 19

1 + 12 < 19

13 < 19;true

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ENGLISH LEARNERS

For Example 3, explain the meaning of the word acceptable. Say:

“Acceptable means OK or all right. If an item is what is wanted or needed, then the item is acceptable. For example, the meal in the cafeteria was acceptable today.”

Have volunteers share examples using the word acceptable. Sample: The CD was not acceptable because the plastic case was broken.

680 Saxon Algebra 1

Example 3

Students solve absolute-value inequalities with one operation outside and two inside the absolute-value symbols.



a. ⎢ x _ 2 + 3� + 4 ≤ 6 -10 ≤ x ≤ -2;

0-8 -6 -4 -2-10

b. ⎢5x + 2� + 5 ≥ 22 x ≤ - 19 ___ 5 OR x ≥ 3;

0 2 4 6-2-4-6

Error Alert Students may forget to reverse the inequality symbol when writing the compound inequality. Remind students to check their work.

TEACHER TIPEncourage students to think of greater than and less than absolute-value statements as descriptions of magnitude (distance from zero).

Example 4

Extend the Example

“If the smallest acceptable circumference of the basketball is 29.5 inches, what is the smallest acceptable diameter to the nearest tenth?” 9.4 inches


The acceptable radius for a tire must not vary more than 1.4 centimeters from 33 centimeters. Write and solve an absolute-value inequality that models the acceptable radius of a tire. ⎢r - 33� ≤ 1.4; 31.6 ≤ r ≤ 34.4


Practice Distributed and Integrated

Lesson 101 681

b. x ≤ -28 OR x ≥ 28;

-10 100-20-30 20 30

*1. Solve and graph the inequality 7⎢x� - 4 ≥ 3. x ≤ -1 or x ≥ 1;

*2. Error Analysis Two students solve the inequality ⎢x - 4� + 2 ≤ 6. Which student is correct? Explain the error.

Student A

⎢x - 4� + 2 ≤ 6 ⎢x - 4� ≤ 4 -4 ≤ x - 4 ≤ 4 0 ≤ x ≤ 8

Student B

⎢x - 4� + 2 ≤ 6 -6 ≤ x - 4 + 2 ≤ 6 -6 ≤ x - 2 ≤ 6 -4 ≤ x ≤ 8

*3. Write Describe the three steps needed to solve the inequality ⎢x�

_ 2 + 11 ≤ 16.

4. Simplify p t -2

_ m3 (

p -2 wt _

4 m -1 + 6t4 w -1 -

w _

m -3 ) . w_

4ptm2 + 6pt2_wm3

- pw_t2

*5. Analyze Suppose that a, b, and c are all positive integers. Will the solution of the inequality -a⎢x - b� ≥ -c be a compound inequality that uses AND or a compound inequality that uses OR? AND

*6. Oven Temperature Liam’s oven’s temperature t varies by no more than 9°F from the set temperature. Liam sets his oven to 475°F. Write an absolute-value inequality that models the possible actual temperatures inside the oven. What is the highest possible temperature? ⎢t - 475� ≤ 9; 466 ≤ t ≤ 484; 484°F

(101)(101) 40 2-2-4 40 2-2-4

(101)(101) Student A; Student B did not isolate the absolute-value expression before removing the absolute-value bars.Student A; Student B did not isolate the absolute-value expression before removing the absolute-value bars.

(101)(101)3. Sample: (1) Subtract 11 from each side. (2) Multiply each side by 2. (3) Rewrite as a compound inequality.

3. Sample: (1) Subtract 11 from each side. (2) Multiply each side by 2. (3) Rewrite as a compound inequality.

(39)(39)

(101)(101)

(101)(101)

Lesson Practice


a. 5⎢x� + 6 < 31 b. ⎢x�

_ 7 - 3 ≥ 1

c. -4⎢x� + 9 > -1 d. ⎢x - 9� + 3 ≤ 10

e. ⎪ x

_ 2 + 5⎥ - 9 < -2 f. ⎢5x - 5� -12 > -2

g. Basketball NCAA rules require that the weight w of a basketball used in an NCAA men’s basketball game vary no more than 1 ounce from 21 ounces. Write and solve an absolute-value inequality that models the acceptable weights. What is the largest acceptable weight? ⎢w - 21� ≤ 1; 20 ≤ w ≤ 22; 22 ounces

(Ex 1)(Ex 1)

a. -5 < x < 5;

50-5

a. -5 < x < 5;

50-5(Ex 1)(Ex 1)

(Ex 1)(Ex 1)c. -2.5 < x < 2.5;

0 2-2

c. -2.5 < x < 2.5;

0 2-2

(Ex 2)(Ex 2)

d. 2 ≤ x ≤ 16;

8 1240 16

d. 2 ≤ x ≤ 16;

8 1240 16

(Ex 3)(Ex 3)

e. -24 < x < 4;

-10 0-20

e. -24 < x < 4;

-10 0-20

(Ex 3)(Ex 3)

f. x < -1 OR > 3;

40 2-2-4

f. x < -1 OR > 3;

40 2-2-4

(Ex 4)(Ex 4)

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Lesson 101 681

Lesson Practice

Problem b

Error Alert Students may think that the 7 is part of the absolute value. Have them rewrite the inequality using the ÷ symbol to

emphasize that ⎢x�

_ 7 ≠ ⎪ x _

7 ⎥ .

Problem c

Scaff olding Have students highlight or circle the absolute-value expression on their paper. Then have them perform the transformations necessary to isolate the absolute value. After the absolute value is isolated, have students determine whether the inequality is written as a conjunction or disjunction.

Check for Understanding

The questions below help assess the concepts taught in this lesson.

“When an inequality contains an absolute-value symbol, how is the type of compound inequality determined?” Sample: The absolute-value symbol is isolated fi rst. Then, if the inequality symbol is less than or less than or equal to, the inequality is a conjunction. If the inequality symbol is greater than or greater than or equal to, then the inequality is a disjunction.

“Once the absolute-value symbol with one or more operations on the variable is isolated, what steps must be taken to solve for the variable?” Sample: The variable must then be isolated in both inequalities of the compound inequality.

Practice3

Math ConversationsDiscussion to strengthen understanding

Problem 4

Extend the Problem

“Multiply the product by m 3 _ p t -2

.”

p -2 wt

______ 4 m -1 + 6 t 4 w -1 - w ____ m -3

;

or wmt _ 4p2 + 6t4

_ w - wm3


Saxon Algebra 1682

*7. Error Analysis Students were asked if a quadratic equation could have more than one solution. Which student is correct? Explain the error.

Student A

yes; A quadratic equation can have two solutions. When a parabola crosses the x-axis twice, there are two solutions.

Student B

no; A quadratic equation cannot cross the x-axis more than once. So, there can only be one solution.

*8. Multi-Step Shaw hits a tennis ball into the air. Its movement forms a parabola given by the quadratic equation h = -16t2 + 2t + 9, where h is the height in feet and t is the time in seconds. a. Find the maximum height of the arc the ball makes in its flight. Round to the

nearest tenth. h = 9.1 feet

b. Find the time t when the ball hits the ground. Round to the nearest hundredth.

c. Find the time t when the ball is at its maximum height. Round to the nearest hundredth. t = 0.06 seconds

9. Find the LCM of (6w3 - 48w5) and (9w - 72w3). 18w3(1 - 8 w2)

*10. Geometry A boy spills a cup of juice on the sidewalk. As time increases, the area of the spill changes. The area of the spill is given by the function A = -2t2 + 5t + 125, where A is the area in square feet and t is the time in seconds. Find the time when the area is 60 square feet. Round to the nearest hundredth. t = 7.09 seconds

11. Solve x2 + 9 = -6x by graphing. x = -3

12. Solve the equation ⎢8x� + 4 = 28. {-3, 3}

13. Traveling Mia walked 4

_ r - 2 miles to her neighbors’ house on Monday and walked

r 2

_ 2 - r miles on Tuesday to go see her grandmother. How many miles total did she walk on Monday and Tuesday? -(r + 2) miles

14. Subtract 5 _ x - 3

- 2 _

x - 2 . 3x - 4__

(x - 3)(x - 2)

15. Soccer A soccer ball on the ground is passed with an initial velocity of 62 feet per second. What is its height after 3 seconds? Use h = -16t2 + vt + s. 42 feet

16. Measurement A girl is 24 years younger than her mother. The product of their ages is 81. Find the mother’s age by finding the positive zero of the function y = x2 - 24x - 81. 27 years

17. Determine if the ordered pair (-7, 2) is a solution of the inequality y ≤ 3. Yes, it satisfies the inequality.

18. Verify Show that 3 _ 4 is a solution to (4x - 3)(5x + 7) = 0.

19. Multiple Choice What are the roots of the equation 0 = x2 - 10x - 39? DA 0, 39 B 10, 0 C 3, -13 D 13, -3

(100)(100)

7. Student A; Sample: A parabola can cross the x-axis once, twice, or not at all.

7. Student A; Sample: A parabola can cross the x-axis once, twice, or not at all.

(100)(100)

t = 0.82 secondst = 0.82 seconds

(57)(57)

(100)(100)

(100)(100)

(94)(94)

(95)(95)

(95)(95)

(96)(96)

3 43 4

(96)(96)

(97)(97)

(98)(98)Sample: (4 · 3_

4- 3)(5 · 3_

4+ 7) = 0 ( 43_

4 ) = 0Sample: (4 · 3_4

- 3)(5 · 3_4

+ 7) = 0 ( 43_4 ) = 0

(98)(98)


CHALLENGE

Have students write an inequality that satisfi es x < 0 OR x > 0 and that does not use absolute value. Sample: x 2 > 0

682 Saxon Algebra 1

Problem 9

Make sure that students factor the binomials completely before they determine the LCM. Remind them to fi nd the least expression that both binomials will divide into.

Problem 13

Error Alert Some students may think that 2 - r and r - 2 are the same expression. Remind them that they are opposites because their sum is 0. So, -1 must be factored out of one of the expressions in order to have a common denominator and in order to add the rational expressions that represent how far Mia walked.


Lesson 101 683

*20. Solve and check: x

_ 11 = 6 _

x - 5 . x = -6, 11

21. Does the graph of -8x2 - 12 = 3 - y open upward or downward? upward

22. Office Management Maria can complete all the copies in 1 hour. It takes Lachelle 2 hours. How long will it take them if they use two identical copiers and work together? 2_3 hours

23. Error Analysis Two students solve x - 8 _

x + 2 = x - 6

_ 3x + 6

. Which student is correct? Explain the error. Student A; Sample: Student B did not check to see that -2 is an extraneous solution.

Student A

(x - 8)(3x + 6) = (x + 2)(x - 6)3x2 - 18x - 48 = x2 - 4x - 12

2x2 - 14x - 36 = 02(x - 9)(x + 2) = 0{9}

Student B

(x - 8)(3x + 6) = (x + 2)(x - 6)3x2 - 18x - 48 = x2 - 4x - 122x2 - 14x - 36 = 0

2(x - 9)(x + 2) = 0{-2, 9}

24. Do the side lengths 3, 3 √ 3 , and 6 form a Pythagorean triple? no

25. Let P = (-2, 1), Q = (0, 2), R = (1, -2), and S = (-1, -3). Use the distance formula to determine whether PQRS is a rhombus. no

26. Multi-Step Find the product of

5x2y2

_ 3x3y3 ·

9xy2

_ 25xy3 using two different methods.

a. Solve the expression by multiplying first and then simplifying.

b. Solve the expression by simplifying each factor and then multiplying.

c. Explain which method you prefer. 3_5xy2 ; Sample: simplifying before multiplying,

because I can cancel out like terms before needing to multiply anything 27. Road Trip Carlos tracks the mileage for a road trip on his car’s odometer. The

total distance is 974.6 miles plus or minus 0.1 miles. Solve and graph the inequality ⎢x - 974.6� ≤ 0.1. 974.5 ≤ x ≤ 974.7;

974.7974.5 974.6

28. Multi-Step Amy skipped for 3x - 6 _

9x hours to get to her grandmother’s house that

was 2x2 - 4x _

7x3 miles away.

a. Find her rate in miles per hour. 6_7x miles per hour

b. If the rate is divided by 1 _ x2 , what is the new rate? 6x_

7 miles per hour

29. How do you write a remainder of 5 for a division problem that has a divisor of (3x2 + 7x + 8)? 5_

3x2 + 7x + 8

*30. What is the parent quadratic function defined to be? What is the shape of its graph and where is it located on the coordinate system? f(x) = x2; Its graph is a parabola opening upward with its vertex at the origin, (0, 0).

(99)(99)

(84)(84)

(99)(99)

(99)(99)

(85)(85)

(86)(86)

(88)(88)

(91)(91)

(92)(92)

(93)(93)

(Inv 10)(Inv 10)


LOOKING FORWARD

Solving multi-step absolute-value inequalities prepares students for

• Lesson 107 Graphing Absolute-Value Functions

Lesson 101 683

Problem 22

Error AlertStudents may not know how to write an equation that describes the situation. Encourage them to use a variable to represent the total number of copies. Then they can fi nd how much work can be done in 1 hour by each girl and can solve for the variable.

1 = x + 1 _ 2 x

1 = 3 _ 2 x

1 ( 2 _ 3 ) =

3 _ 2 x (

2 _ 3 )

2 _ 3 = x

Problem 24

To determine which value is the length of the hypotenuse c, encourage students to estimate the value of 3 √ 3 and compare that value to 6.

Problem 27

Guide the students by asking them the following questions.

“What does the variable represent in this absolute-value inequality?” Sample: It represents the total range of distance that Carlos may have driven, taking into account that his car’s odometer varies plus or minus 0.1 miles.

“To solve the absolute-value inequality, it must be written as what type of compound inequality? a conjunction

Problem 28

Encourage students to think in terms of miles ____

hour to make sure that

they divide correctly.


Saxon Algebra 1684

Warm Up

102LESSON

Solving Quadratic Equations

Using Square Roots

1. Vocabulary The of x is the number whose square is x. square root

Simplify.

2. √ � 81 9 3. - √ � 25 -5

4. √ � 24 2√ � 6 5. √ ��

9 _

49 3_7

Sometimes quadratic equations do not have linear terms. Quadratic equations in the form x2 = a, can be solved by taking the square root of both sides.

Example 1 Solving x2 = a

Solve each equation.

a. x2 = 25

SOLUTION

Find the square root of both terms.

x2 = 25

√ � x2 = ± √ � 25 Take the square root of both sides.

x = 5 or x = -5

You can combine the solutions using the ± symbol.

x = ±5

Check x2 = 25 x2 = 25

52 � 25 (-5)2 � 25

25 = 25 ✓ 25 = 25 ✓

b. x2 = -16

SOLUTION

Find the square root of both terms.

x2 = -16

√ � x2 = ± √ �� -16 Take the square root of both sides.

x ≠ ± √ �� -16 No real number squared can be negative.

There is no real-number solution.

When the quadratic equation is in the form ax2 + c = 0, the square root can be taken after the variable is isolated.

(13)(13)

(13)(13) (13)(13)

(46)(46) (46)(46)


Online Connection


Math Reasoning

Verify Show by factoring that the equation x2 = 25 has the solution ±5.

Math Reasoning

Analyze What is the relationship between squaring a number and taking the square root of a number?

Sample: Squaring and taking the root are inverse operations.

Sample: x2 = 25 x2 - 25 = 0

(x + 5)(x - 5) = 0x + 5 = 0 x - 5 = 0

x = -5 x = 5


MATH BACKGROUND

then x = ± √ � 10 . These are real solutions, not approximations.

Note, however, that solving by using square roots also has limitations. It only works when the equation can be arranged so that one side of the equation is a quadratic term and the other side is a constant. The equation cannot contain any linear terms.

LESSON RESOURCES



Master C102

Students have solved quadratic equations by factoring and graphing, but these methods have limitations. If quadratic equations cannot be factored, or when a graphing calculator can only provide irrational solutions as approximations, then solving quadratic equations by using the defi nition of square roots may be a better method.

For any non-negative real number b, if a 2 = b, then a = ± √ � b . So, if x 2 = 10,

Warm Up1

684 Saxon Algebra 1

102LESSON

Problem 3

Remind students to be mindful of the negative sign.

2 New Concepts

In this lesson, students will solve quadratic equations by isolating the squared term and then taking the square root of each side of the equation. As with solving by factoring or graphing, there can be one, two, or no solutions. If a > 0, then there are two solutions. If a = 0, then there is one solution. If a < 0, then there is no solution. The solutions can be rational or irrational, depending on the value of the non-variable side of the equation after the squared term is isolated.

Example 1

Remind students that when quadratic terms equal non-negative numbers there is a positive and negative solution.


Solve.

a. x 2 = 121 x = ±11

b. x 2 = -400 no solution


Lesson 102 685

Example 2 Solving ax2 + c = 0


a. x2 + 3 = 52

SOLUTION

Isolate the variable and solve.

x2 + 3 = 52

__ -3 __ -3 Subtraction Property of Equality

x2 = 49 Simplify.

√ � x2 = ± √ � 49 Take the square root of both sides.

x = ±7 Simplify.

Check x2 + 3 = 52 x2 + 3 = 52

72 + 3 � 52 (-7)2 + 3 � 52

49 + 3 = 52 ✓ 49 + 3 = 52 ✓

b. 4x2 - 100 = 0

SOLUTION

Isolate the variable and solve.

4x2 - 100 = 0

___+100 = ___+100 Addition Property of Equality

4x2 = 100 Simplify.

4x2

_ 4 =

100 _ 4 Division Property of Equality

x2 = 25 Simplify.

√�x2 = ± √�25 Take the square root of both sides.

x = ±5 Simplify.

Check

4x2 - 100 = 0 4x2 - 100 = 0

4(5)2 - 100 � 0 4(-5)2 - 100 � 0

4(25) - 100 � 0 4(25) - 100 � 0

100 - 100 � 0 100 - 100 � 0

0 = 0 ✓ 0 = 0 ✓

Numbers that are not perfect squares have irrational roots. Irrational solutions can be expressed in square root form: ± √�x . An approximate answer can be found using a calculator. To approximate √�10 on a

graphing calculator, press , and then

press .

Caution

When x2 equals a number other than 0, the equation has two solutions. Use the ± symbol after taking the square root.

Math Reasoning

Estimate How can √�10 be estimated?

Sample: 10 is between the perfect squares 9 and 16. Since it is closer to 9 and √�9 = 3, then √�10 is about 3.1 or 3.2.


The following is useful for auditory and verbal learners.

To help students make the connection that squaring and taking the square root are inverse operations, ask the following questions in this order:

“What is 4 plus 3?” 7“What is 7 minus 3?” 4

“What is 4 times 2?” 8“What is 8 divided by 2?” 4

“What is 4 squared?” 16“What is the positive square root of 16?” 4

Explain to students that they must use both square roots, positive and negative, when solving a quadratic equation, because both make the equation true.

INCLUSION

Lesson 102 685

Example 2

In order to take the square root of each side, one side must contain the squared term only. Use inverse operations to isolate x 2 .

Error Alert Students may rush to answer “no solution” when they see one side of the equation equal to a negative number. Stress the importance of working through inverse operations. Show that though - x 2 = -16 may fi rst appear to have no solutions, when the student has divided both sides by -1, it will become clear that the solutions are -4 and 4.


Solve.

a. 49 = x 2 - 15 x = ±8

b. 2 x 2 + 5 = 77 x = ±6


Saxon Algebra 1686

If no rounding instructions are given, round the approximation to the thousandths place.

Example 3 Approximating Solutions


a. x2 = 40

SOLUTION

x2 = 40

√�x2 = ± √�40 Take the square root of both sides.

Simplify the square root.

√�x2 = ± √��4 · 10 Find a factor that is a perfect square.

√�x2 = ± √�4 · √�10 Product Property of Radicals

x = ±2 √�10 Simplify.

Use a calculator to find the approximate value of √�10 .

x ≈ 2 · (3.16227766) Write the approximate value.

x ≈ ±6.32455532 Multiply.

x ≈ ±6.325 Round to the nearest thousandth.

Check x2 = 40 x2 = 40

(6.325)2 ≈Q 40 (-6.325)2 ≈Q 40

40.006 ≈ 40 ✓ 40.006 ≈ 40 ✓

b. 8x2 - 24 = 100

SOLUTION

Begin by isolating x2.

8x2 - 24 = 100

__+24 __+24 Addition Property of Equality

8x2 = 124 Combine like terms.

8x2

_ 8 =


x2 = 15.5 Simplify.

√ � x2 = ± √ �� 15.5 Take the square root of both sides.

x ≈ ±3.937003937 Find the approximate square root.

x ≈ ±3.937 Round to the nearest thousandth.

Check 8x2 - 24 = 100 8x2 - 24 = 100

8(3.937)2 - 24 ≈Q 100 8(-3.937)2 - 24 ≈Q 100

8(15.499969) - 24 ≈Q 100 8(15.499969) - 24 ≈Q 100

123.999752 - 24 ≈Q 100 123.999752 - 24 ≈Q 100

99.999752 ≈ 100 ✓ 99.999752 ≈ 100 ✓

Caution

Round after all computations have been made.

Caution

Remember to check both solutions.


ALTERNATE METHOD FOR EXAMPLE 3a

The approximation for √ � 40 can be found without using a calculator.

x = ± √ �� 4 · 10

x = ±2 · √ � 10

The number 10 falls between the perfect squares 9 and 16. The square roots of each are 3 and 4, so students can reasonably estimate that the square root of 10 is between 3.1 and 3.2.

686 Saxon Algebra 1

Example 3

The square root of a number that is not a perfect square will be an irrational number; that is, a nonrepeating and nonterminating decimal number.


Solve.

a. 62 = x 2 x ≈ ±7.874

b. 4 x 2 + 9 = 82 x ≈ ±4.272

TEACHER TIPPoint out that in Example 3a, ±2 √ � 10 are the exact solutions and ±6.324 are the approximate solutions. Students should not consider √ � 40 as the fi nal solution because it is not in simplifi ed form.



Lesson 102 687

Example 4 Application: Crafts

Malik covered a cube with exactly 864 square inches of self-stick vinyl. What is the side length of the cube?

SOLUTION

Use the formula to find the surface area of a cube: S = 6s2.

S = 6s2

864 = 6s2 Substitute 864 for S.

864

_ 6 =

6s2

_ 6 Division Property of Equality

144 = s2 Simplify.

± √ �� 144 = √ � s2 Take the square root of both sides.

±12 = s Simplify.

The longest possible side length of the cube is 12 inches.

Check S = 6s2

864 � 6(12)2

864 � 6(144)

864 = 864 ✓

Lesson Practice


a. x2 = 81 x = ±9

b. x2 = -36 no real solution

c. x2 + 5 = 54 x = ±7

d. 3x2 - 75 = 0 x = ±5

e. x2 = 72 x = ±8.485

f. 5x2 - 60 = 0 x = ±3.464

g. A golf ball is dropped from a height of 1600 feet. Use the equation 16t2 - 1600 = 0 to find how many seconds t it takes for the ball to hit the ground. 10 seconds

(Ex 1)(Ex 1)

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)

Simplify.

1. 4(2 p -2 q)2(3p3q)2 144p2q4 2. (7 √ � 8 ) 2 392

(40)(40) (76)(76)

Math Reasoning

Analyze Why is -12 square inch not a possible answer?

Sample: A linear measurement cannot be negative.


In Problem 13, explain the meaning of the word vinyl. Say:

“Vinyl is a type of plastic material that has many uses. It can be used to cover and protect the fl oor.”

Ask students to describe other uses of vinyl. Sample: Toys and furniture are made of vinyl.

ENGLISH LEARNERS

Lesson 102 687

Example 4

Explain that the surface area of a cube equals 6 s 2 because there are six sides, each with an area of s 2 units. Remind students that since distances are always positive, they can disregard the negative solution.

Extend the Example

“Suppose that Malik does not want to cover the bottom of the cube because it will not be seen. Write and solve a quadratic equation to fi nd the side length of the largest cube he can now cover.” 864 = 5 s 2 ; s ≈ 13.145 inches


Sue covers a cube with exactly 1536 square inches of self-stick vinyl. What is the side length of the largest cube that Sue can cover completely? 16 inches

Lesson Practice

Problem d

Scaff olding Ask students how they can isolate 3 x 2 , then how they can isolate x 2 , and fi nally, how they can isolate x.

Problem e

Error Alert Students might divide by 2 and answer x = ±36. A quick check of their answer will show that this is not correct.



“What type of quadratic equation can be solved by using square roots?” Sample: a quadratic that has no linear term

“What must be true about a if x 2 = a does not have two solutions?” Sample: a is either 0 or negative.


Saxon Algebra 1688

*3. Error Analysis Two students want to find the length of the sides of a square with an area that is 720 square meters less than 1161 square meters. Which student is correct? Explain the error.

Student A Student B

x2 + 720 = 1161 ___ -720 ___ -720

x2 = 441 √ � x2 = ± √ �� 441

x = ±21

x2 + 720 = 1161 ___ -720 ___ -720

x2 = 441 √ � x2 = ± √ �� 441

x = ±21The sides of the square are ±21 m.

The sides of the square are 21 m.

*4. Multi-Step Dominic wants to fence the perimeter of his property. The property is in the shape of a square. The area of the yard is 12,600 ft2, and the area of the house is 1800 ft2. a. Write an equation to find the length of the sides of the property.

b. Solve the equation. x = 120 ft

c. How many feet of fencing will Dominic need? 480 ft

*5. Banking Serena places $1000 in an interest-earning account where the interest compounds annually. After two years, there is $1123.60 in the account. Use the formula $1000(1 + r)2 = $1123.60 to find the interest rate of the account. 6%

*6. Verify True or False: If 8x2 - 72 = 0, then x = ±3. If the answer is false, provide the correct answer. true

*7. Estimate Find the length of the side of a square with an area of 680 square kilometers. Round to the nearest thousandth. 26.077 km

8. Solve and graph the inequality ⎢x ⎢

_ 3 + 6 < 13. -21 < x < 21;

*9. Coordinate Geometry One side of a rectangle drawn in the coordinate plane has points whose y-coordinates are 7 and whose x-coordinates are the solutions of the inequality ⎢x + 1⎢ - 8 ≤ -4. Another side has points whose x-coordinates are -5 and whose y-coordinates are solutions of the inequality ⎢y - 4⎢ + 6 ≤ 9. a. Solve the inequality ⎢x + 1⎢ - 8 ≤ -4. -5 ≤ x ≤ 3

b. Solve the inequality ⎢y - 4⎢ + 6 ≤ 9. 1 ≤ y ≤ 7

c. What are the coordinates of the four vertices of the rectangle? (-5, 7), (3, 7),(3, 1), (-5, 1)

10. Find the product of (x - 7)(-7x2 - x + 7) using the vertical method.

11. Graph the function y = 4x2 + 6.

12. Water Balloons A water balloon is dropped from a third-story window. Its height in feet is represented by h = -16t2 + 30. How high is the balloon after 1 second? 14 feet

(102)(102)3. Student B; Sample: Both have worked the problem correctly, but Student A did not realize that a negative measurement is impossible in this situation.

3. Student B; Sample: Both have worked the problem correctly, but Student A did not realize that a negative measurement is impossible in this situation.

(102)(102)

x2 = 12,600 + 1800 x2 = 12,600 + 1800

(102)(102)

(102)(102)

(102)(102)

(101)(101) 200 10-10-20 200 10-10-20

(101)(101)

(58)(58) -7x3 + 48x2 + 14x - 49-7x3 + 48x2 + 14x - 49

(96)(96)

x

y

O

2020

2 4

-10

-2-4

11.

x

y

O

2020

2 4

-10

-2-4

11.

(96)(96)

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688 Saxon Algebra 1

Practice3


Problem 8


“How can you isolate the fraction?” Subtract 6 from both sides.

“How can you isolate the numerator?” Multiply both sides by 3.

“Is 0 contained in the solution set?” yes


Lesson 102 689

*13. Multi-Step When the temperature t of the gas argon is within 1.65 degrees of -187.65°C, it will be in a liquid form. This can be modeled by the absolute-value inequality ⎢t -(-187.65)⎢ < 1.65. a. Solve and graph the inequality ⎢t -(-187.65)⎢ < 1.65. -189.3 < t < -186

b. One endpoint of the graph represents the boiling point of argon—the temperature at which argon changes from liquid to gas. The other endpoint represents the melting point—the temperature at which argon turns from solid to liquid. The higher temperature is the boiling point, and the lower temperature is the melting point. What is the boiling point of argon? What is the melting point? -186°C; -189.3°C

14. Factor. a. x2 + 10x + 25 (x + 5) 2

b. x2 - 25 (x + 5)(x - 5)

15. Measurement The sides of a triangle are labeled 5x inches, 4y inches, and 20 inches. Jonas wrote an inequality that satisfies the Triangle Inequality Theorem: 5x + 4y > 20. Graph the inequality.

16. Art Supplies Tim plans to go shopping for new paper and paint for his students, and he does not want to spend more than $40. Each pack of paper costs $2, and each set of paints costs $10. Write an inequality that describes this situation, and graph it on a graphing calculator. 2x + 10y ≤ 40

17. Solve x(x + 12) = 0. {0, -12}

18. Verify Show that x = 1 is an extraneous solution to 1 _ x - 1

= 3 _

2x - 2 .

19. Multiple Choice Solve 2 _ x - 3

= x _ 9 . B

A {3, 6} B {-3, 6}

C {3, -6} D {6}

20. Add m

_ m2 - 4 + 2 _ 3m + 6

. 5m - 4__3(m + 2)(m - 2)

*21. Error Analysis Students were asked to write a quadratic equation that had no solution. Which student is correct? Explain the error.

Student A

f (x) = x2 - 3x + 12

Student B

f (x) = x2 + 11x + 11

*22. Rocket Malachi shot a rocket for his science project. The path of the rocket’s movement formed a parabola given by the quadratic equation h = -16t2 + 4t + 10, where h is the height in feet and t is the time in seconds. Find the maximum height of the path the rocket makes and the time t when the rocket hits the ground. Round to the nearest hundredth. h = 10.25 feet and t = 0.93 seconds

(101)(101)

-186-188-19013a.

-186-188-19013a.

(Inv 9)(Inv 9)

(97)(97)

3 43 4

x

y

O

8

4

4 8

-4

-4-8

-8

15.

x

y

O

8

4

4 8

-4

-4-8

-8

15.

(97)(97)

16.16.(98)(98)

(99)(99)

18. Sample: 1_1 - 1

=

3_2(1) - 2

; 1_0

= 3_0,

which is undefined. This shows that 1 is an extraneous solution.

18. Sample: 1_1 - 1

=

3_2(1) - 2

; 1_0

= 3_0,

which is undefined. This shows that 1 is an extraneous solution.

(99)(99)

(95)(95)

(100)(100)

21. Student A; Sample: Student B wrote an equation that forms a parabola that crosses the x-axis twice, so it has two solutions.21. Student A; Sample: Student B wrote an equation that forms a parabola that crosses the x-axis twice, so it has two solutions.

(100)(100)


Lesson 102 689

Problem 15

Error AlertStudents may forget to use a dashed line instead of a solid line. Remind them to double- check the inequality symbol to see if the boundary line contains solutions.

Problem 16

Say, “If I do not want to spend more than $40, then I want to spend less than $40, but I’m also willing to spend exactly $40.”

Problem 17

Error AlertStudents often forget to set a single x-factor equal to 0 because the equation formed by using the Zero Product Property does not require further steps. They should always write down x = 0, even if they think they will remember to include it at the end.

Problem 21

Extend the Problem

“Using the discriminant, how can you tell that the solutions of Student B’s function will have a solution?” Sample: Because b 2 - 4ac ≥ 0.


Saxon Algebra 1690

*23. Solve x2 + 12x + 40 = 0 by graphing. no solution

24. Find the midpoint of the line segment with the endpoints (13, -3) and (-7, -3). (3, -3)

25. Factor -3y3 - 9yz + 5y2 + 15z. (-3y + 5)( y2 + 3z)

26. Multi-Step Mr. Tranh’s lawn has an area of 144 square feet. The length of his yard is 7 feet more than the width. What are the dimensions of his yard? a. Write a formula to find the dimensions of the yard and describe how you will

solve it.

b. What are the dimensions of the yard? 16 feet long and 9 feet wide.

27. Do the side lengths 3, 7, and 8 form a Pythagorean triple? no

28. Running It took Wayne x _

2x2 + x - 15 minutes to run to the gym that was

9x _

4x - 10 + 5x2

_ 3x + 9 miles away. Find his rate in miles per minute.

20x2 - 23x + 81__6

29. Multi-Step Raj is measuring the area of his rectangular living room. He determined

that the area is -64x + x3 - 2x2 + 128 square feet. The width is (x2 - 64)

_

(x + 8) feet.

a. Simplify the expression for the width of the living room. (x - 8) feet

b. Find the expression for the length. x2 + 6x - 16 feet

30. Generalize What is the difference between solving an absolute-value equation with operations on the outside and solving absolute-value equations with operations on the inside? Sample: When the operations are on the inside, write two equations to represent the absolute-value equation and solve them. When the operations are on the outside, isolate the absolute value first, then write two equations to represent the absolute-value equation and solve them.

(100)(100)

(86)(86)

(87)(87)

(89)(89)26. x(x + 7) =

144; Sample: the formula needs to be set in the form ax2 + bx +

c = 0 in order to solve for x

26. x(x + 7) =

144; Sample: the formula needs to be set in the form ax2 + bx +

c = 0 in order to solve for x

(85)(85)

(92)(92)

miles per minutemiles per minute

(93)(93)

(94)(94)

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Solving quadratic equations using square roots prepares students for

• Lesson 104 Solving Quadratic Equations by Completing the Square

• Lesson 110 Using the Quadratic Formula

The area of a square is 225 square inches. Write an equation that can be used to fi nd the dimensions of the square. Find the dimensions. x2 = 225; 15 inches by 15 inches

CHALLENGE LOOKING FORWARD

690 Saxon Algebra 1

Problem 26

Extend the Problem

“Write an algebraic expression for the perimeter of the yard in terms of its width. What is the perimeter?” 4x + 14; 50 feet

Problem 30

To help students form their answer, have them write and solve a simple equation of each type, such as ⎢x� + 2 = 6 and ⎢x + 2� = 6.


Lesson 103 691

Warm Up

103LESSON Dividing Radical Expressions

1. Vocabulary The number or expression under a radical symbol is called the __________. radicand

Simplify. All variables represent non-negative numbers.

2. √ �� 150 5√ � 6 3. 3 √ � 72 18√ � 2

4. √ �� 48x3 4x√ � 3x 5. √ � 12 · √ � 15 6√ � 5

When dividing radical expressions, use the Quotient Property of Radicals.

n √ � a _

b =

n √ � a _

n √ � b

, where b ≠ 0.

A radical expression in simplest form cannot have a fraction for a radicand or a radical in the denominator. To rationalize a denominator means to use a method which removes radicals from the denominator of a fraction. Using this method, a fraction is multiplied by another fraction that is equivalent to 1 in order to remove the radical from the denominator.

Example 1 Rationalizing the Denominator

Simplify.

√ � 7 _

3

SOLUTION

Use the quotient property. Then rationalize the denominator.

√ � 7 _

3

= √ � 7

_ √ � 3

Quotient Property of Radicals

= √ � 7

_ √ � 3

· √ � 3

_ √ � 3

Multiply the expression by a factor of 1 that will make the radicand in the denominator a perfect square.

= √ �� 7 · 3

_ √ �� 3 · 3

Multiplication Property of Radicals

= √ � 21

_ √ � 9

Multiply.

= √ � 21

_ 3 Simplify the square root.

(13)(13)

(61)(61) (61)(61)

(61)(61) (76)(76)


Math Language

In the expression n √ � a _

b ,

a _ b

is the radicand and n

is the index number.

Math Reasoning

Verify Multiply √ � 21 _ 3

by √ � 3 _

√ � 3 to show that

the product equals the original expression √ � 7

_ √ � 3

.

Sample: √ �� 21_3

·√ � 3_√ � 3

=√ �� 21 · 3 _3 · √ � 3

=√ �� 63_3√ � 3

=√ � 9 · √ � 7_

3√ � 3=

3√ � 7_3√ � 3

=√ � 7_√ � 3

= √ � 7_3


LESSON RESOURCES



Master C103

When a radical expression is not in simplest form because radicands contain perfect squares and there is a radical in the denominator, it is best to factor out the perfect squares before rationalizing the denominator.

To simplify an expression of the form

a ___ √ � b

, most students will multiply the numerator and denominator by √ � b without much hesitation.

For example, 1 ___ √ � 8

can be simplifi ed by

multiplying by √ � 8

___ √ � 8

. However, multiplying

by √ � 2

___ √ � 2

will also make the radicand in the

denominator a perfect square.

1 ___ √ � 8

· √ � 2 ___ √ � 2

= √ � 2 ____

√ � 16 =

√ � 2 ___ 4

In this instance, using this method saves an extra step of simplifying in the numerator.

MATH BACKGROUND

Warm Up1

103LESSON

Lesson 103 691

Problem 3

Writing the radicand as a product of 36 times 2 is the quickest way to simplify the expression.

2 New Concepts

Radical expressions are in simplest form if there are no perfect square factors in a radicand, no fractions in a radicand, and no radicals in a denominator.

Example 1

The goal in rationalizing a denominator is to make the radicand in the denominator a perfect square. The student can achieve this by multiplying the numerator and denominator by the radical in the denominator.


Simplify √ � 15 ___ 2 . √ �� 30 ____

2


Saxon Algebra 1692

Example 2 Rationalizing a Variable Denominator

Simplify √ � 5 _ x . All variables represent non-negative numbers.

SOLUTION

Use the quotient property. Then rationalize the denominator.

√ � 5 _ x =

√ � 5 _

√ � x Quotient Property of Radicals

= √ � 5

_ √ � x

· √ � x

_ √ � x

Multiply the expression by a factor of 1 that will make the radicand in the denominator a perfect square.

= √ �� 5 · x

_ √ �� x · x

Multiplication Property of Radicals

= √ � 5x

_ √ � x2

Multiply.

= √ � 5x

_ x Simplify the square root.

A radical expression is completely simplified when the radicand contains no perfect square factors other than 1, and there are no fractions in the radicand.

Example 3 Simplifying Before Rationalizing the Denominator

Simplify √ �� 72x4

_ 3 √ �� 20x3

. All variables represent non-negative numbers.

SOLUTION

Simplify the numerator and denominator.

√ �� 72x4

_ 3 √ �� 20x3

= √ �� 36 · 2 · x2 · x2

__ 3 √ �� 4 · 5 · x2 · x

Factor out perfect squares, if possible.

= 6x2 √ � 2

__ 2 · 3 · x √ � 5x

Simplify the radical expressions.

= 6x2 √ � 2

_ 6x √ � 5x

Simplify the denominator.

= x √ � 2

_ √ � 5x

Divide out common factors in the numerator and denominator.

= x √ � 2

_ √ � 5x

· √ � 5x

_ √ � 5x

Rationalize the denominator.

= x √ �� 10x

_ 5x

Simplify.

= √ �� 10x

_ 5 Divide out common factors in the numerator

and denominator. Online Connection



692 Saxon Algebra 1

Example 2

In this example students rationalize a radical in a denominator with a variable as a radicand.


Simplify √ � 7 _ y . All variables represent non-negative

numbers. √ � 7y

____ y

Example 3

Students learn to simplify radicals in the denominator before rationalizing the denominator.

Extend the Example

“Show that the simplifi ed expression has the same value as the original expression by substituting 2 for x in each expression and fi nding the value. Round the value to the thousandths place.” ≈ 0.894


Simplify 2 √ �� 50 y 6

______ √ �� 28 y 5

. All variables

represent non-negative numbers.

5 √ �� 14y

______ 7

Error Alert Students may not extend a radical symbol far enough to completely cover the radicand. It may be helpful to have students draw the radical symbol so that the top right side drops down a bit, “enclosing” the radicand.


Lesson 103 693

The conjugate of an irrational number in the form a + √�b is a - √�b . The conjugate is used to rationalize the denominator of a fraction when the denominator is a binomial with at least one term containing a radical.

Example 4 Using Conjugates to Rationalize the Denominator

Simplify.

a. 3 _

4 + √ � 5

SOLUTION

Find the conjugate of the denominator. Use the conjugate to write a factor equivalent to 1. Multiply the fraction by the factor.

3 _

4 + √ � 5

3 _

4 + √ � 5 ·

(4 - √ � 5 ) _

(4 - √ � 5 ) The conjugate of 4 + √ � 5 is 4 - √ � 5.

= 12 - 3 √ � 5

__ 16 - 4 √ � 5 + 4 √ � 5 - 5

Use the Distributive Property and the FOIL method to multiply numerators and denominators.

= 12 - 3 √ � 5

_ 11

Combine like terms and simplify.

= 12 _ 11

- 3 √ � 5

_ 11

Write the solution as two fractions with the same denominator.

b. 2 _ √ � 3 + 1

SOLUTION

2 _ √ � 3 + 1

2 _ √ � 3 + 1

· ( √ � 3 - 1)

_ ( √ � 3 - 1)

The conjugate of √ � 3 + 1 is √ � 3 - 1.

= 2 √ � 3 - 2

__ 3 - √ � 3 + √ � 3 - 1

Use the Distributive Property and the FOIL method to multiply numerators and denominators.

= 2 √ � 3 - 2

_ 2 Combine like terms and simplify.

= 2( √ � 3 - 1)

_ 2 Factor the numerator. Divide.

= √ � 3 - 1 Simplify.

Math Reasoning

Analyze Why must conjugates be used when rationalizing denominators with radicals containing binomials?

Sample: The denominatoris rationalized to remove any radicals from the denominator. The product of conjugates does not contain any radicals.


Help students see how multiplying a pair of conjugates that contain radicals eliminates radicals by completing this chart with them.

Conjugate F O I L

(3 + 5)(3 - 5) 9 -15 15 -25

(x + 2)(x - 2) x2 -2x 2x -4

2 + √ � 3 2 - √ � 3 4 -2 √ � 3 2 √ � 3 -3

√ � 4 + √ � 3 √ � 4 - √ � 3 4 - √ � 12 √ � 12 -3

Ask, “What is true about all the outer and inner terms?” Their sum is 0.

Ask, “Why do the fi rst and last terms of the product never have a radical sign?” Each is the square of a square root, which is no longer a radical.

INCLUSION

Lesson 103 693

Example 4

The conjugates used to rationalize binomial denominators are called conjugate pairs.

TEACHER TIPIn Example 4b, show students that they do not have to distribute in the numerator until they have seen if the factor that would be distributed can be simplifi ed with the simplifi ed value of the denominator.


Simplify.

a. 4 ________ 2 + √ � 10

- 4 __ 3 +

2 √ �� 10 _____

3

b. 6 _______ √ � 7 + 2

2 √ � 7 - 4



Saxon Algebra 1694

Lesson Practice

Simplify. All variables represent non-negative numbers.

a. √ � 5 _

3

√ �� 15_3 b. √ ��

11 _ x √ �� 11x_

x

c. √ �� 6x6

_ √ �� 27x

x2 √ � 2x_3

d. 3 _

5 - √ � 6 15 + 3 √ � 6_

19 or 15_19

+ 3√ � 6_19

e. 3 _

√ � 7 - 1

√ � 7+ 1_2 or √ � 7_

2+ 1_

2

(Ex 1)(Ex 1) (Ex 2)(Ex 2)

(Ex 3)(Ex 3) (Ex 4)(Ex 4)

(Ex 4)(Ex 4)

*1. Simplify 35 _ √ � 7

. 5√ � 7

2. Solve 8 _

x - 1 =

x _ 7

. x = -7, 8

*3. Error Analysis Two students simplified the following expression. Which student is correct? Explain the error. Student A; Sample: Student B did not use a conjugate to rationalize the denominator.

Student A Student B

1 _ 3 + √ � 2

1 _ 3 + √ � 2

· 3 - √ � 2

_ 3 - √ � 2

3 - √ � 2

_ 7

1 _ 3 + √ � 2

1 _ 3 + √ � 2

· √ � 2

_ √ � 2

√ � 2 _

3 √ � 2 + 2

*4. Skydiving A 150-pound skydiver reaches terminal velocity after free-falling for a number of seconds. The formula for the terminal velocity V of a skydiver

(in feet per second) can be estimated by the formula V = √ �� 2W

_ 0.0063 , where W equals

the weight of the skydiver in pounds. Write a rational expression for the terminal velocity of the skydiver. 1000√ �� 21_

21 ft/s

5. What is 400% of 40? Use a proportion to solve. 160

*6. Write Is 2 √ � 3

_ √ � 2

in simplest form? Explain.

*7. Predict If 2 ÷ √ � 2 is √ � 2 , and 3 ÷ √ � 3 is √ � 3 , what is a good prediction of what the quotient of 239 ÷ √ �� 239 might be? √ �� 239

8. Multi-Step The area of a square is 9x2. The length of one of its sides plus 32 is 47. a. What is the length of one of its sides? 15 units

b. What is the area of the square? 225 square units

c. What is x? 5

(103)(103)

(99)(99)

(103)(103)

(103)(103)

(42)(42)

(103)(103)no; Sample: The radical in the denominator needs to be rationalized.no; Sample: The radical in the denominator needs to be rationalized.

(103)(103)

(102)(102)


Explain the meaning of the word terminal. Say:

“The word terminal often means to stop or to end. For instance, there will be a train terminal at the end of a set of railroad tracks. The word terminal is related to the word terminate.”

Velocity is related to speed. Have students look at Problem 4. Explain that when

something is falling through air, the speed of the fall continues to increase because of gravity. But after a time, the increase stops, or terminates, because of the slowing effect of pushing through the air, and the speed of the fall becomes constant. This speed is the terminal velocity.

ENGLISH LEARNERS

694 Saxon Algebra 1

Lesson Practice

Problem a

Error Alert After students simplify, they may divide the 15 in the numerator by the 3 in the denominator. Have students use their calculators to see that this does not lead to an equivalent expression.

Problem c

Scaff olding Students can use the Quotient Property to write the

expression as √ �� 6 x 6 ___ 27x and reduce

the radicand to √ �� 2 x 5 ___ 9 . Then use

the Quotient Property again to

write the expression as √ �� 2 x 5

____ 3 .



“What conditions must be true for a radical expression to be considered simplifi ed?” Sample: The radicand cannot include a fraction or a perfect square and a radical cannot appear in a denominator.

“What does it mean to rationalize a denominator?” Sample: Multiply the radicand by a form of 1 that eliminates the radical from the denominator.

Practice3


Problem 5


“Is 40 a part or a whole?” a whole

“What is 400% as a fraction?” 400 ___ 100

or 4 __

1

“How do you solve a proportion?” by cross-multiplying


Lesson 103 695

*9. Time and Distance A stone is dropped from a height of 450 feet. Use the equation 25t2 - 450 = 0 to find how many seconds it takes for the stone to hit the ground. ≈4.243 seconds

10. Solve 4x + 2y = 22

6x - 5y = 9

by substitution. (4, 3)

*11. Verify True or False: 5x2 + 125 = 0; x = ±5. If the answer is false, provide the correct answer. false; The correct answer is ±5√ �� -1.


12. -6⎢x⎢ + 20 ≥ 2 -3 ≤ x ≤ 3 40 2-2-4

13. 14x + 2y > 6

*14. Error Analysis Two students solve the inequality -12⎢x⎢ - 15 > -39. Which student is correct? Explain the error. Student B; Sample: Student A did not reverse the inequality symbol when dividing each side by -12.

Student A Student B

-12⎢x⎢ - 15 > -39-12⎢x⎢ > -24

⎢x⎢ > 2x < -2 OR x > 2

-12⎢x⎢ - 15 > -3912⎢x⎢ + 15 < 39

12⎢x⎢ < 24⎢x⎢ < 2

-2 < x < 2

*15. Geometry Find the length of the line segment that is the graph of the inequality ⎢ x

_ 7 + 6 - 5 ≤ 4. 126

*16. Tennis The diameter d of a tennis ball should vary no more than 1

_ 16 inch from 2 5

_ 16 inches. Write and solve an absolute-value inequality that models the

acceptable diameters. What is the greatest acceptable diameter?

17. Graph the function y = 10x2 - 20.

18. Soccer The height h in meters of a kicked soccer ball is represented by the function h = -5t2 + 20t, where t stands for the number of seconds after the ball is kicked. When is the ball on the ground? It is on the ground at 0 and 4 seconds.

19. Data Analysis A teacher graphed the test grades. He found that the distribution formed a parabola. Solve the equation 0 = x2 - 170x + 7000 to find its roots. 70, 100

20. Write What are the other names for the x-intercepts of a function? Sample: zeros or roots

21. Multiple Choice What is the equation of the parabola that passes through the points (0, 2), (-2, 6), and (6, 14)? D A y = x2 - x + 2 B y = -

1 _ 2 x2 - x + 2

C y = 1 _ 2 x2 + x - 2 D y = 1 _

2 x2 - x + 2

(102)(102)

(59)(59)

(102)(102)

(101)(101)

(97)(97)

13.

x

y

O4

-4

-4

13.

x

y

O4

-4

-4

(101)(101)

(101)(101)

(101)(101)

16. ⎢d -4 44_ ⎢ ≤ _

_4

≤ d ≤4 44_8

4 44_84

16. ⎢d -4 44_ ⎢ ≤ _

_4

≤ d ≤4 44_8

4 44_84

(96)(96)

17.

x

y

O2 4-2-4

10

2017.

x

y

O2 4-2-4

10

20

(98)(98)

(98)(98)

(100)(100)

(100)(100)


Lesson 103 695

Problem 15

Error AlertStudents may subtract the smaller absolute value from the larger when fi nding the length of the segment and get an answer of 105 - 21 = 79. Tell them to subtract the value of the coordinate to the left of zero from the value of the coordinate to the right of zero: 21 - (-105) = 126.

Problem 18

Extend the Problem

“When will the ball be at its maximum height? What is that height, given that it is being measured in meters?” 2 seconds, 20 meters

Problem 21

Students can eliminate choice C because the constant is the y-intercept, which is 2, not -2. They can also eliminate choice B by quickly sketching the three points to see that the parabola opens upward.


Saxon Algebra 1696

22. Factor 2x2y + 4xy - 7xyz - 14yz. (2xy - 7yz)(x + 2)

23. Multiply 90

_ 24a • 6a2b2

_ 25b . 9ab_

10

24. Do the side lengths 15, 36, and 39 form a Pythagorean triple? yes

25. Multi-Step A plane left the airport and traveled west, with a tailwind, at a cruising speed of 230 miles per hour for 300 miles. After dropping passengers off, the plane traveled east at the same cruising speed, but into a headwind, for 220 miles before landing for fuel. a. Justify Write an expression for the time going west and another expression for

the time going east. Explain how you came up with these expressions.

b. Add the expressions and simplify. 119,600 - 80w__(230 + w)(230 - w)

c. What does the simplified expression represent? Sample: the total time the plane flew in both directions

26. Gardening Jasmine has a rectangular garden with an area of (x2 - 14x + 45) square feet and a length of (x - 5) feet. What is the width of her garden? (x - 9) feet

27. Multi-Step A bike rental company charges $8 for each bike rental plus $10 for each hour it is rented. A couple has budgeted $66 for both of them to rent bikes. They hope that the total cost is within $10 of their budget. a. Write an absolute-value equation for the minimum and maximum number of

hours the couple can ride bikes. ⎢2(8 + 10x) - 66⎢ = 10

b. What is the minimum and maximum number of hours the couple can ride bikes? 2 hours, 3 hours

28. Write How do you find 4y - 5

_ 6 as the difference of two rational expressions?

29. Can x2 + x + 1 be factored? Explain.

30. If a quadratic function has been vertically stretched, does that mean the parabola is wider or narrower than the parent quadratic function, f(x) = x2? narrower

(87)(87)

(88)(88)

(85)(85)

(90)(90)

(93)(93)

(94)(94)

(95)(95)

(Inv 9)(Inv 9)

(Inv 10)(Inv 10)

29. It cannot be factored. The only whole-number factors of 1 are ±1, and neither will produce a middle term of x.

25a. west: 300_230 + w

; east: 220_230 - w

; Sample: The numerators represent distance and the denominators represent rate. Add the wind speed to the rate when the plane is going with the wind and subtract it from the rate when the plane is going against the wind.

28. Write each term in the numerator separately over the common denominator, and then simplify, if possible, to get

4y_6

- 5_6 or

2y_3

- 5_6.


Dividing radical expressions prepares students for

• Lesson 106 Solving Radical Equations

• Lesson 114 Graphing Square-Root Functions

Have advanced students simplify these expressions by making the radicand a perfect cube.

4 ___ 3 √ � 2

2 3 √ � 4

1 ___ 3

√ � 3

3 √ � 9 ___

3


696 Saxon Algebra 1

Problem 23

Because the coeffi cients are large, encourage students to fi rst divide out common factors before multiplying.

Problem 27


“What is the expression for the cost of one bike?” 8 + 10x

“What is the expression for the cost of two bikes?” 2(8 + 10x)

“Which goes inside the absolute-value bars with 2(8 + 10x): 66 or 10? Why?” 66; Sample: $66 is the budget, and the difference between the actual cost and the budget is what is limited.


Lesson 104 697

Warm Up

104LESSON

1. Vocabulary A is a trinomial that is the square of a binomial.perfect-square trinomial

Simplify.

2. ( 8 _

3 )

2

64_9 3. ( 12 _

3 )

2

- 5 11

Solve.

4. x2 + 6x + 9 = 0 x = -3 5. x2 - 18x + 81 = 0 x = 9

The product of a binomial square is a perfect-square trinomial.

Binomial Square Perfect-Square Trinomial

(x - 7) 2 x2 - 14x + 49

(x + 3) 2 x2 + 6x + 9

Completing the square is a process used to form a perfect-square trinomial.

Completing the Square

Complete the square of x2 + bx by adding ( b _ 2 )

2 to the expression.

x2 + bx + Example: x2 + 6x +

x2 + bx + ( b _ 2 )

2

x2 + 6x + ( 6 _ 2 )

2

x2 + bx + ( b2

_ 22

) x2 + 6x + (3)2

(x + b _ 2 )

2

x2 + 6x + 9

(x + 3) 2

Example 1 Completing the Square

Complete the square.

x2 + 8x

SOLUTION

x2 + 8x +

x2 + 8x + ( 8 _ 2 )

2

Add the square of 8 divided by 2.

x2 + 8x + (4)2 Simplify the fraction.

x2 + 8x + 16 Simplify.

(60)(60)

(3)(3) (4)(4)

(98)(98) (98)(98)


Solving Quadratic Equations by

Completing the Square

Online Connection


Hint

The last term of the binomial is doubled to get the coefficient of the middle term of the trinomial and squared to get the last term of the trinomial.


LESSON RESOURCES



Master C104, E104

Students have already learned that when solving quadratic equations there is a potential for more than one value of x. They have learned to factor trinomials.

For any real number n, there exists a value x, where n + x will equal a perfect-square. If the expression ax 2 + bx is treated the same as a real number, it can be assumed that there exists a value of c where ax 2 + bx + c is a perfect-square trinomial. Turning the expression ax 2 + bx into a perfect square

makes it easier to fi nd the square root and thus to isolate the variable x.

In Lesson 110, students learn that the quadratic formula is found by solving the general quadratic equation ax2 + bx + c = 0 by completing the square. This lesson is prerequisite to the understanding of the derivation of the quadratic formula. Since any quadratic can be solved using the quadratic formula, it can also be solved by completing the square.

MATH BACKGROUND

Warm Up1

104LESSON

Lesson 104 697

Problem 5

Encourage students to try factoring the trinomial before using the quadratic formula.

2 New Concepts

In this lesson, students learn how to form a perfect-square trinomial to solve quadratic equations.

Example 1

Make sure students are dividing the value for b by 2 before squaring.



x 2 - 12x x 2 - 12x + 36

TEACHER TIPFor students who are having trouble understanding why they

need to add ( b __ 2 ) 2

to complete the

square, have them fi nd (x + b __ 2 )

2

to get x 2 + bx + ( b __ 2 ) 2

.


Saxon Algebra 1698

Exploration Exploration Modeling Completing the Square

Algebra tiles are used to visualize the process of completing the square. Use algebra tiles to model x2 + 4x.

+ + + + +

a. Can these tiles be used to form a square? Explain. no; Sample: There is no way to organize the tiles to form a perfect square without 1-tiles.

b. What type of tile could be added to form a square? How many are needed? Make a drawing. four 1-tiles

c. What is the value of the tiles in this square? x2 + 4x + 4

d. What is the factored form of the trinomial? What is the length of a side of the square? (x + 2)2; x + 2

e. Write Does the sign of the coefficient of the x-term determine the sign of the constant? Explain. no; Sample: The last term will always be positive because the square of any nonzero real number is always positive.

Completing the square is used to solve quadratic equations. Once the square is completed, the equation is solved by finding the square root of both sides.

Example 2 Solving x2 + bx = c by Completing the Square

Solve by completing the square.

a. x2 + 10x = 11

SOLUTION


x2 + 10x = 11

x2 + 10x + = 11

x2 + 10x + ( 10

_ 2 )

2

= 11 + ( 10

_ 2 )

2

Complete the square. Add the missing value to both sides of the equation.

x2 + 10x + (5)2 = 11 + (5)2 Simplify the fraction.

x2 + 10x + 25 = 11 + 25 Simplify.

(x + 5)2 = 36 Factor the left side. Simplify the right side.

Solve using square roots.

√ �� (x + 5) 2 = ± √ � 36 Take the square root of both sides of the equation.

x + 5 = ± 6 Simplify.

x + 5 = -6 or x + 5 = 6 Write as two equations.

__ -5 = __ -5 __ -5 = __ -5 Subtraction Property of Equality

x = -11 or x = 1 Simplify.

b.b.

Hint

The square root of a number squared is that number. So, √ �� (x + 5)2 = √ �� (x + 5)(x + 5)

= (x + 5).

Math Reasoning

Analyze How would the drawing be different for completing the square of x2 - 4x?

The x-squared tiles and 1-tiles would be the same, but the x-tiles have a “-” to indicate being negative.

Math Reasoning

Analyze Why is 25 added to both sides of the equation?

Sample: to keep it equivalent to the original equation


ENGLISH LEARNERS

For the exploration, explain the meaning of the word model. If possible, show students a road or city map. Say:

“A model is a pattern or example. A city map is a model of a city showing the location of homes and businesses.”

Discuss other models such as those used in hobbies, architecture, and museums. Have volunteers give examples of models. Sample: model airplanes

698 Saxon Algebra 1

ExplorationExploration

In this Exploration, students use algebra tiles to see how a square

is completed by adding ( b _ 2 ) 2 .

Extend the Exploration

Have students use algebra tiles to complete the square for x 2 + 6x.

+ + + +

+

+

+

+

+

+

+

+

+

+

+

+


Lesson 104 699

Check x2 + 10x = 11

(-11)2 + 10(-11) � 11 Substitute -11 for x.

121 - 110 � 11 Simplify using the order of operations.

11 = 11 ✓ Subtract.

x2 + 10x = 11

(1)2 + 10(1) � 11 Substitute 1 for x.

1 + 10 � 11 Simplify using the order of operations.

11 = 11 ✓ Add.

b. x2 - 8x = 9

SOLUTION

x2 - 8x = 9

x2 - 8x + ( 8 _ 2 )

2

= 9 + ( 8 _ 2 )

2


x2 - 8x + (4)2 = 9 + (4)2 Simplify the fraction.

x2 - 8x + 16 = 9 + 16 Simplify.

(x - 4)2 = 25 Factor the left side. Simplify the right side.

√ �� (x - 4)2 = ± √ � 25 Take the square root of both sides of the equation.

x - 4 = ±5 Simplify.

x - 4 = -5 or x - 4 = 5 Write as two equations.

__ +4 = __ +4 __ +4 = __ +4 Addition Property of Equality

x = -1 or x = 9 Simplify.

Check x2 - 8x = 9

(-1)2 - 8(-1) � 9 Substitute -1 for x.

1 + 8 � 9 Simplify using the order of operations.

9 = 9 ✓ Add.

x2 - 8x = 9

(9)2 - 8(9) � 9 Substitute 9 for x.

81 - 72 � 9 Simplify using the order of operations.

9 = 9 ✓ Add.

In Example 2 the coefficient of each quadratic term is 1. The coefficient of the quadratic term must be 1 in order to use the completing-the-square method for solving quadratic equations. However, the coefficient of the quadratic term is often not 1. In which case, each term must be divided by the coefficient a.

Math Language

The quadratic term is the x2 term.


It may be helpful to provide students with a template for completing the square for equations given in the form x2 + ax = c.

=

+ ( ___ 2 ) 2

= + ( __ 2 ) 2

+ = +

( )2 =

Solve the equation x2 - 12x = 13.

x2 - 12x = 13

x2 - 12x + ( 12 ___ 2 ) 2

= 13 + ( 12 __ 2 )

2

x2 - 12x + 36 = 13 + 36

(x - 6)2 = 49

Once students have a perfect square on both sides of the equal sign, they can take the square root and solve for x.

INCLUSION

Lesson 104 699

Example 2

Remind students to add ( b __ 2 ) 2

to both sides of the equation.

Error Alert Students may think that √ �� (x + 5) 2 = ±(x + 5). Remind students that for the real number value of x, x 2 will always be a positive real number, so √ � x 2 will always be positive.



a. x 2 + 12x = 13 x = 1 or x = -13

b. x 2 - 16x = 36 x = -2 or x = 18


Saxon Algebra 1700

Example 3 Solving ax2 + bx = c by Completing the Square


a. 4x2 + 16x = 8

SOLUTION

Write the equation so that the coefficient of x2 is 1. Then complete the square.

4x2 + 16x = 8

4x2 + 16x _ 4 =

8 _ 4 Divide both sides by the coefficient of x2.

x2 + 4x = 2 Simplify.

x2 + 4x + ( 4 _ 2 )

2

= 2 + ( 4 _ 2 )

2


x2 + 4x + (2)2 = 2 + (2)2 Simplify the fraction.

x2 + 4x + 4 = 2 + 4 Simplify.

(x + 2)2 = 6 Factor the left side. Simplify the right side.

√ �� (x + 2)2 = ± √ � 6 Take the square root of both sides.

x + 2 = ± √ � 6 Simplify.

x + 2 = - √ � 6 or x + 2 = √ � 6 Write as two equations.

__ -2 = __ -2 __ -2 = __ -2 Subtraction Property of Equality

x = -2 - √ � 6 or x = -2 + √ � 6 Simplify.

x ≈ -4.450 or x ≈ 0.450 Use a calculator to find approximate values.

Check

4x2 + 16x = 8

4(-4.450)2 + 16(-4.450) ≈ 8 Substitute -4.450 for x.

4(19.8025) + 16(-4.450) ≈ 8 Square (-4.450).

79.21 - 71.2 ≈ 8 Multiply.

8.01 ≈ 8 ✓ Subtract.

4x2 + 16x = 8

4(0.450)2 + 16(0.450) ≈ 8 Substitute 0.450 for x.

4(0.2025) + 16(0.450) ≈ 8 Square (0.450).

0.81 + 7.2 ≈ 8 Multiply.

8.01 ≈ 8 ✓ Subtract.

QQ

QQ

QQ

QQ

QQ

QQ


700 Saxon Algebra 1

Example 3

Remind students to divide both sides of the equation by a to get it into the form x 2 + bx = c.



a. 3 x 2 + 12x = 42 -2 ± 3 √ � 2 ; -x = -6.243

or x = 2.243

b. 5 x 2 - 40x = -85 Ø; There is no real number for x

to solve the polynomial.


Lesson 104 701

b. 3x2 - 12x = -54

SOLUTION

3x2 - 12x = -54

3x2 - 12x _ 3 =

-54 _ 3 Divide both sides by the coefficient

of x2.

x2 - 4x = -18 Simplify.

x2 - 4x + ( -4 _ 2 )

2

= -18 + ( -4 _ 2 )

2


x2 - 4x + 4 = -18 + 4 Simplify.

(x - 2)2 = -14 Factor the left side. Simplify the right side.

√ �� (x - 2)2 = ± √ �� -14 Take the square root of both sides of the equation.

x - 2 = ± √ �� -14 Simplify.

x = 2 ± √ �� -14 Ø; No real number is the square root of a negative value.

Example 4 Finding Dimensions of a Rectangle

The length of a rectangle is 12 feet more than its width. The total area of the rectangle is 64 square feet. What are the dimensions of the rectangle?

SOLUTION

Write and solve an equation to find the dimensions.

x = width; x + 12 = length Assign values for the length and width.

w · l = A Use the area formula.

x(x + 12) = 64 Substitute the width, length, and area.

x2 + 12x = 64 Distribute.

x2 + 12x + ( 12 _ 2 )

2

= 64 + ( 12 _ 2 )

2

Complete the square. Add the missing value to both sides.

x2 + 12x + 36 = 64 + 36 Simplify.

(x + 6)2 = 100 Factor and simplify.

√ �� (x + 6)2 = ± √ �� 100 Take the square root of both sides.

x + 6 = ±10 Simplify.

x + 6 = -10 or x + 6 = 10 Write as two equations.

x = -16 or x = 4 Subtract 6 from both sides.

A negative length is not possible, so 4 feet is the solution. This means that the width of the rectangle is 4 feet and the length is 4 + 12, or 16 feet.

Math Reasoning

Verify Show that w =4 feet and l = 16 feet are the correct dimensions.

Sample: The dimensions are correct because the area of a 4-foot-by-16-foot rectangle is 64 f t2.

Reading Math

A symbol for no solution is Ø.


Remind students that completing the square is not always the easiest method. Factoring may be a quicker method.

x 2 + 12x = 64 x 2 + 12x - 64 = 0

The factors of 64 are 1, 2, 4, 8, 16, 32, and 64.

Since -4 + 16 = 12, the equation x2 + 12x - 64 = 0 can be factored as (x - 4)(x + 16) = 0.

ALTERNATE METHOD FOR EXAMPLE 4

Lesson 104 701

Example 4

In application problems, solutions may be correct mathematically but may not make sense in the situation described in the problem.

Extend the Problem

Suppose that the total area is increased by 21 square feet. What are the new dimensions of the rectangle? w = 5 ft; l = 17 ft


The length of a rectangle is 8 inches more than its width. The total area of the rectangle is 384 square inches. What are the dimensions of the rectangle? l = 24 in.; w = 16 in.



Saxon Algebra 1702

*1. Find the missing term of the perfect-square trinomial: c2 + 100c + . 2500

*2. Find the missing term of the perfect-square trinomial: y2 - 26y + . 169

*3. Multiple Choice What is the missing value for the perfect-square trinomial?

x2 - 30x + DA -225 B -15 C 15 D 225

*4. Justify Solve 6x2 - 12x - 18 = 0 by completing the square. Justify each step in your solution. Then check the answer(s). See Additional Answers.

*5. Design The diagram shows the cutout for an open box. The height of the box is 3 inches. The length is 5 inches greater than the width. The area of the base of the box is 24 square inches. What are the dimensions of the box? width: 3 in.; length: 8 in.; height: 3 in.

6. Simplify √ � 3

_ √ � 11

. √ �� 33_11

7. A deck has 6 green cards and 2 yellow cards in it. What is the probability of drawing a green card, keeping it, and then drawing a yellow card? 3_

14

*8. Multi-Step A circle has an area of 6 square meters. Find the radius of the circle. Use 22

_ 7 for π. (Hint: Area of a circle = πr2)

a. Write the formula for finding the radius of the circle. r = √ � A_π

b. Write the equation for finding the radius after substituting in 22

_ 7 for π. r = √ �� 7A_22

c. What is the radius of the circle? √ �� 231_

11

*9. Coordinate Geometry A right triangle is plotted at points A ( √ � 5

_ 3 , 3 √ � 3

_ 4 ) , B (

√ � 5 _ 3 ,

√ � 3

_ 4 ) ,

and C ( 2 √ � 5

_ 3 , √ � 3

_ 4 ) , and line segment AC forms the hypotenuse of the triangle. What

is the length of the hypotenuse of triangle ABC? √ �� 47_

6

(104)(104)

(104)(104)

(104)(104)

(104)(104)

3

3

x

x + 5

3

3 3

3

x

x + 5

3

3

(104)(104)

(103)(103)

(33)(33)

(103)(103)

(86)(86)

Lesson Practice

a. Complete the square: x2 + 24x. x2 + 24x + 144


b. x2 + 2x = 8 x = 2 or x = -4

c. x2 - 14x = 15 x = -1 or x = 15

d. 3x2 + 24x = -27 x = -4 + √ � 7 or x = -4 - √ � 7 ; -6.646 or -1.354

e. 2x2 + 6x = -6 Ø

f. The base of a parallelogram is 8 centimeters more than its height. If the total area of the parallelogram is 20 square centimeters, what are the dimensions of the parallelogram? h = 2 cm, b = 10 cm

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)


702 Saxon Algebra 1

Lesson Practice

Problems b–d

Scaff olding Remind students to check their answers by testing solutions in the original polynomial.

Problems b–e

Error Alert Students may forget to add ( b _ 2 )

2 to both sides of the

equation. Remind them to think of an equation as a balance. If one number is added to one side, the same number needs to be added to the other side to maintain the balance.



“Explain why there are two values of x when solving a x 2 + bx = c.” Sample: When you take the square root of any positive real number, it can be positive or negative, so you have to use both values to solve for x.

“Could the completing the square method be used to solve x2 + 9x = 15 ? Why or why not ?” yes; Sample: You would not be adding an integer to both sides of the equation. You would be adding a fraction.

Practice3


Problem 7

Guide students by asking them the following questions.

“What is the probability of drawing a green card?” 6 __

8

“If I remove one green card, what is left in the deck?” 5 green cards and 2 yellow cards

“What is the probability of drawing a yellow card out of the new deck?” 2 __ 7


Lesson 104 703

10. Printing A photographer has printing paper that is 8 inches by 10 inches with a half-inch margin on the left and right side and a one-inch margin on the top and bottom. He can print out six square images. What are the dimensions of the image?3 in. × 3 in.

*11. Geometry The volume of a cylindrical container is 339.12 cubic meters. The formula representing the volume of the container is (18π)r2 = 339.12. Find r, the radius of the container. Use 3.14 for π. r ≈ 2.449 m

12. Error Analysis Two students were asked to find the LCD of 4x2

_ x2 + 15x + 56

+ 7x + 1 _ -3x + 21

. Which student is correct? Explain the error.

Student A Student B

4x2

__ x2 + 15x + 56

+ 7x + 1

_ -3x + 21

4x2

__ x2 + 15x + 56

+ 7x + 1

_ -3x + 21

4x2

__ (x + 7)(x + 8)

+ 7x + 1

_ -3(x - 7)

4x2

__ (x + 7)(x + 8)

+ 7x + 1

_ -3(x + 7)

LCD = -3(x - 7)(x + 7)(x + 8) LCD = -3(x + 7)(x + 8)

13. For which values is the rational expression x - 6

_ x undefined? x = 0

14. Find the roots of 32x - 3x = 24 - 4x2. ⎧ ⎨ ⎩ 3_4, -8⎫

⎬ ⎭

15. Solve x2 = 100. x = ±10

16. Football The height of a punted ball at time t is represented by the function -32t2 + 12t + 2 = h, where t stands for the number of seconds after the ball is kicked. When does the ball land on the ground? 1_2 second

17. Masonry Pedro can build a brick fence in 10 hours. His partner can build the same brick fence in 12 hours. How long would it take them to do the masonry work together? 60_

11 hours

18. Solve x2 + 81 = 18x by graphing. x = 9

19. Measurement A student uses indirect measurement to find the height of a flagpole. She writes a proportion relating the heights and lengths of the shadows. The equation she must solve is x

_ 10 = x - 20

_ 2 ; where x is the height of the flagpole in feet.

Find the height of the flagpole. 25 feet

*20. a. Solve the inequality -8 ⎪x + 7⎥ ≥ -24. -10 ≤ x ≤ -4

b. Verify Choose two x-values in the solution set you found in part a. Verify that each x-value satisfies the original inequality. See Additional Answers.

*21. Multiple Choice Suppose a number n is a solution of the inequality ⎪5x - 2⎥ < 9. Which of the following inequalities does not have n as a solution? BA 5x - 2 > -9 B 5x - 2 > 9 C -9 < 5x - 2 D 5x - 2 < 9

(102)(102)

(102)(102)

12. Student A; Sample: Student B didn’t correctly factor the GCF of -3 in the second denominator.

12. Student A; Sample: Student B didn’t correctly factor the GCF of -3 in the second denominator.

(95)(95)

(43)(43)

(98)(98)

(102)(102)

(98)(98)

(99)(99)

(100)(100)

3 43 4

(99)(99)

(101)(101)

(101)(101)


Lesson 104 703

Problem 10

Error AlertWithout the benefi t of having the sides labeled in the art, students might forget they have to subtract the correct margin from each side. It may help students to draw a picture of the art on a separate piece of paper and to label the sides.

Problem 16

Extend the Problem

“At what time will the height of the ball be 2.5 feet?” t = 0.048 second and t = 0.327 second

Problem 18

“What is an easier way to solve for x than graphing?” Sample: Factor it; x 2 - 18x + 81 is a perfect-square trinomial.


Saxon Algebra 1704

22. Find the zeros of the function shown. -6 and 2

x

y

O

8

4

4 8

-4

-8

y =1_2

x2 + 2x - 6

23. Find the quotient of 49x2 + 21xy

_ 5x2 ÷ 14x

_ 25xy2 .

35xy2 + 15y3__2x

24. Find the product (6y - 3)(6y + 3). 36y2 - 9

25. Factor 4x4 - 64 completely. 4( x2 + 4)(x + 2)(x - 2)

26. Multi-Step When budgeting to purchase a new car, a student is willing to spend $3000 plus or minus $200. a. Write an absolute-value inequality to show the range of prices the student is

willing to consider. ⎪x - 3000⎥ ≤ 200

b. Solve and find the range of the actual price the student might pay.

27. Find the midpoint of the line segment with the endpoints (-4, 3) and (2, 4). (-1, 7_2 )

28. Cell Phone A student budgets $25 for his cell phone each month. He pays $10 for the service and $0.05 per minute. He knows that his budget can be off by $5 in either direction. What is the maximum and minimum number of minutes he can talk each month? 200 minutes, 400 minutes

29. Justify a

_ x

+ a _ b

≠ 2a _

x + b

Sample: The common denominator should be xb. Also, you have to have like denominators to be able to add the numerators without writing equivalent fractions.

30. Generalize When does a quadratic function only have one zero? Sample: It only has one zero when its vertex is on the x-axis.

(89)(89)

(88)(88)

(60)(60)

(Inv 9)(Inv 9)

(91)(91)

2800 ≤ x ≤ 32002800 ≤ x ≤ 3200

(86)(86)

(94)(94)

(95)(95)

(96)(96)


Problem 28

Have students write an inequality before trying to solve.

Solving quadratic equations by completing the square prepares students for

• Lesson 110 Using the Quadratic Formula

• Lesson 112 Graphing and Solving Systems of Linear and Quadratic Equations

• Lesson 113 Interpreting the Discriminant

Have students complete the square to solve the quadratic equation x2 + 3x = 4. Then have them check the answer using factoring. {1, -4}


704 Saxon Algebra 1


Lesson 105 705

Warm Up

105LESSON

1. Vocabulary A(n) __________ is a list of numbers that often follows a rule. sequence

Simplify

2. -25 -32 3. (-3 ) 4 81

4. 15(-0.4 ) 2 2.4 5. 3 -3 1_27

A geometric sequence is a sequence with a constant ratio between consecutive terms. The ratio between consecutive terms is known as the common ratio. In a geometric sequence, the ratio of any term divided by the previous term is the same for any two consecutive terms.

geometric sequence: 3, 6, 12, 24, …

ratios: 6 _ 3 =

12 _ 6 =

24 _ 12

Example 1 Finding Common Ratios

Find the common ratio for each geometric sequence.

a. 4, 12, 36, 108, …

SOLUTION

4 12 36 108

12 _ 4

= 3 36

_ 12

= 3 108

_ 36

= 3

The common ratio is 3.

b. 320, -80, 20, -5, …

SOLUTION

320 -80 20 -5

-80

_ 320

= - 1 _ 4

20 _

-80 = -

1 _ 4

-5 _

20 = -

1 _ 4

The common ratio is - 1 _ 4

.

c. 0.4, 1, 2.5, 6.25, …

SOLUTION

0.4 1 2.5 6.25

1 _ 0.4

= 2.5 2.5

_ 1 = 2.5

6.25 _

2.5 = 2.5

The common ratio is 2.5.

(34)(34)

(3)(3) (3)(3)

(4)(4) (32)(32)


Recognizing and Extending Geometric

Sequences

Online Connection


Math Language

A sequence is a list of numbers that often follows a rule.


LESSON RESOURCES



Master C105, E105

A sequence of numbers is an ordered set or list of numbers arranged in such a way that you can determine the number preceding and following any term in the sequence according to a specifi ed set of rules or a pattern. The pattern may be arithmetic, geometric, or neither.

In this lesson, students learn to recognize geometric sequences in which the ratio of successive terms is the same number, r, known as the common ratio.

The explicit formula for a geometric sequence (an = a1 · r n-1 ) introduces students to exponential functions. The graph of a geometric sequence is exponential—increasing or decreasing rapidly.

Understanding different types of sequences also prepares students for more advanced algebra, when they will study series and learn to fi nd the sum of a sequence of numbers.

MATH BACKGROUND

Warm Up1

105LESSON

Lesson 105 705

Problem 5

Point out that the exponent in the problem is negative and help students recall that a -n = 1 __ a n .

2 New Concepts

Remind students that a sequence is a number pattern and that each number in the pattern is called a term of the sequence.

Discuss the meaning of a common ratio and how it defi nes a geometric sequence.

Example 1

Error Alert Some students may fi nd only the ratio of the fi rst two terms. Encourage them to check their work with each term.



a. 5, 10, 20, 40, ... 2

b. 270, -90, 30, -10, ... - 1 __ 3

c. 3, 4.5, 6.75, 10.125, ... 1.5


Saxon Algebra 1706

Example 2 Extending Geometric Sequences

Find the next four terms in the geometric sequence.

a. 2, 8, 32, 128, …

SOLUTION

The common ratio is 4. Each term of the sequence is 4 times the previous term. Use the common ratio to find the next 4 terms.

× 4 × 4 × 4 × 4

128 512 2048 8192 32,768

The next 4 terms of the sequence are 512, 2048, 8192, and 32,768.

b. 250, -50, 10, -2, …

SOLUTION

The common ratio is - 1_5

. Use the common ratio to find the next 4 terms.

× (- 1_5 ) × (-

1_5 ) × (-

1_5 ) × (-

1_5 )

-2 2_5 -

2_25

2_125

- 2_

625

The next 4 terms of the sequence are 2 _ 5 , - 2 _

25 , 2 _

125 , and - 2 _

625 .

Examine the terms of the sequence in Example 2a.

Term 1: 2 = 2 · 1 = 2 · 40 Term 2: 8 = 2 · 4 = 2 · 41

Term 3: 32 = 2 · 4 · 4 = 2 · 42 Term 4: 128 = 2 · 4 · 4 · 4 = 2 · 43

The exponent on the common ratio 4 is 1 less than the number of the term. In general, the nth term of the sequence is 2 · 4

n-1 .

Finding the nth Term of a Geometric Sequence

Let A(n) equal the nth term of a geometric sequence, then

A(n) = a r n-1

where a is the first term of the sequence and r is the common ratio.

Example 3 Finding the nth Term of a Geometric Sequence

a. The first term of a geometric sequence is 7 and the common ratio is -3. Find the 6th term in the sequence.

SOLUTION

A(n) = a r n-1 Use the formula.

A(6) = 7(-3 ) 6-1 Substitute 6 for n, 7 for a, and -3 for r.

= 7(-3 ) 5 Simplify the exponent.

= 7(-243) Raise -3 to the 5th power.

= -1701 Multiply.

The 6th term in the sequence is -1701.

Math Reasoning

Generalize When will the common ratio be negative?

Sample: when the sign of each term is the opposite of the term before and after it

Math Reasoning

Analyze Which operation is equivalent to multiplying by - 1_5 ?

Sample: dividing by -5

Reading Math

In the expression 4 n-1 , n represents the integers 1, 2, 3, 4, and so on.


Make a list of the terms of the sequence and their factors and look for a pattern.

Term Factors250 -(2 · - 5 3 )

-50 -(2 · - 5 2 )

10 -(2 · - 5 1 )

-2 -(2 · - 5 0 )

Continue the pattern to fi nd the next four terms in the sequence.

-(2 · - 5 -1 ) = - (2 · - 1 __ 5 ) = 2 __

5

-(2 · - 5 -2 ) = - (2 · 1 ___ 25 ) = - 2 ___ 25

-(2 · - 5 -3 ) = - (2 · - 1 ____ 125

) = 2 ____ 125

-(2 · - 5 -4 ) = - (2 · 1 ____ 625

) = - 2 ____ 625

ALTERNATE METHOD FOR EXAMPLE 2b

706 Saxon Algebra 1

Example 2

Have a volunteer give an example of an arithmetic sequence. Discuss how it differs from a geometric sequence.


Find the next four terms in the geometric sequence.

a. 36, 12, 4, 4 __ 3 , ...

4 __ 9 , 4 ___

27 , 4 ___

81 , 4 ____

243

b. 2, -10, 50, -250, ...1250, –6250, 31,250, –156,250

Example 3

As with an arithmetic sequence, a student can use a formula to fi nd the value of the nth term when the previous term is unknown.

Extend the Example

“Suppose the common ratio in Example 3a is 1 __ 2 . What would be the sixth term of the sequence?” 7

___ 32

“Use the same common ratio of1 __ 2 . What would be the sixth term

of the sequence if the fi rst term was 32?” 1


a. The fi rst term of a geometric sequence is 1 __

8 and the

common ratio is 4. Find the eighth term in the sequence. 2048

b. Find the fi fth term of the geometric sequence.

-288, 48, -8, ... - 2 __ 9

c. Find the twelfth term of the geometric sequence.

2, 4, 8, 16, ... 4096

d. Find the sixth term of the geometric sequence.

2.3, 16.1, 112.7, 788.9, ... 38,656.1


Lesson 105 707

b. Find the 7th term of geometric sequence.

1 _ 3 , -

1 _ 9 , 1 _

27 , ...

SOLUTION

Find the common ratio: - 1 _ 9 ÷ 1 _

3 = - 1 _

9 · 3 _

1 = - 1 _

3 .


A(7) = 1 _ 3 (-

1 _ 3

) 7-1

Substitute 7 for n, 1_3 for a, and - 1_3 for r.

= 1 _ 3 (-

1 _ 3 )

6

Simplify the exponent.

= ( 1 _ 3 ) (

1 _ 729

) Raise - 1_3 to the 6th power.

= 1 _

2187 Multiply.

The 7th term in the sequence is 1 _ 2187

.

c. Find the 9th term in the geometric sequence.

17, 8 1 _ 2 , 4 1 _

4 , 2 1 _

8 , …

SOLUTION

Find the common ratio: 8 1 _ 2 ÷ 17 = 17

_ 2 ÷ 17 = 17

_ 2 · 1 _

17 = 1 _

2 .


A(9) = 17 ( 1 _ 2 )

9-1

Substitute 9 for n, 17 for a, and 1_2 for r.

= 17 ( 1 _ 2

) 8

Simplify the exponent.

= 17 ( 1 _ 256

) Raise 1_2 to the 8th power.

= 17 _

256 Multiply.

The 9th term of the sequence is 17

_ 256 .

d. Find the 5th term of the geometric sequence.

1.2, 7.2, 43.2, …

SOLUTION

Find the common ratio: 7.2 ÷ 1.2 = 6.


A(5) = 1.2 (6) 5-1 Substitute 5 for n, 1.2 for a, and 6 for r.

= 1555.2 Simplify.

The 5th term of the sequence is 1555.2.

Hint

Choose the two terms that are the easiest for finding the common ratio. Then use that ratio to check.


Make a table of each number’s position in the sequence n, up to the ninth term in the sequence, and fi ll in the fi rst four given terms.

Find the common ratio and multiply each value by the ratio to fi nd the next term in the sequence until you have found the ninth term.

Position (n) 1 2 3 4 5 6 7 8 9

Term 17 8 1 __ 2 4 1 __

4 2 1 __

8 1 1 ___

16 17 ___

32 17 ___

64 17 ____

128 17 ____

256

ALTERNATE METHOD FOR EXAMPLE 3c

Lesson 105 707

TEACHER TIPEmphasize that a sequence may be arithmetic, geometric, or neither. Suggest that students look for a common difference or a common ratio when fi rst presented with a sequence.

Error Alert For students who multiply the ratio to the nth power instead of to n - 1, suggest that they make a table of values. Place the fi rst few term numbers in one column and evaluate a1 · r n in the second column for each term number n. Now evaluate a1 · r n-1 in a third column for the same values of n. Students can compare the value of each term to the original sequence to see that the fi rst term in the sequence is left out when they only evaluate a1 · r n .

17, 8 1 __ 2 , 4 1 __

4 , 2 1 __

8 , …

a1 = 17; r = 1 __ 2

n a1 · r n a1 · r n-1

1 8 1 __ 2 17

2 4 1 __ 4 8 1 __

2

3 2 1 __ 8 4 1 __

4

4 1 1 ___ 16

2 1 __ 8


Saxon Algebra 1708

Example 4 Application: Bounce Height

A ball is dropped from a height of 2 yards. The height of each bounce is 85% of the previous height. What is the height of the ball after 10 bounces?

SOLUTION

Understand A ball is dropped from a height of 2 yards. The height of each bounce is 85% of the height of the previous bounce. The common ratio is 85%, or 0.85.

? yds.

2 yds.

1stbounce

2nd 3rd 4th 5th 6th 7th 8th 9th 10th

The heights of the bounces form a geometric sequence.

Plan Multiply the drop height of 2 yards by the common ratio 0.85 to find the height of the first bounce. This product is the 1st term of the sequence. Then use the formula A(n) = a r

n–1 to find the height of the 10th bounce.

This is the 10th term in the sequence.

Solve Find the height of the 1st bounce: 2 · 0.85 = 1.7 yards.

So, the first term of the sequence is 1.7.

Use the formula A(n) = a r n-1 to find the height of the 10th bounce.

A(n) = a r n-1

A(10) = 1.7(0.85 ) 10-1 Substitute 1.7 for a, 10 for n, and 0.85 for r.

= 1.7(0.85 ) 9 Simplify the exponent.

≈ 0.39 yards Simplify and round to the nearest hundredth.

The height of the 10th bounce is about 0.39 yards.

Check Multiply the height of the first bounce by 0.85 nine times.

1.7 × 0.85 × 0.85 × 0.85 × 0.85 × 0.85 × 0.85 × 0.85 × 0.85 × 0.85 ≈ 0.39 ✓

Lesson Practice


a. 2, 16, 128, 1024, ... 8

b. -162, 54, -18, 6, ... - 1_3

c. 0.7, 4.9, 34.3, 240.1, ... 7

Find the next four terms of each sequence.

d. 5, -15, 45, -135, ... 405, -1215, 3645, -10,935

e. 336, 168, 84, 42, ... 21, 10 1_2, 5 1_

4, 2 5_

8

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

Caution

The height of the first bounce is 1.7 yards. The first term of the sequence is 1.7, not 2. The height of the drop is 2 yards.

Math Reasoning

Analyze Why is 1.7 multiplied by 0.85 nine times instead of ten times?

Sample: The height of the first bounce is 1.7 yards. There are 9 more bounces to get to the 10th bounce.


Students can draw a diagram of the problem by starting where the ball drops from a height of 2 yards and sketching the path of 10 bounce heights.


2 yards

× 0.85× 0.85

× 0.85× 0.85

× 0.85× 0.85× 0.85

× 0.85

× 0.85

× 0.85

1.70 1.45 1.23 1.05 0.90 0.65 0.55 0.47 0.400.77

Then they can multiply 2 · 0.85 and label the height of the fi rst bounce: 1.7. They can continue multiplying each bounce height by 0.85 until they fi nd the height of the tenth bounce. Have students explain why the answer differs from the example.

708 Saxon Algebra 1

Example 4

Explain that drawing a diagram can help students better understand this problem and problems similar to it.

Extend the Example

“Suppose the ball is now dropped from a height of 15 feet. A student standing nearby is 5 feet tall. After how many bounces will the ball height be less than the student’s height?” after 6 bounces


A ball is dropped from a height of 3 meters. The height of each bounce is 90% of the height of the previous bounce. What is the height of the ball after 7 bounces? about 1.43 meters

Lesson Practice

Problem e

Error Alert Students may mistakenly treat a geometric sequence as an arithmetic sequence and use the difference between two terms to extend the pattern. Remind them that the difference between each term in an arithmetic sequence is constant. In a geometric sequence, the difference between each consecutive term varies.



Lesson 105 709

*1. Find the common ratio of the geometric sequence -80, 20, -5, 1 1 _ 4 , .... - 1_

4

*2. Multiple Choice Which rule can you use to find the nth term of the sequence 4, 6, 9, 13.5, …? BA A(n) = 4(1.5 ) n B A(n) = 4(1.5 ) n-1

C A(n) = 4(2 ) n-1 D A(n) = 3(1.5 ) n

*3. Depreciation Harper buys a car in 2007 for $20,000. Each year, the car decreases in value by 18%. How much will the car be worth in 2012? Round to the nearest cent. $7414.80

*4. Write The third term of a sequence is 0. The first two terms are not 0. Can this be a geometric sequence? Explain.

5. Write a recursive formula for the arithmetic sequence with a1 = 1 _ 2 and common

difference d = 1 _ 2 . Then find the first four terms of the sequence.

*6. Analyze Write two possible rules for the nth term of the geometric sequence with a first term of 5 and a third term of 605. 5(11) n-1 or 5(-11) n-1

7. Find the missing term of the perfect-square trinomial: x2 + 18x + ____. 81

8. Error Analysis Dominic stated that the missing value for completing the square of g2 + 28g + is 14. Is this correct? Explain.

*9. Multi-Step A design for a rectangular flower bed is shown. The total area of the flower bed is 880 square feet. a. Write an equation to represent the problem.

b. Write the quadratic equation in the form x2 + bx = c.

c. What is the width of the interior of the flower bed? 18 ft

d. What is the area of the border? 232 ft2

(105)(105)

(105)(105)

(105)(105)

(105)(105)

4. no; Sample: The formula to fi nd the third term of a geometric series is A(n) = ar 2.If the fi rst term is not 0, then the only way any term of the series could equal 0 would be if r = 0. Since the second term is not 0, this cannot be true. So the sequence cannot be geometric.

4. no; Sample: The formula to fi nd the third term of a geometric series is A(n) = ar 2.If the fi rst term is not 0, then the only way any term of the series could equal 0 would be if r = 0. Since the second term is not 0, this cannot be true. So the sequence cannot be geometric.

(34)(34)5. a1 = 1_

2,

an = an-1 + 1_2;

1_2, 1, 1 1_

2, 2

5. a1 = 1_2,

an = an-1 + 1_2;

1_2, 1, 1 1_

2, 2

(105)(105)

(104)(104)

(104)(104)

8. no; Sample: The correct value is 196; 14 is the constant value in the factored form.

8. no; Sample: The correct value is 196; 14 is the constant value in the factored form.

(104)(104) 2

2

22x

2x

2

2

22x

2x

9a. (x + 4)(2x + 4) = 8809b. x2 + 6x = 4329a. (x + 4)(2x + 4) = 8809b. x2 + 6x = 432

f. The first term of a geometric sequence is -3 and the common ratio is 4. Find the 6th term in the sequence. -3072

g. Find the 7th term in the geometric sequence.

- 1 _ 2

, 1 _ 8

, - 1 _ 32

, 1 _ 128

, - 1 _

512 , ... - 1_

8192

h. Find the 8th term in the geometric sequence. -544

4 1 _ 4 , -8 1 _

2 , 17, -34, ...

i. Find the 6th term of the geometric sequence 40, 32, 25.6, .... 13.1072

j. A fish tank is 9 _

10 full. Every minute, 1 _

3 of the water leaks out of the tank.

After 5 minutes, how full is the tank? 16_135

(Ex 3)(Ex 3)

(Ex 3)(Ex 3)

(Ex 3)(Ex 3)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)


For this lesson, discuss the meaning of the word sequence. Say:

“A sequence is a listing of items or objects that follow a set order or pattern.”

Discuss other examples of sequences, such as the sequence of events students follow to get ready for school. Ask them if they could come to school before getting dressed in the morning or eat their breakfast before getting out of bed. There is a sequence they must follow to get from one event to the next.

ENGLISH LEARNERS

Lesson 105 709

Problem h

Scaff olding Before students solve, have them decide which two terms to use to determine the common ratio. The terms 17 and -34 make computation easy. After fi nding the common ratio, use it to evaluate the formula.



“Write four terms of a geometric sequence. What is the formula that gives the value of an for your sequence?” Sample: 3, 12, 48, 192, ...; an = 3 · 4 n-1

“How are a common difference and a common ratio similar? How are they different?” Sample: Neither changes within a sequence, but consecutive terms in an arithmetic sequence share a common difference and consecutive terms in a geometric sequence share a common ratio.

Practice3


Problem 8

Remind students that the relationship to complete the square is x 2 + bx + ( b __

2 )

2 .


Saxon Algebra 1710

Simplify.

10. 11x + 22 _

22x2 + 44x 1_

2x, x ≠ 0 - 2 11.

√ � 63 _

√ � 18

√ �� 14_2

*12. Geometry The base of a right triangle is 14 units longer than its height. The hypotenuse is 26 units. What are the base and height measurements of the triangle? base = 24 units; height = 10 units

*13. Error Analysis Two students simplified the given expression. Which student is correct? Explain the error. Student B; Sample: When rationalizing the denominator, Student A did not multiply by a factor of 1.

Student A Student B

5 _

√ � 8

5 _

√ � 8

5 _

2 √ � 2 ·

1 _ √ � 2

5 _

2 √ � 2 ·

√ � 2 _

√ � 2

5 _

4

5 √ � 2 _

4

*14. Architecture In the city of Rotterdam, Netherlands, architect Piet Blom designed a group of cube-shaped houses that each sit upon its vertex. If the surface area of each cube measures 337 1

_ 2 square meters, write a rational expression representing

the edge length of the cube. (Hint: edge length = √ � A _ 6 ) 15_

2 meters

15. Find the roots: 14x2 - 2x = 3 - 21x. { 1_7 , - 3_2 }

16. Construction It takes a woman 3 hours to build a doghouse. Her husband can build it in 4 hours. How long will it take them if they build the doghouse together? 12_

7 hours

17. Use mental math to find the product of 292. 841

18. Horseshoes Shannon plays a game of horseshoes. The horseshoe’s movement forms a parabola given by the quadratic equation h = -16t2 + 6t + 6 where h is the height in feet and t is the time in seconds. Find the maximum height of the path the horseshoe makes and the time t when the horseshoe hits the ground. Round to the nearest hundredth. h = 6.56 feet and t = 0.83 seconds

19. Measurement A puddle of water creates a shape on the ground. As time increases, the area of the puddle changes. The area of the puddle is given by the function A = 3t2 + 8t - 70, where A is the area in square feet and t is the time in seconds. Find the time when the area is 55 square feet. Round to the nearest hundredth.

20. Solve and graph the inequality 2⎢x� - 12 > -5.

21. Verify True or False: 4x2 - 64 = 0; x = 4. Verify that the answer is true. If the answer is false, provide the correct answer. false; The correct answer is ±4.

(43)(43) (103)(103)

(104)(104)

(103)(103)

(103)(103)

(98)(98)

(99)(99)

(60)(60)

(100)(100)

3 43 4

(100)(100)

t = 5.26 secondst = 5.26 seconds

(101)(101)x < -3.5 or x > 3.5;

0-2-4 2 4x < -3.5 or x > 3.5;

0-2-4 2 4

(102)(102)


Materials: large right isosceles triangles, circles or squares cut from sheets of construction paper, scissors

Have each student take a shape and sketch a picture of it. Ask them to label the drawing “a1.” Students will now fold their shapes in half, creating two congruent shapes, and cut the shapes apart on the fold. Tell them to make another sketch of the new shapes they have and to label this drawing “a2.” Place the shapes one on top of the other and repeat the process, labeling the next drawing “a3.”

Have students repeat the process once more and label the last drawing “a4.” Ask them to count and label how many shapes were in each drawing they made.

“What is the value of a1 ? a2 ? a3 ? a4 ?” 1, 2, 4, 8

“Is the difference between one number and the next always the same?” No, it changes.

“What pattern do you notice?” Each number is double the one before it.

INCLUSION

710 Saxon Algebra 1

Problem 11

Error Alert A common error students may make is to leave the rational

expression in the form √ � 7

____ √ � 2

or √ � 7 __ 2 . Point out that a rational

expression in simplest form will not have a radical as a denominator, nor have a fraction as a radicand. Guide students to fi nd a fraction equivalent to 1 that can help them rationalize the denominator correctly.

Problem 15


“What must a quadratic equation equal in order to fi nd the roots?” 0

“How can the equation be written so that it is equal to 0?” 14 x 2 + 19x - 3 = 0

Problem 17

One way to mentally solve the problem is to square (30 - 1) using the FOIL method.

(30 - 1)(30 - 1) = 900 - 60 + 1 = 841

Have a volunteer who found a different method demonstrate how he or she came to the solution.

Problem 19

Extend the Problem

“The water is collecting on Silvia’s kitchen fl oor. The fl oor has an area of 100 square feet. If the area of the puddle becomes greater than the area of the kitchen fl oor, the room will fl ood. Using the function given in Problem 19, how long will it take for the kitchen to fl ood ?” 6.31 s

“Silvia reacts quickly and begins to mop up the water after 4.2 seconds have passed. What was the area of the puddle of water as Silvia began to mop it up?” 16.52 square feet


Lesson 105 711

*22. Error Analysis Student A and Student B want to find the length of the sides of a square with an area 460 square meters less than 685 square meters. Which student is correct? Explain the error.

Student A Student B

x2 + 460 = 685 x2 + 460 = 685 -460 -460 +460 +460 x2 = 225 x2 = 1145 √ � x2 = √ �� 225 √ � x2 = √ �� 1145 x = ±15 x ≈ ±3.838

The sides of the square are 15 m. The sides of the square are about 3.838 m.

23. Give the coordinates of the parabola’s vertex. Then give the minimum or maximum value. (0, 0); maximum: 0

x

y

O

8

4

4 8

-4

-4-8

-8

y = 1_5

x2

24. Add d _

d - 10 +

10 _ 10 - d

. 1

25. Solve and check: 18 _ 2x

- 4 = 15 _ 3x

. x = 1

26. Multi-Step Louis walked for 6x2 - 24x _

6x minutes to get to a grocery store that was

4x - 16 _

x3 miles away.

a. Find his rate in miles per minute. 4_x3 miles per minute

b. If the rate is divided by 1 _ x , what is the new rate? 4_

x2 miles per minute

27. Football The school football team is going to a camp that is 8x2

_ x2 - 11x + 18 miles

away. The team traveled 2

_ 8x - 72 miles on the first day. How many miles are left to travel?

32x2 - x + 2__4(x - 2)(x - 9)

miles

28. Find the midpoint of the line segment with the endpoints (-5, 0) and (1, 14).(-2, 7)

29. Multi-Step A ball is thrown into the air from the top of a cliff at an initial velocity of 32 feet per second. (Use h = -16t2 + vt + 0.) a. How high is the ball after 2 seconds? 0 feet

b. What does this height represent? Sample: the height the ball was thrown from

c. After 3 seconds, the ball is -48 feet. What does this height represent? Sample: 48 feet below the top of the cliff

30. Justify List the inequality symbols that result in graphs with dashed boundary lines and list the inequality symbols that result in graphs with solid boundary lines.

(102)(102)

Student A; Sample: Student B made a transformation error when attempting to isolate the variable and arrived at the wrong answer.Student A; Sample: Student B made a transformation error when attempting to isolate the variable and arrived at the wrong answer.

(89)(89)

(90)(90)

(99)(99)

(92)(92)

(95)(95)

(86)(86)

(96)(96)

(97)(97)

30. Sample: < and > are graphed with dashed lines and ≤and ≥ are graphed with solid lines.

30. Sample: < and > are graphed with dashed lines and ≤and ≥ are graphed with solid lines.


The fi rst three stages of Sierpinski’s Triangle are shown below.

Stage 1 Stage 2 Stage 3

If the fi rst triangle has an area of 1 square unit, what is the area of the shaded part of the second triangle? the third triangle? What is the area of the shaded part of a triangle at Stagen? 3 __

4 ; 9 ___

16 ; Stagen = ( 3 __

4 )

n-1

Recognizing and extending geometric sequences prepares students for

• Lesson 108 Identifying and Graphing Exponential Functions

• Investigation 11 Investigating Exponential Growth and Decay


• Lesson 115 Graphing Cubic Functions


Lesson 105 711

Problem 27

Point out that students will need to fi nd a common denominator so that they can subtract and fi nd the miles left to travel. Have them fi rst factor out any common factors, and then factor the trinomial into two binomials, if possible.


Saxon Algebra 1712

Warm Up

106LESSON

1. Vocabulary Which of the following expressions is a radical expression? B

A 2x + 4 B √ �� x + 1 + 2 C x2 + 5 D 2x2 + 3x + 6

Simplify.

2. ( √ � 5 ) 2 5 3. ( √ �� x + 2 )

2 x + 2

Solve.

4. x2 + 5x + 6 = 0 x = -2, x = -3 5. x2 - 5x = 14 x = -2, x = 7

An equation containing a variable in a radicand is called a radical equation. Inverse operations are used to solve radical equations. The inverse of finding the square root of a term is squaring the term.

Example 1 Solving Simple Radical Equations


a. √ � x = 7

SOLUTION

Use inverse operations.

√ � x = 7

( √ � x ) 2 = 72 Square both sides.

x = 49 Simplify.

Check √ � x = 7

√ � 49 � 7

7 = 7 ✓

b. √ � 4x = 12

SOLUTION


√ � 4x = 12

( √ � 4x ) 2 = 122 Square both sides.

4x = 144 Simplify.

4x _ 4 =


x = 36 Simplify.

(61)(61)

(76)(76) (76)(76)

(98)(98) (98)(98)


Solving Radical Equations

Math Language

Inverse operations are

operations that undo each other.

Math Reasoning

Verify Show that simplifying √�4x before squaring will result in the same solution.

Sample:√�4x = 12

2 √�x = 12 (2√�x ) 2 = 1 22

4x = 144 x = 36


MATH BACKGROUND

Radical equations may have extraneous solutions. The solution to √ �� x - 1 = x - 3 is only x = 5, even though the extraneous solution x = 2 may have been found while solving for x. Since the principal square root of a real number is equal to a positive number, x = 2 does not make sense because it makes the right side of the equation negative. This misleading solution is the effect of squaring both sides of the equation when solving for x.

To verify that the solution is only x = 5, graph both sides of the equation as separate equations and fi nd the point where the lines intersect.

LESSON RESOURCES



Master C106

Warm Up1

712 Saxon Algebra 1

106LESSON

Problems 2 and 3

Check to see if students remember that the square root of a number raised to the second power is equal to the number under the radical sign.

2 New Concepts

In this lesson, students learn to solve radical equations using inverse operations.

Discuss the defi nition of inverse operations and radical equations.

Example 1



a. √ � x = 8 x = 64

b. √ � 2x = 10 x = 50

c. √ �� x + 7 = 11 x = 114

d. √ �� x __ 3 + 4 = 5 x = 63


Lesson 106 713

Check √�4x = 12

√��4(36) � 12

√ �� 144 � 12

12 = 12 ✓

c. √ �� x + 2 = 12

SOLUTION


√ �� x + 2 = 12

( √ �� x + 2 ) 2 = 122 Square both sides.

x + 2 = 144 Simplify.

__ -2 = __ -2 Subtraction Property of Equality

x = 142 Simplify.

Check √ �� x + 2 = 12

√ �� 142 + 2 � 12

√ �� 144 � 12

12 = 12 ✓

d. √��x_2

- 6 = 8

SOLUTION


√��x_2

- 6 = 8

(√��x_2

- 6 )2

= 8 2 Square both sides.

x _ 2 - 6 = 64 Simplify.

__ +6 __ +6 Addition Property of Equality

x _ 2 = 70 Simplify.

2 · x _ 2 = 70 · 2 Multiplication Property of Equality

x = 140 Simplify.

Check √ �� x _ 2 - 6 = 8

√ ��

140 _

2 - 6 � 8

√ �� 70 - 6 � 8

√ � 64 � 8

8 = 8 ✓ Online Connection



Lesson 106 713

Extend the Example

Have the students graph each equation in each part of the example with a graphing calculator. Tell them to use to enter one side of the equation as Y1 and the other side of the equation as Y2. Then have them press to fi nd the x-coordinate of the point of intersection. They can then verify that the solutions calculated by solving the equation algebraically and graphically are the same.

TEACHER TIPRemind students that to solve the equation, they have to isolate the variable.


Saxon Algebra 1714

Sometimes the radical is not isolated in a radical equation. In those cases, inverse operations for addition, subtraction, multiplication, and division can be used to isolate the radical. Once the radical has been isolated, both sides of the equation can be squared.

Example 2 Solving by Isolating the Square Root


a. √ � x - 5 = 8

SOLUTION

b. √ � x + 5 = 8

SOLUTION

Use inverse operations. Use inverse operations.

√ � x - 5 = 8

__ +5 = __ +5

√ � x = 13

( √ � x ) 2 = 132

x = 169

Addition Property of Equality

Simplify.

Square both sides.

Simplify.

√ � x + 5 = 8

__ -5 = __ -5

√ � x = 3

( √ � x ) 2 = 32

x = 9

SubtractionProperty of Equality

Simplify.

Square both sides.

Simplify.

Check √ � x - 5 = 8 √ � x + 5 = 8

√ �� 169 - 5 � 8 √ � 9 + 5 � 8

13 - 5 � 8 3 + 5 � 8

8 = 8 ✓ 8 = 8 ✓

c. 3 √ � x = 21

SOLUTION

d. √ � x

_ 2 = 18

SOLUTION

√ � x

_ 2 = 18

√ � x

_ 2 · 2 _

1 = 18 · 2 _

1

√ � x = 36

( √ � x ) 2 = 362

x = 1296

MultiplicationProperty of Equality

Simplify.

Square both sides.

Simplify.


3 √ � x = 21

3 √ � x

_ 3 =

21 _ 3

√ � x = 7

( √ � x ) 2 = 72

x = 49

Division Property of Equality

Simplify.

Square both sides.

Simplify.

Check 3 √ � x = 21 √ � x

_ 2 = 18

3 √ �49 � 21 √ �� 1296

_ 2 � 18

3 · 7 � 21 36

_ 2

� 18

21 = 21 ✓ 18 = 18 ✓

Math Reasoning

Analyze Is squaring first helpful in solving the equation √ � x - 5 = 8 ?

Sample: No, the left side of the equation still has a radical: x - 10 √ � x + 25 = 64.


INCLUSION

Students who have diffi culty with mental processing may not remember how to approach equations where the radical is not already isolated. These students may square each side before isolating, which results in a dead end. Encourage students to develop a process tree to help them remember the process.

RadicalEquation

RadicalIsolated

Square Both Sides

Simplify

IsolateRadical

RadicalNot Isolated

714 Saxon Algebra 1

Example 2

Students learn to isolate the radical before they square both sides.



a. √ � x - 6 = 5 x = 121

b. √ � x + 8 = 15 x = 49

c. 4 √ � x = 24 x = 36

d. √ � x

____ 3 = 12 x = 1296

TEACHER TIPFor equations with one radical that is not isolated, have students write the problem and then use the eraser end of a pencil to cover the radical term. Any terms not covered will need to be moved to the opposite side of the equation and then simplifi ed.


Lesson 106 715

Some equations contain more than one radical expression. If possible, it is helpful to put the radical expressions on opposite sides of the equal sign.

Example 3 Solving With Square Roots on Both Sides


a. √ �� x + 2 = √ �� 2x + 4

SOLUTION


√ ��x + 2 = √ �� 2x + 4

( √ �� x + 2 ) 2 = ( √ �� 2x + 4 )

2 Square both sides.

x + 2 = 2x + 4 Simplify.

__ -2 = __ -2 Subtraction Property of Equality

x = 2x + 2 Simplify.

__ -2x = __ -2x Subtraction Property of Equality

-x = 2 Simplify.

x = - 2 Multiply by -1.

Check √ �� x + 2 = √ �� 2x + 4

√ �� -2 + 2 � √ �� 2(-2) + 4

√ � 0 � √ �� -4 + 4

√ � 0 � √ � 0

0 = 0 ✓

b. √ �� x + 2 - √ � 2x = 0

SOLUTION


√ �� x + 2 - √ � 2x = 0

___ + √ � 2x = ___ + √ � 2x Addition Property of Equality

√ �� x + 2 = √ � 2x Simplify.

( √ �� x + 2 ) 2 = ( √ � 2x )

2 Square both sides.

x + 2 = 2x Simplify.

__ -x = __ -x Subtraction Property of Equality

2 = x Simplify.

Check √ �� x + 2 - √ � 2x = 0

√ �� 2 + 2 - √ �� 2(2) � 0

√ � 4 - √ � 4 � 0

0 = 0 ✓

Hint

When a single radical is on each side, begin by writing the equation without radical symbols.


Lesson 106 715

Example 3

Explain to students that they must eliminate the radical symbols by squaring both sides.



a. √ �� x + 6 = √ �� 2x + 4 x = 2

b. √ �� x + 4 - √ � 3x = 0 x = 2

TEACHER TIPAs a class, it may be helpful to have students try to solve Example 2b by keeping the radicals on the same side of the equation. It not only will demonstrate to them that the equation becomes more complicated, it will sharpen skills that involve the FOIL method and multiplying radical expressions.


Saxon Algebra 1716

When both sides of an equation are squared to solve an equation, the resulting equation may have solutions that do not satisfy the original equation. Recall that an extraneous solution is a solution of a derived equation that does not satisfy the original equation.

Example 4 Determining Extraneous Solutions


a. √ �� x - 1 = x - 3

SOLUTION The radical expression is isolated. Use inverse operations.

√ �� x - 1 = x - 3

( √ �� x - 1 ) 2 = (x - 3)2 Square both sides.

x - 1 = x2 - 6x + 9 Simplify.

__ +1 = __ +1 Addition Property of Equality

x = x2 - 6x + 10

-x = -x Subtraction Property of Equality

0 = x2 - 7x + 10 Simplify.

0 = (x - 2)(x - 5) Factor.

x - 2 = 0 x - 5 = 0 Write two equations.

x = 2 x = 5 Use inverse operations to simplify.

Check √ �� x - 1 = x - 3; x = 2 √ �� x - 1 = x - 3; x = 5

√ �� 2 - 1 � 2 - 3 √ �� 5 - 1 � 5 - 3

√ � 1 � -1 √ � 4 � 2

1 ≠ -1 ✗ 2 = 2 ✓

The solution x = 2 is extraneous, so x = 5 is the only solution.

b. √ � x + 5 = -2

SOLUTION Use inverse operations to isolate the radical.

√�x + 5 = -2

__-5 = __-5 Subtraction Property of Equality

√�x = -7 Simplify.

(√�x )2 = (-7)2 Square both sides.

x = 49 Simplify.

Check √�x + 5 = -2

√�49 + 5 � -2

7 + 5 � -2

12 ≠ -2 ✗

The solution x = 49 is extraneous. There is no solution.

Math Language

The derived equation is a new equation that results from squaring the original equation.

Hint

Remember that positive numbers have two square roots. By convention, √ � returns the positive square root.


716 Saxon Algebra 1

Example 4

Error Alert Students may forget to square the right side of the equation. Remind them that in this case, they have to use the FOIL method.



a. √ �� 7x + 15 = x + 1 x = 7

b. √ � x + 6 = -3 no solution

TEACHER TIPOnce students have solved a few problems of the type in Example 4b, ask them if they can identify under what conditions a radical equation will have no solution. They should respond that if the radical is isolated and equal to a negative number, then no solution will be possible as the square root of a real number is always positive.



Lesson 106 717

Example 5 Application: Architecture

An architect is designing a performance center. If the area of the center is 30 square decameters, what is the area of the gallery?

SOLUTION

total area = area of auditorium + area of gallery

30 = ( √ � x ) 2 + √ � x (6 - √ � x )

30 = x + 6 √ � x - x Simplify.

30 = 6 √ � x Combine like terms.

5 = √ � x Divide both sides by 6.

52 = ( √ � x ) 2 Square both sides.

25 = x Simplify.

The area of the auditorium is 25 square decameters.

To find the area of the gallery, subtract the area of the auditorium from the total area.

30 - 25 = 5

The area of the gallery is 5 square decameters.

Lesson Practice


a. √ � x = 6 x = 36 b. √ � 5x = 15 x = 45

c. √ �� x + 3 = 12 x = 141 d. √ �� 4x - 15 = 7 x = 16

e. √ � x - 8 = 5 x = 169 f. √ � x + 8 = 15 x = 49

g. 6 √ � x = 24 x = 16 h. √ � x

_ 3

= 15 x = 2025

i. √ �� x + 4 = √ �� 2x - 1 x = 5 j. √ �� x + 5 - √ � 6x = 0 x = 1

k. √ �� x - 2 = x - 4 x = 6 l. √ � x + 8 = -3 no solution

m. A breakfast nook has a planter along one side. The entire area of the nook is 42 square yards. What is the area of the planter? 6 yd2

Auditorium Gallery√x

√x 6 √x

Auditorium Gallery√x

√x 6 √x

(Ex 1)(Ex 1) (Ex 1)(Ex 1)

(Ex 1)(Ex 1) (Ex 1)(Ex 1)

(Ex 2)(Ex 2) (Ex 2)(Ex 2)

(Ex 2)(Ex 2) (Ex 2)(Ex 2)

(Ex 3)(Ex 3) (Ex 3)(Ex 3)

(Ex 4)(Ex 4) (Ex 4)(Ex 4)

(Ex 5)(Ex 5)

Breakfast Nook

Planter

√x

√x (7 √x )

Breakfast Nook

Planter

√x

√x (7 √x )

1. Solve x2 = 64. x = ±8

2. Factor x2 - 9x + 20. (x - 5)(x - 4)

(102)(102)

(Inv 9)(Inv 9)

Math Reasoning

Analyze Is the answer reasonable?

yes; Sample: the sum of the areas of the gallery and the auditorium is the same as the area of the center: 5 dkm2 + 25 dkm2

= 30 dkm2.


Lesson 106 717

Example 5

Extend the Example

“If the architect decreases the area of the center by 1 _ 3 , what is the new area of the gallery?” 80

_ 9 dkm2


Linda is making a dollhouse for her little sister. If the area of the dollhouse is 216 square inches, what is the area of the kitchen in the dollhouse? 72 square inches

KitchenLiving Area

18 inches

√x

√x 18 √x

Lesson Practice

Problem i

Scaff olding Remind students to isolate the variables under the radical symbols by squaring both sides and combining like terms.

Problem k

Error Alert Make sure students check for extraneous solutions. In this case, x = 3 is extraneous.



“Why is it necessary to check every apparent solution of a radical equation in the original equation?” Sample: Sometimes there is an extraneous solution that does not work when it is substituted into the original equation.

“How do you solve a radical equation?” Sample: Isolate the radical, raise both sides to the same power to eliminate the radical, and then solve the equation.


Saxon Algebra 1718

3. Translate the inequality 3z + 4 < 10 into a sentence. Sample: The sum of three times a number and 4 is less than 10.

*4. Multiple Choice Which of the following radical equations will require the use of division to isolate the radical? DA √ � x - 12 = 2 B √ � x + 12 = 13

C √ � x

_ 7 = 5 D 14 √ � x = 70

*5. Verify Solve √ �� x - 1 = √ �� 3x + 2 . Check your answer.

*6. Justify Solve √ � x

_ 4 = 32. Justify your answer. x = 16,384; √ � x_

4= 32; √ � x = 128,

multipled both sides by 4; x = 1282, squared both sides; x = 16,384. *7. Find the common ratio of the geometric sequence 18, -9, 4 1

_ 2 , -2 1 _ 4 , .... - 1_

2

*8. Find the 6th term in the geometric series that has a common ratio of 2 and an initial term of 5. 160

9. Multi-Step Leila drops a ball from a height of 1 meter. The height of each bounce is 75% of the previous height. a. What is the ball’s height after the first bounce? 0.75 meter

b. What rule can be used to find the ball’s height after n bounces? 0.7 5n

c. What is the height of the sixth bounce? Round your answer to the nearest hundredth. 0.18 meter

10. Geometry Each unit square in the figure represents 5 square feet. If the pattern continues, what will the area of the ninth figure be? 327,680 square feet

*11. Botany The growth of an ivy plant in feet can be described by 2 √ �� x - 4 . How many days x will it take for the ivy to reach a length of 20 feet? x = 104 days

12. Solve -5x + 4y = -37

3x - 6y = 33

. (5, -3)

*13. Fractals Fractals are geometric patterns that repeat themselves at smaller scales. The pattern shows fractals of equilateral triangles. How many unshaded triangles will be in the sixth figure? 243

*14. Solve x2 + 9x = 4.75 by completing the square. x = 0.5 or x = -9.5

15. Error Analysis Two students started solving the equation 2x2 + 20x = -18 as shown below. Which student is correct? Explain the error.

Student A Student B

2x2 + 20x = -18x2 + 10x = -9

x2 + 10x + 25 = -9 + 25(x + 5 )2 = 16

2x2 + 20x = -18x2 + 10x = -18

x2 + 10x + 25 = -18 + 25( x + 5 )2 = 7

(45)(45)

(106)(106)

(106)(106)

5. no solution; When x = - 3_2, the

radicand is negative: √ �� - 3_2

- 1 = √ �� - 5_2

.

5. no solution; When x = - 3_2, the

radicand is negative: √ �� - 3_2

- 1 = √ �� - 5_2

.

(106)(106)

(105)(105)

(105)(105)

(105)(105)

(105)(105)

(106)(106)

(63)(63)

(105)(105)

(104)(104)

(104)(104)Student A; Sample: Student B did not divide all terms by 2 in the initial step.

Student A; Sample: Student B did not divide all terms by 2 in the initial step.


Explain the meaning of the word bounce. Say:

“A ball has bounced if a ball has been dropped or thrown, hit a surface, and traveled back in the opposite direction.”

Ask students to name objects that can bounce. Sample: balls, putty, erasers

ENGLISH LEARNERS

718 Saxon Algebra 1

Practice3


Problem 4

Ask, “What operation is the inverse of division?” multiplication

Problem 5

Ask, “What is the fi rst step needed to solve this problem?” Square both sides to eliminate the radical signs.

Problem 10

Extend the Problem

Ask students, “What will be the area of the fi fteenth fi gure?” 1,342,177,280 square feet

Problem 12

“Why should the equations be setup so that the coeffi cients of the x- or y-terms are opposites?” Sample: so that the terms will cancel when the equations are added

“What should the fi rst equation be multiplied by so that the x-terms are opposites? 3

“What should the second equation be multiplied by so that the x-terms are opposites? 5


Lesson 106 719

*16. Business The marketing group for a cosmetics company determined that the expression u2 - 0.8u represents the profit for every 1000 units u of mascara sold. How many units need to be sold to have a profit margin of $0.33? 1100 units

Solve each equation. Check your answer.

17. 60

_ 4x

+ 45 _ 5x

= 3 *18. √ � x = 9

19. Egg Toss Tyrese and Jameka were playing an egg-toss game. The egg’s movement through the air formed a parabola given by the quadratic equation h = -16t2 + 9t + 4, where h is the height in feet and t is the time in seconds. Find the maximum height of the path the egg makes and the time t when the egg hits the ground. Round to the nearest hundredth. h = 5.27 feet and t = 0.85 seconds

20. Solve x2 - 16 = 0 by graphing. x = 4 and x = -4

21. Tennis The weight w of a tennis ball should vary no more than 1

_ 12 ounce from 2 1

_ 12 ounces. Write an absolute-value inequality that models the acceptable weights. What is the least acceptable weight?

22. Multiple Choice Which is the simplest form of 18 √ � 7 _

3 √ � 28 ? B

A 3 _

2 B 3 C

6 √ � 7 _

√ � 28 D

6 √ � 7 _ 7

23. Write Anton wants to estimate the quotient of √ �� 145 _

2 √ � 9 . How should he do this?

24. Subtract 2r _ r - 4

- 6 _

12 - 3r .

2(r + 1) _

r - 4

25. Solve and graph ⎢x - 16⎢ ≤ 12. 4 ≤ x ≤ 28; 20 300 10

26. Martha built a new playroom. She determined that the rectangular reading area is (9x2 + 44x - 5) square feet. The width is (x + 5) feet. What is the length? (9x - 1) feet

27. Volleyball A server’s hand is 3 feet above the floor when it hits the volleyball. After the volleyball is hit, it has an initial velocity of 23 feet per second. What is its height after 1 second? Use h = -16t2 + vt + s. 10 feet

28. Multi-Step Tickets for the Valley High School production of Romeo and Juliet are $5 for adults and $4 for students. In order to cover expenses, at least $2500 worth of tickets must be sold. a. Write an inequality that describes this situation. 5x + 4y ≥ 2500

b. Graph the inequality. See Additional Answers.

c. If 200 adult and 400 student tickets are sold, will the expenses be covered? yes

29. Generalize Consider the equation (x - 5)(x + 8) = 0. How can you quickly tell what the roots are? Sample: The roots are the opposite of the constant term in each factor.

30. The graph of f(x) = x2 + bx + 3 has an axis of symmetry x = 4. What is the value of b? b = -8

(104)(104)

(99)(99)

17. x = 8; 60_4(8)

+ 45_5(8)

= 15_8

+ 9_8

= 24_8

= 317. x = 8; 60_4(8)

+ 45_5(8)

= 15_8

+ 9_8

= 24_8

= 3

(106)(106)x = 81; √ �� 81 = 9x = 81; √ �� 81 = 9

(100)(100)

(100)(100)

(101)(101)

⎪w - 2 1_12 ⎥ ≤ 1_

12; 2 ≤ w ≤ 2 1_

6; 2 ounces⎪w - 2 1_

12 ⎥ ≤ 1_12

; 2 ≤ w ≤ 2 1_6; 2 ounces

(103)(103)

(103)(103)

23. Sample: Since√ �� 145 is close to √ �� 144 , Anton should fi nd the square root of 144 for the numerator (12) and multiply the square root of nine(3) by 2 in the denominator (6).The estimated quotient would be 2.

23. Sample: Since√ �� 145 is close to √ �� 144 , Anton should fi nd the square root of 144 for the numerator (12) and multiply the square root of nine(3) by 2 in the denominator (6).The estimated quotient would be 2.

(90)(90)

(91)(91)

(93)(93)

(96)(96)

(97)(97)

(98)(98)

(Inv 10)(Inv 10)

SM_A1_NLB_SBK_L106.indd Page 719 5/19/08 4:16:01 PM user-s191 /Volumes/ju115/HCAC057/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L106


Solve √ �� (x + 1)(x + 4 ) =

√ �� (x - 2)(x + 8) .x = 20

Solving radical equations prepares students for



Lesson 106 719

Problem 22

Error Alert Students may factor a 4 out of √ � 28 instead of 2. Remind them to pull out the square root and not a factor of the radicand when factoring radicals.

Problem 28

Extend the Problem

“How much money would the drama club make if 200 adult and 400 student tickets were sold?” $2600

Problem 29

Remind students that if either term is equal to zero, then the whole equation is equal to zero.


Saxon Algebra 1720

Graphing Absolute-Value Functions

Warm Up

107LESSON

1. Vocabulary A is the simplest function of a particular type, or family. parent function

Simplify.

2. 3 · 2 + 2⎪-5⎥ 16 3. 5 · 8 - 4⎪-6⎥ 16

4. 4⎪x - 2⎥ = 60 -13, 17 5. -3⎪x + 4⎥ = 36 no solution

A function whose rule has one or more absolute-value expressions is called an absolute-value function. The absolute-value parent function is f(x) = ⎪x⎥.

Example 1 Graphing the Absolute-Value Parent Function

Graph the absolute-value parent function f(x) = ⎪x⎥.

SOLUTION

Use a table to graph the function.

x y

-2 2

-1 1

0 0

1 1

2 2

x

y8

4

4 8

-4

-4-8

-8

The absolute-value parent function forms the shape of a “V.” The equation of the axis of symmetry of the absolute-value parent function is x = 0. The point on the axis of symmetry of the absolute-value graph, or the “corner” of the graph, is the vertex of an absolute-value graph.

The absolute-value function has two slopes. If the graph opens upward, the slope of the graph on the left of the axis of symmetry is -1. The slope of the graph on the right side of the axis of symmetry is 1.

y4

2

2 4

-2

-2-4

-4

vertex: (0, 0)

axis of symmetry: x = 0

slope = -1 slope = 1x

(Inv 6)(Inv 6)

(5)(5) (5)(5)

(74)(74) (74)(74)


Math Reasoning

Write Why is “axis of symmetry” an appropriate name?

Sample: One side of the graph is the mirror image of the other side of the graph over the axis of symmetry. The graph is symmetrical about the axis.


MATH BACKGROUNDLESSON RESOURCES



Master C107

First-degree equations generally only have one solution, but those with an absolute-value expression in them have more than one. Similar to quadratic equations, absolute-value equations have two solutions, which results in a nonlinear function when graphed.

The absolute-value function is defi ned by

⎢x� = ⎧

⎨

⎩ x if x ≥ 0.

-x if x < 0.

This function is a composition of two other functions. If you take a number, square it,

and then take its square root, you will fi nd the absolute value of the number.

So, if f (x) = x2 and g(x) = √ � x , then g( f (x)) = √ � x2 . The composition gives the graph of the function.

The domain of an absolute-value parent function is all real numbers, while the range is the interval (0, ∞).

Absolute-value equations force a solution to be positive and are useful in computing standard deviations and distances.

Warm Up1

720 Saxon Algebra 1

107LESSON

Problem 5

If students do not see that there is no solution to this equation, ask them if ⎢x� can ever equal a negative number.

2 New Concepts

In this lesson, students learn how to graph absolute-value functions, using what they learn about the graph of the parent function to graph absolute value functions in other forms.

Example 1

Although an absolute-value function is nonlinear, its graph is two straight lines with opposite slopes that come to a shared point.


Graph the absolute-value function f (x) = ⎢-x�. The graph will be the same as the graph in Example 1.

Have students explain why y = ⎢-x� will have the same table as the one shown in the solution of Example 1. Taking the absolute value of x or negative x will result in the same answer.


Lesson 107 721

Translations of Absolute-Value Graphs

The absolute-value parent function can be translated by adding or subtracting constants.Vertical Translation

If a constant k is added outside the absolute-value bars, the graph is translated up or down k units. For f(x) = ⎢x� + k:

• Graph translates up if k > 0.

• Graph translates down ifk < 0.

• Coordinate of vertex is (0, k).

The graph of f (x) = ⎢x� + k, where k = 1, is shown.

x

y8

4

4 8

-4

-4-8

-8

The graph of f(x) = ⎢x� + k, where k = -1, is shown.

x

y8

4

4 8

-4

-4-8

-8

Horizontal Translation

If a constant h is subtracted inside the absolute-value bars, the graph is translated right or left h units. For f(x) = ⎢x - h�:

• Graph translates right if h > 0.

• Graph translates left if h < 0.

• Coordinate of vertex is (h, 0).

The graph of f(x) = ⎢x - h�, where h = 1, is shown.

x

y8

4

4 8

-4

-4-8

-8

The graph of f(x) = ⎢x - h�, where h = -1, is shown.

x

y8

4

4 8

-4

-4-8

-8 Online Connection


Reading Math

For positive h values, the graph moves right relative to the graph of the parent function and f(x) = ⎢x - h�. For negative h values, the graph moves left andf(x) = ⎢x + h�.

SM_A1_NL_SBK_L107.indd Page 721 1/14/08 5:49:13 PM user /Volumes/ju102/HCAC044/SM_A1_SBK_indd%0/SM_A1_NL_SBK_Lessons/SM_A1_SBK_L10

For this lesson, explain the meaning of the word translation. Say:

“Translation comes from the word ‘across’ or ‘change.’ When a book written in English is then written in Spanish, a translation occurs. The language in which the book was written has changed, but the meaning of the words remains the same.”

Discuss how ‘translation’ relates to shifts of the absolute-value parent function.

ENGLISH LEARNERS

Lesson 107 721

Error Alert Students may choose to evaluate only positive values of x when they create a table of values for an absolute-value function. Make sure that they use a selection of numbers that include both negative and positive values so they correctly graph the function.

TEACHER TIPEmphasize that a translation is simply a shift of a graph horizontally, vertically, or both. When students translate an absolute-value function, the result is a graph the same size and shape as the parent function, but in a different position. This can easily be demonstrated with a transparency on an overhead.


Saxon Algebra 1722

Example 2 Translating Absolute-Value Graphs

Graph the function and give the coordinates of the vertex.

a. f(x) = ⎢x� - 2

SOLUTION


x -3 -2 -1 0 1 2 3y 1 0 -1 -2 -1 0 1

The graph of the parent function is translated down 2 units. The vertex is (0, -2).

b. f(x) = ⎢x + 3�

SOLUTION


x -6 -3 0 3 6y 3 0 3 6 9

The graph of the parent function is translated left 3 units. The vertex is (-3, 0).

Multiple Translations of Absolute-Value GraphsVertical and Horizontal Translation

If a constant h is subtracted inside the absolute-value bars and a constant k is added outside the bars, as in f(x) = ⎢x - h� + k. The graph is translated both vertically and horizontally. The vertex is at (h, k).

The graph of f(x) = ⎢x - h� + k, where h = 1 and k = 1, is shown.

x

y8

4

4 8

-4

-4-8

-8

O

Example 3 Graphing Multiple Translations


f(x) = ⎢x - 4� + 1

SOLUTION

The graph of the function is determined by translating the parent function. Evaluate how the function is different from the parent function.

The vertex is (4, 1).

x

y8

4

4 8

-4

-4-8

-8

O x

y8

4

4 8

-4

-4-8

-8

O

x

y8

4

4 8

-4

-4-8

-8

O x

y8

4

4 8

-4

-4-8

-8

O

x

y

O

8

4

4 8

-4

-4-8

-8

x

y

O

8

4

4 8

-4

-4-8

-8

Hint

Knowing how to translate the graph of y = ⎢x� using h and k can replace the use of a table of values to find points on the graph.


Write and graph two linear equations using the defi nition of absolute value.

First isolate the absolute value.

y = ⎢x - 4� + 1y - 1 = ⎢x - 4�

Now write one equation for when x - 4 ≥ 0, and another for when x - 4 < 0.

x - 4 ≥ 0 x - 4 < 0 y - 1 = x - 4 and y - 1 = -(x - 4)

y = x - 3 y = -x + 5

Graph each equation using the domain x - 4 ≥ 0, or x ≥ 4, for y = x - 3, and x - 4 < 0, or x < 4, for y = -x + 5.

xO

4

2

4

-2


722 Saxon Algebra 1

Example 2

Remind students that they can check their work by substituting ordered pairs from the graph into the equation.



a. f (x) = ⎢x + 3

x

y

O

4

6

2

2 4

-2

-2-4

vertex: (0, 3)

b. f (x) = ⎢x - 4�

x

y

O

2

2 4 6

-2

-2

-4

vertex: (4, 0)

Example 3

Remind students that they can check their work by substituting ordered pairs from the graph into the equation.



f (x) = ⎢x + 2� - 1

x

y

2

2

-2

-4

-4

vertex: (-2, -1)


Lesson 107 723

Reflections, Stretches, and Compressions of Absolute-Value GraphsThe absolute-value parent function can be reflected, stretched, and compressed by multiplying by a constant a.

If a < 0, then the graph is reflected across the x-axis.

If ⎢a� > 1, then the graph is stretched vertically, or away from the x-axis.

If ⎢a� < 1, then the graph is compressed vertically, or toward the x-axis.

Example 4 Reflecting, Stretching, and Compressing

Absolute-Value Graphs

Describe the graph of each function.

a. f(x) = 3⎢x�

SOLUTION

a = 3, so ⎢a� = 3.

Since ⎢a� > 1, the graph is stretched vertically.

b. f(x) = -4⎢x�

SOLUTION

a = -4, so ⎢a� = 4.

Since a < 0, the graph is reflected across the x-axis. Since ⎢a� > 1, the graph is stretched vertically.

c. f(x) = -0.2⎢x�

SOLUTION

a = -0.2, so ⎢a� = 0.2.

Since a < 0, the graph is reflected across the x-axis. Since ⎢a� < 1, the graph is compressed vertically.

Example 5 Application: Travel

A helicopter pilot is flying from town A to town B at 60 miles per hour. To make sure he is on course, he will fly over a landmark that he knows is 20 miles from town A. Write and graph the distance from the landmark as a function of minutes of flight time.

SOLUTION

Let a = rate = 60 mph = 1 mile per minute

Let h = time from landmark = 20

_ 1 = 20 minutes

Let k = closest distance to landmark = 0 miles

f(x) = 1⎢x - 20� + 0

f(x) = 1⎢x - 20�

x

y8

4 8

-4

-4-8

-8

x

y8

4 8

-4

-4-8

-8

x

y8

4

4 8-4-8O x

y8

4

4 8-4-8O

x

y8

4

O

-4

-8

x

y8

4

O

-4

-8

x

y

O

80

-40

-40-80

-80

40 80

x

y

O

80

-40

-40-80

-80

40 80

Math Reasoning

Formulate What values are described by the inequality⎢a� > 1? ⎢a� < 1?

Sample:a > 1 or a < -1; -1 < a < 1

Hint

A function that is reflected can also be stretched or compressed.

Math Reasoning

Connect What other function is stretched or compressed vertically by changing the a value?

the quadratic function


Graph the parent function f (x) = ⎢x� on a graphing calculator by using the absolute-value function and entering Y1 = abs(X). Now graph the function f (x) = -0.2⎢x� using Y1 = -0.2abs(X). Compare the two graphs. Sample: The second graph is fl ipped over the x-axis and it is compressed vertically.

ALTERNATE METHOD FOR EXAMPLE 4c

Lesson 107 723

Example 4

Absolute-value functions may be transformed in more than one way. Each transformation relates back to the parent function, f (x) = ⎢x�.

Extend the Example

Challenge students to write an absolute-value function for f (x) = ⎢x� that meets the following parameters:

translated 2 units right and 3 units down and stretched vertically. any function f(x) = a⎢x - 2� - 3, where a > 1



a. f (x) = 1 _ 4 ⎢x� a = 1 _

4 , so ⎢a� =

1 _ 4 . Since ⎢a� < 1, the graph is

compressed vertically.

b. f (x) = 8⎢x� a = 8, so ⎢a� = 8. Since ⎢a� > 1, the graph is stretched vertically.

c. f (x) = -2.5⎢x� a = -2.5, so ⎢a� = 2.5. Since a < 0, the graph is refl ected across the x-axis. Since ⎢a� > 1, the graph is stretched vertically.

Example 5


The path of a cue ball hitting the sidewall of a pool table is described by the function f (x) = |x|. For the next shot, the cue ball follows the same path except 12 inches to the right. Write and graph the function of the new path.

x

y

O

2

4

4 8 12

-2

-4

f(x) = ⎢x - 12�



Saxon Algebra 1724

Lesson Practice

Graph each function and give the coordinates of the vertex.

a. f(x) = ⎢x� + 2. (0, 2)

b. f(x) = ⎢x + 2�. (-2, 0); See Additional Answers.

c. f(x) = ⎢x - 1� + 2 (1, 2); See Additional Answers.


d. f(x) = 4⎢x� Since ⎢a� > 1, the graph is stretched vertically.

e. f(x) = -2⎢x�

f. f(x) = -0.5⎢x�

g. The distance of a truck to a manhole cover is given by the function f(t) = ⎢t� + 25. Write the function representing the distance of a truck starting at the same location, but traveling twice as fast. f(t) = 2⎢t� + 25

(Ex 2)(Ex 2)a.

x

y

O

8

4

4 8-4-8

-4

-8

a.

x

y

O

8

4

4 8-4-8

-4

-8

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)

Since a < 0, the graph is reflected across the x-axis. Since ⎢a� > 1, the graph is stretched vertically.Since a < 0, the graph is reflected across the x-axis. Since ⎢a� > 1, the graph is stretched vertically.

(Ex 5)(Ex 5)

*1. Estimate Without graphing the function, which direction would the function f(x) = ⎢x� - 6 shift the parent function? down

2. Solve √ � 2x = 14. Check your answer. x = 98; √ �� 2(98) = √ �� 196 = 14

3. Write 5y - 29 = -14x in standard form. 14x + 5y = 29

*4. Multiple Choice What absolute-value function is shown by the graph? CA f(x) = 2⎢x� B f(x) = 0.5⎢x�

C f(x) = -5⎢x� D f(x) = -0.5⎢x�

5. Translate the inequality 3b + 2 _ 5 ≥ 1 3

_ 5 into a sentence. Sample: The sumof 3 times an unknown and 2_

5 is greater than or equal to 1 3_

5.

*6. Boating The path of a sailboat is represented by the function

f(x) = ⎪ 3 _ 5 x - 30⎥ + 30. At what point does the sailboat tack (turn)? (50, 30)

*7. Write Why does the graph of an absolute-value function not extend past the vertex?

8. Solve the system of linear equations: 4y = -3x - 4

4x + 6 = -5y

. (-4, 2)

*9. Geometry The perimeter of the square is 20 centimeters. Solve for x. x = 25 cm

√x

(107)(107)

(106)(106)

(35)(35)

(107)(107)

x

y

O

4

2

2 4-2-4

x

y

O

4

2

2 4-2-4

(45)(45)

(107)(107)

(107)(107)7. Sample: The absolute-valuefunction has a minimum value and that is the y-value at the vertex.

7. Sample: The absolute-valuefunction has a minimum value and that is the y-value at the vertex.

(63)(63)

(106)(106)

f. Since a < 0, the graph is reflected across the x-axis. Since ⎢a� < 1, the graph is compressed vertically.


Have students who are struggling with the lesson’s concepts and abstractions graph the line y = x, but tell them that it cannot go below the x-axis. Then have them graph the line y = -x, and again give the condition that the line cannot go below the x-axis.

“At which point do these two lines meet?” (0, 0)

“In an absolute-value function, what is this point called?” the vertex

“What do all the values of y have in common?” They are all positive.

INCLUSION

724 Saxon Algebra 1

Lesson Practice

Problem b

Error Alert If students mistakenly graph the vertex at (2, 0), remind them that they need to fi nd a value for x that makes the expression within the absolute-value brackets equal to 0.

Problem f

Scaff olding First have students fi nd the values for a and ⎢a�, and then ask them where the graph is refl ected if a < 0. Then ask how the value of a further changes the graph of the absolute-value parent function.



“How is the vertex of an absolute-value function found?” Sample: Find the values of h and k in the function and use (h, k).

“What are two different absolute-value functions that share the same vertex?” Sample: y = ⎢x�; y = ⎢-x�

Practice3


Problem 8


“What is the best method for solving this system of linear equations, and why?” elimination; Sample: Solving by substitution would require computing with fractions.

“Are both equations in standard form?” no

“Does either equation need to be multiplied by a constant in order to eliminate a variable?” Yes, both equations need to be multiplied by a constant.


Lesson 107 725

*10. Error Analysis Two students found the solution for a radical equation. Which student is correct? Explain the error. Student A; Sample: Student B squared incorrectly and should have subtracted seven from both sides, first.

Student A

√ � x + 7 = 14 √ � x = 7 x = 49

Student B

√ � x + 7 = 14 √ � x + 49 = 196 √ � x = 147 x = 21,609

11. Jason built a new deck with an area of (-20x + 100 + x2) square feet. The width is (x - 10) feet. What is the length? (x - 10) feet

*12. Multi-Step A triangular brace is constructed in the shape of a right triangle. The two legs of the brace are √ �� x + 5 and √ � x units long. a. What expression could be used to solve for the length, l, of the third side

of the brace? ( √ �� x + 5 ) 2

+ ( √ � x) 2 = l 2

b. Simplify the equation so it does not contain any radicals. 2x + 5 = l 2

c. Find the value of x for which the length of the third side of the brace is equal to 10. x = 95_

2

13. Coordinate Geometry Find the coordinates of the point(s) at which the graphs of y = x and y = √ � x intersect. (0, 0), (1, 1)

14. Find the next 3 terms of the sequence 125, 25, 5, 1. 0.2, 0.04, 0.008

*15. Carbon Dating Scientists can use the ratio of radioactive carbon-14 to carbon-12 to find the age of organic objects. Carbon-14 has a half-life of about 5730 years, which means that after 5730 years, half the original amount remains. Carbon dating can date objects to about 50,000 years ago, or about 9 half-lives. About what percent of the original amount of carbon-14 remains in objects about 50,000 years old? about 0.2%

*16. Error Analysis Two students find the 5th term in a geometric series that has a common ratio of 1

_ 2 and a first term of 6. Which student is correct? Explain the error.

Student A

A(n) = ar n-1

= 6 · ( 1 _ 2 )

4

= 3 _ 8

Student B

A(n) = ar n-1

= 6 · 1 _ 2 · 4

= 12

17. Solve by graphing on a graphing calculator. Round to the nearest tenth.

-11x2 + x = -4 x = -0.6 and 0 .7


18. ⎢x - 4� + 15 ≥ 21 x ≤ -2 OR x ≥ 10 19. ⎢x� + 45 ≤ 34 {∅};

(106)(106)

(93)(93)

(106)(106)

(106)(106)

(105)(105)

(105)(105)

(105)(105)

Student A; Student B incorrectly multiplied by 4 rather than using 4 as an exponent.

Student A; Student B incorrectly multiplied by 4 rather than using 4 as an exponent.

(100)(100)

(101)(101)

80 4-4 1280 4-4 12

(91)(91) 40 2-2-4 40 2-2-4


x

y

O

4

2

2 4

-2

-2

Have students graph the function f (x) = ⎢x2 - 2x - 4� using a table of values.

Describe how this graph differs from the graph of the function f (x) = x2 - 2x - 4.Sample: Because there cannot be a negative value for y, the points that fall below the x-axis are “fl ipped,” or refl ected over the x-axis.

CHALLENGE

Lesson 107 725

Problem 11

Have students write the area as a trinomial in standard form, and point out that they can use what they learned about special product patterns to factor the trinomial into a perfect-square binomial.

Students may need a reminder that the square root of any quantity squared is the quantity.

Use ( √ � 25 ) 2 = (5)2 = 25 as an

example.


Saxon Algebra 1726

20. Football NCAA rules require that the circumference c of a football, measured around its widest part, 21 inches, to vary by no more than 0.25 inches. Write and solve an absolute-value inequality that models the acceptable circumferences. What is the least acceptable circumference? ⎢c - 21� ≤ 0.25; 20.75 ≤ c ≤ 21.25; 20.75 inches

21. Area of a Pool Maria wants to increase the radius of a pool by 3 meters.The new area of the pool is 200.96 square meters. a. Write a formula to find the original radius of the pool.

b. Solve the formula. r = 5 m

c. What will the new diameter of the pool be? 16 m

*22. Graph the function f(x) = ⎢x� + 3.

23. Multiple Choice Solve -3x2 + 24x = 36. CA x = -8 or 0 B x = -6 or 2 C x = 2 or 6 D x = 0 or 8

24. Analyze Determine what values of c would make the equation x2 - 50x = c have no solution. c < -625

Simplify.

25.

4x _

2x + 12 +

x _ 3x + 18

__

8x2

__ x2 + 8x + 12

7(x + 2)

_24x 26. √ ��

20

_ 3

2√ �� 15_3

27. Multi-Step A businessman makes $50 profit on each item sold. He would like to make $950 plus or minus $100 total each week. a. Write an absolute-value equation for the minimum and maximum profit he

desires. ⎢50x - 950� = 100

b. What is the minimum and maximum number of items he needs to sell each week? 17 items, 21 items

28. School Dance Tickets for the school dance are $4 for middle school students and $6 for high school students. In order to cover expenses, at least $600 worth of tickets must be sold. Write an inequality that models this situation and graph it.

29. Multi-Step A painting is 5 inches by 4 inches. The frame around it is x inches wide. a. Write expressions for the length and width of the picture with the frame.

b. The total area of the picture and frame is 42 square inches. What is the width of the frame? 1 inch

30. Justify Explain how to transform x _

x - 3 = 4 _

x to x2 = 4x - 12.

Sample: cross multiply

(101)(101)

(102)(102)

r 3r 3

π(r + 3)2 = 200.96π(r + 3)2 = 200.96

(107)(107)22.

x

y

44

6

2

-2

2 4-2-4O

22.

x

y

44

6

2

-2

2 4-2-4O

(104)(104)

(104)(104)

(92)(92) (103)(103)

(94)(94)

(97)(97)

28. 4x + 6y ≥ 600;

x

y

O

200

100

28. 4x + 6y ≥ 600;

x

y

O

200

100

(98)(98)

29a. 5 + 2x, 4 + 2x29a. 5 + 2x, 4 + 2x(99)(99)

x in.

x in.

x in.5 in.

4 in.

x in.


Graphing absolute-value functions prepares students for




• Lesson 119 Graphing and Comparing Linear, Quadratic, and Exponential Functions

LOOKING FORWARD

726 Saxon Algebra 1

Problem 21

Extend the Problem

“Maria wants to build a circular deck around her new pool that has a width of x meters. Write an expression to determine the area of the deck.” π (x + 8)2 - 200.96

“If the area of the deck is equal to 113.04 square meters, what is the width of the deck?” 2 meters

Problem 25

Error Alert Students may make errors if they miss a step while simplifying. Suggest they use the checklist: 1) Factor each denominator completely. 2) Find the LCD. 3) Write equivalent fractions. 4) Add the terms in the numerator. 5) Divide the expression in the numerator by the expression in the denominator.

Problem 28

Discuss with students the domain and range of the situation. Use only Quadrant 1 since a negative number of tickets cannot be sold. Since only a whole number of tickets can be sold, all the points with whole-number coordinates on or above the line are the different combinations of tickets that can be sold.


Lesson 108 727

Warm Up

108LESSON

1. Vocabulary In the expression 35, 5 is the . exponent

Simplify.

2. 42 16 3. 6 -3 1_216

4. 2 · 5 -2 2_25 5. 5 · 2 -1 5_2

In a geometric sequence, any term, except the first, can be found by multiplying the previous term by the common ratio. In the geometric sequence 2, 6, 18, 54, 162, …, the common ratio is 3.

The sequence can also be written like this: 2, 2(3)1, 2(3)2, 2(3)3, 2(3)4, …. Or, with a1 representing the first term and r representing the common ratio, it can be written as a1, a1(r)1, a1(r)2, a1(r)3, a1(r)4, ….

Using n as the term number, observe that the nth term of a geometric sequence can be found by using the rule an = a1rn-1.

Notice that the independent variable n occurs in the exponent of the function rule. Any function for which the independent variable is an exponent is an exponential function.

Exponential FunctionAn exponential function is a function of the form f (x) = a b x , where a and b are nonzero constants and b is a positive number not equal to 1.

Example 1 Evaluating an Exponential Function

Evaluate each function for the given values.

a. f (x) = 5 x for x = -3, 0, and 4.

SOLUTION

Use the order of operations.

f (-3) = 5 -3 = 1 _ 53

= 1 _

125 , f (0) = 50 = 1, f (4) = 54 = 625

b. f (x) = 2(4 ) x for x = -1, 1, and 2.

SOLUTION

Use the order of operations. Evaluate exponents before multiplying.

f (-1) = 2(4 ) -1 = 2 · 1 _ 4 =

2 _ 4 =

1 _ 2

f (1) = 2(4)1 = 2(4) = 8

f (2) = 2(4)2 = 2(16) = 32

(3)(3)

(3)(3) (3)(3)

(3)(3) (3)(3)


Online Connection


Identifying and Graphing Exponential

Functions

Reading Math

The value of b in an exponential function is comparable to r in a geometric sequence.

Hint

a-n = 1 _ a n


LESSON RESOURCES



Master C108Technology Lab Master 108

By restricting the domain to the whole numbers, exponential functions can represent geometric sequences where each successive term is multiplied by the same value.

Exponential functions are nonlinear functions that have a variety of applications, including tracking growth, population changes, compound interest, and predicting decay, such as that studied in carbon dating and other half-life calculations.

MATH BACKGROUND

Warm Up1

108LESSON

Lesson 108 727

Problems 3–5

Remind students that a base with a negative exponent can be expressed as the reciprocal of the base to the opposite exponential value.

2 New Concepts

In this lesson, students will investigate exponential patterns and graph them. They will also look at applications of exponential functions, such as growth.

Example 1

In an exponential function, the value of the base of the exponent is constant and the exponent is the term that changes.



a. f (x) = (-3 ) x for x = -2, 0, and 3. 1 __

9 , 1, -27

b. f (x) = 2(2 ) x for -2, 2, and 7. 1 __ 2 , 8, 256


Saxon Algebra 1728

The common ratio of a geometric sequence is comparable to the base of an exponential function. For any exponential function, as the x-values change by a constant amount, the y-values change by a constant factor. For f (x) =

4(2 ) x , as each x-value increases by 1, each y-value increases by a factor of 2.

x -1 0 1 2 3f (x) 2 4 8 16 32

Change: +1

Change: ×2

The base 2 of the exponential function f (x) = 4(2 ) x is the common ratio of

the sequence 2, 4, 8, 16, 32, ….

Example 2 Identifying an Exponential Function

Determine if each set of ordered pairs satisfies an exponential function. Explain your answer.

a. ⎧

⎨

⎩ (0, -3), (-2, -

1 _ 3 ) , (1, -9), (-1, -1)

⎫ ⎬

⎭

SOLUTION

Arrange the ordered pairs so that the x-values are increasing.

⎧ ⎨

⎩ (-2, -

1 _ 3 ) , (-1, -1), (0, -3), (1, -9)

⎫ ⎬

⎭

The x-values increase by the constant amount of 1.

Divide each y-value by the y-value before it.

-1 ÷ - 1 _ 3 = -1 × -3 = 3

-3 ÷ -1 = 3

-9 ÷ -3 = 3

Because each ratio is the same, 3, the base b = 3. The set of ordered pairs satisfies an exponential function.

b. {(6, 150), (4, 100), (8, 200), (2, 50)}

SOLUTION

Arrange the ordered pairs so that the x-values are increasing.

{(2, 50), (4, 100), (6, 150), (8, 200)}

The x-values increase by the constant amount of 2.

Divide each y-value by the y-value before it.

100 ÷ 50 = 2

150 ÷ 100 = 1 1 _ 2

200 ÷ 150 = 1 1 _ 3

Because the ratios are not the same, the ordered pairs do not satisfy an exponential function.

Reading Math

In the expression f(x) =

4(2 ) x , 2 is the base and x is the exponent.

Math Reasoning

Analyze What type of function do the ordered pairs in Example 2b satisfy, and why?

linear function; The range values increase by the constant amount of 50.


A table of values can be useful in helping visual learners identify any patterns in the values of the functions. If the pattern in the values is multiplicative, the function is exponential. Have students make a table like the one shown for Example 2b. Show them that the y-value does not change by the same ratio, and therefore is not an exponential function.

x 2 4 6 8f (x) 50 100 150 200

Change: +2

Change: +50

INCLUSION FOR EXAMPLE 2b

728 Saxon Algebra 1

Example 2

Point out that the pattern of the outputs of the terms in an exponential function is multiplicative. If students were to look for the differences, they would not fi nd a constant difference between successive terms.

Extend the Example

Using the formula for a geometric sequence, fi nd the ordered pair with an x-coordinate of 5. -729


Determine if each set of ordered pairs satisfi es an exponential function. Explain your answer.

a. {(2, 4), (1, 2), (4, 128), (3, 16)} Because the ratios are not the same, the ordered pairs do not satisfy an exponential function.

b. {(12, 64), (4, 4), (16, 256), (8, 16)} Because each ratio is the same, 4, the base b = 4. The set of ordered pairs satisfi es an exponential function.


Lesson 108 729

To graph an exponential function, make a table of ordered pairs and plot the points. The graph will always form a curve that comes close to, but never touches, the x-axis.

Example 3 Graphing y = a b x

Graph each function by making a table of ordered pairs.

a. y = 5(2 ) x

SOLUTION

Choose both positive and negative x-values.

x y

-2 1.25

-1 2.5

0 5

1 10

2 20

x

y

O

8

16

24

2 4-2-4

(-2, 1.25)

(-1, 2.5)(0, 5)

(1, 10)

(2, 20)

b. y = -(3 ) x

SOLUTION

x y

-1 - 1 _ 3

0 -1

1 -3

2 -9

3 -27

xy

O

-8

-16

-24

2 4-2-4

(3, -27)

(2, -9)(1, -3)(0, -1)

( 1, 1_3)

c. y = 6 ( 1 _ 2 )

x

SOLUTION

x y

-2 24

-1 12

0 6

1 3

2 1.5

x

y

O

16

24

2 4-2-4

(-2, 24)

(-1, 12)

(0, 6) (1, 3)

(2, 1.5)

Math Reasoning

Generalize Compare the domains and ranges of the functions in Examples 3a and 3b.

The domains for both are the same: all real numbers. In 3a, the range is real numbers greater than 0, in 3b it is real numbers less than 0.

Caution

Due to limitations of scale, graphs of exponential functions often appear to touch the x-axis. The graph will approach but never touch the x-axis. Since a ≠ 0 and b ≠ 0, then y ≠ 0.


Students may need additional help in organizing their work. Have them create a table of values. Offer them blank copies of the table at right to help show the substitution step. Give them space to simplify the expressions.

The function y = 5 (2) x is evaluated for

x = -2 as an example.

INCLUSION

x Substitute Simplify y

-2 y = 5 (2) -2 y = 5 ( 1 __ 4 ) 1.25

Lesson 108 729

Example 3

Creating a table of values is the best method for graphing an exponential equation without technology.

TEACHER TIPRemind students to use the order of operations to simplify the expressions.



a. y = - 4 x

x -1 0 1 2 3

y - 1 __ 4 -1 -4 -16 -64

21 3 xyO

-16

-32

-48

-64

b. y = 8(2 ) x

x -2 -1 0 1 2

y 2 4 8 16 32

x

y

O

16

24

8

2 4-2-4

c. y = 4 ( 1 __ 4 ) x

x -1 0 1 2

y 16 4 1 1 __ 4

x

y

O

4

2 4-2-4


Saxon Algebra 1730

A graphing calculator is helpful with comparing graphs of functions and formulating rules based on the values of a and b.

Example 4 Comparing Graphs

Using a graphing calculator, graph each pair of functions on the same screen. Tell how the graphs are alike and how they are different.

a. y = 3(2 ) x and y = -3(2 ) x

SOLUTION

Use to enter the equations. Use to graph the equations.

Alike: Both graphs are symmetric about the x-axis. For any x-value, the absolute values of the corresponding y-values are the same.

Different: When a = 3, the y-values increase from left to right. Whena = -3, the y-values decrease from left to right.

b. y = 3(2 ) x and y = 3 ( 1 _ 2 )

x

SOLUTION

Alike: Both graphs are above the x-axis and symmetric about the y-axis. For any y-value, the absolute values of the corresponding x-values are the same.

Different: When b = 2, the y-values increase from left to right. When b = 1 _

2 , the y-values decrease from left to right.

Example 5 Application: Population

The exponential function y = 12.28(1.00216 ) x models the approximate population of Pennsylvania from 2000 to 2006, where x is the number of years after 2000 and y represents millions of people. Use a graphing calculator to find the approximate population of Pennsylvania in 2005. Assuming the model does not change, when will the population reach 13 million?

Hint

Use 6 to set the intervals on the x-axis and y-axis from -10 to 10.

Math Reasoning

Generalize For which values of b, 2 or 1 _ 2 , do the y-values approach 0 as x increases? As x decreases?

b = 1_2; b = 2


Students can also compare pairs of functions by making tables instead of graphing.

y = 3(2 ) x

x -2 -1 0 1 2

y 0.75 1.5 3 6 12

y = -3(2 ) x

x -2 -1 0 1 2

y -0.75 -1.5 -3 -6 -12

ALTERNATE METHOD FOR EXAMPLE 4a

730 Saxon Algebra 1

Example 4

Changes to the values of the coeffi cient or base of an exponential equation affect the shape and characteristics of the graph.

Error Alert Students may think that any function with an exponential term is an exponential function. Have them look carefully at the function and determine if the exponent has a variable term in it. If the exponent is constant, then it is not an exponential function.



a. y = 2(2 ) x and y = 1 __ 2 (2 ) x

Sample: Alike: Both graphs increase from left to right and all values of y are positive. Different: The graph of the second equation does not increase as steeply as the fi rst.

b. y = 3 ( 1 __ 2 ) x

and y = -3 ( 1 __ 2 ) x

Sample: Alike: Both graphs are symmetric about the x-axis. For any x-value, the absolute value of the corresponding y-values are the same. Different: When a = 3, the y-values decrease from left to right. When a = -3, the y-values increase from left to right.



Lesson 108 731

SOLUTION Enter the function rule into the Y= editor. Access the Table

function by pressing . Since 2005 is 5 years after 2000, find the y-value for x = 5. The population was about 12,413,000. To find when the population will reach 13 million, scroll down until y equals 13 or more. It occurs during the 27th year after 2000, or 2027.

Lesson Practice


a. Evaluate f (x) = 2 x for x = -4, 0, and 5. f(-4) = 1_16

, f(0) = 1, f(5) = 32

b. Evaluate f (x) = -3(3 ) x for x = -3, 1, and 3. f(-3) = - 1_9, f(1) = -9,

f(3) = -81Determine whether each set of ordered pairs satisfies an exponential function. Explain your answer.

c. {(3, -12), (6, -24), (12, -48), (9, -36)} No, the y-values do not have a common ratio.

d. {(3, 108), (1, 12), (2, 36), (4, 324)} Yes, as x increases by 1, the ratio of the y-values = 3.


e. y = 2(3 ) x f. y = -4(2 ) x g. y = 2 ( 1 _ 4

) x


h. y = ( 1 _ 3

) x

and y = - ( 1 _ 3 )

x

i. y = -2(3 ) x and y = -2 ( 1 _ 3

) x

j. The exponential function y = 8.05(1.01683 ) x models the approximate population of North Carolina from 2000 to 2006, where x is the number of years after 2000 and y represents millions of people. Use a graphing calculator to find the approximate population of North Carolina in 2006. Assuming the model does not change, when will the population reach 10 million? 8,897,900; 2013

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

See Additional Answers.See Additional Answers. See Additional Answers.See Additional Answers. See Additional Answers.See Additional Answers.

(Ex 4)(Ex 4)h. Sample: Alike: Both are symmetric about the x-axis. For any x-value, the absolute values of the y-valuesare the same. Different: When a is positive, all the range values are positive. When a is negative, all the range values are negative.

h. Sample: Alike: Both are symmetric about the x-axis. For any x-value, the absolute values of the y-valuesare the same. Different: When a is positive, all the range values are positive. When a is negative, all the range values are negative.

(Ex 4)(Ex 4)

i. Sample: Alike: Both graphs are below thex-axis and symmetic about the y-axis.Different: When b is 3, the y-values decrease as the x-values increase. When b is 1_

3, the

y-values increase as the x-values increase.

i. Sample: Alike: Both graphs are below thex-axis and symmetic about the y-axis.Different: When b is 3, the y-values decrease as the x-values increase. When b is 1_

3, the

y-values increase as the x-values increase.

(Ex 5)(Ex 5)

*1. Evaluate the function f (x) = 2(5 ) x for x = -2, 0, and 2. 2_25, 2, 50

2. Graph the function f (x) = ⎢x - 2⎢. See Additional Answers.

(108)(108)

(107)(107)

Caution

The variable y represents millions of people. The table entry y1 = 12.413 means 12.413 million.


Students could use a graphing calculator to solve the problem in Example 5.

They should graph the equation given as Y1. Then they should graph y = 13 as Y2, and use the CALC function to fi nd the intersection and get an exact value. Since the x-value of the point of intersection is approximately 26.4, the population reaches 13 million during the 27th year.


Lesson 108 731

Example 5

Remind students that since the variable represents the years after 2000, they must determine a value and add 2000 to it to fi nd the actual year. Also point out that y represents millions of people, so 12.28 means 12,280,000 people.


Assuming the model does not change, when will the population reach 15 million? 2092

Lesson Practice

Problem b

Error Alert Remind students to evaluate the exponent before multiplying by the coeffi cient -3.

Problem c

Scaff olding Remind students to organize the data so that the fi rst coordinates are in increasing order.



“Use the number 2 and the variable x to create a linear equation. Describe its graph.” Sample: f(x) = 2x; The graph would be a straight line with a slope of 2, and a constant difference of 2 in the y-values.

“Use the number 2 and the variable x to create a quadratic equation. Describe its graph.” Sample: f(x) = x2; The graph is a parabola and there is no constant difference or ratio in the y-values.

“Use the number 2 and the variable x to create an exponential equation. Describe its graph.” Sample: f(x) = 2x; The graph is nonlinear and the constant ratio of the y-values is 2.


Saxon Algebra 1732

*3. Justify Why is f (x) = 4(1 ) x not an exponential function?

*4. Multiple Choice Which could be the function graphed? B

A y = - ( 1 _ 2 )

x

B y = ( 1 _ 2 )

x

C y = -(2 ) x D y = 2 x

*5. Population The exponential function y = 20.85(1.0212 ) x can model the approximate population of Texas from 2000 to 2006, where x is the number of years after 2000 and y represents millions of people. Assuming the model does not change, what is the difference in expected populations for 2010 and 2020? 6,002,563

*6. Verify Show that the set {(3, -4), (2, -1), (5, -64), (4, -16)} is an exponential function when b = 4.

7. Name the corresponding sides and angles if ΔRST ∼ ΔNVQ. −−

RS and−−

NV ;−−

STand

−− VQ;

−− RT and

−−− NQ; ∠R and ∠N; ∠S and ∠V; ∠T and ∠Q

*8. Multi-Step Graph the parent function f (x) = ⎢x�. Translate the function down by 2. Then reflect the function across the x-axis. What is the new function? f(x) = -⎢x� - 2

9. Is the graph an absolute-value function? Explain. no; Sample: There is no axis of symmetry.

x

y

O

4

2 4

-2

-2-4

-4

10. Evaluate 3 √ � x when x = (-4 ) 3 . -4

*11. Geometry Describe why the function f (x) = ⎢x� is in the shape of a “V”. Sample: The output is the same for x and -x.

12. Error Analysis Two students found the solution to a radical equation. Which student is correct? Explain the error.

Student A

√ �� x + 3 = 6x + 9 = 36

x = 27

Student B

√ �� x + 3 = 6x + 3 = 36

x = 33

13. Solve √ � x - 2 = 8. Check your answer. x = 100; √ �� 100 - 2 = 10 - 2 = 8

(108)(108)

(108)(108)

(108)(108)

(108)(108)

6. Sample: When the ordered pairs are arranged so that the x-values are 2, 3, 4, and 5, then the y-values are -1, -4,-16, and -64;-64 ÷ -16 = 4,-16 ÷ -4 = 4 and

-4 ÷ -1 = 4.Because thex-values increase by the constant amount of 1, the common ratio is the value of b.

6. Sample: When the ordered pairs are arranged so that the x-values are 2, 3, 4, and 5, then the y-values are -1, -4,-16, and -64;-64 ÷ -16 = 4,-16 ÷ -4 = 4 and

-4 ÷ -1 = 4.Because thex-values increase by the constant amount of 1, the common ratio is the value of b.

(36)(36)

(107)(107)

(107)(107)

(46)(46)

(107)(107)

(106)(106)

12. Student B; Sample: Student A squared the number 3 within the radicand instead of squaring the expression √ �� x + 3.

12. Student B; Sample: Student A squared the number 3 within the radicand instead of squaring the expression √ �� x + 3.

(106)(106)

3. Sample: Because 1 raised to any power is 1, and 4 would be multiplied by 1 for every value of x, the resulting constant linear function is f(x) = 4


732 Saxon Algebra 1

Practice3


Problem 7

Have students draw each triangle before they identify corresponding sides and angles.

Problem 10

Error Alert Some students may want to drop the negative sign when they solve because they are thinking about the principal square root. Remind them that the root they are looking for is odd, which maintains the negative value.

Problem 11

Students may need to graph a few points of the function f (x) in order to determine why the function is in the shape of a “V”.


Lesson 108 733

*14. Write the equation of the function graphed. f(x) = -2⎢x - 2⎢ + 3

x

y

O

4

2

2-2-4

-4

15. Assuming that y varies inversely as x, what is y when x = 8, if y = 55 whenx = 11.6? y = 79.75

16. Meteorology In the mountains snow will accumulate quickly in winter. If the average accumulation can be described using the expression 12 √ � x , find the value of x when the accumulation is equal to 108 inches? x = 81

17. Solve and graph the inequality ⎢x�

_ 8 - 10 < -9. -8 < x < 8;

80 4-4-8

Solve.

18. x2 = -9 no solution 19. 12⎢x + 9⎢ - 11 = 1 {-10, -8}

20. Building Tom’s house has two square rooms. He knocks down a wall separating the rooms. The area of the new room is 338 square feet. What were the dimensions of the original rooms? 13 ft × 13 ft

x

x

21. Simplify 24a2b _

7c2 _

8ab2

_ 49c2

. 21a_b

22. Find the missing term of the perfect-square trinomial: x2 + 7x + . 49_4

= 12.25 *23. Multiple Choice What is the common ratio of the geometric sequence - 5 _

8 , - 5 _

16 ,

- 5 _ 32

, - 5 _ 64

, …? C

A -2 B - 1 _ 2

C 1 _ 2 D 2

*24. Landscaping Li is designing a triangular flower bed in one corner of her rectangular yard. She plans on making one leg of the triangle 1 11

_ 12 meters long

and the other leg 2 5 _ 12 meters long. She wants to know how much edging material

she needs to buy to place along the hypotenuse of the triangle. Write a rational expression to show how much material Li needs to buy.

√ �� 1370_

12 meters 2 5_12

m

111_12

m?

(107)(107)

(64)(64)

(106)(106)

(101)(101)

(102)(102) (94)(94)

(102)(102)

(92)(92)

(104)(104)

(105)(105)

(103)(103)


Explain the meaning of accumulate and accumulation in Problem 16. Say:

“Accumulate is a verb that means to build or pile up. Accumulation is a noun and means something that is gathered over a period of time.”

Use snow as an example of accumulation. Explain that, when referring to snow, people say it accumulates, or piles up. Money that gathers interest in a bank is also an example of accumulation.

ENGLISH LEARNERS

Lesson 108 733

Problem 15

Watch for students who work this problem as a direct variation. Discuss the differences in a direct and inverse variation. The constant in a direct variation is equal to the ratio of two numbers. The constant in an inverse variation is equal to the product of two numbers.

Problem 17

Error AlertSome students may think that there is no solution since there is a negative number on the right side of the inequality symbol. Remind them that the absolute value must be isolated before solving.

Problem 20

Extend the Problem

The area of Tom’s house is 2x2 + 30x. What is the area of Tom’s house minus the area of the two rooms? 390 ft2


Saxon Algebra 1734

25. Analyze Is the sequence -72, -57.6, 46.08, 36.864, … geometric? Explain.

26. Find the quotient of (36x + 12x2 + 15) ÷ (2x + 1). 3(2x + 5)

27. Multi-Step Amber drove 7x2

_ x2 - 49

miles on Monday and x - 1 _

4x + 28 miles on Tuesday

while delivering pizzas. a. What is the total distance she drove? 29x2 - 8x + 7__

4(x + 7)(x - 7) miles

b. If her rate was 7

_ 7x + 49 miles per hour, how much time did it take her to deliver pizzas on Monday and Tuesday? 29x2 - 8x + 7_

4(x - 7) hours

28. Construction A box needs to be built so that its rectangular top has a length that is 3 more inches than the width, and so that its area is 88 square inches. Find the length and the width. The width is 8 inches, and the length is 11 inches.

29. Multi-Step Sherry can enter all weekly data into the computer in 16 hours. When she works with Kim, they complete the data entry in 9 hours 36 minutes. a. Convert 9 hours 36 minutes to hours. 9.6 hours

b. Write an equation to find how long it would take Kim to enter the same data. 9.6_

16 + 9.6_

K = 1

c. How long would it take Kim to enter the data alone? 24 hours

30. Analyze If the y-coordinate of the ordered pair represents the maximum height of the path of a ball thrown into the air, what does the x-coordinate represent? Sample: It represents the time it takes for the ball to reach that height.

(105)(105)

25. no; Sample: The absolute values of the terms have a common ratio of 4_

5,

but the signs of the terms do not follow a geometric pattern.

25. no; Sample: The absolute values of the terms have a common ratio of 4_

5,

but the signs of the terms do not follow a geometric pattern.

(93)(93)

(95)(95)

(98)(98)

(99)(99)

(100)(100)


Have students do research to fi nd some common exponential functions, such as half-life and compound interest. Then have them choose one function to create and solve their own problem. For example, the half-life of radioactive cobalt is 30 years. After 90 years, how much of 100 grams of cobalt remain? 12.5 grams

Identifying and graphing exponential functions prepares students for





734 Saxon Algebra 1

Problem 26

Students should note that they can factor 3 out of the dividend fi rst to make the division of the polynomial easier.

Problem 27


“What must be found before adding the fractions?” Sample: Find the least common denominator

“How can the LCD be determined?” Sample: Factor each denominator, then use each factor that occurs in either denominator in the LCD. Factors common to both denominators should be used the greatest number of times that the factor occurs.

“How can Amber’s time be found?” Sample: Use the formula r t = d. Use the total found in part a for the distance and the rate given in part b for the rate.

Problem 29


“How much of the work can Sherry do in 1 hour?” 1

___ 16

“How much of the work can Kim do in 1 hour?” 1 __ x

“How can the total amount of work done by either girl be determined?” Sample: Multiply the rate of work by the amount of time worked.


Lesson 109 735

Warm Up

109LESSON

1. Vocabulary A(n) (inequality, equality) is a mathematical statement comparing quantities that are not equal. inequality

2. Graph y < 2x + 3.

3. Is the boundary of the graph of y ≤ 3x + 5 solid or dashed? solid

4. Is the shading above or below the boundary line on the graph of y ≥ 2x - 6? above

Recall that a system of linear equations is a set of two or more equations with the same variables.

The solution of the system below is (1, 2) because the ordered pair (1, 2) makes both equations true.

y = x + 1

y = 2x

y = x + 1 y = 2x

2 = 1 + 1 2 = 2(1)

2 = 2 2 = 2

The coordinates also identify the point of intersection of the two lines.

Likewise, a system of linear inequalities is a set of linear inequalities with the same variables.

In the system shown below, all of the ordered pairs in the overlapping region satisfy both inequalities. For example, (3, 2) lies in the overlapping region and makes both inequalities true.

y ≤ x + 1

y ≤ 2x

y ≤ x + 1 y ≤ 2x

2 ≤ 3 + 1 2 ≤ 2(3)

2 ≤ 4 2 ≤ 6

A solution of a system of linear inequalities is an ordered pair or set of ordered pairs that satisfy all the inequalities in the system. Therefore, all the ordered pairs in the overlapping region make up the solution of the system.

(45)(45)

(97)(97)

2.

-4

-2

x

y

2 4-4

4

O

2.

-4

-2

x

y

2 4-4

4

O(97)(97)

(97)(97)


x

y4

2

2 4-2-4

-4

y = x + 1

y = 2x

(1, 2)

x

y4

2

2 4-2-4

-4

y = x + 1

y = 2x

(1, 2)

x

y4

2

2 4

solution set

-2-4

-4

x

y4

2

2 4

solution set

-2-4

-4

Graphing Systems of Linear Inequalities

Online Connection


Math Reasoning

Verify Show that (-4, -4) is not a solution of the system.

It does not satisfy y ≤ 2x because -4 is not less than or equal to -8.

SM_A1_NL_SBK_L109.indd Page 735 1/14/08 2:05:40 PM user /Users/user/Desktop/Anil_14-01-08/current

LESSON RESOURCES



Master C109, E109Technology Lab Master 109

Solutions of systems of inequalities consist of an infi nite set of points common to both inequalities. Generally, both inequalities are graphed on the coordinate plane, and the solution consists of all points in the region where the shading overlaps. Suggest that students use vertical lines to shade one of the regions and horizontal lines to shade the other. Then the cross-hatched region will show the solution set.

Remind students that points along a dashed line are not included in the solution set. Substituting a test point from a dashed line into both inequalities will show that the point does not satisfy at least one of the inequalities.

MATH BACKGROUND

Warm Up1

109LESSON

Lesson 109 735

Problem 4

Remind students that they can check the direction of the inequality by substituting values.

2 New Concepts

In this lesson, students learn to graph systems of linear inequalities. It may be helpful to begin with a brief review of graphing linear inequalities–when to shade above or below the line and when to make the line solid or dashed.

Use of two different colors of lead pencils or a pen and pencil to graph the systems may help students see the solution set of the system more easily.


Saxon Algebra 1736

Example 1 Solving by Graphing

Graph each system.

a. y > 1 _ 4 x - 3

y ≤ 3x + 4.

SOLUTION

Graph each inequality on the same plane. Every point in the overlapping region is a solution.

Check Substitute a point in the overlapping region to see that it satisfies both inequalities. The point (0, 0) is convenient to substitute.

y > 1 _ 4 x - 3 y ≤ 3x + 4

0 > 1 _ 4 (0) - 3 0 ≤ 3(0) + 4

0 > -3 ✓ 0 ≤ 4 ✓

b. y < 4

2y + 2 > -6x.

SOLUTION

Write the second inequality in slope-intercept form.

2y + 2 > -6x

2y > -6x - 2

y > -3x - 1

Check See if (0, 0) satisfies both inequalities.

y < 4 2y + 2 > -6x

0 < 4 ✓ 2(0) + 2 > -6(0)

2 > 0 ✓

Example 2 Solving with a Graphing Calculator

Graph the system on a graphing calculator.

y < 3 _ 4 x + 2

y ≥ - 1 _ 5 x + 4

SOLUTION Enter both functions. Use the arrow keys to move to the symbol to the left of Y1 and press enter until the symbol shows the lower half of a plane shaded. For Y2, select the symbol with the upper half shaded.

Note that for many graphing calculators, the option to choose between a strict and non-strict inequality does not exist.

x

y4

2

2 4-2-4

-4

-2

O x

y4

2

2 4-2-4

-4

-2

O

x

y

2

2 4-2-4

-4

-2

x

y

2

2 4-2-4

-4

-2

Caution

Do not forget to use a dashed line for the boundary line when the inequality has < or >.

Graphing

Calculator Tip

For help with graphing inequalities, refer to the graphing calculator keystrokes in Graphing Calculator Lab 9 on p. 645.


For Example 1 explain the meaning of convenient. Say:

“Convenient means easy. Usually, it means ‘easy’ at a particular time. If we are near a mall, it is convenient to shop there due to the variety of stores.”

Ask a volunteer to use the word convenient in a sentence. Sample: It’s convenient to do my homework while waiting for the bus.

ENGLISH LEARNERS

736 Saxon Algebra 1

Example 1


Graph each system.

a. y < 3x + 2 y ≤ 1 __

3 x + 10.

y16

168-8-16O

b. y < 2x + 1 3y + 6 > -9x.

x

y

5 10-5-10

Example 2


Graph the system on a graphing calculator.

y < 1 __

5 x + 2

y ≥ 1 ___ 10

x + 5

x

y80

40

60 120

-40

-80


Lesson 109 737

Remember that a system of equations is inconsistent when there are no solutions. This occurs when the slopes of the lines are the same and the y-intercepts are different.

The system has no solutions because the lines are parallel.

y = - 2 _ 3 x - 2

y = - 2 _ 3 x + 3

When the equal signs in these equations are replaced with inequality symbols, the system may or may not have a solution set.

Example 3 Solving Systems of Inequalities with Parallel

Boundary Lines

Graph each system.

a. y ≤ - 2 _ 3

x - 2

y ≥ - 2 _ 3

x + 3

SOLUTION

The two solution sets do not intersect, so the system has no solution.

b. y ≥ - 2 _ 3

x - 2

y ≤ - 2 _ 3

x + 3

SOLUTION

The solution set is the region between the parallel lines.

c. y ≥ - 2 _ 3

x - 2

y ≥ - 2 _ 3

x + 3

SOLUTION

The solutions of y ≥ - 2 _ 3 x + 3 are a subset of

the solutions of y ≥ - 2 _ 3 x - 2.

The solutions of the system are the same as the solutions of y ≥ - 2 _ 3 x + 3.

x

y

O

4

2

2 4-4

-4

x

y

O

4

2

2 4-4

-4

x

y

O

4

2

2 4-4

-4

x

y

O

4

2

2 4-4

-4

x

y

O

4

2

2 4-4

-4

x

y

O

4

2

2 4-4

-4

x

y

O

4

2

2 4-4

-4

x

y

O

4

2

2 4-4

-4

Math Reasoning

Generalize When the boundary lines are parallel, what must be true about the inequality symbols for one graph to be a subset of the other?

They must both include greater than or both include less than.


Have students determine the solution using test points from each region of the grid created by the boundary lines. In Example 3b use the test point (3, -4) to determine if the region between the two lines is part of the solution set. Substitute the point into both inequalities.

y ≥ - 2 __ 3 x - 2 y ≤ - 2 __

3 x + 3

-4 ≥ - 2 __ 3 (3) - 2 -4 ≤ - 2 __

3 (3) + 3

-4 ≥ -2 - 2 -4 ≤ -2 + 3

-4 ≥ -4 True -4 ≤ 1 True

Since the inequalities are true, then all of the points in the region containing the test point are also solutions of the system. This region can be shaded. Students then select a checkpoint from each of the other two regions and substitute it into both inequalities. Any region for which the checkpoint makes both inequalities true is shaded. Any shaded area is part of the solution set for the system of inequalities.

ALTERNATE METHOD FOR EXAMPLE 3b

Lesson 109 737

Example 3

Error Alert Students may think of the solution to these systems as any point that satisfi es either inequality. Remind students that the solution is the set of points that satisfi es both systems, where the graphs overlap.


Graph each system.

a. y < 3x - 1y > 3x + 1

xO

4

2

2 4-2-4

b. y > 3x - 1y < 3x + 1

x

y

O

2

4

2 4-2-4

c. y ≤ 3x - 1y ≤ 3x + 1

x

y

O

2

4

2 4-2-4


Saxon Algebra 1738

Example 4 Application: Employment

Lena has to earn at least $210 per week from two part-time summer jobs. She can work up to 15 hours per week at Job A, which pays $12 per hour, and can work up to 35 hours per week at Job B, which pays $10 per hour. She is not allowed to work more than 40 hours per week. Graph the possible combinations of hours Lena can work per week.

SOLUTION

Write a system of inequalities where x is the number of hours worked per week at Job A, and y is the number of hours worked per week at Job B.

x ≤ 15 no more than 15 hours at Job A

y ≤ 35 no more than 35 hours at Job B

12x + 10y ≥ 210 must earn at least $210 per week

x + y ≤ 40 cannot work more than 40 hours per week

The region where all four solution sets intersect shows the possible combinations of hours at each job. One possible combination is 9 hours at Job A and 20 hours at Job B.

Lesson Practice

Graph each system.

a. y > -2x - 1 b. 6y + 6 > -2x

y ≤ 1 _ 5 x + 4 y < 2

c. Graph the system on a graphing calculator.

y ≥ x -6

y ≤ -x + 3

Graph each system.

d. y > 1 _ 2 x - 4 e. y <

1 _ 2 x - 4 f. y >

1 _ 2

x - 4

y > 1 _ 2 x y >

1 _ 2 x y <

1 _ 2

x

g. Brett has $30 with which to buy dried strawberries and dried pineapple for a hiking trip. The dried strawberries cost $3 per pound and the dried pineapple costs $2 per pound. Brett needs at least 2 pounds of strawberries and 3.5 pounds of pineapple. Graph the possible combinations of pounds of each dried fruit that Brett can buy.See Additional Answers.

x

y

Hours at Job A

Ho

urs

at J

ob

B

10 20 30 40

10

20

30

40

0x

y

Hours at Job A

Ho

urs

at J

ob

B

10 20 30 40

10

20

30

40

0

(Ex 1)(Ex 1)a.

x

y

2

2 4

-2

-2-4

-4

a.

x

y

2

2 4

-2

-2-4

-4

b.

x

y4

2 4

-2

-4

-4

O

b.

x

y4

2 4

-2

-4

-4

O

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

c.c.

See Additional Answers.See Additional Answers.

(Ex 4)(Ex 4)

Math Reasoning

Verify Verify that Lena can make at least $210 working 9 hours at Job A and 20 hours at Job B.

9(12) + 20(10) =

308


Students who struggle with directionality may have diffi culty with Example 4. Suggest to students that they think of the lines that slant up or down to the right as a tabletop. Have them associate above the line with the greater than symbol and below the line with the less than symbol. Help them determine “above” or “below” the line by telling them that if an object such as a paper clip could rest on the line, then that object is above the line. Otherwise, the object is below the line.

INCLUSION

738 Saxon Algebra 1

Example 4

Extend the Example

Lena worked 5 hours at Job A and 10 hours at Job B. Explain why this is not a solution to the system. Sample: The number of hours worked does not produce the minimum amount money of $210 that she has to earn.


Doug must earn at least $300 per week from two part-time summer jobs. Doug can work up to 20 hours per week at Job A, which pays $11 per hour. He can work up to 30 hours per week at Job B, which pays $9 per hour. He cannot work more than 40 hours per week. Graph the possible combinations of hours Doug can work per week.

⎧

⎨

⎩

x ≤ 20 y ≤ 30

11x + 9y ≥ 300

x + y ≤ 40

x

y

O

20

-20

-20

-40

Lesson Practice

Problems a–f

Scaff olding Remind students to test their solutions for each inequality.

Problem g

Error Alert Students may use the x-axis to represent price and the y-axis to represent quantity. Remind students that they can write the quantity of pineapple in terms of the quantity of strawberries using the inequality: 3x + 2y ≤ 30.



Lesson 109 739

*1. Multiple Choice Which system is represented in the graph? A

y

O

4

2

2 4-4 -2

-4

-2

A y ≤ -0.5x + 3

y ≥ -0.5x - 1

B y ≤ -0.5x + 3

y ≤ -0.5x - 1

C y ≥ -0.5x + 3

y ≥ -0.5x - 1

D y ≥ -0.5x + 3

y ≤ -0.5x - 1

*2. Sports The requirements for a major league baseball are shown in the graph. Write the system of inequalities that matches the graph.

x

y

Circumference in Inches

Wei

ght

in O

unce

s

8 9

4

5

0

3. Graph the function f(x) = -3⎢x�.

*4. Write Explain how to represent the solution set of y ≤ -3x + 4

y < 2x - 1

.

*5. Verify Graph the solution set of y ≥ -x

y ≤ 2x

to verify that (1, -2) is not a solution of the system.

*6. Evaluate the function f(x) = 3 ( 1 _ 3 )

x

for x = -2, 0, and 2. 27, 3, 1_3

7. If the original price was increased 44% to a new price of $900, what was the original price? $625

Simplify.

8. 10 √ �� 8x2y3 - 5y √ �� 98x2y -15xy√ � 2y 9. √ ��

24y8

_ 6x3

2y4 √ � x_x2

(109)(109)

(109)(109)

x ≥ 9x ≤ 9.25y ≥ 5y ≤ 5.25

x ≥ 9x ≤ 9.25y ≥ 5y ≤ 5.25

(107)(107)

3.

x

y

O

4

2

2 4-2-4

-4

3.

x

y

O

4

2

2 4-2-4

-4

(109)(109)

4. Sample: Graph y = -3x + 4 with a solid line and shade below the line. On the same plane, graph y = 2x - 1with a dashed line and shade below it. The solution set is represented by the region where the shadings overlap.

4. Sample: Graph y = -3x + 4 with a solid line and shade below the line. On the same plane, graph y = 2x - 1with a dashed line and shade below it. The solution set is represented by the region where the shadings overlap.

(109)(109)5.

-4

x

y

2 4-4 -2

2

45.

-4

x

y

2 4-4 -2

2

4

(108)(108)

(47)(47)

(69)(69) (103)(103)


Lesson 109 739


The question below help assess the concepts taught in this lesson.

“Explain how to use a graph to solve systems of inequalities.” Graph each inequality and identify the overlapping region.

“How is it determined whether the points on the boundary line are part of the solution set?” Sample: If the inequality symbols are < or >, the points on the boundary line are not part of the solution set. If the inequality symbols are ≤ or ≥ the points on the boundary line are part of the solution set.

Practice3


Problem 7

Error Alert

Students may not realize that the new price is 144% of the original price instead of 44% of the original price. Encourage them to determine if 44% of the original value increases or decreases the price.


Saxon Algebra 1740

10. Error Analysis Student A said that the following set satisfies an exponential function because there is a common ratio of 3 among the y-values. Student B said that this is not so. Which student is correct? Explain the error. Student B; Sample: The x-values do not increase by a constant amount.

{(3, 1), (5, 3), (6, 9), (7, 27)}

*11. Multi-Step Niall has a baseball card whose value, in dollars, x years after he acquired it, is represented by the function f(x) = 4.8(1.25)x. If Niall bought the card in the year 2000, how much more is it worth in 2010 than it was in 2005? $30.06

*12. Geometry Mr. Flores gives the length of a rectangle, in inches, as f(x) = 16 ( 1 _ 2 )

x

, where x is the number of times he cuts the length in half. What is the length of the rectangle after Mr. Flores has cut it in half 4 times? 6 times? 0 times? 1 inch, 1_

4 inch, 16 inches

*13. Probability For the function f(x) = 7 (5) x , what is the probability that for a randomly chosen x-value from the domain of {0, 1, 2, 3, 4, 5}, f(x) is a number between 100 and 1000? 1_3

14. Is the graph an absolute-value function? Explain. No; Sample: It does not make a V.

x

y

O4 8

-4

-8

-8 -4

*15. Graph the system y > 1 _

4 x + 3

y > - 1 _ 4 x + 3

.

16. Baseball An outfielder catches a ball 120 feet from the pitcher’s mound and throws it to home. If d = ⎢90t - 120� represents the ball’s distance from the pitcher’s mound, how would the graph change if the outfielder caught the ball 100 feet from the pitcher’s mound? The graph would shift to the left.

17. Renovations Nadia is using 48 tiles to cover a floor. The tiles come in 6-inch,12-inch, and 13-inch sizes. If the total area of the floor is 6912 square inches, which tile size will fit best? 48x2 = 6,912; x2 = 144; x = 12. The 12 in square tiles will work best.

18. Projectile Motion The equation for the time in seconds (t) it takes an object to strike the ground is -4.9t2 - 53.9t = -127.4. When will the object strike the ground? 2 seconds

19. Find the next 3 terms of the sequence 5, 4.5, 4.05, 3.645, …. 3.2805, 2.95245, 2.657205

*20. Multiple Choice Which of the following radical equations has no solution? CA √ x - 3 = x - 9 C √ x + 7 = -2

B 13 √ x = 65 D √ x + 10 = √ 2x + 8

21. Write Why is it important to isolate the radical in a radical equation?

(108)(108)

(108)(108)

(108)(108)

(108)(108)

(107)(107)

(109)(109)

15.

x

y

O

4

2

2 4

-2

-2-4

-4

15.

x

y

O

4

2

2 4

-2

-2-4

-4

(107)(107)

(102)(102)

(104)(104)

(105)(105)

(106)(106)

(106)(106)

21. Sample: The equation is easier to solve if the radical is by itself, because squaring the equation then eliminates the radical.

21. Sample: The equation is easier to solve if the radical is by itself, because squaring the equation then eliminates the radical.


CHALLENGE

Have students create a system of inequalities whose solution is (1, 1).

Sample:

⎧

⎨

⎩

y ≥ 1

y ≤ 1

x ≥ 1 x ≤ 1

740 Saxon Algebra 1

Problem 16

Extend the Problem

“Curt can change the speed of his pitch by ⎢8� mph. If his slowest pitch is 72 mph, what is the speed of his fastest pitch?” 88 mph

Problem 18


“What should be the coeffi cient of the quadratic term when completing the square? 1

“What formula is used to fi nd the constant term for a perfect square trinomial?” ( b __

2 )

2

Extend the Problem

Explain why one of the solutions to the equation does not make sense. Sample: The situation described in the problem involves time, which cannot be negative. The other solution is negative


Lesson 109 741

22. Jim’s rectangular home gym has an area of (x2 - 144) square feet. The length is (x - 12) feet. What is the width? (x + 12) feet

Solve.

23. 4|x + 2| - 9 = 19 24. x2 = -49 no solution 25. 2 ⎪ x

_ 4 - 6⎥ = 8 {8, 40}

26. Multi-Step A pitcher throws a softball. The height in feet is represented by the function h = -16t2 + 47t + 5.a. How high is the ball after 1 second? 36 feet

b. How high is the ball when it is released? 5 feet

c. What is the initial velocity of the ball? 47 feet per second

27. Gardening It takes a boy 2 hours to pull all the weeds in the garden. It takes his sister 4 hours. How long will it take them if they pull weeds together? 4_3 hours

28. Multi-Step Andrew hits a golf ball into the air. Its movement forms a parabola given by the quadratic equation h = -16t2 + 31t + 7, where h is the height in feet and t is the time in seconds.a. Find the time t when the ball is at its maximum height. Round to the nearest

hundredth. t = 0.97 seconds

b. Find the time t when the ball hits the ground. Round to the nearest hundredth. t = 2.14 seconds

c. Find the maximum height of the arc the ball makes in its flight. Round to the nearest hundredth. h = 22.02 feet

29. Write Describe the similarities and differences between solving the inequality 2⎢x� + 1 < 7 and solving the inequality ⎢2x + 1� < 7.

30. If the area of a rectangle is represented by the expression 3x2 + 22x - 45 and the width by the expression (x + 9), what would the length be? (3x - 5)

(93)(93)

(94)(94){-9, 5}{-9, 5}

(102)(102) (94)(94)

(96)(96)

(99)(99)

(100)(100)

(101)(101)29. Sample: In both cases, subtract 1 from each side and divide each by 2. When solving 2⎢x� + 1 < 7, do these operations before removing the absolute-value bars, but when solving ⎢2x + 1� < 7, do these operations after writing as a compound inequality.

29. Sample: In both cases, subtract 1 from each side and divide each by 2. When solving 2⎢x� + 1 < 7, do these operations before removing the absolute-value bars, but when solving ⎢2x + 1� < 7, do these operations after writing as a compound inequality.

(Inv 9)(Inv 9)


Graphing systems of linear inequalities prepares students for





LOOKING FORWARD

Lesson 109 741

Problem 29

“How does subtracting the 1 at a different stage impact the inequality?” It changes the solution from -3 < x < 3 to -4 < x < 3.

Problem 30

Encourage students to be systematic as they search for the factors that work. Tell them to keep a written record of the factors that they try, otherwise they may spend time unnecessarily retrying the same factors.


Saxon Algebra 1742

Using the Quadratic Formula

Warm Up

110LESSON

1. Vocabulary A equation can be written in the form ax2 + bx + c = 0, where a is not equal to 0. quadratic

Find the value of c to complete the square for each expression.

2. x2 + 8x + c 16 3. x2 + 9x + c 81_4

4. Solve x2 + 10x = 24 by completing the square. Check your answer. 2, -12

Different methods are used to solve quadratic equations. One method is applying the quadratic formula. The quadratic formula is derived by completing the square of the standard form of the quadratic equation ax2 + bx + c = 0.

ax2 + bx + c = 0

ax2

_ a + bx _ a + c _ a = 0 Divide by the coefficient of x2.

x2 + bx _ a = - c _ a Subtract the constant c_

a from both sides.

x2 + bx _ a + (

b _ 2a

) 2

= - c _ a + (

b _ 2a

) 2

Add ( b_2a )

2 to complete the square.

x2 + bx _ a + b2

_ 4a2

= - c _ a + b2

_ 4a2

Simplify.

(x + b _

2a )

2

= b2 - 4ac _

4a2 Write the left side as a squared

binomial and the other side with the LCD.

√ �� (x +

b _

2a )

2

= ± √ ��

b2 - 4ac _

4a2 Take the square root.

x + b _

2a = ±

√ �� b2 - 4ac _

2a Simplify.

x = -b ± √ �� b2 - 4ac

__ 2a

Solve.

Quadratic Formula

For the quadratic equation ax2 + bx + c = 0,

x = -b ± √ �� b2 - 4ac

__ 2a

when a ≠ 0.

The quadratic formula can be used to solve any quadratic equation.

(84)(84)

(104)(104) (104)(104)

(104)(104)


Online Connection


Math Language

A quadratic equation is an equation whose graph is a parabola.


LESSON RESOURCES



Master C110Technology Lab Master 110

The proof of the quadratic formula is a common algorithm used to show how the formula was developed. The quadratic formula is useful for all quadratic equations, especially when the equation is unfactorable or when the values of a and b make completing the square complicated. Although graphing calculators provide solutions to quadratic equations, sometimes the solutions are not exact numeric answers, but approximations. The quadratic formula

MATH BACKGROUND

will always give precise numeric answers that the student can round if necessary. This method may be preferred over factoring because there is no trial and error involved.

It may interest students to know that in Latin, quadrum means square, and in Middle English, quadrat means something square.

Warm Up1

742 Saxon Algebra 1

110LESSON

Problem 4

Remind students to subtract the constant on both sides of the equation before completing the square.

2 New Concepts

In this lesson students will learn how the quadratic formula is derived and then apply it to solve quadratic equations. It is not always appropriate to apply other solution methods (graphing, factoring, or completing the square). Students can use the quadratic formula to solve any quadratic equation.

TEACHER TIPPoint out that if a = 0, the equation is not quadratic, so it makes sense that the quadratic formula is undefi ned when a = 0.


Lesson 110 743

Example 1 Solving a Quadratic Equation in Standard Form

Use the quadratic formula to solve x2 - 9x + 20 = 0 for x.

SOLUTION

x = -b ± √ �� b2 - 4ac

__ 2a

Use the quadratic formula.

= -(-9) ± √ �� (-9)2 - 4(1)(20)

___ 2(1)

Substitute 1 for a, -9 for b,and 20 for c.

= 9 ± √ �� 81 - 80

__ 2

= 9 ± √ � 1

_ 2 =

9 ± 1 _

2 Simplify.

x = 5 and 4

Check Verify that 5 and 4 make the original equation true.

x2 - 9x + 20 = 0 x2 - 9x + 20 = 0

(5)2 - 9(5) + 20 � 0 (4)2 - 9(4) + 20 � 0

25 - 45 + 20 � 0 16 - 36+ 20 � 0

0 = 0 ✓ 0 = 0 ✓

Example 2 Rearranging Quadratic Equations before Solving

Use the quadratic formula to solve -18x + x2 = -32 for x.

SOLUTION Rearrange the equation into the standard form ax2 + bx + c = 0.

x2 - 18x + 32 = 0 Write the equation in standard form.

x =-b ± √ �� b2 - 4ac __

2aUse the quadratic formula.

= -(-18) ± √ �� (-18)2 - 4(1)(32)

___ 2(1)

Substitute 1 for a, -18 for b,and 32 for c.

= 18 ± √ �� 324 - 128

__ 2

= 18 ± √ �� 196

_ 2 =

18 ± 14 _ 2 Simplify.

x = 16 and 2

Check Verify the solutions for x.

-18x + x2 = -32 -18x + x2 = -32

-18(16) + (16)2 � -32 -18(2) + (2)2 � -32

-288 + 256 � -32 -36 + 4 � -32

-32 = -32 ✓ -32 = -32 ✓

Hint

Rearrange terms and their corresponding signs to match the form ax2 + bx + c = 0.


Explain the meaning of the word rearrange. Say:

“To arrange means to move or position items. To rearrange means to move or position the same items in a different way.”

Ask:

“Why would someone rearrange a room of furniture?” Sample: because they may not like the way it was arranged the fi rst time

Have students arrange a set of objects, and then have other students rearrange the objects.

ENGLISH LEARNERS

Lesson 110 743

Example 1

Error Alert Students may not recognize both solutions of a quadratic equation. Remind students to simplify so that there are two distinct answers when using the quadratic formula.


Use the quadratic formula to solve for x. 2x2 + 6x + 4 = 0 x = -1 or -2

Example 2

Students should equate quadratic equations to zero before identifying a, b, and c.


Use the quadratic formula to solve for x.

3x2 - x = 2 x = 1 or - 2 __ 3


Saxon Algebra 1744

Example 3 Finding Approximate Solutions

Use the quadratic formula to solve for x. Then use a graphing calculator to find approximate solutions and verify them.

5x2 - 3x - 1 = 0

SOLUTION

5x2 - 3x - 1 = 0

x = -b ± √ �� b2 - 4ac __

2aUse the quadratic formula.

= -(-3) ± √ �� (-3)2 - 4(5)(-1)

___ 2(5)

Substitute the values for a, b, and c.

x = 3 ± √ �� 9 + 20

__ 10

= 3 ± √ � 29

_ 10

To find the approximate solutions, use a calculator with a square root key. Round the solutions to the nearest ten thousandth.

The solutions are 3 + √ � 29 _

10 ≈ 0.8385 and 3 - √ � 29

_ 10

≈ -0.2385.

Check

On a graphing calculator, graph the related function y = 5x2 - 3x - 1 to check that the approximate solutions are the zeros of the graph.

Example 4 Recognizing a Quadratic Equation With No

Real Solutions

Use the quadratic formula to solve 2x2 + 3x + 4 = 0 for x.

SOLUTION

x = -b ± √ �� b2 - 4ac

__ 2a

= -(3) ± √ �� (3)2 - 4(2)(4)

___ 2(2)

Substitute the values for a, b, and c.

x = -3 ± √ �� 9 - 32

__ 4 =

-3 ± √ �� -23 __

4

The square root of a negative number cannot be taken, so there are no real solutions.

Graphing

Calculator Tip

For help with graphing quadratic equations, see the graphing calculator keystrokes in Lab 8 on p. 583.


To ensure that students account for all negative signs when using the quadratic formula, have them write the quadratic formula with parentheses.

x = -( ) + √ �� ( )2 -4 ( )( )

___________________ 2( )

Have students write the values for a, b, and c respectively inside the parentheses. Then have them simplify the equation.

INCLUSION

744 Saxon Algebra 1

Example 3

Not all quadratic equations have integer or rational solutions. Irrational solutions can be approximated using a calculator.


Use the quadratic formula to solve for x. Then use a graphing calculator to fi nd approximate solutions and verify them. -x2 - 10x + 5 = 0

10 ± √ �� 120 _________

-2 ; x ≈ 0.477 or -10.477

Example 4

Extend the Example

Have students graph the equation.

x

y

O4 8

-4

-4-8

-8

“How can you determine from the graph that there are no real solutions?” Sample: The graph does not intersect the x-axis.


Use the quadratic formula to solve 5x2 + 4x + 2 = 0 for x. Ø


Lesson 110 745

Example 5 Application: Object in Motion

From an initial height s of 70 meters in a stadium, Luis tosses a ball up at an initial velocity v of 5 meters per second. Use the equation -4.9t2 + vt + s = 0 to find the time t when the ball hits the ground.

SOLUTION

Substitute the values into the quadratic formula. Then solve.

-4.9t2 + 5t + 70 = 0

t = -b ± √ �� b2 - 4ac

__ 2a

= -(5) ± √ �� (5)2 - 4(-4.9)(70)

___ 2(-4.9)

= -5 ± √ �� 25 + 1372

__ -9.8

= -5 ± √ �� 1397

__ -9.8

≈ -5 ± 37.3765 __

-9.8

t ≈ -3.3037 and t ≈ 4.3241

Check

-4.9(4.3241)2 + 5(4.3241) + 70 ≈ -91.6194 + 21.6205 + 70 ≈ 0 ✓

The ball will land on the ground in approximately 4.3241 seconds.

Lesson Practice

a. Use the quadratic formula to solve for x.

x2 + 3x - 18 = 0 -6 and 3

b. Use the quadratic formula to solve for x.

-72 - 14x + x2 = 0 -4 and 18

c. Use the quadratic formula to solve for x.

x2 + 80 = 21x 5 and 16

d. Use the quadratic formula to solve for x. Then use a graphing calculator to find approximate solutions and verify them. Round the solutions to the nearest ten thousandth.

9x2 + 6x - 1 = 0 -1 ± √ � 2_3 ≈ 0.1381 or -0.8047

e. Use the quadratic formula to solve 4x2 + 5x + 3 = 0 for x.

f. From an initial height s of 50 meters on a cliff, Janet tosses a ball upward at an initial velocity v of 6 meters/second. At what point does the ball fall back to the ground? Round the solution to nearest ten thousandth.3.8648 seconds

(Ex 1)(Ex 1)

(Ex 2)(Ex 2)

(Ex 2)(Ex 2)

(Ex 3)(Ex 3)

(Ex 4)(Ex 4)no real solutionno real solution

(Ex 5)(Ex 5)

Hint

When the solutions deal with time, we only consider positive values for solutions.

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Lesson 110 745

Example 5

Remind students that some solutions are not appropriate. In this example, a negative time has no meaning.


From an initial height s of 45 meters, Maria tosses a ball up at an initial velocity v of 7 meters/second. Find the time t when the ball hits the ground. Use the equation -4.9t 2 + vt + s = 0. about 3.8278 seconds

Lesson Practice

Problem a

Scaff olding Remind students to identify the values for a, b, and c including their signs.

Problem b

Error Alert Students may try to solve before ordering the terms. Remind students that the terms should be written in descending order by degree.



“Will all solutions to quadratic equations be integer values? Explain.” No; Sample: Rational values may occur because of the fraction. Irrational values may occur because of the square root.

“What does the solution to a quadratic equation tell about the graph of the equation?” It tells where on the coordinate plane the graph crosses the x-axis.



Saxon Algebra 1746

Use the quadratic formula to solve for x. Check the solutions.

*1. x2 - 2x - 35 = 0 -5, 7 *2. x2 - 10x + 25 = 0 5

*3. Multi-Step Determine why 16h2 + 25 = 40h has only 1 solution using the quadratic formula. a. Rearrange the equation into the ax2 + bx + c = 0 form. 16h2 - 40h + 25 = 0

b. What is different about b2 - 4ac? Sample: It equals zero.

c. Generalize When will the equation ax2 + bx + c = 0 have only 1 solution? when b2 = 4ac

4. Compare: 12,000 1.2 × 103. 12,000 > 1.2 × 103

5. Find the zeros of the function. y = x2 + 12x + 36 -6

6. Describe the graph of an indirect variation when the constant of variation is positive. a hyperbola with the x- and y-axes as asymptotes

7. Identify the outlier or outliers in the data set.

number of cars for sixteen households: 3, 2, 2, 1, 2, 3, 6, 2, 1, 1, 3, 2, 2, 2, 1, 3 6

*8. Predict Use mental math to predict whether the quadratic formula is necessary to solve 3b2 + 15b - 20 = 0. Solve. quadratic formula is necessary; -15 ± √ �� 465_

6

*9. Soccer A 1.5-meter-tall soccer player bounces a soccer ball off his head at a velocity of 7 meters per second upward. Use the formula h = -4.9t2 + v0t + h0 to estimate how many seconds it will take the ball to hit the ground. about 1.6 seconds

*10. Error Analysis For the system of inequalities graphed, Student A said that (1, -4) is a solution of the system and Student B said that (4, 2) is a solution of the system. Which student is correct? Explain the error. Student A; Sample: The ordered pair (4, 2) is a solution to one of the inequalities, but not to both of them.

x

y

O

2

-2

-2-4

-4

4

11. Graph the system y ≤ 2

x ≥ 2

. See Additional Answers.

*12. Multi-Step A student group is planning on washing cars in an effort to raise at least $300. They want to charge $5 for a basic wash, which will take about 10 minutes, and $15 for a detailed wash, which will take about 30 minutes. They have the car-wash lot rented for 8 hours. Write and graph a system of linear inequalities to describe this situation. Explain your findings. See Additional Answers.

(110)(110) (110)(110)

(110)(110)

(37)(37)

(96)(96)

(Inv 7)(Inv 7)

(48)(48)

(110)(110)

(110)(110)

(109)(109)

(109)(109)

(109)(109)


746 Saxon Algebra 1

Practice3


Problem 7

Extend the Problem

Have students draw a stem-and-leaf plot to identify outliers.

Sample:

Stem Leaf

000000

1 1 1 12 2 2 2 2 2 23 3 3 3

6

Legend: 0 | 1 means 1


Lesson 110 747

13. Geometry Suppose the perimeter of a rectangle must be less than 50 units and the width must be greater than 5 units. Graph a system of linear inequalities to describe this situation. Give one set of possible dimensions for the rectangle.

14. Evaluate the function f(x) = -3(6) x for x = -2, 0, and 2. - 1_12

, -3, -108

15. Error Analysis Which student correctly evaluated f(x) = 2(3) x for x = 2? Explain the error.

Student A

f (x) = 2(3) x = 6 x = 62 = 36

Student B

f(x) = 2(3) x = 2(3)2

= 2(9) = 18

*16. Chemistry Amaro uses f(x) = 10 ( 1 _ 2 )

x

to give the amount remaining from 10 grams of a radioactive substance after x number of half-lives. Which graph represents this function? Graph A

Graph A Graph B Graph C

x

y

8

4

2 4 6O

x

y

2

4

6

-2-4-6

xy

-4

-8

2 6O

17. Simplify √ �� 15xy

_ 3 √ �� 10xy3

. √ � 6_6y 18. Subtract 5x2

_ 10x - 30

- 2x - 5

_ x2 - 9

.

19. Astronomy Astronomers can use the formula T = √ � d 3 to find the time T it takes a planet to orbit the Sun (in earth years), knowing the distance d of the planet from the Sun (in astronomical units, AU). If Mars is about 3

_ 2 AU from the Sun, about how long does it take Mars to orbit the Sun in earth years? Give your answer as a rational expression. 3

√ � 6_4

20. Multiple Choice What is the absolute-value function of the graph? B

x

y

O

4

2

2 4

-2

-2-4

-4

A f(x) = |x + 2| B f(x) = |x - 2|

C f(x) = |x| + 2 D f(x) = |x| - 2

21. Solve p2 + 13p = -50 by completing the square. no real solutions

(109)(109)


(108)(108)

(108)(108)

Student B; Sample: Student A should not multiply 2 and 3 because 3 is the base of an exponent.

Student B; Sample: Student A should not multiply 2 and 3 because 3 is the base of an exponent.

(108)(108)

(103)(103) (95)(95)

x3 + 3 x2 - 4x + 10__2(x - 3)(x + 3)

x3 + 3 x2 - 4x + 10__2(x - 3)(x + 3)

(103)(103)

(107)(107)

(104)(104)


Lesson 110 747

Problem 14

Make sure that students are correctly applying the order of operations when simplifying.

Problem 18

Error Alert Students may not simplify each fraction before fi nding a common denominator. Encourage them to factor the denominators as much as possible before fi nding a common denominator.

Problem 20

Extend the Problem

Have students describe a scenario that the graph and correct equation might represent.


Saxon Algebra 1748

*22. Compound Interest The formula for a fund that compounds interest is

An = P (1 + r _ n ) nt , where A is the balance, P is the initial amount deposited, r is

the annual interest rate, t is the number of years, and n is the number of times the interest is compounded per year. Gretchen deposits $1500 into an account that pays 4.5% interest compounded annually. Write the first 4 terms of the sequence representing Gretchen’s balance after t years. Round to the nearest cent. $1567.50, $1638.04, $1711.75, $1788.78

23. Solve √ �� x + 11 = 16. Check your answer. x = 245; √ �� 245 + 11 = √ �� 256 = 16

*24. Analyze Are the graphs for f(x) = 5|x| and f(x) = |5x| the same? Explain.

25. Solve the equation 9 ⎪ x _ 2 - 6⎥ = 27. 26. Factor x2 + 42 + 13x. (x + 6)(x + 7)

27. Multi-Step Lisa plans to shop for books and magazines and she plans to spend no more than $32. Each book costs $14 and each magazine costs $4. a. Write an inequality that describes this situation. 14x + 4y ≤ 32

b. Graph the inequality. See Additional Answers.

c. If Lisa wants to spend exactly $32, what is a possible number of each she can spend her money on? Sample: 2 books and 1 magazine

28. Volleyball Diego hits a volleyball into the air. The ball’s movement forms a parabola given by the quadratic equation h = -16t2 + 3t + 14 where h is the height in feet and t is the time in seconds. Find the maximum height of the path the volleyball makes and the time when the volleyball hits the ground. Round to the nearest hundredth. h = 14.14 feet and t = 1.03 seconds

29. Multi-Step When the temperature (t) of the gas neon is within 1.25° of -247.35°C it will be in a liquid form. This can be modeled by the absolute-value inequality |t - (-247.35)| < 1.25. a. Solve and graph the inequality |t - (-247.35)| < 1.25.

b. One endpoint of the graph represents the boiling point of neon, the temperature at which neon changes from liquid to gas. The other endpoint represents the melting point, at which neon turns from solid to liquid. The higher temperature is the boiling point and the lower temperature is the melting point. What is the boiling point of neon? What is the melting point? -246.1°C;-248.6°C

30. Measurement The following formula represents the area of circle A: πr2 - 165.05 m2 = 0. What is the approximate measurement, in meters, of the radius r? Use 3.14 for π. ≈7.249 m

(105)(105)

(106)(106)

(107)(107)24. Yes; Sample: Multiplication does not change the absolute value like addition and subtraction do.

24. Yes; Sample: Multiplication does not change the absolute value like addition and subtraction do.

(94)(94) 6, 186, 18 (72)(72)

(97)(97)

(100)(100)

(101)(101)

29a. -248.6 < t < -246.1;

-246-248-250

29a. -248.6 < t < -246.1;

-246-248-250

3 43 4

r

A

r

A

(102)(102)


Have students write their own quadratic equation. It should have two real solutions. Then have them solve their equation using the quadratic formula. See student work.

Using the quadratic formula prepares students for


• Lesson 116 Solving Simple and Compound Interest Problems



748 Saxon Algebra 1

Problem 22

Be sure students use the order of operations after they substitute the values into the equation.

Problem 28

Make sure that students realize that they are identifying two distinct events, the maximum height and the time when the ball hits the ground where its height is 0.


749Investigation 11

INVESTIGATION

11Water Flow Rates

Water flows from a crack in the side of a swimming pool, initially releasing one gallon of water. The crack continues to widen as water continues to flow from the pool. For every second after that, the amount flowing from the pool doubles. The table below shows the relationshipbetween time and the amount of water flowing.

Time (s)Amount of Water (gal)

0 11 22 43 8

1. Create a graph of the data. See Additional Answers.

2. Predict How many gallons of water flow from the pool in the fourth second? 16 gallons

Near the origin the graph looks similar to a parabola, however it grows much more quickly. The graph models exponential growth. Exponential growth is a situation where a quantity always increases by the same percent for a given time period.

Stock Exchange

The annual number of shares S in billions traded on the New York Stock Exchange from 1990 to 2000 can be approximated by the model S = 39(1.2 ) x , where x is the number of years since 1990.

3. Create a table of values like the one below. Round each share to the nearest billion.

x S0 392 564 81

6 116

8 168

10 241

4. Plot the coordinates. Connect the points with a smooth curve.

5. Use the graph to estimate the number of shares traded in 1997. about 140 billion shares

6. Verify Use the equation to calculate the exact number of shares traded in 1997 algebraically. 139.74 billion shares


Investigating Exponential Growth and Decay

Online Connection


Math Reasoning

Analyze What characteristics of the data and the graph indicate that this data does not model a linear function?

Math Reasoning

Analyze In the exponential growth equation f(x) = k b x what is the domain? Why?

Math Reasoning Sample: The data values for the amount of water do not increase by a constant amount. The graph of the data is not a line.

Math ReasoningSample: x represents time, so it must be 0 or positive. The domain is x ≥ 0.

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INVESTIGATION RESOURCES

Reteaching Master Investigation 11

Technology Lab Master Investigation 11

Banks pay interest to depositors in return for the use of the depositors’ money. The interest depositors earn from a bank is usually compound interest, which means the bank pays based not only on the amount deposited in the bank, but also on the interest it has already paid.

For instance, if Marla deposits $500 into a bank that has an interest rate of 5% that is compounded annually, at the end of the fi rst

year she will have $525. This is 1.05 times Marla’s $500 deposit.

To compound the interest, a bank will use an exponential growth formula, where k is the original deposit, b is the interest, and the exponent is the number of years the money stays in the bank:

y = k b x y = 500 · 1.05 x

MATH BACKGROUND

Investigation 11 749

11INVESTIGATION

Materials

• several sheets of notebook paper

• graph paper

Discuss

In this investigation, students learn to fi nd exponential growth and decay by graphing data and creating data tables. Remind them that they learned how to identify and graph exponential functions in Lesson 108.

Defi ne exponential growth.

Water Flow Rates

Extend the Problem

Have students extend the table to show the amount of water that fl ows from the pool in the fourth through eighth seconds.

Time (s)Amount of Water (gal)

4 16

5 32

6 64

7 128

8 256

Stock Exchange

Extend the Problem

Have students use the equation to estimate the number of shares traded in 1995. about 97 billion

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Saxon Algebra 1750

Exponential growth is modeled by the function f (x) = k b x , where k > 0. The percent of growth b, expressed as a decimal number, is greater than 1.

Exploration Exploration Analyzing Different Values of k in the Exponential Growth Function

Step 1: Take one sheet of notebook paper. Fold it in half. Unfold the paper and count the number of rectangular regions formed. Record the number of folds and regions in a table like the one below.

Folds Regions0 11 22 4

3 8

4 16

Refold the paper along the initial crease you made and fold it in half again. Continue counting regions and folding in half at least four times.

Step 2: Take three sheets of notebook paper and stack them. Repeat Step 1. Create and complete a table like the one below.

Folds Regions0 31 62 12

3 24

4 48

Step 3: Take five sheets of notebook paper and stack them. Repeat Step 1. Create and complete a table like the one below.

Folds Regions0 51 10

2 20

3 40

4 80

7. Plot the points on one coordinate plane. Let x = the number of folds. Let y = the number of regions. Connect the point for each set of data with a smooth curve. See Additional Answers.

The data in the Exploration are included in the graphs of the functions f (x) = 2 x , g(x) = 3(2 ) x , and h(x) = 5(2 ) x , respectively. All three functions are of the form y = k(b ) x .

8. What is the y-intercept of each function? Compare the y-intercept of each equation to y = k b x . Name the y-intercept of y = k b x . (0, 1); (0, 3); (0, 5);(0, k)

Materials

• several sheets of notebook paper

Math Reasoning

Analyze Why are each of the three functions named using function notation?

Sample: It is convenient to refer to each function as f, g, or h. It would not be clear to call each exponential equation “y.”

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Explain to students the meaning of a share of stock. Students already know that to share means to divide something among people, but may not know about a share of stock. Say:

“A share of stock is a part of a company and its profi ts that people can buy. Shares of stock are divided among people who invest in a company.”

Discuss businesses with which the students may be familiar that sell shares of stock in the company.

750 Saxon Algebra 1

Analyzing Diff erent Values

of k in the Exponential

Growth Function

Discuss

Ask the students if they notice a pattern in the number of regions each time that they fold the paper. Discuss that the number of regions on the paper doubles with each fold. Connect this with the base of the exponential equation f(x) = k2x .

Error AlertStudents may try to jump to quickly from the concrete to the abstract by fi lling in the tables without folding the paper, resulting in errors in their data. Encourage them to complete the paper folding for a thorough understanding of the concept.

ENGLISH LEARNERS



9. Generalize How does changing the value of k affect the graph of the function? Sample: The y-intercept is k. Each graph has the same shape, but larger values of k cause the graph to curve upward more sharply.

10. Formulate As the number of folds increase, what happens to the number of regions on the folded paper? What is the b-value for each equation? Write an equation in the form y = k(b ) x to model situations in which y doubles as x increases.

11. For any function y = k(b ) x , what does k represent in any situation when x = 0? k represents the initial amount present.

The period of time required for a quantity to double in size or value is called doubling time. The equation will be of the form y = k(2 ) x .

Just as data can grow exponentially, some data can model exponential decay. Exponential decay is a situation where a quantity always decreases by the same percent in a given time period.

Carbon-14 dating is used to find the approximate age of animal and plant material after it has decomposed. The half-life of carbon-14 is 5730 years. So, every 5730 years half of the carbon-14 in a substance decomposes. Find the amount remaining from a sample containing 100 milligrams of carbon-14 after four half lives.

12. How many years are there in four half-lives? 22,920 years

13. Create and complete a table like the one below.

Number of Half-Lives Number of YearsAmount of Carbon-14

Remaining (mg)0 0 1001 5730 50

2 11,460 25

3 17,190 12.5

4 22,920 6.25

14. How much of the sample remains after 22,920 years? 6.25 mg

Exponential decay is modeled by the function f (x) = k b x , where k > 0 and 0 <

b < 1. Since the value of b is a positive number less than 1, as x increases, the value of f(x) decreases by b.

An exponential decay function can model the amount of a substance in the body over time. Many diabetes patients take insulin. The exponential

function f (x) = 100 ( 1 _ 2 )

x

describes the percent of insulin in the body after x half-lives. The half-life of a substance is the time it takes for one-half of the substance to decay into another substance.

15. About what percent of insulin would be left in the body after 8 half-lives? 0.39%

16. Write Describe the effect that the b-value has on the amount of substance remaining as the number of half-lives x increases.

10. Sample: Every time the paper is folded in half, the number of regions doubles; 2; y = k(2)

x

10. Sample: Every time the paper is folded in half, the number of regions doubles; 2; y = k(2)

x

16. Sample: In each interval x, the amount of y remaining decreases by half.

16. Sample: In each interval x, the amount of y remaining decreases by half.

Caution

Do not divide the original amount of a substance by 3 to calculate the amount of a substance left after three half-lives.

Hint

Since x usually represents time in decay equations, x > 0.

Math Reasoning

Analyze Why does f(x)

decrease as x increases?

Sample: Since b is a fraction between 0 and 1, f(x)

decreases in value as b is raised to greater values of x.

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The half-life of cobalt-60, an element used in some medical radiation therapies, is 5.26 years. If a sample of cobalt-60 is 550 milligrams, how many milligrams of cobalt-60 will remain after 31 14

___ 25 years? Round to the nearest hundredths place if necessary. 8.59 mg


Problem 10

Extend the Problem

Have students use the formula y = k (2) x to fi nd the amount of regions 6 pieces of paper would have if they were folded 7 times. 768 regions

Discuss

Defi ne doubling time and exponential decay. Ask students why they think the value of b in the exponential decay formula f(x) = k b x is a positive number that is less than one.

Problem 15

Error AlertStudents may move the decimal place too far and get 39% as an answer. Remind them that they need to move the decimal only two decimal places to the right to get their answer.

CHALLENGE


Saxon Algebra 1752

17. Predict Graph the functions f (x) = 100 ( 1 _ 2 )

x

and g(x) = 50 ( 1 _ 2 )

x

. How does the value of k in each equation compare to the y-intercept? How does the k-value affect the graph of the function? See Additional Answers.

Match the following exponential growth and decay equations to the graphs shown. Explain your choices.

18. y = 2(0.5)x See Additional Answers.

19. y = 2(3)x See Additional Answers.

20. y = (0.25)x See Additional Answers.

21. y = 0.25(2)x See Additional Answers.

Graph A Graph B

x

y

O

1

2

1 2

(0, 1)

x

y

O

2

4

1

3

2 41 3

(0, 2)

Graph C Graph D

x

y

O

1

2

3

4

21

(0, 2)

-1-2 x

y

O

0.5

1

21

(0, 0.25)

-1-2

Investigation Practice

a. Formulate Alex invested $500 in an account that will double his balance every 8 years. How many times will the amount in the account double in 32 years? Write an equation to model the account balance y after x doubling times. What will his balance be in 32 years? 4 times; y = 500(2)x; $8000

b. Formulate Radioactive glucose is used in cancer detection. It has a half-life of 100 minutes. How many half-lives are in 24 hours? Write an equation to model the amount y remaining of a 100 milligram sample after x half-lives. How much of a 100 milligram sample remains after 24 hours?

Use the equation f(x) = (

1 _ 2 ) x to answer each problem.

c. Does the equation model exponential growth or exponential decay? Explain. decay; The value of b is 0.5, which is between 0 and 1.

d. How does the graph of f (x) = ( 1 _ 2 )

x compare to the graph of g(x) = ( 1 _

3 )

x ?

e. How does the graph of f (x) = ( 1 _ 2 )

x compare to the graph of h(x) = 2 x ? See

Additional Answers.

d. 14.4 half-lives; y = 100 (1_2)

x;

about 0.0046 mgd. 14.4 half-lives; y = 100 (1_

2)x;

about 0.0046 mg

d. Both graphs have y-intercept (0, 1) and neither crosses the x-axis and both graphs show exponential decay. The graph of y = ( 1_

3)x curves

downward more sharply than the graph of y = ( 1_

2)x.

d. Both graphs have y-intercept (0, 1) and neither crosses the x-axis and both graphs show exponential decay. The graph of y = ( 1_

3)x curves

downward more sharply than the graph of y = ( 1_

2)x.

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752 Saxon Algebra 1

Investigation Practice


Problem a

Scaff olding Guide students by asking them the following questions.

“What is the formula for fi nding exponential growth?” f(x) = k b x

“What number will become the variable k in this problem? What number will become b?” 500; 2

“What number will become the exponent in this problem?” the number of times the account doubles in 32 years: 4

Problem e

Error AlertSome students may not realize that h(x) = 2x is exponential growth. Remind students that the percent of decay is a positive number that is less than one. In exponential growth, the percent of growth is a number greater than one.



Match the following exponential growth and decay equations to the graphs shown. Explain your choices.

f. y = 3(0.5)x See Additional Answers.

g. y = 3(2)x See Additional Answers.

h. y = (4)x See Additional Answers.

i. y = 2(0.25)x See Additional Answers.

Graph A Graph B

x

y

O

1

2

3

4

21

(0, 3)

-1-2 x

y

O

1

2

3

4

21

(0, 1)

-1-2

Graph C Graph D

x

y

O

1

2

3

4

21

(0, 3)

-1-2 x

y

O

1

2

4

21

(0, 2)

-1-2

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Investigating exponential growth and decay prepares students for





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