Chapter 3 Resource Masters - Commack Schools

Chapter 3 Resource Masters

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Copyright © by The McGraw-Hill Companies, Inc.

All rights reserved. The contents, or parts thereof, may be reproduced in print form for non-profit educational use with Glencoe Algebra 1, provided such reproductions bear copyright notice, but may not be reproduced in any form for any other purpose without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, network storage or transmission, or broadcast for distance learning.

Send all inquiries to:McGraw-Hill Education8787 Orion PlaceColumbus, OH 43240

ISBN: 978-0-07-661383-0MHID: 0-07-661383-6

Printed in the United States of America.

1 2 3 4 5 6 7 8 9 DOH 16 15 14 13 12 11

CONSUMABLE WORKBOOKS Many of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks in both English and Spanish.

MHID ISBNStudy Guide and Intervention Workbook 0-07-660292-3 978-0-07-660292-6Homework Practice Workbook 0-07-660291-5 978-0-07-660291-9

Spanish VersionHomework Practice Workbook 0-07-660294-X 978-0-07-660294-0

Answers For Workbooks The answers for Chapter 3 of these workbooks can be found in the back of this Chapter Resource Masters booklet.

ConnectED All of the materials found in this booklet are included for viewing, printing, and editing at connected.mcgraw-hill.com.

Spanish Assessment Masters (MHID: 0-07-660289-3, ISBN: 978-0-07-660289-6) These masters contain a Spanish version of Chapter 3 Test Form 2A and Form 2C.

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Teacher’s Guide to Using the Chapter 3 Resource Masters ..............................................iv

Chapter Resources Chapter 3 Student-Built Glossary ...................... 1Chapter 3 Anticipation Guide (English) ............. 3Chapter 3 Anticipation Guide (Spanish) ............ 4

Lesson 3-1Graphing Linear EquationsStudy Guide and Intervention ............................ 5Skills Practice .................................................... 7Practice .............................................................. 8Word Problem Practice ...................................... 9Enrichment ...................................................... 10Spreadsheet Activity .........................................11

Lesson 3-2Solving Linear Equations by GraphingStudy Guide and Intervention .......................... 12Skills Practice .................................................. 14Practice ............................................................ 15Word Problem Practice .................................... 16Enrichment ...................................................... 17

Lesson 3-3Rate of Change and SlopeStudy Guide and Intervention .......................... 18Skills Practice .................................................. 20Practice ............................................................ 21Word Problem Practice .................................... 22Enrichment ...................................................... 23

Lesson 3-4Direct VariationStudy Guide and Intervention .......................... 24Skills Practice .................................................. 26Practice ............................................................ 27Word Problem Practice .................................... 28Enrichment ...................................................... 29

Lesson 3-5Arithmetic Sequences as Linear FunctionsStudy Guide and Intervention .......................... 30Skills Practice .................................................. 32Practice ............................................................ 33Word Problem Practice .................................... 34Enrichment ...................................................... 35

Lesson 3-6Proportional and Nonproportional RelationshipsStudy Guide and Intervention .......................... 36Skills Practice .................................................. 38Practice ............................................................ 39Word Problem Practice .................................... 40Enrichment ...................................................... 41

AssessmentStudent Recording Sheet ............................... 43Rubric for Scoring Extended Response .......... 44Chapter 3 Quizzes 1 and 2 ............................. 45Chapter 3 Quizzes 3 and 4 ............................. 46Chapter 3 Mid-Chapter Test ............................ 47Chapter 3 Vocabulary Test ............................... 48Chapter 3 Test, Form 1 .................................... 49Chapter 3 Test, Form 2A ................................. 51Chapter 3 Test, Form 2B ................................. 53Chapter 3 Test, Form 2C ................................. 55Chapter 3 Test, Form 2D ................................. 57Chapter 3 Test, Form 3 .................................... 59Chapter 3 Extended-Response Test ................ 61Standardized Test Practice .............................. 62

Answers ........................................... A1–A29

Contents

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iv

Teacher’s Guide to Using the Chapter 3 Resource Masters

The Chapter 3 Resource Masters includes the core materials needed for Chapter 3. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing, printing, and editing at connectED.mcgraw-hill.com.

Chapter Resources

Student-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 3-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.

Anticipation Guide (pages 3–4) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed.

Lesson ResourcesStudy Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.

Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.

Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson.

Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.

Enrichment These activities may extend the concepts of the lesson, offer an historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. They are written for use with all levels of students.

Graphing Calculator, TI-Nspire, or Spreadsheet Activities These activities present ways in which technology can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presentation.

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Assessment OptionsThe assessment masters in the Chapter 3 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.

Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.

Extended Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions.

Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.

Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 10 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.

Leveled Chapter Tests

• Form 1 contains multiple-choice ques-tions and is intended for use with below grade level students.

• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in

format to offer comparable testing situations.

• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Form 3 is a free-response test for use with above grade level students.

All of the above mentioned tests include a free-response Bonus question.

Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation.

Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.

Answers• The answers for the Anticipation Guide

and Lesson Resources are provided as reduced pages.

• Full-size answer keys are provided for the assessment masters.

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Chapter 3 1 Glencoe Algebra 1

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 3. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary TermFound

on PageDefi nition/Description/Example

arithmetic sequence

constant

constant of variation

direct variation

family of functions

inductive reasoning

linear equation

parent function

rate of change

root

slope

3 Student-Built Glossary

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Vocabulary TermFound

on PageDefi nition/Description/Example

standard form

terms

x-intercept

y-intercept

3 Student-Built Glossary (continued)


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Before you begin Chapter 3

• Read each statement.

• Decide whether you Agree (A) or Disagree (D) with the statement.

• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

STEP 1A, D, or NS

StatementSTEP 2A or D

1. The equation 6x + 2xy = 5 is a linear equation because each variable is to the first power.

2. The graph of y = 0 has more than one x-intercept.

3. The zero of a function is located at the y-intercept of the function.

4. All horizontal lines have an undefined slope.

5. The slope of a line can be found from any two points on the line.

6. A direct variation, y = kx, will always pass through the origin.

7. In a direct variation y = kx, if k < 0 then its graph will slope upward from left to right.

8. A sequence is arithmetic if the difference between all consecutive terms is the same.

9. Each number in a sequence is called a factor of that sequence.

10. Making a conclusion based on a pattern of examples is called inductive reasoning.

After you complete Chapter 3

• Reread each statement and complete the last column by entering an A or a D.

• Did any of your opinions about the statements change from the first column?

• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

Step 1

Step 2

3 Anticipation GuideLinear Functions


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Capítulo 3 4 Álgebra 1 de Glencoe

Antes de comenzar el Capítulo 3

• Lee cada enunciado.

• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.

• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a)).

PASO 1A, D o NS

EnunciadoPASO 2A o D

1. La ecuación 6x + 2xy = 5 es una ecuación lineal porque cada variable está elevada a la primera potencia.

2. La gráfica de y = 0 tiene más de una intersección x.

3. El cero de una función está ubicado en la intersección y de la función.

4. Todas las rectas horizontales tienen una pendiente indefinida.

5. La pendiente de una recta se puede encontrar a partir de cualquier par de puntos en la recta.

6. Una variación directa, y = kx, siempre pasará por cero.

7. En una variación directa y = kx, si k < 0, entonces su gráfica se inclinará hacia arriba de izquierda a derecha.

8. Una sucesión es aritmética si la diferencia entre todos los términos consecutivos es la misma.

9. Cada número en una sucesión se llama factor de la sucesión.

10. Sacar una conclusión en base a un patrón de ejemplos se llama razonamiento inductivo.

Después de completar el Capítulo 3

• Vuelve a leer cada enunciado y completa la última columna con una A o una D.

• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?

• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.

PASO 1

PASO 2

Ejercicios preparatoriosFunciones lineales

3


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Identify Linear Equations and Intercepts A linear equation is an equation that can be written in the form Ax + By = C. This is called the standard form of a linear equation.

Determine whether y = 6 - 3x is a linear equation. Write the equation in standard form.

First rewrite the equation so both variables are on the same side of the equation. y = 6 - 3x Original equation

y + 3x = 6 - 3x + 3x Add 3x to each side.

3x + y = 6 Simplify.

The equation is now in standard form, with A = 3, B = 1 and C = 6. This is a linear equation.

Determine whether 3xy + y = 4 + 2x is a linear equation. Write the equation in standard form.

Since the term 3xy has two variables, the equation cannot be written in the form Ax + By = C. Therefore, this is not a linear equation.

ExercisesDetermine whether each equation is a linear equation. Write yes or no. If yes, write the equation in standard form.

1. 2x = 4y 2. 6 + y = 8 3. 4x - 2y = -1

4. 3xy + 8 = 4y 5. 3x - 4 = 12 6. y = x2 + 7

7. y - 4x = 9 8. x + 8 = 0 9. -2x + 3 = 4y

10. 2 + 1 − 2 x = y 11. 1 −

4 y = 12 - 4x 12. 3xy - y = 8

13. 6x + 4y - 3 = 0 14. yx - 2 = 8 15. 6x - 2y = 8 + y

16. 1 − 4 x - 12y = 1 17. 3 + x + x2

= 0 18. x2 = 2xy

Standard Form of a Linear EquationAx + By = C, where A ≥ 0, A and B are not both zero, and A, B, and

C are integers with a greatest common factor of 1

Example 1 Example 2

3-1 Study Guide and InterventionGraphing Linear Equations


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Graph Linear Equations The graph of a linear equations represents all the solutions of the equation. An x-coordinate of the point at which a graph of an equation crosses the x-axis in an x-intercept. A y-coordinate of the point at which a graph crosses the y-axis is called a y-intercept.

Graph 3x + 2y = 6 by using the x- and y-intercepts.

To find the x-intercept, let y = 0 and solve for x. The x-intercept is 2. The graph intersects the x-axis at (2, 0).To find the y-intercept, let x = 0 and solve for y. The y-intercept is 3. The graph intersects the y-axis at (0, 3).Plot the points (2, 0) and (0, 3) and draw the line through them.

ExercisesGraph each equation by using the x- and y-intercepts.

1. 2x + y = -2 2. 3x - 6y = -3 3. -2x + y = -2

x

y

Ox

y

Ox

y

O

Graph each equation by making a table.

4. y = 2x 5. x - y = -1 6. x + 2y = 4

x

y

Ox

y

Ox

y

O

x

y

O

Graph y - 2x = 1 by making a table.

Solve the equation for y. y - 2x = 1 Original equation

y - 2x + 2x = 1 + 2x Add 2x to each side.

y = 2x + 1 Simplify.

Select five values for the domain and make a table. Then graph the ordered pairs and draw a line through the points.

y

xO

(2, 0)

(0, 3)

x 2x + 1 y (x, y)

-2 2(-2) + 1 -3 (-2, -3)

-1 2(-1) + 1 -1 (-1, -1)

0 2(0) + 1 1 (0, 1)

1 2(1) + 1 3 (1, 3)

2 2(2) + 1 5 (2, 5)

Example 1 Example 2

3-1 Study Guide and Intervention (continued)

Graphing Linear Equations


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Determine whether each equation is a linear equation. Write yes or no. If yes, write the equation in standard form.

1. xy = 6 2. y = 2 - 3x 3. 5x = y - 4

4. y = 2x + 5 5. y = -7 + 6x 6. y = 3x2 + 1

7. y - 4 = 0 8. 5x + 6y = 3x + 2 9. 1 − 2 y = 1

Find the x- and y-intercepts of each linear function.

10. 11. 12.

Graph each equation by making a table. 13. y = 4 14. y = 3x 15. y = x + 4

Graph each equation by using the x- and y-intercepts.

16. x - y = 3 17. 10x = -5y 18. 4x = 2y + 6

x

y

O

x

y

Ox

y

O

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O

3-1 Skills PracticeGraphing Linear Equations


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Determine whether each equation is a linear equation. Write yes or no. If yes, write the equation in standard form and determine the x- and y-intercepts.

1. 4xy + 2y = 9 2. 8x - 3y = 6 - 4x 3. 7x + y + 3 = y

4. 5 - 2y = 3x 5. x − 4 - y −

3 = 1 6. 5 − x -

2 − y = 7

Graph each equation.

7. 1 − 2 x - y = 2 8. 5x - 2y = 7 9. 1.5x + 3y = 9

10. COMMUNICATIONS A telephone company charges $4.95 per month for long distance calls plus $0.05 per minute. The monthly cost c of long distance calls can be described by the equation c = 0.05m + 4.95, where m is the number of minutes.

a. Find the y-intercept of the graph of the equation.

b. Graph the equation.

c. If you talk 140 minutes, what is the monthly cost?

11. MARINE BIOLOGY Killer whales usually swim at a rate of 3.2–9.7 kilometers per hour, though they can travel up to 48.4 kilometers per hour. Suppose a migrating killer whale is swimming at an average rate of 4.5 kilometers per hour. The distance d the whale has traveled in t hours can be predicted by the equation d = 4.5t.

a. Graph the equation.

b. Use the graph to predict the time it takes the killer whale to travel 30 kilometers.

Time (minutes)

Long Distance

Co

st (

$)

0 40 80 120 160

14

12

10

8

6

4

2

x

y

O

x

y

Ox

y

O

3-1 PracticeGraphing Linear Equations


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1. FOOTBALL One football season, the Carolina Panthers won 4 more games than they lost. This can be represented by y = x + 4, where x is the number of games lost and y is the number of games won. Write this linear equation in standard form.

2. TOWING Pick-M-Up Towing Company charges $40 to hook a car and $1.70 for each mile that it is towed. The equation y = 1.7x + 40 represents the total cost y for x miles towed. Determine the y-intercept. Describe what the value means in this context.

3. SHIPPING The OOCL Shenzhen, one of the world’s largest container ships, carries 8063 TEUs (1280 cubic feet containers). Workers can unload a ship at a rate of a TEU every minute. Using this rate, write and graph an equation to determine how many hours it will take the workers to unload half of the containers from the Shenzhen.

4. BUSINESS The equation y = 1000x - 5000 represents the monthly profits of a start-up dry cleaning company. Time in months is x and profit in dollars is y. The first date of operation is when time is zero. However, preparation for opening the business began 3 months earlier with the purchase of equipment and supplies. Graph the linear function for x-values from -3 to 8.

5. BONE GROWTH The height of a woman can be predicted by the equation h = 81.2 + 3.34r, where h is her height in centimeters and r is the length of her radius bone in centimeters.

a. Is this is a linear function? Explain.

b. What are the r- and h-intercepts of the equation? Do they make sense in the situation? Explain.

c. Use the function to find the approximate height of a woman whose radius bone is 25 centimeters long.

y

x2 4 6 8O

2000

-2000

-4000

-6000

-8000

Time (hours)20100 4030

y

x50 60 70 80

TEU

s o

n S

hip

(th

ou

san

ds)

3

4

2

1

5

8

9

7

6

3-1 Word Problem PracticeGraphing Linear Equations


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Translating Linear Graphs Linear graphs can be translated on the coordinate plane. This means that the graph moves up, down, right, or left without changing its direction.Translating the graphs up or down affects the y-coordinate for a given x value. Translating the graph right or left affects the x-coordinate for a given y-value.

Translate the graph of y = 2x + 2, 3 units up.

ExercisesGraph the function and the translation on the same coordinate plane.

1. y = x + 4, 3 units down 2. y = 2x – 2, 2 units left

3. y = -2x + 1, 1 unit right 4. y = -x - 3, 2 units up

Add 3 to

each

y-value.

y

xO

y = 2x + 2

y

xO

y

xO

y

xO

y

xO

y = 2x + 2

x y

-1 0

0 2

1 4

2 6

Translation

x y

-1 3

0 5

1 7

2 9

Example

3-1 Enrichment


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In addition to organizing data, a spreadsheet can be used to represent data graphically.

Exercises 1. A photo printer offers a subscription for digital photo finishing. The subscription costs

$4.99 per month. Each standard size photo a subscriber prints costs $0.19. Use a spreadsheet to graph the equation y = 4.99 + 0.19x, where x is the number of photos printed and y is the total monthly cost.

2. A long distance service plan includes a $8.95 per month fee plus $0.05 per minute of calls. Use a spreadsheet to graph the equation y = 8.95 + 0.05x, where x is the number of minutes of calls and y is the total monthly cost.

An internet retailer charges $1.99 per order plus $0.99 per item to ship books and CDs. Graph the equation y = 1.99 + 0.99x, where x is the number of items ordered and y is the shipping cost.

Step 1 Use column A for the numbers of items and column B for the shipping costs.

Step 2 Create a graph from the data. Select the data in columns A and B and select Chart from the Insert menu. Select an XY (Scatter) chart to show the data points connected with line segments.

A1

4

56

78

910

11

12

2

3

B

Shipping.xls

Items Shipping Cost

1

2

3

4

5

6

7

8

9

10

$2.98

$3.97

$4.96

$5.95

$6.94

$7.93

$8.92

$9.91

$10.90

$11.89

Sheet 1 Sheet 2 Sheet

Example

3-1 Spreadsheet ActivityLinear Equations


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Solve by Graphing You can solve an equation by graphing the related function. The solution of the equation is the x-intercept of the function.

Solve the equation 2x - 2 = -4 by graphing.

First set the equation equal to 0. Then replace 0 with f(x). Make a table of ordered pair solutions. Graph the function and locate the x-intercept.

2x - 2 = -4 2x - 2 + 4 = -4 + 4 2x + 2 = 0 f(x) = 2x + 2

To graph the function, make a table. Graph the ordered pairs.

x f(x) = 2x + 2 f(x) [x, f(x)]

1 f(1) = 2(1) + 2 4 (1, 4)

-1 f(-1) = 2(-1) + 2 0 (-1, 0)

-2 f(-2) = 2(-2) + 2 -2 (-2, -2)

y

x

The graph intersects the x-axis at (-1, 0).The solution to the equation is x = -1.

ExercisesSolve each equation.

1. 3x - 3 = 0 2. -2x + 1 = 5 - 2x 3. -x + 4 = 0 y

xO

y

xO

y

xO

4. 0 = 4x - 1 5. 5x - 1 = 5x 6. -3x + 1 = 0

y

x

y

xO

y

xO

Original equation

Add 4 to each side.

Simplify.

Replace 0 with f(x).

Example

Study Guide and InterventionSolving Linear Equations by Graphing

3-2


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Estimate Solutions by Graphing Sometimes graphing does not provide an exact solution, but only an estimate. In these cases, solve algebraically to find the exact solution.

WALKING You and your cousin decide to walk the 7-mile trail at the state park to the ranger station. The function d = 7 – 3.2t represents your distance d from the ranger station after t hours. Find the zero of this function. Describe what this value means in this context.

Make a table of values to graph the function.

t d = 7 - 3.2t d (t, d)

0 d = 7 - 3.2(0) 7 (0, 7)

1 d = 7 - 3.2(1) 3.8 (1, 3.8)

2 d = 7 - 3.2(2) 0.6 (2, 0.6)

Mile

s fr

om R

ange

r Sta

tion

2

3

1

0

4

5

6

7

8

y

Time (hours)21 x3

The graph intersects the t–axis between t = 2 and t = 3, but closer to t = 2. It will take you and your cousin just over two hours to reach the ranger station.

You can check your estimate by solving the equation algebraically.

Exercises 1. MUSIC Jessica wants to record her favorite songs to

one CD. The function C = 80 - 3.22n represents the recording time C available after n songs are recorded. Find the zero of this function. Describe what this value means in this context.

2. GIFT CARDS Enrique uses a gift card to buy coffee at a coffee shop. The initial value of the gift card is $20. The function n = 20 – 2.75c represents the amount of money still left on the gift card n after purchasing c cups of coffee. Find the zero of this function. Describe what this value means in this context.

Tim

e Av

aila

ble

(min

)

10

0

20

30

40

50

60

70

80

90

Number of Songs5 10 15 20 25 30

Example

Study Guide and Intervention (continued)

Solving Linear Equations by Graphing

3-2

Valu

e Le

ft on

Car

d ($

)

4

0

8

12

16

20

24

Coffees Bought2 4 6 8 10 12


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Solve each equation.

1. 2x - 5 = -3 + 2x 2. -3x + 2 = 0 3. 3x + 2 = 3x - 1 y

x

y

x

y

x

4. 4x - 1 = 4x + 2 5. 4x - 1 = 0 6. 0 = 5x + 3 y

x

y

x

y

x

7. 0 = -2x + 4 8. -3x + 8 = 5 - 3x 9. -x + 1 = 0 y

x

y

x

y

x

10. GIFT CARDS You receive a gift card for trading cards from a local store. The function d = 20 - 1.95c represents the remaining dollars d on the gift card after obtaining c packages of cards. Find the zero of this function. Describe what this value means in this context.

Amou

nt R

emai

ning

on

Gift

Card

($)

2

0

4

6

8

10

12

14

16

18

20

d

Packages of Cards Bought1 2 3 4 5 6 7 8 9 10 c

Skills PracticeSolving Linear Equations by Graphing

3-2


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PracticeSolving Linear Equations by Graphing

Solve each equation.

1. 1 − 2 x - 2 = 0 2. -3x + 2 = -1 3. 4x - 2 = -2

y

xO

O

y

x

y

x

4. 1 − 3 x + 2 = 1 −

3 x - 1 5. 2 −

3 x + 4 = 3 6. 3 −

4 x + 1 = 3 −

4 x - 7

y

x

y

x

y

x

Solve each equation by graphing. Verify your answer algebraically

7. 13x + 2 = 11x - 1 8. -9x - 3 = -4x - 3 9. - 1 − 3 x + 2 = 2 −

3 x - 1

y

x

y

x

y

x

10. DISTANCE A bus is driving at 60 miles per hour toward a bus station that is 250 miles away. The function d = 250 – 60t represents the distance d from the bus station the bus is t hours after it has started driving. Find the zero of this function. Describe what this value means in this context.

Dist

ance

from

Bus

Stat

ion

(mile

s)

50

0

100

150

200

250

300

Time (hours)1 2 3 4 5 6

3-2


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Word Problem PracticeSolving Linear Equations by Graphing

1. PET CARE You buy a 6.3-pound bag of dry cat food for your cat. The function c = 6.3 – 0.25p represents the amount of cat food c remaining in the bag when the cat is fed the same amount each day for p days. Find the zero of this function. Describe what this value means in this context.

2. SAVINGS Jessica is saving for college using a direct deposit from her paycheck into a savings account. The function m = 3045 - 52.50t represents the amount of money m still needed after t weeks. Find the zero of this function. What does this value mean in this context?

3. FINANCE Michael borrows $100 from his dad. The function v = 100 - 4.75p represents the outstanding balance v after p weekly payments. Find the zero of this function. Describe what this value means in this context.

4. BAKE SALE Ashley has $15 in the Pep Club treasury to pay for supplies for a chocolate chip cookie bake sale. The function d = 15 – 0.08c represents the dollars d left in the club treasury after making c cookies. Find the zero of this function. What does this value represent in this context?

5. DENTAL HYGIENE You are packing your suitcase to go away to a 14-day summer camp. The store carries three sizes of tubes of toothpaste.

TubeSize

(ounces)

Size

(grams)

A 0.75 21.26

B 0.9 25.52

C 3.0 85.04

Source: National Academy of Sciences

a. The function n = 21.26 - 0.8brepresents the number of remaining brushings n using b grams per brushing using Tube A. Find the zero of this function. Describe what this value means in this context.

b. The function n = 25.52 – 0.8brepresents the number of remaining brushings n using b grams per brushing using Tube B. Find the zero of this function. Describe what this value means in this context.

c. Write a function to represent the number of remaining brushings n using b grams per brushing using Tube C. Find the zero of this function. Describe what this value means in this context.

d. If you will brush your teeth twice each day while at camp, which is the smallest tube of toothpaste you can choose? Explain your reasoning.

3-2


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3-2 Enrichment

Composite Functions Three things are needed to have a function—a set called the domain, a set called the range, and a rule that matches each element in the domain with only one element in the range. Here is an example.

Rule: f(x) = 2x + 1

35

-5

f(x)x

12

-3

f(x) = 2x + 1

f(1) = 2(1) + 1 = 2 + 1 = 3

f(2) = 2(2) + 1 = 4 + 1 = 5

f(-3) = 2(-3) + 1 = 26 + 1 = -5

Suppose we have three sets A, B, and C and two functions described as shown below.

Rule: f(x) = 2x + 1 Rule: g( y) = 3y - 4A B C

f(x)

3 5

x

1

g[f(x)]

g( y) = 3y - 4

g(3) = 3(3) - 4 = 5

Let’s find a rule that will match elements of set A with elements of set C without finding any elements in set B. In other words, let’s find a rule for the composite function g[f(x)].

Since f(x) = 2x + 1, g[ f(x)] = g(2x + 1).

Since g( y) = 3y - 4, g(2x + 1) = 3(2x + 1) - 4, or 6x - 1.

Therefore, g[ f(x)] = 6x - 1.

Find a rule for the composite function g[f(x)].

1. f(x) = 3x and g( y) = 2y + 1 2. f(x) = x2 + 1 and g( y) = 4y

3. f(x) = -2x and g( y) = y2 - 3y 4. f(x) = 1 − x - 3

and g( y) = y-1

5. Is it always the case that g[ f(x)] = f[ g(x)]? Justify your answer.


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Study Guide and InterventionRate of Change and Slope

3-3

Rate of Change The rate of change tells, on average, how a quantity is changing over time.

POPULATION The graph shows the population growth in China.

a. Find the rates of change for 1950–1975 and for 2000–2025.

1950–1975: change in population

−− change in time

= 0.93 - 0.55 − 1975 - 1950

= 0.38 − 25

or 0.0152

2000–2025: change in population

−− change in time

= 1.45 - 1.27 − 2025 - 2000

= 0.18 − 25

or 0.0072

b. Explain the meaning of the rate of change in each case.From 1950–1975, the growth was 0.0152 billion per year, or 15.2 million per year.From 2000–2025, the growth is expected to be 0.0072 billion per year, or 7.2 million per year.

c. How are the different rates of change shown on the graph?There is a greater vertical change for 1950–1975 than for 2000–2025. Therefore, the section of the graph for 1950–1975 has a steeper slope.

Exercises1. LONGEVITY The graph shows the predicted life

expectancy for men and women born in a given year.

a. Find the rates of change for women from 2000–2025 and 2025–2050.

b. Find the rates of change for men from 2000–2025 and 2025–2050.

c. Explain the meaning of your results in Exercises 1 and 2.

d. What pattern do you see in the increase with each 25-year period?

e. Make a prediction for the life expectancy for 2050–2075. Explain how you arrived at your prediction.

Predicting Life Expectancy

Year Born

Ag

e

2000

WomenMen

100

95

90

85

80

75

70

65

2050*2025*

*Estimated

Source: USA TODAY

8084

8178

74

87

Population Growth in China

Year

Peo

ple

(b

illio

ns)

1950 1975 2000

2.0

1.5

1.0

0.5

02025*

*Estimated

Source: United Nations Population Division

0.550.93

1.271.45

Example


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Rate of Change and Slope

3-3

Find Slope The slope of a line is the ratio of change in the y- coordinates (rise) to the change in the x- coordinates (run) as you move in the positive direction.

Slope of a Linem = rise

− run

or m = y 2 - y 1

− x 2 - x 1 , where (x

1, y

1) and (x

2, y

2) are the coordinates

of any two points on a nonvertical line

Find the slope of the line that passes through (-3, 5) and (4, -2).

Let (-3, 5) = (x1, y1) and (4, -2) = (x2, y2).

m = y2 - y1 − x2 - x1

Slope formula

= -2 - 5 − 4 - (-3)

y 2 = -2, y 1 = 5, x 2 = 4, x 1 = -3

= -7 − 7 Simplify.

= -1

Exercises Find the slope of the line that passes through each pair of points.

1. (4, 9), (1, 6) 2. (-4, -1), (-2, -5) 3. (-4, -1), (-4, -5)

4. (2, 1), (8, 9) 5. (14, -8), (7, -6) 6. (4, -3), (8, -3)

7. (1, -2), (6, 2) 8. (2, 5), (6, 2) 9. (4, 3.5), (-4, 3.5)

Find the value of r so the line that passes through each pair of points has the given slope.

10. (6, 8), (r, -2), m = 1 11. (-1, -3), (7, r), m = 3 − 4

12. (2, 8), (r, -4) m = -3

13. (7, -5), (6, r), m = 0 14. (r, 4), (7, 1), m = 3 − 4

15. (7, 5), (r, 9), m = 6

Find the value of r so that the line through (10, r) and (3, 4) has a slope of - 2 −

7 .

m = y2 - y1 − x2 - x1

Slope formula

- 2 − 7 = 4 - r −

3 - 10 m = - 2 −

7 , y

2 = 4, y

1 = r, x

2 = 3, x

1 = 10

-

2 − 7 = 4 - r −

-7 Simplify.

-2(-7) = 7(4 - r) Cross multiply.

14 = 28 - 7r Distributive Property

-14 = -7r Subtract 28 from each side.

2 = r Divide each side by -7.

Example 1 Example 2


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Skills PracticeRate of Change and Slope

3-3

Find the slope of the line that passes through each pair of points.

1.

(0, 1)

(2, 5)

x

y

O

2.

(0, 0)

(3, 1)

x

y

O

3.

(0, 1)

(1, -2)x

y

O

4. (2, 5), (3, 6) 5. (6, 1), (-6, 1)

6. (4, 6), (4, 8) 7. (5, 2), (5, -2)

8. (2, 5), (-3, -5) 9. (9, 8), (7, -8)

10. (-5, -8), (-8, 1) 11. (-3, 10), (-3, 7)

12. (17, 18), (18, 17) 13. (-6, -4), (4, 1)

14. (10, 0), (-2, 4) 15. (2, -1), (-8, -2)

16. (5, -9), (3, -2) 17. (12, 6), (3, -5)

18. (-4, 5), (-8, -5) 19. (-5, 6), (7, -8)


20. (r, 3), (5, 9), m = 2 21. (5, 9), (r, -3), m = -4

22. (r, 2), (6, 3), m = 1 − 2 23. (r, 4), (7, 1), m = 3 −

4

24. (5, 3), (r, -5), m = 4 25. (7, r), (4, 6), m = 0


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PracticeRate of Change and Slope

3-3

Find the slope of the line that passes through each pair of points.

1.

(–1, 0)

(–2, 3)

x

y

O

2.

(3, 1)

(–2, –3)

x

y

O

3. (–2, 3)

(3, 3)

x

y

O

4. (6, 3), (7, -4) 5. (-9, -3), (-7, -5)

6. (6, -2), (5, -4) 7. (7, -4), (4, 8)

8. (-7, 8), (-7, 5) 9. (5, 9), (3, 9)

10. (15, 2), (-6, 5) 11. (3, 9), (-2, 8)

12. (-2, -5), (7, 8) 13. (12, 10), (12, 5)

14. (0.2, -0.9), (0.5, -0.9) 15. ( 7 − 3 , 4 −

3 ) , (- 1 −

3 , 2 −

3 )


16. (-2, r), (6, 7), m = 1 − 2 17. (-4, 3), (r, 5), m = 1 −

4

18. (-3, -4), (-5, r), m = - 9 − 2 19. (-5, r), (1, 3), m = 7 −

6

20. (1, 4), (r, 5), m undefined 21. (-7, 2), (-8, r), m = -5

22. (r, 7), (11, 8), m = - 1 − 5 23. (r, 2), (5, r), m = 0

24. ROOFING The pitch of a roof is the number of feet the roof rises for each 12 feet horizontally. If a roof has a pitch of 8, what is its slope expressed as a positive number?

25. SALES A daily newspaper had 12,125 subscribers when it began publication. Five years later it had 10,100 subscribers. What is the average yearly rate of change in the number of subscribers for the five-year period?


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Word Problem Practice Rate of Change and Slope

3-3

1. HIGHWAYS Roadway signs such as the one below are used to warn drivers of an upcoming steep down grade that could lead to a dangerous situation. What is the grade, or slope, of the hill described on the sign?

2. AMUSEMENT PARKS The SheiKra roller coaster at Busch Gardens in Tampa, Florida, features a 138-foot vertical drop. What is the slope of the coaster track at this part of the ride? Explain.

3. CENSUS The table shows the population density for the state of Texas in various years. Find the average annual rate of change in the population density from 2000 to 2009.

4. REAL ESTATE A realtor estimates the median price of an existing single-family home in Cedar Ridge is $221,900. Two years ago, the median price was $195,200. Find the average annual rate of change in median home price in these years.

5. COAL EXPORTS The graph shows the annual coal exports from U.S. mines in millions of short tons.

Source: Energy Information Association

a. What was the rate of change in coal exports between 2001 and 2002?

b. How does the rate of change in coal exports from 2005 to 2006 compare to that of 2001 to 2002?

c. Explain the meaning of the part of the graph with a slope of zero.

Mill

ion

Sh

ort

To

ns

40

50

30

0

60

70

80

90

100

2004

2003

2000

2001

2002

2006

2005

Total Exports

Population Density

Year People Per Square Mile

1930 22.1

1960 36.4

1980 54.3

2000 79.6

2009 96.7

Source: Bureau of the Census, U.S. Dept. of Commerce


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3-3 Enrichment

Treasure Hunt with Slopes

Using the definition of slope, draw segments with the slopes listed below in order. A correct solution will trace the route to the treasure.

1. 3 2. 1 − 4 3. - 2 −

5 4. 0

5. 1 6. -1 7. no slope 8. 2 − 7

9. 3 − 2 10. 1 −

3 11. -

3 − 4 12. 3

Start Here

Treasure


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Study Guide and InterventionDirect Variation

3-4

Direct Variation Equations A direct variation is described by an equation of the form y = kx, where k ≠ 0. We say that y varies directly as x. In the equation y = kx, k is the constant of variation.

Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points.

For y = 1 − 2 x, the constant of variation is 1 −

2 .

m = y 2 - y 1 − x 2 - x 1

Slope formula

= 1 - 0 − 2 - 0

(x1, y

1) = (0, 0), (x

2, y

2) = (2, 1)

= 1 − 2 Simplify.

The slope is 1 − 2 .

Suppose y varies directly as x, and y = 30 when x = 5.

a. Write a direct variation equation that relates x and y.Find the value of k. y = kx Direct variation equation

30 = k(5) Replace y with 30 and x with 5.

6 = k Divide each side by 5.

Therefore, the equation is y = 6x.b. Use the direct variation equation to

find x when y = 18. y = 6x Direct variation equation

18 = 6x Replace y with 18.

3 = x Divide each side by 6.

Therefore, x = 3 when y = 18.

ExercisesName the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points.

1. 2.

(1, 3)

(0, 0) x

y

O

y = 3x

3.

Suppose y varies directly as x. Write a direct variation equation that relates x to y. Then solve.

4. If y = 4 when x = 2, find y when x = 16.

5. If y = 9 when x = -3, find x when y = 6.

6. If y = -4.8 when x = -1.6, find x when y = -24.

7. If y = 1 − 4 when x = 1 −

8 , find x when y = 3 −

16 .

(–2, –3)

(0, 0)x

y

O

y = 32x(–1, 2)

(0, 0)x

y

O

y = –2x

(2, 1)(0, 0) x

y

Oy = 12x

Example 1 Example 2


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Direct Variation

3-4

Direct Variation Problems The distance formula d = rt is a direct variation equation. In the formula, distance d varies directly as time t, and the rate r is the constant of variation.

TRAVEL A family drove their car 225 miles in 5 hours.

a. Write a direct variation equation to find the distance traveled for any number of hours. Use given values for d and t to find r.

d = rt Original equation

225 = r(5) d = 225 and t = 5

45 = r Divide each side by 5.

Therefore, the direct variation equation is d = 45t. Automobile Trips

Time (hours)

Dis

tan

ce (

mile

s)

10 2 3 4 5 6 7 8 t

d

360

270

180

90(1, 45)

(5, 225)d = 45t

b. Graph the equation. The graph of d = 45t passes through the origin with slope 45.

m = 45 − 1 rise

− run

�CHECK (5, 225) lies on the graph.

c. Estimate how many hours it would take the family to drive 360 miles.

d = 45t Original equation

360 = 45t Replace d with 360.

t = 8 Divide each side by 45.

Therefore, it will take 8 hours to drive 360 miles.

Exercises 1. RETAIL The total cost C of bulk jelly beans is

$4.49 times the number of pounds p.

a. Write a direct variation equation that relates the variables.

b. Graph the equation on the grid at the right.

c. Find the cost of 3 − 4 pound of jelly beans.

2. CHEMISTRY Charles’s Law states that, at a constant pressure, volume of a gas V varies directly as its temperature T. A volume of 4 cubic feet of a certain gas has a temperature of 200 degrees Kelvin.

a. Write a direct variation equation that relates the variables.

b. Graph the equation on the grid at the right.

c. Find the volume of the same gas at 250 degrees Kelvin.

Cost of Jelly Beans

Weight (pounds)

Co

st (

do

llars

)

20 4 p

C

18.00

13.50

9.00

4.50

Charles’s Law

Temperature (K)

Vo

lum

e (c

ub

ic f

eet)

1000 200 T

V

4

3

2

1

Example


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Skills PracticeDirect Variation

3-4

Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points.

1.

(3, 1)(0, 0)

x

y

O

y = 13 x

2. 3.


4. y = 3x 5. y = - 3 −

4 x 6. y = 2 −

5 x

Suppose y varies directly as x. Write a direct variation equation that relates x and y. Then solve.

7. If y = -8 when x = -2, find x 8. If y = 45 when x = 15, find x when y = 32. when y = 15.

9. If y = -4 when x = 2, find y 10. If y = -9 when x = 3, find y when x = -6. when x = -5.

11. If y = 4 when x = 16, find y 12. If y = 12 when x = 18, find x when x = 6. when y = -16.

Write a direct variation equation that relates the variables. Then graph the equation.

13. TRAVEL The total cost C of gasoline 14. SHIPPING The number of delivered toys Tis $3.00 times the number of gallons g. is 3 times the total number of crates c.

Gasoline Cost

Gallons

Co

st (

$)

20 4 6 71 3 5 g

C

28

24

20

16

12

8

4

Toys Shipped

Crates

Toys

20 4 6 71 3 5 c

T

21

18

15

12

9

6

3

x

y

Ox

y

O x

y

O

(–2, 3)

(0, 0)x

y

O

y = – 32 x

(-1, 2)

(0, 0)x

y

O

y = -2x


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3-4 PracticeDirect Variation

Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points.

1.

(4, 3)

(0, 0)x

y

O

y = 3 x4

2. (3, 4)

(0, 0)x

y

O

y = 43 x

3. (–2, 5)

(0, 0)x

y

O

y = - 52 x


4. y = -2x 5. y = 6 −

5 x 6. y = - 5 −

2 x

Suppose y varies directly as x. Write a direct variation equation that relates x and y. Then solve.

7. If y = 7.5 when x = 0.5, find y when x = -0.3.

8. If y = 80 when x = 32, find x when y = 100.

9. If y = 3 −

4 when x = 24, find y when x = 12.

Write a direct variation equation that relates the variables. Then graph the equation.

10. MEASURE The width W of a 11. TICKETS The total cost C of tickets isrectangle is two thirds of the length �. $4.50 times the number of tickets t.

12. PRODUCE The cost of bananas varies directly with their weight. Miguel bought

3 1 −

2 pounds of bananas for $1.12. Write an equation that relates the cost of the bananas

to their weight. Then find the cost of 4 1 −

4 pounds of bananas.

x

y

O

x

y

O x

y

O

Rectangle Dimensions

Length

Wid

th

40 8 122 6 10 �

W

10

8

6

4

2


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Word Problem PracticeDirect Variation

3-4

1. ENGINES The engine of a chainsaw requires a mixture of engine oil and gasoline. According to the directions, oil and gasoline should be mixed as shown in the graph below. What is the co nstant of variation for the line graphed?

2. RACING In a recent year, English driver Lewis Hamilton won the United States Grand Prix at the Indianapolis Motor Speedway. His speed during the race averaged 125.145 miles per hour. Write a direct variation equation for the distance d that Hamilton drove in h hours at that speed.

3. CURRENCY The exchange rate from one currency to another varies every day. Recently the exchange rate from U.S. dollars to British pound sterling (£) was $1.58 to £1. Write and solve a direct variation equation to determine how many pounds sterling you would receive in exchange for $90 of U.S. currency.

4. SALARY Henry started a new job in which he is paid $9.50 an hour. Write and solve an equation to determine Henry’s gross salary for a 40-hour work week.

5. SALES TAX Amelia received a gift card to a local music shop for her birthday. She plans to use the gift card to buy some new CDs.

a. Amelia chose 3 CDs that each cost $16. The sales tax on the three CDs is $3.96. Write a direct variation equation relating sales tax to the price.

b. Graph the equation you wrote in part a.

c. What is the sales tax rate that Amelia is paying on the CDs?

Oil

(fl

oz)

3

4

2

1

5

9

10

8

7

6

Gasoline (gal)3210 54 x

y

6


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Enrichment 3-4

nth Power Variation

An equation of the form y = kxn, where k ≠ 0, describes an nth power variation. The variable n can be replaced by 2 to indicate the second power of x (the square of x) or by 3 to indicate the third power of x (the cube of x).Assume that the weight of a person of average build varies directly as the cube of that person’s height. The equation of variation has the form w = kh3.The weight that a person’s legs will support is proportional to the cross-sectional area of the leg bones. This area varies directly as the square of the person’s height. The equation of variation has the form s = kh2.

Answer each question.

1. For a person 6 feet tall who weighs 200 pounds, find a value for k in the equation w = kh3.

2. Use your answer from Exercise 1 to predict the weight of a person who is 5 feet tall.

3. Find the value for k in the equation w = kh3 for a baby who is 20 inches long and weighs 6 pounds.

4. How does your answer to Exercise 3 demonstrate that a baby is significantly fatter in proportion to its height than an adult?

5. For a person 6 feet tall who weighs 200 pounds, find a value for k in the equation s = kh2.

6. For a baby who is 20 inches long and weighs 6 pounds, find an “infant value” for k in the equation s = kh2.

7. According to the adult equation you found (Exercise 1), how much would an imaginary giant 20 feet tall weigh?

8. According to the adult equation for weight supported (Exercise 5), how much weight could a 20-foot tall giant’s legs actually support?

9. What can you conclude from Exercises 7 and 8?


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Recognize Arithmetic Sequences A sequence is a set of numbers in a specific order. If the difference between successive terms is constant, then the sequence is called an arithmetic sequence.

Write an equation for the nth term of the sequence 12, 15, 18, 21, . . . .

In this sequence, a1 is 12. Find the common difference.12

+ 3 + 3 + 3

15 18 21

The common difference is 3.Use the formula for the nth term to write an equation. a n = a 1 + (n - 1)d Formula for the nth term

a n = 12 + (n - 1)3 a 1 = 12, d = 3

a n = 12 + 3n - 3 Distributive Property

a n = 3n + 9 Simplify.

The equation for the nth term is a n = 3n + 9.

ExercisesDetermine whether each sequence is an arithmetic sequence. Write yes or no. Explain.

1. 1, 5, 9, 13, 17, . . . 2. 8, 4, 0, -4, -8, . . . 3. 1, 3, 9, 27, 81, . . .

Find the next three terms of each arithmetic sequence.

4. 9, 13, 17, 21, 25, . . . 5. 4, 0, -4, -8, -12, . . . 6. 29, 35, 41, 47, . . .

Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms of the sequence.

7. 1, 3, 5, 7, . . . 8. -1, -4, -7, -10, . . . 9. -4, -9, -14, -19, . . .

Determine whether the sequence 1, 3, 5, 7, 9, 11, . . . is an arithmetic sequence. Justify your answer.

If possible, find the common difference between the terms. Since 3 - 1 = 2, 5 - 3 = 2, and so on, the common difference is 2.Since the difference between the terms of 1, 3, 5, 7, 9, 11, . . . is constant, this is an arithmetic sequence.

3-5 Study Guide and InterventionArithmetic Sequences as Linear Functions

Arithmetic Sequencea numerical pattern that increases or decreases at a constant rate

or value called the common difference

Terms of an Arithmetic Sequence If a

1 is the fi rst term of an arithmetic sequence with common

difference d, then the sequence is a1, a

1 + d, a

1 + 2d, a

1 + 3d, . . . .

nth Term of an Arithmetic Sequence an = a1 + (n - 1)d

Example 1 Example 2


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Arithmetic Sequences as Linear Functions

Arithmetic Sequences and Functions An arithmetic sequence is a linear function in which n is the independent variable, a n is the dependent variable, and the common difference d is the slope. The formula can be rewritten as the function a n = a 1 + (n - 1)d, where n is a counting number.

SEATING There are 20 seats in the first row of the balcony of the auditorium. There are 22 seats in the second row, and 24 seats in the third row.

a. Write a function to represent this sequence. The first term a 1 is 20. Find the common difference.

20

+ 2 + 2

22 24

The common difference is 2.

a n = a 1 + (n - 1)d = 20 + (n - 1)2 = 20 + 2n - 2 = 18 + 2n

The function is a n = 18 + 2n.

b. Graph the function.The rate of change is 2. Make a table and plot points.

n a n

1 20

2 22

3 24

4 26

nO 1

20

22

24

26

28

2 3 4

an

Exercises 1. KNITTING Sarah learns to knit from her grandmother. Two days ago, she measured

the length of the scarf she is knitting to be 13 inches. Yesterday, she measured the length of the scarf to be 15.5 inches. Today it measures 18 inches. Write a function to represent the arithmetic sequence.

2. REFRESHMENTS You agree to pour water into the cups for the Booster Club at a football game. The pitcher contains 64 ounces of water when you begin. After you have filled 8 cups, the pitcher is empty and must be refilled.

a. Write a function to represent the arithmetic sequence.

b. Graph the function.

3-5

Formula for the nth term

a 1 = 20 and d = 2

Distributive Property

Simplify.

an

nO 1 82 4 6

816

32

48

64

24

40

56

72

3 5 7

Example


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Skills PracticeArithmetic Sequences as Linear Functions

Determine whether each sequence is an arithmetic sequence. Write yes or no. Explain.

1. 4, 7, 9, 12, . . . 2. 15, 13, 11, 9, . . .

3. 7, 10, 13, 16, . . . 4. -6, -5, -3, -1, . . .

5. -5, -3, -1, 1, . . . 6. -9, -12, -15, -18, . . .

7. 10, 15, 25, 40, . . . 8. -10, -5, 0, 5, . . .


9. 3, 7, 11, 15, . . . 10. 22, 20, 18, 16, . . .

11. -13, -11, -9, -7 . . . 12. -2, -5, -8, -11, . . .

13. 19, 24, 29, 34, . . . 14. 16, 7, -2, -11, . . .

15. 2.5, 5, 7.5, 10, . . . 16. 3.1, 4.1, 5.1, 6.1, . . .


17. 7, 13, 19, 25, . . . 18. 30, 26, 22, 18, . . . 19. -7, -4, -1, 2, . . .

20. VIDEO DOWNLOADING Brian is downloading episodes of his favorite TV show to play on his personal media device. The cost to download 1 episode is $1.99. The cost to download 2 episodes is $3.98. The cost to download 3 episodes is $5.97. Write a function to represent the arithmetic sequence.

n

an

2 4 6

4

-4

-8

O

n

an

O 2 4 6

30

20

10

n

an

O 2 4 6

30

20

10

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PracticeArithmetic Sequences as Linear Functions

Determine whether each sequence is an arithmetic sequence. Write yes or no. Explain.

1. 21, 13, 5, -3, . . . 2. -5, 12, 29, 46, . . . 3. -2.2, -1.1, 0.1, 1.3, . . .

4. 1, 4, 9, 16, . . . 5. 9, 16, 23, 30, . . . 6. -1.2, 0.6, 1.8, 3.0, . . .


7. 82, 76, 70, 64, . . . 8. -49, -35, -21, -7, . . . 9. 3 − 4 , 1 −

2 , 1 −

4 , 0, . . .

10. -10, -3, 4, 11 . . . 11. 12, 10, 8, 6, . . . 12. 12, 7, 2, -3, . . .


13. 9, 13, 17, 21, . . . 14. -5, -2, 1, 4, . . . 15. 19, 31, 43, 55, . . .

16. BANKING Chem deposited $115.00 in a savings account. Each week thereafter, he deposits $35.00 into the account.

a. Write a function to represent the total amount Chem has deposited for any particular number of weeks after his initial deposit.

b. How much has Chem deposited 30 weeks after his initial deposit?

17. STORE DISPLAYS Tamika is stacking boxes of tissue for a store display. Each row of tissues has 2 fewer boxes than the row below. The first row has 23 boxes of tissues.

a. Write a function to represent the arithmetic sequence.

b. How many boxes will there be in the tenth row?

n

an

O

60

40

20

2 4 6

n

an

2 4 6

8

4

-4

O

n

an

O 2 4 6

30

20

10

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1. POSTAGE The price to send a large envelope first class mail is 88 cents for the first ounce and 17 cents for each additional ounce. The table below shows the cost for weights up to 5 ounces.

Source: United States Postal Servcie

How much did a large envelope weigh that cost $2.07 to send?

2. SPORTS Wanda is the manager for the soccer team. One of her duties is to hand out cups of water at practice. Each cup of water is 4 ounces. She begins practice with a 128-ounce cooler of water. How much water is remaining after she hands out the 14th cup?

3. THEATER A theater has 20 seats in the first row, 22 in the second row, 24 in the third row, and so on for 25 rows. How many seats are in the last row?

4. NUMBER THEORY One of the most famous sequences in mathematics is the Fibonacci sequence. It is named after Leonardo de Pisa (1170–1250) or Filius Bonacci, alias Leonardo Fibonacci. The first several numbers in the Fibonacci sequence are:1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .Does this represent an arithmetic sequence? Why or why not?

5. SAVINGS Inga’s grandfather decides to start a fund for her college education. He makes an initial contribution of $3000 and each month deposits an additional $500. After one month he will have contributed $3500.

a. Write an equation for the nth term of the sequence.

b. How much money will Inga’s grandfather have contributed after 24 months?

Word Problem PracticeArithmetic Sequences as Linear Functions

3-5

Weight

(ounces)1 2 3 4 5

Postage

(dollars)0.88 1.05 1.22 1.39 1.56


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Arithmetic SeriesAn arithmetic series is a series in which each term after the first may be found by adding the same number to the preceding term. Let S stand for the following series in which each term is 3 more than the preceding one.S = 2 + 5 + 8 + 11 + 14 + 17 + 20The series remains the same if we S = 2 + 5 + 8 + 11 + 14 + 17 + 20reverse the order of all the terms. So S = 20 + 17 + 14 + 11 + 8 + 5 + 2let us reverse the order of the terms 2S = 22 + 22 + 22 + 22 + 22 + 22 + 22and add one series to the other, term 2S = 7(22)by term. This is shown at the right. S =

7(22) −

2 = 7(11) = 77

Let a represent the first term of the series.Let � represent the last term of the series.Let n represent the number of terms in the series.In the preceding example, a = 2, � = 20, and n = 7. Notice that when you add the two series, term by term, the sum of each pair of terms is 22. That sum can be found by adding the first and last terms, 2 + 20 or a + �. Notice also that there are 7, or n, such sums. Therefore, the value of 2S is 7(22), or n(a + �) in the general case. Since this is twice the sum of the series, you can use the formula S = n(a + �)

− 2 to find the sum of any arithmetic series.

Find the sum: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9.

a = 1, � = 9, n = 9, so S = 9(1 + 9) −

2 =

9 . 10 − 2 = 45

Find the sum: -9 + (-5) + (-1) + 3 + 7 + 11 + 15.

a = 29, � = 15, n = 7, so S = 7(-9 + 15)

− 2 =

7.6 − 2 = 21

ExercisesFind the sum of each arithmetic series.

1. 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24

2. 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50

3. -21 + (-16) + (-11) + (-6) + (-1) + 4 + 9 + 14

4. even whole numbers from 2 through 100

5. odd whole numbers between 0 and 100

Enrichment 3-5

Example 1

Example 2


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Study Guide and InterventionProportional and Nonproportional Relationships

Proportional Relationships If the relationship between the domain and range of a relation is linear, the relationship can be described by a linear equation. If the equation passes through (0, 0) and is of the form y = kx, then the relationship is proportional.

COMPACT DISCS Suppose you purchased a number of packages of blank compact discs. If each package contains 3 compact discs, you could make a chart to show the relationship between the number of packages of compact discs and the number of discs purchased. Use x for the number of packages and y for the number of compact discs.

Make a table of ordered pairs for several points of the graph.

Number of Packages 1 2 3 4 5

Number of CDs 3 6 9 12 15

The difference in the x values is 1, and the difference in the y values is 3. This pattern shows that y is always three times x. This suggests the relation y = 3x. Since the relation is also a function, we can write the equation in function notation as f (x) = 3x.The relation includes the point (0, 0) because if you buy 0 packages of compact discs, you will not have any compact discs. Therefore, the relationship is proportional.

Exercises 1. NATURAL GAS Natural gas use is often measured in “therms.” The total amount a gas

company will charge for natural gas use is based on how much natural gas a household uses. The table shows the relationship between natural gas use and the total cost.

Gas Used (therms) 1 2 3 4

Total Cost ($) $1.30 $2.60 $3.90 $5.20

a. Graph the data. What can you deduce from the pattern about the relationship between the number of therms used and the total cost?

b. Write an equation to describe this relationship.

c. Use this equation to predict how much it will cost if a household uses 40 therms.

3-6

Example

Tota

l Cos

t ($)

2

3

1

0

4

5

6

y

Gas Used (therms)321 x4


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Proportional and Nonproportional Relationships

Nonproportional Relationships If the ratio of the value of x to the value of y is different for select ordered pairs on the line, the equation is nonproportional.

Write an equation in functional notation for the relation shown in the graph.

Select points from the graph and place them in a table.

x -1 0 1 2 3

y 4 2 0 -2 -4

y

x

The difference between the x–values is 1, while the difference between the y-values is –2. This suggests that y = –2x.If x = 1, then y = –2(1) or –2. But the y–value for x = 1 is 0.

x 1 2 3

-2x -2 -4 -6

y 0 -2 -4

y is always 2 more than -2x

This pattern shows that 2 should be added to one side of the equation. Thus, the equation is y = -2x + 2.

ExercisesWrite an equation in function notation for the relation shown in the table. Then complete the table.

1. x -1 0 1 2 3 4

y -2 2 6

2. x -2 -1 0 1 2 3

y 10 7 4

Write an equation in function notation for each relation.

3.

x

y

O

4.

x

y

O

3-6

Example


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Skills PracticeProportional and Nonproportional Relationships


1. 2.

3. 4.

5. 6.

7. GAMESHOWS The table shows how many points are awarded for answering consecutive questions on a gameshow.

Question answered 1 2 3 4 5

Points awarded 200 400 600 800 1000

a. Write an equation for the data given.

b. Find the number of points awarded if 9 questions were answered.

xO

f (x)

xO

f (x)

xO

f (x)

xO

f (x)

xO

f (x)

xO

f (x)

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PracticeProportional and Nonproportional Relationships

1. BIOLOGY Male fireflies flash in various patterns to signal location and perhaps to ward off predators. Different species of fireflies have different flash characteristics, such as the intensity of the flash, its rate, and its shape. The table below shows the rate at which a male firefly is flashing.

Times (seconds) 1 2 3 4 5

Number of Flashes 2 4 6 8 10

a. Write an equation in function notation for the relation.

b. How many times will the firefly flash in 20 seconds?

2. GEOMETRY The table shows the number of diagonals that can be drawn from one vertex in a polygon. Write an equation in function notation for the relation and find the number of diagonals that can be drawn from one vertex in a 12-sided polygon.


3. 4. 5.

For each arithmetic sequence, determine the related function. Then determine if the function is proportional or nonproportional. Explain.

6. 1, 3, 5, . . . 7. 2, 7, 12, . . . 8. -3, -6, -9, . . .

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y

O

x

y

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x

y

O

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Sides 3 4 5 6

Diagonals 0 1 2 3


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Word Problem PracticeProportional and Nonproportional Relationships

1. ONLINE SHOPPING Ricardo is buying computer cables from an online store. If he buys 4 cables, the total cost will be $24. If he buys 5 cables, the total cost will be $29. If the total cost can be represented by a linear function, will the function be proportional or nonproportional? Explain.

2. FOOD It takes about four pounds of grapes to produce one pound of raisins. The graph shows the relation for the number of pounds of grapes needed, x, to make y pounds of raisins. Write an equation in function notation for the relation shown.

3. PARKING Palmer Township recently installed parking meters in their municipal lot. The cost to park for h hours is represented by the equation C = 0.25h.

a. Make a table of values that represents this relationship.

b. Describe the relationship between the time parked and the cost.

4. MUSIC A measure of music contains the same number of beats throughout the song. The table shows the relation for the number of beats counted after a certain number of measures have been played in the six-eight time. Write an equation to describe this relationship.

Source: Sheet Music USA

5. GEOMETRY A fractal is a pattern containing parts which are identical to the overall pattern. The following geometric pattern is a fractal.

a. Complete the table.

b. What are the next three numbers in the pattern?

c. Write an equation in function notation for the pattern.

Pounds of Grapes210 43

f(x)

x5 6 7 8

Pou

nd

s o

f R

aisi

ns

1.5

2.0

1.0

0.5

2.5

4.0

3.5

3.0

3-6

Measures

Played (x)1 2 3 4 5 6

Total Number

of Beats (y)6 12 18 24 30 36

Term x 1 2 3 4

Number of

Smaller

Triangles

y 1


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Enrichment

Taxicab GraphsYou have used a rectangular coordinate system to graph

x

y

O

y = x - 1

y = -2

equations such as y = x - 1 on a coordinate plane. In a coordinate plane, the numbers in an ordered pair (x, y) can be any two real numbers.A taxicab plane is different from the usual coordinate plane. The only points allowed are those that exist along the horizontal and vertical grid lines. You may think of the points as taxicabs that must stay on the streets. The taxicab graph shows the equations y = -2 and y = x - 1. Notice that one of the graphs is no longer a straight line. It is now a collection of separate points.

Graph these equations on the taxicab plane at the right.

x

y

O

1. y = x + 1 2. y = -2x + 3

3. y = 2.5 4. x = -4

Use your graphs for these problems.

5. Which of the equations has the same graph in both the usual coordinate plane and the taxicab plane?

6. Describe the form of equations that have the same graph in both the usual coordinate plane and the taxicab plane.

In the taxicab plane, distances are not measured diagonally, but along the streets. Write the taxi-distance between each pair of points.

7. (0, 0) and (5, 2) 8. (0, 0) and (-3, 2) 9. (0, 0) and (2, 1.5)

10. (1, 2) and (4, 3) 11. (2, 4) and (-1, 3) 12. (0, 4) and (-2, 0)

Draw these graphs on the taxicab grid at the right.

13. The set of points whose taxi-distance from (0, 0) is 2 units.

14. The set of points whose taxi-distance from (2, 1) is 3 units.

x

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3

Read each question. Then fill in the correct answer.

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

8. F G H J

9. A B C D

Multiple Choice

Student Recording SheetUse this recording sheet with pages 210–211 of the Student Edition.

Short Response/Gridded Response

Record your answer in the blank.

For gridded response questions, also enter your answer in the grid by writing each number or symbol in a box. Then fill in the corresponding circle for that number or symbol.

10. ————————

11. ————————

12. ————————

13. ———————— (grid in)

14. ————————

15. ———————— (grid in) 9

8

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9

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Extended Response

Record your answers for Question 16 on the back of this paper.

13.

9

8

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5

4

3

2

1

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9

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043_053_ALG1_A_CRM_C03_AS_661383.indd 43043_053_ALG1_A_CRM_C03_AS_661383.indd 43 12/21/10 12:34 PM12/21/10 12:34 PM

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3 Rubric for Scoring Extended Response

General Scoring Guidelines • If a student gives only a correct numerical answer to a problem but does not show how

he or she arrived at the answer, the student will be awarded only 1 credit. All extended-response questions require the student to show work.

• A fully correct answer for a multiple-part question requires correct responses for all parts of the question. For example, if a question has three parts, the correct response to one or two parts of the question that required work to be shown is not considered a fully correct response.

• Students who use trial and error to solve a problem must show their method. Merely showing that the answer checks or is correct is not considered a complete response for full credit.

Exercise 16 Rubric

Score Specifi c Criteria

4

For part a, the explanation must demonstrate an understanding that because each element of the domain is paired with exactly one element of the range, the table must represent a function. For part b, the explanation must demonstrate an understanding that it cannot be represented by an equation of the form y = kx.

3 A generally correct solution, but may contain minor flaws in reasoning or computation.

2 A partially correct interpretation and/or solution to the problem.

1 A correct solution with no evidence or explanation.

0 An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given.


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3

3

SCORE

1. Determine whether y = 2x - 1 is a linear equation. If so, write the equation in standard form.

2. MULTIPLE CHOICE What is the y-intercept when 3x - 2y = -6 is graphed?

A -3 B -2 C 2 D 3

3. The distance d in miles that a car travels in t hours at a rate of 58 miles per hour is given by the equation d = 58t. What is the best estimate of how far a car travels in 7 hours?

4. Graph 3x - y = 3.

5. Solve 3x - 45 = 0.

Find the slope of the line passing through each pair of points.

1. (5, 8) and (-4, 6) 2. (9, 4) and (5, -3)

3. MULTIPLE CHOICE Which value of r gives the line passing through (3, 2) and (r, -4) a slope of 3 −

2 ?

A -6 B -1 C 7 D 12

4. Determine whether the function is linear. x y

0 1

-2 -1

-4 1

Write linear or not linear.

5. In 2004, there were approximately 275 students in the Delaware High School band. In 2010, that number increased to 305. Find the annual rate of change in the number of students in the band.

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Chapter 3 Quiz 2 (Lesson 3-3)

Chapter 3 Quiz 1 (Lessons 3-1 and 3-2)

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3.

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3

3

SCORE

For Questions 1–3, find the function that represents the relationship.

1. x 0 1 2 3 4

y 3 6 9 12 15

2. x 0 1 2 3 4

y 4 9 14 19 24

3. x 0 1 2 3 4

y -2 7 16 25 34

4. Is the following relationship proportional? Write yes or no.

x 0 1 2 3 4

y 13 26 39 52 65

5. What values for a and b will make this linear relationship proportional?

x 0 1 2 3 4

y a 11 22 b 44

Chapter 3 Quiz 3(Lessons 3-4 and 3-5)

1. MULTIPLE CHOICE Suppose y varies directly as x. If y = 5 when x = 8, find y when x = 64.

A 25 B 40 C 61 D 102.4

2. Chris’s wages vary directly as the time she works. If her wages for 20 hours are $150, what are her wages for 38 hours?

3. Determine whether 2, 5, 9, 14, . . . is an arithmetic sequence.

4. Find the next three terms of the arithmetic sequence 5, 9, 13, 17, . . . .

5. Find the 15th term of the arithmetic sequence if a1 = - 3 and d = 2.

Chapter 3 Quiz 4 (Lesson 3-6)

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.


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3 SCORE

Write the letter for the correct answer in the blank at the right of each question.

1. Find the slope of the line through (6, -7) and (4, -8). A - 1 −

2 B 2 C 1 −

2 D -2

2. Find the slope of the line through (0, 5) and (5, 5). F 0 G 1 H 2 J undefined

3. It is expected that 563 quadrillion thermal units of Btu (British thermal units) of energy will be consumed worldwide in 2015. In 2003, worldwide consumption was 421 quadrillion Btu. What is the expected rate of change in consumption from 2003 to 2015?

A about 0.069 quadrillion Btu per year B about 11.83 quadrillion Btu per year C about 142 quadrillion Btu per year D about 0.085 years per quadrillion Btu

4. The total price of a bag of peaches varies directly with the cost per pound. If 3 pounds of peaches cost $3.60, how much would 5.5 pounds cost?

F $1.20 G $6.60 H $6.00 J $1.96

5. A carpenter can build 8 cabinets in 3 hours. At this rate, how long will it take for the carpenter to build 26 cabinets?

A 8 hours C 21 hours B 9 hours 45 minutes D 69 hours 20 minutes

6. Solve 3x - 12 = 3x + 4.

7. Find the value of r so the line that passes through (-5, 2)

and (3, r) has a slope of - 1 − 2 .

8. Determine whether y = 2x - 3 is a linear equation. If so, write the equation in standard form.

9. Graph 3y - x = 6 by using the x- and y-intercepts.

Part II

Chapter 3 Mid-Chapter Test (Lessons 3-1 through 3-3)

Part I

1.

2.

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5.

6.

7.

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3 SCORE Chapter 3 Vocabulary Test

Choose from the terms above to complete each sentence.

1. In an arithmetic sequence the constant rate of increase or decrease between successive terms is called the .

2. The zero of a function is its .

3. The of an equation is any value that makes the equation true.

4. An equation in the form Ax + By = C is called the

of a linear equation.

5. In a(n) , the difference betweenterms is constant.

6. For a(n) Ax + By = C, the numbers A, B, and C cannot all be zero.

7. A specific number in a sequence is calleda(n) .

8. The act of using a pattern to find a general rule is known as .

Define each term in your own words.

9. y-intercept

10. rate of change

arithmetic sequence

constant

constant of variation

direct variation

family of functions

inductive reasoning

linear equation

parent function

rate of change

root

slope

standard form

term

x-intercept

y-intercept

1.

2.

3.

4.

5.

6.

7.

8.


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3 SCORE Chapter 3 Test, Form 1

Write the letter for the correct answer in the blank at the right of each question .

1. Where does the graph of y = -3x - 18 intersect the x-axis? A (0, 6) B (0, - 6) C (6, 0) D (- 6, 0)

2. Tickets to see a movie cost $5 for children and $8 for adults. The equation 5x + 8y = 80 represents the number of children (x) and adults (y) who can see the movie with $80. If no adults see the movie, how many children can see the movie with $80?

F 6 G 10 H 13 J 16

For Questions 3 – 5, find the slope of each line described.

3. the line through (3, 7) and (-1, 4) A 4 −

3 B 3 −

4 C 11 −

2 D 2 −

11

4. the line through (-3, 2) and (6, 2)

F 4 − 9 G 4 −

3 H 0 J undefined

5. a vertical line A 1 B 0 C -1 D undefined

6. Which graph has a slope of -3?

F y

xO

G y

xO

H y

xO

J y

xO

7. COMMUNICATION In 1996, there were 171 area codes in the United States. In 2007, there were 215. Find the rate of change from 1996 to 2007.

A 44 B 4 C 1 − 4 D -4

For Questions 8 and 9, use the arithmetic sequence 12, 15, 18, 21, . . . .

8. Which is an equation for the nth term of the sequence? F an = 3n + 9 H an = 12n + 3 G an = 9n + 3 J an = n + 3

9. What is the 12th term in the sequence? A 38 B 42 C 45 D 48

1.

2.

3.

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5.

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7.

8.

9.


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3 Chapter 3 Test, Form 1 (continued)

10. Suppose y varies directly as x, and y = 26 when x = 8. Find x when y = 65.

F 3.25 G 20 H 47 J 211.25

For Questions 11 and 12, use the graph shown.

11. Which line has an y-intercept of -2? A � C t B p D both � and t

12. Which line is the graph of y = 2x + 4? F � H the x-axis G p J t

13. Which arithmetic sequence has a proportional related function? A -4, -1, 2, . . . B 0, -2, -4, . . . C 1, 2, 3, . . . D -

1 − 2 , 0, 1 −

2 , . . .

14. Write y + 1 = -2x - 3 in standard form. F 2x + y = -4 G y = -2x - 4 H -2x - y = 4 J x + 1 −

2 y = -2

15. Find the root of 5x - 20 = 0.

A -20 B 0 C 4 D 5

16. Determine which sequence is an arithmetic sequence. F 3, 6, 12, 24, . . . H -7, -3, 1, 5, . . . G 1 __ 5 , 1 __ 7 , 1 __ 9 , 1 ___ 11 , . . . J -10, 5, -

5 − 2 , 5 −

4 , . . .

17. Find the next three terms of the arithmetic sequence 5, 9, 13, 17, . . . A 21, 23, 25 B 21, 25, 29 C 41, 45, 49 D 21, 41, 61

18. Find the function that represents the relationship.

F y = 8x H y = 14x + 8

G y = 8x + 14 J y = 14x + 14

x 0 1 2 3 4

y 14 22 30 38 46

19. Which equation is a linear equation?

A 4m2 = 6 C 2 __ 3 xy – 3 −

4 y = 0

B 3a + 5b = 3 D x2 + y2

= 0

20. Write an equation in function notation for the relation at the right.F f (x) = 2x H f (x) = 1 - xG f (x) = x + 1 J f (x) = - x

Bonus Graph y = x – 3 by using the x- and y-intercepts.

xO

f (x)

y

xO

p

t

�

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:


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3 SCORE


1. LANDSCAPING The equation 3x + 7y = 105 represents the number of bags of sand x and bags of mulch y that can be bought with $105. If no bags of sand are bought, how many bags of mulch can be bought with $105?

A 35 B 17 C 15 D 10

2. If (a, -5) is a solution to the equation 3a = -2b - 7, what is a?

F -1 G 0 H 1 J 4

3. What is the slope of the line through (1, 9) and (-3, 16)?

A - 7

− 4

B - 4 − 7 C -

25 − 2 D -

2 − 25

4. Which equation is not a linear equation?

F -4v + 2w = 7 G x __ 4 = y H x = -5 J 2 __ x + 3 __ y = 6

5. What is the slope of the line through (-4, 3) and (5, 3)?

A 0 B undefined C 9 D 1

6. In 2005, there were 12,000 students at Beacon High. In 2010, there were 12,250. What is the rate of change in the number of students?

F 250/yr G 50/yr H 42/yr J 200/yr

7. Which is the graph of y = 2 − 3 x?

A B C D

8. If y varies directly as x and y = 3 when x = 10, find x when y = 8.

F 80 − 3 G 12 −

5 H 15 −

4 J none of these

9. DRIVING A driver’s distance varies directly as the amount of time traveled. After 6 hours, a driver had traveled 390 miles. How far had the driver traveled after 4 hours?

A 130 miles B 220 miles C 260 miles D 650 miles

For Questions 10 and 11, use the following information.

The number of seats in each row of a theater form an arithmetic sequence, as shown in the table.

10. How many seats are in the 12th row?F 68 G 74 H 96 J 114

11. Which formula can be used to find the number of seats in any given row? A an = 6n + 2 B an = 2n + 6 C an = n + 6 D an = 5n + 3

Chapter 3 Test, Form 2A

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2.

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4.

5.

6.

7.

8.

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10.

11.

Row 1 2 3 4

Number of Seats 8 14 20 26


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3


F y = x - 3 H y = 6x - 3

G y = 3x - 3 J y = 6x

For Questions 13 and 14, use the relation shown in the table.

13. Which equation describes this relationship?A f(x) = 3x C f(x) = x + 2B f(x) = 4x - 1 D f(x) = 2x + 1

14. What is the value of y when x = 43?F 87 H 49G 85 J 45

15. Which line shown at the right is the graph ofx - 2y = 4?

A � C p B m D t

16. Which equation has a graph that is a vertical line?

F 2x = y H 3x - 2 = 0

G y + 5 = 3 J x - y = 0

17. What is the standard form of y - 7 = - 2 − 3 (x + 1)?

A -2x + 3y = 23 B -3x + 2y = 17 C 2x + 3y = 19 D 3x + 2y = 11

18. Determine which sequence is not an arithmetic sequence.

F -7, 0, 7, 14, . . . H 10, 6, 2, -2, . . .

G 0, 1 __ 2 , 1, 3 __ 2 , . . . J 2, 4, 8, 16, . . .

19. Which equation describes the nth term of the arithmetic sequence 7, 10, 13, 16, . . .?

A an = 3n + 4 B an = 7 + 3n C an = - 4n + 3 D an = 3n - 4

20. Write an equation in function notation for the relation shown at the right.

F f (x) = -2x H f (x) = x - 2 G f (x) = 2x + 2 J f (x) = -x + 2

Bonus The table shows the schedule for a theme park roller coaster.

Run Number 1 2 3 4

Time 9:30 A.M. 9:37 A.M. 9:44 A.M. 9:51 A.M.

At what time will the roller coaster make its 15th run?

Chapter 3 Test, Form 2A (continued)

y

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p t

m

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f (x)

x 0 1 2 3 4

y -3 3 9 15 21

x y

1 3

2 5

3 7

4 9

5 11

B:

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3 SCORE


1. PHOTOCOPIES Black and white copies cost $0.05 each, and color copies cost $0.49 each. The equation 0.05x + 0.49y = 5 represents the number of black and white copies x and color copies y that can be made with $5. If no color copies are made,how many black and white copies can be made with $5?

A 200 B 100 C 25 D 10

2. If (a, -7) is a solution to the equation 8a = -3b - 5, what is a? F -17 G 2 H 8 J 17

3. What is the slope of the line through (2, -8) and (4, 1)? A - 2 −

9 B - 6 −

7 C - 7 −

6 D 9 −

2

4. Which equation is not a linear equation? F 2x + 5y = 3 G y = - 10 H 5 = 3xy J y =

x __

7 + 4

5. What is the slope of the line through (-4, -6) and (9, -6)? A - 12 −

5 B - 5 −

12 C 0 D undefined

6. In 2005, MusicMart sold 14,550 CDs. In 2010, they sold 12,000 CDs. What is the rate of change in the number of CDs sold?

F -2550 per yr G -510 per yr H -425 per yr J -2400 per yr

7. Which is the graph of y = - 1 − 2 x?

A B C D

8. If y varies directly as x and y = 5 when x = 8, find y when x = 9.

F 72 − 5 G 45 −

8 H 40 −

9 J 6

9. SPRINGS The amount a spring stretches varies directly as the weight of the object attached to it. If an 8-ounce weight stretches a spring 10 centimeters, how much weight will stretch it 15 centimeters?

A 16 oz B 6 oz C 10 oz D 12 ozFor Questions 10 and 11, use the following information.

The number of students seated in each row of an auditorium form an arithmetic sequence, as shown in the table. 10. How many students are seated in the 15th row? F 46 G 58 H 59 J 195

11. Which formula can be used to find the number of students seated in any given row?

A an = 4n B an = 4n - 1 C an = n + 3 D an = 3n + 1

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Chapter 3 Test, Form 2B

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y

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2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Row 1 2 3 4

Number of students 4 7 10 13


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3


F y = x - 5 H y = 7x

G y = 7x - 5 J y = 7x + 7

For Questions 13 and 14, use the relation shown in the table.

13. Which equation describes this relationship?A f (x) = x C f (x) = x + 1

B f(x) = 2x - 1 D f(x) = x + 2

14. What is the value of y when x = 26? F 51 H 37 G 48 J 28

15. Which line shown at the right is the graph of x + 2y = 6?

A r C t B n D v

16. Which equation has a graph that is a horizontal line?

F x - 7 = 0 H x = y G 2y + 3 = 4 J x + y = 0

17. What is the standard form of y + 2 = 1 − 2 (x - 4)?

A x + 2y = 0 B x - 2y = 8 C 2x - y = 10 D 4x - 2y = 0

18. Determine which sequence is an arithmetic sequence. F -16, -12, -8, -4, . . . H 1, 4, 2, 5, 3, . . . G 4, 8, 16, 32, . . . J 1, 1, 2, 3, 5, . . .

19. Which equation describes the nth term of the arithmetic sequence -12, -14, -16, -18, . . .?

A an = -2n - 10 C an = 10n - 2

B an= -12 - 2n D an = -2n + 10

20. Write an equation in function notation for the relation.

F f (x) = -2x H f (x) = -x + 2

G f (x) = x - 2 J f (x) = 2x + 2

Bonus The table shows the arrival times for a commuter train.

Train Number 1 2 3 4

Time 5:15 A.M. 5:29 A.M. 5:43 A.M. 5:57 A.M.

At what time will the 12th train arrive?

Chapter 3 Test, Form 2B (continued)

r n

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t v

y

xO

f (x)

x 0 1 2 3 4

y -5 2 9 16 23

x y

1 1

2 3

3 5

4 7

5 9

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13.

14.

15.

16.

17.

18.

19.

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3 SCORE

1. Tickets for a spaghetti dinner cost $4 for children and $6 for adults. The equation 4x + 6y = 36 represents the number of children x and adults y who can eat at the dinner for $36. If no children are eating at the dinner, how many adults can eat for $36?

2. If (a, 9) is a solution to the equation -4a = b - 21, what is a?

3. Find the x-intercept of x - 2y = 9.

4. Solve -3 = -2x + 1 by graphing.


For Questions 6 – 8, find the slope of the line passing through each pair of points. If the slope is undefined, write undefined.

6. (2, 5) and (3, 6)

7. (6, -4) and (-3, 7)

8. (-1, 3) and (6, 3)

9. In 1972, federal vehicle emission standards allowed 3.4 hydrocarbons released per mile driven. By 2007, the standards allowed only 0.8 hydrocarbons per mile driven. What was the rate of change from 1972 to 2007?

10. If a shark can swim 27 miles in 9 hours, how many mileswill it swim in 12 hours?

For Questions 11 and 12, determine whether each equation is a linear equation. If so, write the equation in standard form.

11. xy = 6 12. 2x + 3y + 7 = 3

13. Graph the equation x – 4y = 2.

Chapter 3 Test, Form 2C

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

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3

14. Graph y = - 1 − 2 x.

15. Solve 1 − 2 x + 7 −

2 = 5 by graphing.

16. Determine whether the sequence -10, -7, -4, -1, . . . is an arithmetic sequence. Write yes or no. If so, state the common difference.

17. Find the next three terms of the arithmetic sequence 8, 15, 22, 29, . . ..

18. Write an equation for the nth term of the sequence 12, 5, -2, -9, . . ..

For Questions 19 and 20, use the table below that shows the amount of gasoline a car consumes for different distances driven.

Distance (mi) 1 2 3 4 5

Gasoline (gal) 0.04 0.08 0.12 0.16 0.20

19. Write an equation in function notation for the relationship between distance and gasoline used.

20. How many gallons will the car consume after driving for 150 miles?

Bonus Graph x = 3, y = -1, and 2x - 2y = 0 on a coordinate plane. Give the vertices of the figure formed by the three lines.

Chapter 3 Test, Form 2C (continued)

y

xO

y

x

14 . y

xO

15.

16.

17.

18.

19.

20.

B:


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3 SCORE

1. Tickets for a school play cost $7 for students and $10 for adults. The equation 7x + 10y = 80 represents the number of students x and adults y who can attend the play for $80. If no students attend, how many adults can see the play for $80?

2. If (a, -8) is a solution to the equation -a = 4b - 7, what is a?

3. Find the x-intercept of 3x - y = 7.

4. Solve -3 = -2x - 1 by graphing.


For Questions 6 –8, find the slope of the line passing through each pair of points. If the slope is undefined, write “undefined.”

6. (4, 1) and (-4, 1)

7. (-6, 7) and (-6, -2)

8. (-2, 1) and (3, -2)

9. In 1995, 57.5% of students at Gardiner University graduated in 4 or fewer years of study. In 2009, that number had fallen to 52.8%. What was the rate of change for percent of students graduating within 4 years from 1995 to 2009?

10. If a snail travels 200 inches in 2 hours, how long will it take the snail to travel 50 inches?

For Questions 11 and 12, determine whether each equation is a linear equation. Write yes or no. If so, write the equation in standard form.

11. 1 __ x + 1

__

y = 2 − 3

12. 4x = 2y

13. Graph the equation x + 4y - 3 = 0.

Chapter 3 Test, Form 2D

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13. y

xO

y

x


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3

14. Graph y = 2 − 3 x.

15. Solve 3 = 1 − 2 x + 5 −

2 by graphing.

16. Determine whether the sequence 7, 11, 15, 19, . . . is an arithmetic sequence. Write yes or no. If so, state the common difference.

17. Find the next three terms of the arithmetic sequence -25, -22, -19, -16, . . ..

18. Write an equation for the nth term of the sequence 3, 12, 21, 30, . . ..

For Questions 19 and 20, use the table below that shows the amount of gasoline a car consumes for different distances driven. Distance (mi) 1 2 3 4 5

Gasoline (gal) 0.05 0.10 0.15 0.20 0.25

19. Write an equation in function notation for the relationship between distance and gasoline used.

20. How many gallons will the car consume after driving for 140 miles?

Bonus Graph the points (4, -2), (4, 3), (-2, 3). Find a fourth point that completes a rectangle with the given three points, then graph the rectangle on the coordinate plane.

Chapter 3 Test, Form 2D (continued)

1 4.

15.

16.

17.

18.

19.

20.

B:

y

xO

y

xO

y

x


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3 SCORE

1. Determine whether 3x - 4y + 7 = 3y + 1 is a linear equation. Write yes or no. If so, write the equation in standard form.

2. The equation 300x + 50y = 600 represents the number of premium tickets x and the number of discount tickets y for a horse race that can be bought with $600. If no premium tickets are purchased, how many discount tickets can be purchased with $600?

3. If (a, -7) is a solution to the equation 5a - 7b = 28, what is a?

4. Find the x-intercept of 4x - 5y = 15.

5. Solve - 5 − 3 x + 7 = 9 −

2 by graphing.

6. Find the root of 14x + 5 = 61.

7. Graph 2x __ 3 = y −

2 - 1.

For Questions 8 and 9, find the slope of the line passing through each pair of points. If the slope is undefined, write undefined.

8. (-8, 7) and (5, -2)

9. (5, 9) and (5, -3)

10. Five years ago there were approximately 35,000 people living in Lancaster. Now the population is 38,452. Find the rate of change in the population.

11. If an ostrich can run 15 kilometers in 15 minutes, how many kilometers can it run in an hour?

Chapter 3 Test, Form 3

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

y

xO

y

x


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3

12. Graph y = - 3 − 4 x.

13. Find the value of r so that the line through (-4, 3) and (r, -3)

has a slope of 2 − 3 .

14. Find the value of r so that the line through (r, 5) and (6, r)

has a slope of 5 − 8 .

15. Determine whether the sequence 0, 1 __ 2 , 1, 3 __ 2 , . . . is an

arithmetic sequence. If it is, state the common difference.

16. Find the value of y that makes 9, 4, y, -6, . . . an arithmetic sequence.

17. Write an equation for the nth term of the arithmetic sequence -15, -11, -7, -3, . . .. Then graph the first five terms of the sequence.

For Questions 18 and 19, use the table below that shows the value of a vending machine over the first five years of use.

Number of Years 0 1 2 3 4 5

Value (dollars) 2000 1810 1620 1430 1240 1050

18. Write an equation in function notation for the relationship between years of use t and value v(t).

19. When will the value of the vending machine reach 0?

20. Brian collects baseball cards. His father gave him 20 cards to start his collection on his tenth birthday. Each year Brian adds about 15 cards to his collection. About how many years will it take to fill his collection binder if it holds 200 cards?

Bonus A gym membership costs $29.99 per month. Write a direct variation equation to find the total cost c to own a gym membership for y years.

Chapter 3 Test, Form 3 (continued)

12. y

xO

13.

14.

15.

16.

17.

18.

19.

20.

B:

an

n1

2

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

-10-12-14


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3 SCORE

Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.

1. Draw a line on a coordinate plane so that you can determine at least two points on the graph.

a. Describe how you would determine the slope of the graph and justify the slope you found.

b. Explain how you could use the slope to write various forms of the equation of that line. Then write three forms of the equation.

2. Suppose your total cost C at a home improvement store varies directly as f, the number of feet of lumber you buy.

a. Write a direct variation equation that might model the situation.

b. What is the meaning of the constant of variaton in the equation?

c. Suppose you also wanted to by $10 worth of nails. Could the situation still be modeled by a direct variation equation? Why or why not?

3. a. Write a sequence that is an arithmetic sequence. State the common difference, and find a6.

b. Write a sequence that is not an arithmetic sequence. Determine whether the sequence has a pattern, and if so describe the pattern.

c. Determine if the sequence 1, 1, 1, 1, . . . is an arithmetic sequence. Explain your reasoning.

4. Draw a line on the coordinate grid below. Identify the x- and y-intercepts. Then write an equation in function notation for the line.

xO

f (x)

Chapter 3 Extended-Response Test


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3 SCORE Standardized Test Practice(Chapters 1–3)

1. The student council is selling candy bars to earn money towards their budget for the school dance. Identify the independent and dependent variables. (Lesson 1-6)

A I: student council; C I: candy bars sold; D: money earned D: money earned B I: budget; D I: candy bars sold; D: school dance D: school dance

2. Dion owns a delivery service. He charges his customers $15.00 for each delivery. His expenses include $7000 for the motorcycle he drives and $0.42 for gasoline per trip. Which equation could Dion use to calculate his profit p for d deliveries? (Lesson 1-7)

F p = 15 - 0.42d H p = 14.58d - 7000 G p = 7000 + 15d J p = 0.42d + 7000

3. Evaluate 60 ÷ 5 · 6 - 32. (Lesson 1-2)

A -7 B -4 C 63 D 4761

4. Jim’s new car has 150 miles on the odometer. He takes a trip and drives an average of m miles each day for three weeks. Which expression represents the mileage on Jim’s car after his trip? (Lesson 2-4)

F 150m + 3 G 150 + 3m H 150m + 21 J 150 + 21m

5. Translate the sentence into an equation. (Lesson 2-1)

Five times the sum of m and t is as much as four times r. A 5m + t = 4 B 5m + t = r C 5(m + t) = 4r D m + t = 5(4r)

6. Solve 8(x – 5) = 12(4x – 1) + 12. (Lesson 2-4)

F - 7 ___

10 G -

5 __ 7 H -2 J -1

7. Paul and Charlene are 420 miles apart. They start toward each other with Paul driving 16 miles per hour faster than Charlene. They meet in 5 hours. Find Charlene’s speed. (Lesson 2-9)

A 34 mph B 50 mph C 40.4 mph D 68 mph

8. Determine which equation is a linear equation. (Lesson 3-1)

F x2 + y = 4 G x + y = 4 H xy = 4 J 1 __ x + y = 4

9. If f(x) = 7 - 2x, find f(3) + 6. (Lesson 1-7)

A 11 B 7 C 14 D –11

10. Chapa is beginning an exercise program that calls for 30 push-ups each day for the first week. Each week thereafter, she has to increase her push-ups by 2. Which week of her program will be the first one in which she will do 50 push-ups a day? (Lesson 3-5)

F 9th week G 10th week H 11th week J 12th week

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

8. F G H J

9. A B C D

10. F G H J

Part 1: Multiple Choice

Instructions: Fill in the appropriate circle for the best answer.


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3 Standardized Test Practice (continued)

11. Which property of equality is illustrated below? (Lesson 1-3)

If 7 + 9 = 11 + 5 and 11 + 5 = 16, then 7 + 9 = 16.

A Transitive C Substitution

B Reflexive D Symmetric

12. Which expression represents the missing second step of simplifying the algebraic expression? (Lesson 1-4)

Step 1 4(x - 3y) + 6 + 5(x + 1)

Step 3 9x - 12y + 11

F 4x - 3y + 6 + 5x + 1 H 4x - 12y + 6 + 5x + 5

G 12(x - y) + 6 + x + 5 J x - 3y + 15 + x + 1

13. Solve 48 = -8r. (Lesson 2-2)

A r = 8 B r = 6 C r = -6 D r = -40

14. Solve 4 - (-h) = 12. (Lesson 2-2)

F h = 16 G h = 8 H h = -8 J h = -16

For Questions 15 and 16, use the arithmetic sequence 2, 5, 8, 11, . . . .

15. Which is an equation for the nth term of the sequence? (Lesson 3-5)

A an = 2n + 1 C an = n + 3

B an = 4n - 2 D an = 3n - 1

16. What is the 20th term in the sequence? (Lesson 3-5)

F 59 G 60 H 78 J 80

11. A B C D

12. F G H J

13. A B C D

14. F G H J

15. A B C D

16. F G H J

17. The ratio of a to b is 4 __ 7 . If a is 16, find the value of b. (Lesson 2-6)

18. The equation C = F - 32 −

1.8 relates the

temperature in degrees Fahrenheit F to degrees Celsius C. If the temperature is 25°C, what is the temperature in degrees Fahrenheit? (Lesson 3-1)

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

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3

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8

7

6

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6

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Part 2: Gridded Response

Instructions: Enter your answer by writing each digit of the answer in a column box

and then shading in the appropriate circle that corresponds to that entry.


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3 Standardized Test Practice (continued)

19. Find the solution of y + 2 − 3 =

22 ___ 15 if the replacement set is 2 __ 5 , 3 __ 5 , 4 __ 5 , 1, 1 1 −

5 . (Lesson 1-5)

20. Simplify 5m + 8p + 3m + p. (Lesson 1-3)

21. Determine the slope of the line passing through (1, 4) and (3, -1). (Lesson 3-3)

22. Translate the following equation into a verbal sentence.

x − 4 - y = -2 ( x − y ) (Lesson 2-1)

23. Find the discounted price. clock: $15.00 discount: 15% (Lesson 2-7)

24. Solve -7x + 23 = 37. (Lesson 2-3)

25. Use cross products to determine whether the ratios 4 __ 7

and 11 ___ 15 form a proportion. Write yes or no. (Lesson 2-6)

For Questions 26 and 27, use the graph.

26. Express the relation as a set of ordered pairs. Then determine the domain and range. (Lesson 1-6)

27. Determine whether the relation isa function. (Lesson 1-7)

28. Find the x-intercept of the graph of 4x = 5 + y. (Lesson 3-1)

29. Graph 2x - 3y = 6. (Lesson 3-1)

30. The table below shows the average amount of gas Therese’s truck uses depending on how many miles she drives.

Gallons of Gasoline 1 2 3 4 5

Miles Driven 18 36 54 72 90

a. Does the table of values represent a function? Explain. (Lesson 3-6)

b. Is this a proportional relationship? Explain. (Lesson 3-6)

y

xO

BE

C D

A

G

F

Part 3: Short Response

Instructions: Write your anwers in the space provided at the right.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30a.

30b.

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Chapter 3 A1 Glencoe Algebra 1

Lesson 3-1

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

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10. 2

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2 x =

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and

com

plet

e th

e la

st c

olu

mn

by

ente

rin

g an

A o

r a

D.

•

Did

an

y of

you

r op

inio

ns

abou

t th

e st

atem

ents

ch

ange

fro

m t

he

firs

t co

lum

n?

•

For

th

ose

stat

emen

ts t

hat

you

mar

k w

ith

a D

, use

a p

iece

of

pape

r to

wri

te a

n

exam

ple

of w

hy

you

dis

agre

e.

Step

1

Step

2

3A

ntic

ipat

ion

Gui

deLin

ear

Fu

ncti

on

s

D A D D A A D A D A

001_

014_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

313

/12/

10

3:53

PM

Answers (Anticipation Guide and Lesson 3-1)

A01_A19_ALG1_A_CRM_C03_AN_661383.indd A1A01_A19_ALG1_A_CRM_C03_AN_661383.indd A1 12/21/10 12:45 PM12/21/10 12:45 PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

6 G

lenc

oe A

lgeb

ra 1

Gra

ph

Lin

ear

Equ

atio

ns

The

gra

ph o

f a

line

ar e

quat

ions

rep

rese

nts

all t

he s

olut

ions

of

the

equa

tion

. An

x-co

ordi

nate

of

the

poin

t at

whi

ch a

gra

ph o

f an

equ

atio

n cr

osse

s th

e x-

axis

in

an

x-in

terc

ept.

A y

-coo

rdin

ate

of t

he p

oint

at

whi

ch a

gra

ph c

ross

es t

he y

-axi

s is

cal

led

a y-

inte

rcep

t.

G

rap

h 3

x +

2y

= 6

by

usi

ng

the

x- a

nd

y-i

nte

rcep

ts.

To

fin

d th

e x-

inte

rcep

t, l

et y

=

0

and

solv

e fo

r x.

Th

e x-

inte

rcep

t is

2. T

he

grap

h i

nte

rsec

ts t

he

x-ax

is a

t (2

, 0).

To

fin

d th

e y-

inte

rcep

t, l

et x

=

0

and

solv

e fo

r y.

T

he

y-in

terc

ept

is 3

. Th

e gr

aph

in

ters

ects

th

e y-

axis

at

(0, 3

).P

lot

the

poin

ts (

2, 0

) an

d (0

, 3)

and

draw

th

e li

ne

thro

ugh

th

em.

Exer

cise

sG

rap

h e

ach

eq

uat

ion

by

usi

ng

the

x- a

nd

y-i

nte

rcep

ts.

1. 2

x +

y

= -

2 2.

3x

- 6y

=

-

3 3.

-2x

+

y

= -

2

x

y

Ox

y Ox

y O

Gra

ph

eac

h e

qu

atio

n b

y m

akin

g a

tab

le.

4. y

=

2x

5.

x -

y

= -

1 6.

x +

2y

=

4

x

y

Ox

y

Ox

y O

x

y O

G

rap

h y

-

2x

=

1

by

mak

ing

a ta

ble

.

Sol

ve t

he

equ

atio

n f

or y

.

y -

2x

=

1

Ori

gin

al equation

y -

2x

+

2x

=

1 +

2x

A

dd 2

x to

each s

ide.

y

= 2x

+

1

Sim

plif

y.

Sel

ect

five

val

ues

for

th

e do

mai

n a

nd

mak

e a

tabl

e.

Th

en g

raph

th

e or

dere

d pa

irs

and

draw

a l

ine

thro

ugh

th

e po

ints

.

y

xO

( 2, 0

)

( 0, 3

)

x2x

+

1

y(x

, y)

-2

2(-

2)

+ 1

-3

(-2

, -

3)

-1

2(-

1)

+ 1

-1

(-1

, -

1)

0

2(0

) +

1

1

(0,

1)

1

2(1

) +

1

3

(1,

3)

2

2(2

) +

1

5(2

, 5

)

Exam

ple

1Ex

amp

le 2

3-1

Stud

y G

uide

and

Inte

rven

tion

(c

onti

nu

ed)

Gra

ph

ing

Lin

ear

Eq

uati

on

s

001_

014_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

612

/15/

10

11:3

2 P

M

Lesson 3-1


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

7 G

lenc

oe A

lgeb

ra 1

Det

erm

ine

wh

eth

er e

ach

eq

uat

ion

is

a li

nea

r eq

uat

ion

. Wri

te y

es o

r n

o.

If y

es, w

rite

th

e eq

uat

ion

in

sta

nd

ard

for

m.

1. x

y =

6

2. y

=

2

- 3

x 3.

5x

= y

- 4

4. y

=

2x

+

5

5. y

=

-

7 +

6x

6.

y =

3x

2 +

1

7. y

-

4

= 0

8. 5

x +

6y

=

3x

+

2

9. 1 −

2 y =

1

Fin

d t

he

x- a

nd

y-i

nte

rcep

ts o

f ea

ch l

inea

r fu

nct

ion

.

10.

11.

12.

Gra

ph

eac

h e

qu

atio

n b

y m

akin

g a

tab

le.

13. y

=

4

14. y

=

3x

15

. y =

x

+ 4

Gra

ph

eac

h e

qu

atio

n b

y u

sin

g th

e x-

an

d y

-in

terc

epts

.

16. x

-

y

= 3

17. 1

0x =

-

5y

18. 4

x =

2y

+

6

x

y

O

x

y

Ox

y

O

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O

3-1

Skill

s Pr

acti

ceG

rap

hin

g L

inear

Eq

uati

on

s

n

o

yes

; 3x

+

y

=

2

yes

; 5x

-

y

=

-

4

y

es;

2x -

y

=

-

5 y

es;

6x -

y

=

7

n

o

y

es;

y =

4

yes

; x

+

3y

=

1

yes

; y

=

2

x

-in

terc

ept:

2,

x-i

nte

rcep

t: 4

, x

-in

terc

ept:

2,

y

-in

terc

ept:

-2

y-i

nte

rcep

t: 4

y

-in

terc

ept:

4

001_

014_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

713

/12/

10

3:53

PM

Answers (Lesson 3-1)


An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

8 G

lenc

oe A

lgeb

ra 1

Det

erm

ine

wh

eth

er e

ach

eq

uat

ion

is

a li

nea

r eq

uat

ion

. Wri

te y

es o

r n

o. I

f ye

s,

wri

te t

he

equ

atio

n i

n s

tan

dar

d f

orm

an

d d

eter

min

e th

e x-

an

d y

-in

terc

epts

.

1. 4

xy +

2y

=

9

2. 8

x -

3y

=

6

- 4x

3.

7x

+ y

+ 3

= y

4. 5

-

2y

=

3x

5.

x −

4 - y −

3 = 1

6. 5 −

x -

2 −

y

= 7

Gra

ph

eac

h e

qu

atio

n.

7.

1 −

2 x -

y

= 2

8.

5x

- 2y

=

7

9. 1

.5x

+ 3y

=

9

10. C

OM

MU

NIC

ATI

ON

S A

tel

eph

one

com

pan

y ch

arge

s $4

.95

per

mon

th f

or l

ong

dist

ance

cal

ls p

lus

$0.0

5 pe

r m

inu

te. T

he

mon

thly

cos

t c

of l

ong

dist

ance

cal

ls c

an b

e de

scri

bed

by t

he

equ

atio

n c

=

0.

05m

+

4.

95, w

her

e m

is

the

nu

mbe

r of

min

ute

s.

a. F

ind

the

y-in

terc

ept

of t

he

grap

h o

f th

e eq

uat

ion

.

b.

Gra

ph t

he

equ

atio

n.

c. I

f yo

u t

alk

140

min

ute

s, w

hat

is

the

mon

thly

cos

t?

11. M

AR

INE

BIO

LOG

Y K

ille

r w

hal

es u

sual

ly s

wim

at

a ra

te o

f 3.

2–9.

7 ki

lom

eter

s pe

r h

our,

thou

gh t

hey

can

tra

vel

up

to 4

8.4

kilo

met

ers

per

hou

r. S

upp

ose

a m

igra

tin

g ki

ller

w

hal

e is

sw

imm

ing

at a

n a

vera

ge r

ate

of 4

.5 k

ilom

eter

s pe

r h

our.

Th

e di

stan

ce d

th

e w

hal

e h

as t

rave

led

in t

hou

rs c

an

be p

redi

cted

by

the

equ

atio

n d

=

4.

5t.

a. G

raph

th

e eq

uat

ion

.

b.

Use

th

e gr

aph

to

pred

ict

the

tim

e it

tak

es t

he

kill

er

wh

ale

to t

rave

l 30

kil

omet

ers.

Ti

me

(ho

urs

)

Kil

ler

Wh

ale

Tra

vels

Distance (km)

02

46

89

13

57

40 35 30 25 20 15 10 5

Tim

e (m

inu

tes)

Lon

g D

ista

nce

Cost ($)

040

8012

016

0

14 12 10 8 6 4 2

x

y

O

x

y

Ox

y

O

3-1

Prac

tice

Gra

ph

ing

Lin

ear

Eq

uati

on

s

n

o

yes

; 4x

-

y

=

2;

y

es;

7x =

-

3;

x

: 1 −

2 ; y:

-2

x:

-3

−

7 ; y

: n

on

e

y

es;

3x +

2y

=

5;

y

es;

3x -

4y

=

12

;

no

x

: 5 −

3 ; y:

5 −

2 x

: 4;

y:

-3

(0, 4

.95)

$11.

95

bet

wee

n 6

h a

nd

7 h

001_

014_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

813

/12/

10

3:53

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 3-1

Cha

pte

r 3

9 G

lenc

oe A

lgeb

ra 1

1.FO

OTB

ALL

On

e fo

otba

ll s

easo

n, t

he

Car

olin

a P

anth

ers

won

4 m

ore

gam

es

than

th

ey l

ost.

Th

is c

an b

e re

pres

ente

d by

y =

x

+ 4,

wh

ere

x is

th

e n

um

ber

of

gam

es l

ost

and

y is

th

e n

um

ber

of g

ames

w

on. W

rite

th

is l

inea

r eq

uat

ion

in

st

anda

rd f

orm

.

2. T

OW

ING

Pic

k-M

-Up

Tow

ing

Com

pan

y ch

arge

s $4

0 to

hoo

k a

car

and

$1.7

0 fo

r ea

ch m

ile

that

it

is t

owed

. Th

e eq

uat

ion

y

= 1.

7x +

40

rep

rese

nts

th

e to

tal

cost

y

for

x m

iles

tow

ed. D

eter

min

e th

e y-

inte

rcep

t. D

escr

ibe

wh

at t

he

valu

e m

ean

s in

th

is c

onte

xt.

3. S

HIP

PIN

G T

he

OO

CL

Sh

enzh

en, o

ne

of

the

wor

ld’s

lar

gest

con

tain

er s

hip

s,

carr

ies

8063

TE

Us

(128

0 cu

bic

feet

co

nta

iner

s). W

orke

rs c

an u

nlo

ad a

sh

ip

at a

rat

e of

a T

EU

eve

ry m

inu

te. U

sin

g th

is r

ate,

wri

te a

nd

grap

h a

n e

quat

ion

to

dete

rmin

e h

ow m

any

hou

rs i

t w

ill

take

th

e w

orke

rs t

o u

nlo

ad h

alf

of t

he

con

tain

ers

from

th

e S

hen

zhen

.

4. B

USI

NES

S T

he

equ

atio

n

y =

10

00x

- 50

00 r

epre

sen

ts t

he

mon

thly

pro

fits

of

a st

art-

up

dry

clea

nin

g co

mpa

ny.

Tim

e in

mon

ths

is

x an

d pr

ofit

in

dol

lars

is

y. T

he

firs

t da

te o

f op

erat

ion

is

wh

en t

ime

is z

ero.

H

owev

er, p

repa

rati

on f

or o

pen

ing

the

busi

nes

s be

gan

3 m

onth

s ea

rlie

r w

ith

th

e pu

rch

ase

of e

quip

men

t an

d su

ppli

es.

Gra

ph t

he

lin

ear

fun

ctio

n f

or x

-val

ues

fr

om -

3 to

8.

5. B

ON

E G

RO

WTH

Th

e h

eigh

t of

a

wom

an c

an b

e pr

edic

ted

by t

he

equ

atio

n

h =

81

.2 +

3.

34r,

wh

ere

h i

s h

er h

eigh

t in

ce

nti

met

ers

and

r is

th

e le

ngt

h o

f h

er

radi

us

bon

e in

cen

tim

eter

s.

a. I

s th

is i

s a

lin

ear

fun

ctio

n?

Exp

lain

.

b.

Wh

at a

re t

he

r- a

nd

h-i

nte

rcep

ts o

f th

e eq

uat

ion

? D

o th

ey m

ake

sen

se i

n

the

situ

atio

n?

Exp

lain

.

c. U

se t

he

fun

ctio

n t

o fi

nd

the

appr

oxim

ate

hei

ght

of a

wom

an w

hos

e ra

diu

s bo

ne

is 2

5 ce

nti

met

ers

lon

g.

y

x2

46

8O

2000

-20

00

-40

00

-60

00

-80

00

Tim

e (h

ou

rs)

2010

040

30

y

x50

6070

80

TEUs on Ship (thousands)34 2 1589 7 6

3-1

Wor

d Pr

oble

m P

ract

ice

Gra

ph

ing

Lin

ear

Eq

uati

on

s

x -

y

=

-

4

T

he

y-i

nte

rcep

t is

40,

wh

ich

is t

he

fee

to h

oo

k th

e ca

r.

y

= 80

63 -

60

x;

abo

ut

67.4

ho

urs

, o

r 67

ho

urs

an

d 2

1.5

min

ute

s

ye

s; t

he

equ

atio

n c

an b

e w

ritt

en

in s

tan

dar

d f

or

wh

ere

A =

1,

B

=

-

3.34

, an

d C

=

-

81.2

.

y-i

nte

rcep

t = 81

.2;

x-i

nte

rcep

t ≈

-24

.3;

no

, w

e w

ou

ld e

xpec

t a

wo

man

81.

2 cm

tal

l to

hav

e ar

ms,

an

d a

neg

ativ

e ra

diu

s le

ng

th h

as

no

rea

l mea

nin

g.

165

cm

001_

014_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

912

/15/

10

11:3

2 P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

10

Gle

ncoe

Alg

ebra

1

Tran

sla

tin

g L

inear

Gra

ph

s

Lin

ear

grap

hs

can

be

tran

slat

ed o

n t

he

coor

din

ate

plan

e. T

his

mea

ns

that

th

e gr

aph

mov

es

up,

dow

n, r

igh

t, o

r le

ft w

ith

out

chan

gin

g it

s di

rect

ion

.T

ran

slat

ing

the

grap

hs

up

or d

own

aff

ects

th

e y-

coor

din

ate

for

a gi

ven

x v

alu

e. T

ran

slat

ing

the

grap

h r

igh

t or

lef

t af

fect

s th

e x-

coor

din

ate

for

a gi

ven

y-v

alu

e.

T

ran

slat

e th

e gr

aph

of

y =

2x

+

2,

3 u

nit

s u

p.

Exer

cise

sG

rap

h t

he

fun

ctio

n a

nd

th

e tr

ansl

atio

n o

n t

he

sam

e co

ord

inat

e p

lan

e.

1. y

=

x

+ 4,

3 u

nit

s do

wn

2.

y =

2x

– 2

, 2 u

nit

s le

ft

3. y

=

-

2x +

1,

1

un

it r

igh

t 4.

y =

-

x -

3,

2

un

its

up

Ad

d 3

to

ea

ch

y-va

lue.

y

xO

y =

2x

+ 2

y

xO

y =

x +

4

y

xO

y =

2x

- 2

y

xO

y =

-2x

+ 1

y

xO

y =

-x

- 3

y =

2x +

2

xy

-1

0

02

14

26

Tra

nsla

tio

n

xy

-1

3

05

17

29

Exam

ple

3-1

Enri

chm

ent

001_

014_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

1013

/12/

10

3:53

PM

Lesson 3-1


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

11

Gle

ncoe

Alg

ebra

1

In a

ddit

ion

to

orga

niz

ing

data

, a s

prea

dsh

eet

can

be

use

d to

rep

rese

nt

data

gra

phic

ally

.

Exer

cise

s 1

. A p

hot

o pr

inte

r of

fers

a s

ubs

crip

tion

for

dig

ital

ph

oto

fin

ish

ing.

Th

e su

bscr

ipti

on c

osts

$4

.99

per

mon

th. E

ach

sta

nda

rd s

ize

phot

o a

subs

crib

er p

rin

ts c

osts

$0.

19. U

se a

sp

read

shee

t to

gra

ph t

he

equ

atio

n y

=

4.

99 +

0.

19x,

wh

ere

x is

th

e n

um

ber

of p

hot

os

prin

ted

and

y is

th

e to

tal

mon

thly

cos

t.

2. A

lon

g di

stan

ce s

ervi

ce p

lan

in

clu

des

a $8

.95

per

mon

th f

ee p

lus

$0.0

5 pe

r m

inu

te o

f ca

lls.

Use

a s

prea

dsh

eet

to g

raph

th

e eq

uat

ion

y =

8.

95 +

0.

05x,

wh

ere

x is

th

e n

um

ber

of m

inu

tes

of c

alls

an

d y

is t

he

tota

l m

onth

ly c

ost.

A

n i

nte

rnet

ret

aile

r ch

arge

s $1

.99

per

ord

er p

lus

$0.9

9 p

er i

tem

to

ship

boo

ks

and

CD

s. G

rap

h t

he

equ

atio

n y

=

1.

99 +

0.

99x,

wh

ere

x is

th

e n

um

ber

of

item

s or

der

ed a

nd

y i

s th

e sh

ipp

ing

cost

.

Ste

p 1

U

se c

olu

mn

A f

or t

he

nu

mbe

rs o

f it

ems

and

colu

mn

B f

or t

he

ship

pin

g co

sts.

Ste

p 2

C

reat

e a

grap

h f

rom

th

e da

ta. S

elec

t th

e da

ta

in c

olu

mn

s A

an

d B

an

d se

lect

Ch

art

from

th

e In

sert

men

u. S

elec

t an

XY

(S

catt

er)

char

t to

sh

ow t

he

data

poi

nts

con

nec

ted

wit

h l

ine

segm

ents

.

A1 4 5 6 7 8 9 10 11 122 3

B

Sh

ipp

ing

.xls

Ite

ms

Sh

ipp

ing

Co

st

1 2 3 4 5 6 7 8 9

10

$2

.98

$3

.97

$4

.96

$5

.95

$6

.94

$7

.93

$8

.92

$9

.91

$1

0.9

0

$1

1.8

9

Sh

eet

1S

hee

t 2

Sh

eet

Exam

ple

3-1

Spre

adsh

eet

Act

ivit

yLin

ear

Eq

uati

on

s

See

stu

den

ts’ w

ork

.

See

stu

den

ts’ w

ork

.

001_

014_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

1113

/12/

10

3:53

PM



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

12

Gle

ncoe

Alg

ebra

1

Solv

e b

y G

rap

hin

g Y

ou c

an s

olve

an

equ

atio

n b

y gr

aph

ing

the

rela

ted

fun

ctio

n. T

he

solu

tion

of

the

equ

atio

n i

s th

e x-

inte

rcep

t of

th

e fu

nct

ion

.

S

olve

th

e eq

uat

ion

2x

- 2

= -

4 b

y gr

aph

ing.

Fir

st s

et t

he

equ

atio

n e

qual

to

0. T

hen

rep

lace

0 w

ith

f(x

). M

ake

a ta

ble

of o

rder

ed p

air

solu

tion

s. G

raph

th

e fu

nct

ion

an

d lo

cate

th

e x-

inte

rcep

t.

2x

- 2

= -

4

2x -

2 +

4 =

-4

+ 4

2x

+ 2

= 0

f(x

) =

2x

+ 2

To

grap

h t

he

fun

ctio

n, m

ake

a ta

ble.

Gra

ph t

he

orde

red

pair

s.

xf(

x)

= 2

x +

2f(

x)

[x, f

(x)]

1f(

1)

= 2

(1)

+ 2

4(1

, 4

)

-1

f(-

1)

= 2

(-1

) +

20

(-1

, 0

)

-2

f(-

2)

= 2

(-2

) +

2 -

2(-

2,

-2

)

y

x

Th

e gr

aph

in

ters

ects

th

e x-

axis

at

(-1,

0).

Th

e so

luti

on t

o th

e eq

uat

ion

is

x =

-1.

Exer

cise

sS

olve

eac

h e

qu

atio

n.

1. 3

x -

3 =

0 1

2.

-2x

+ 1

= 5

- 2

x �

3.

-x

+ 4

= 0

4

y

xO

y

xO

y

xO

4. 0

= 4

x -

1 1

−

4 5.

5x

- 1

= 5x

�

6. -

3x +

1 =

0 1

−

3

y

x

y

xO

y

xO

Ori

gin

al equation

Add 4

to e

ach s

ide.

Sim

plif

y.

Repla

ce 0

with f

(x).

Exam

ple

Stud

y G

uide

and

Inte

rven

tion

So

lvin

g L

inear

Eq

uati

on

s b

y G

rap

hin

g

3-2

001_

014_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

1213

/12/

10

3:53

PM

Lesson 3-2


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

13

Gle

ncoe

Alg

ebra

1

Esti

mat

e So

luti

on

s b

y G

rap

hin

g S

omet

imes

gra

phin

g do

es n

ot p

rovi

de a

n e

xact

so

luti

on, b

ut

only

an

est

imat

e. I

n t

hes

e ca

ses,

sol

ve a

lgeb

raic

ally

to

fin

d th

e ex

act

solu

tion

.

WA

LKIN

G Y

ou a

nd

you

r co

usi

n d

ecid

e to

wal

k t

he

7-m

ile

trai

l at

th

e st

ate

par

k t

o th

e ra

nge

r st

atio

n. T

he

fun

ctio

n d

= 7

– 3

.2t

rep

rese

nts

you

r d

ista

nce

d f

rom

th

e ra

nge

r st

atio

n a

fter

t h

ours

. Fin

d t

he

zero

of

this

fu

nct

ion

. D

escr

ibe

wh

at t

his

val

ue

mea

ns

in t

his

con

text

.

Mak

e a

tabl

e of

val

ues

to

grap

h t

he

fun

ctio

n.

t

d =

7 -

3.2

td

(t, d

)

0d

= 7

- 3

.2(0

) 7

(0

, 7

)

1d

= 7

- 3

.2(1

)3

.8(1

, 3

.8)

2d

= 7

- 3

.2(2

)0

.6(2

, 0

.6)

Miles from Ranger Station 23 1

0

45678y

Tim

e (h

ours

)2

1x

3T

he

grap

h i

nte

rsec

ts t

he

t–ax

is b

etw

een

t =

2 a

nd

t =

3,

but

clos

er t

o t

= 2

. It

wil

l ta

ke y

ou a

nd

you

r co

usi

n ju

st

over

tw

o h

ours

to

reac

h t

he

ran

ger

stat

ion

.

You

can

ch

eck

you

r es

tim

ate

by s

olvi

ng

the

equ

atio

n a

lgeb

raic

ally

.

Exer

cise

s 1

. MU

SIC

Jes

sica

wan

ts t

o re

cord

her

fav

orit

e so

ngs

to

one

CD

. Th

e fu

nct

ion

C =

80

- 3

.22n

rep

rese

nts

th

e re

cord

ing

tim

e C

ava

ilab

le a

fter

n s

ongs

are

rec

orde

d.

Fin

d th

e ze

ro o

f th

is f

un

ctio

n. D

escr

ibe

wh

at t

his

val

ue

mea

ns

in t

his

con

text

.

ju

st u

nd

er 2

5; o

nly

24

son

gs

can

be

reco

rded

on

o

ne

CD

2. G

IFT

CA

RD

S E

nri

que

use

s a

gift

car

d to

bu

y co

ffee

at

a co

ffee

sh

op. T

he

init

ial

valu

e of

th

e gi

ft c

ard

is $

20. T

he

fun

ctio

n n

= 2

0 – 2

.75c

rep

rese

nts

th

e am

oun

t of

mon

ey s

till

le

ft o

n t

he

gift

car

d n

aft

er p

urc

has

ing

c cu

ps o

f co

ffee

. Fin

d th

e ze

ro o

f th

is f

un

ctio

n. D

escr

ibe

wh

at t

his

val

ue

mea

ns

in

this

con

text

.

j

ust

ove

r 7;

En

riq

ue

can

buy

7 c

up

s o

f co

ffee

w

ith

th

e g

ift c

ard

Time Available (min)

10 02030405060708090

Num

ber o

f Son

gs5

1015

2025

30

Exam

ple

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

So

lvin

g L

inear

Eq

uati

on

s b

y G

rap

hin

g

3-2

Value Left on Card ($)

4 0812162024

Coffe

es B

ough

t2

46

810

12

001_

014_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

1313

/12/

10

3:53

PM



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

14

Gle

ncoe

Alg

ebra

1

Sol

ve e

ach

eq

uat

ion

.

1. 2

x -

5 =

-3

+ 2

x

2. -

3x +

2

= 0

3. 3

x +

2 =

3x

- 1

y

x

y

x

y

x

4. 4

x -

1 =

4x

+ 2

5.

4x

- 1

= 0

6. 0

= 5

x +

3

y

x

y

x

y

x

7. 0

= -

2x +

4

8. -

3x +

8

= 5

- 3

x 9.

-x

+ 1

= 0

y

x

y

x

y

x

10. G

IFT

CA

RD

S Yo

u r

ecei

ve a

gif

t ca

rd f

or

trad

ing

card

s fr

om a

loc

al s

tore

. Th

e fu

nct

ion

d

= 2

0 -

1.9

5c r

epre

sen

ts t

he

rem

ain

ing

doll

ars

d o

n t

he

gift

car

d af

ter

obta

inin

g c

pack

ages

of

car

ds. F

ind

the

zero

of

this

fu

nct

ion

. Des

crib

e w

hat

th

is v

alu

e m

ean

s in

th

is c

onte

xt.

≈

10.2

6; y

ou

can

pu

rch

ase

10 p

acka

ges

o

f tr

adin

g c

ard

s w

ith

th

e g

ift c

ard

.

Amount Remaining on Gift Card ($)

2 0468101214161820d

Pack

ages

of C

ards

Bou

ght

12

34

56

78

910

c

d =

20

- 1

.95c

no

so

luti

on

2 −

3 n

o s

olu

tio

n

no

so

luti

on

1 −

4 -

3 −

5

no

so

luti

on

21

Skill

s Pr

acti

ceS

olv

ing

Lin

ear

Eq

uati

on

s b

y G

rap

hin

g

3-2

001_

014_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

1413

/12/

10

3:53

PM

Lesson 3-2


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

15

Gle

ncoe

Alg

ebra

1

Prac

tice

So

lvin

g L

inear

Eq

uati

on

s b

y G

rap

hin

g

Sol

ve e

ach

eq

uat

ion

.

1.

1 −

2 x -

2 =

0

4 2

. -3x

+ 2

= -

1 1

3. 4

x -

2 =

-2

0

y

xO

O

y

x

y

x

4.

1 −

3 x +

2 =

1 −

3 x -

1

5.

2 −

3 x +

4

= 3

6.

3 −

4 x +

1

= 3 −

4 x -

7

n

o s

olu

tio

n

- 3 −

2 n

o s

olu

tio

n

y

x

y

x

y

x

Sol

ve e

ach

eq

uat

ion

by

grap

hin

g. V

erif

y yo

ur

answ

er a

lgeb

raic

ally

7. 1

3x +

2

= 11

x -

1

8. -

9x -

3

= -

4x -

3

9. -

1 −

3 x +

2 =

2 −

3 x -

1

-

3 −

2 0

3

y

x

y

x

y

x

10. D

ISTA

NC

E A

bu

s is

dri

vin

g at

60

mil

es p

er h

our

tow

ard

a bu

s st

atio

n t

hat

is

250

mil

es a

way

. Th

e fu

nct

ion

d =

250

– 6

0t r

epre

sen

ts t

he

dist

ance

d f

rom

th

e bu

s st

atio

n t

he

bus

is t

hou

rs a

fter

it

has

sta

rted

dr

ivin

g. F

ind

the

zero

of

this

fu

nct

ion

. Des

crib

e w

hat

th

is v

alu

e m

ean

s in

th

is c

onte

xt.

Distance from BusStation (miles)

50 0

100

150

200

250

300

Tim

e (h

ours

)1

23

45

6

3-2 ≈ 4

.17

hr;

th

e bu

s w

ill a

rive

at

the

stat

ion

in

ap

pro

xim

atel

y 4.

17 h

ou

rs.

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

1513

/12/

10

5:26

PM



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

16

Gle

ncoe

Alg

ebra

1

Wor

d Pr

oble

m P

ract

ice

So

lvin

g L

inear

Eq

uati

on

s b

y G

rap

hin

g

1. P

ET C

AR

E Yo

u b

uy

a 6.

3-po

un

d ba

g of

dry

cat

foo

d fo

r yo

ur

cat.

Th

e fu

nct

ion

c

= 6

.3 –

0.2

5p r

epre

sen

ts t

he

amou

nt

of

cat

food

c r

emai

nin

g in

th

e ba

g w

hen

th

e ca

t is

fed

th

e sa

me

amou

nt

each

day

for

p

days

. Fin

d th

e ze

ro o

f th

is f

un

ctio

n.

Des

crib

e w

hat

th

is v

alu

e m

ean

s in

th

is c

onte

xt.

2. S

AV

ING

S Je

ssic

a is

sav

ing

for

coll

ege

usi

ng

a di

rect

dep

osit

fro

m h

er p

aych

eck

into

a s

avin

gs a

ccou

nt.

Th

e fu

nct

ion

m

= 3

045

- 5

2.50

t re

pres

ents

th

e am

oun

t of

mon

ey m

sti

ll n

eede

d af

ter

t w

eeks

. Fin

d th

e ze

ro o

f th

is f

un

ctio

n.

Wh

at d

oes

this

val

ue

mea

n i

n t

his

co

nte

xt?

3. F

INA

NC

E M

ich

ael

borr

ows

$100

fro

m

his

dad

. Th

e fu

nct

ion

v =

100

- 4

.75p

re

pres

ents

th

e ou

tsta

ndi

ng

bala

nce

v

afte

r p

wee

kly

paym

ents

. Fin

d th

e ze

ro o

f th

is f

un

ctio

n. D

escr

ibe

wh

at

this

val

ue

mea

ns

in t

his

con

text

.

4. B

AK

E SA

LE A

shle

y h

as $

15 i

n t

he

Pep

C

lub

trea

sury

to

pay

for

supp

lies

for

a

choc

olat

e ch

ip c

ooki

e ba

ke s

ale.

Th

e fu

nct

ion

d =

15

– 0

.08c

rep

rese

nts

th

e do

llar

s d

lef

t in

th

e cl

ub

trea

sury

aft

er

mak

ing

c co

okie

s. F

ind

the

zero

of

this

fu

nct

ion

. Wh

at d

oes

this

val

ue

repr

esen

t in

th

is c

onte

xt?

5.

DEN

TAL

HY

GIE

NE

You

are

pac

kin

g yo

ur

suit

case

to

go a

way

to

a 14

-day

su

mm

er c

amp.

Th

e st

ore

carr

ies

thre

e si

zes

of t

ube

s of

too

thpa

ste.

Tub

eS

ize

(ou

nce

s)S

ize

(gra

ms)

A0

.75

21.

26

B0

.92

5.5

2

C3

.08

5.0

4

So

urc

e: N

ational A

cadem

y o

f S

cie

nces

a. T

he

fun

ctio

n n

= 2

1.26

- 0

.8b

repr

esen

ts t

he

nu

mbe

r of

rem

ain

ing

bru

shin

gs n

usi

ng

b gr

ams

per

bru

shin

g u

sin

g T

ube

A. F

ind

the

zero

of

th

is f

un

ctio

n. D

escr

ibe

wh

at t

his

va

lue

mea

ns

in t

his

con

text

.

b.

Th

e fu

nct

ion

n=

25.

52 –

0.8

bre

pres

ents

th

e n

um

ber

of r

emai

nin

g br

ush

ings

n u

sin

g b

gram

s pe

r br

ush

ing

usi

ng

Tu

be B

. Fin

d th

e ze

ro

of t

his

fu

nct

ion

. Des

crib

e w

hat

th

is

valu

e m

ean

s in

th

is c

onte

xt.

c. W

rite

a f

un

ctio

n t

o re

pres

ent

the

nu

mbe

r of

rem

ain

ing

bru

shin

gs

n u

sin

g b

gram

s pe

r br

ush

ing

usi

ng

Tu

be C

. Fin

d th

e ze

ro o

f th

is f

un

ctio

n.

Des

crib

e w

hat

th

is v

alu

e m

ean

s in

th

is c

onte

xt.

d.

If y

ou w

ill

bru

sh y

our

teet

h t

wic

e ea

ch

day

wh

ile

at c

amp,

wh

ich

is

the

smal

lest

tu

be o

f to

oth

past

e yo

u c

an

choo

se?

Exp

lain

you

r re

ason

ing.

3-2 25.2

; Th

ere

are

25 f

ull

serv

ing

s o

f ca

t fo

od

in t

heb

ag.

58;

It w

ill t

ake

58 w

eeks

fo

r Je

ssic

a to

sav

e th

e m

on

ey s

he

nee

ds.

$21.

05; A

fter

21

wee

ks, h

e w

ill

hav

e p

aid

bac

k 21

× 4.

75, o

r $9

9.75

. He

pay

s $0

.25

on

wee

k 22

.

187.

5, S

he

bre

aks

even

at

188

coo

kies

.

n =

85.

04 -

0.8

b;

106;

Tu

be

C

will

pro

vid

e 10

6 b

rush

ing

s.

31.9

; Tu

be

B w

ill p

rovi

de

31 b

rush

ing

s.

26.5

75; T

ub

e A

will

pro

vid

e 26

bru

shin

gs.

Tub

e B

; Yo

u n

eed

28

bru

shin

gs.

Tu

be

A is

no

t en

ou

gh

an

d

Tub

e C

is t

oo

mu

ch.

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

1613

/12/

10

3:53

PM

Lesson 3-2


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

17

Gle

ncoe

Alg

ebra

1

3-2

Enri

chm

ent

Co

mp

osit

e F

un

cti

on

s

Th

ree

thin

gs a

re n

eede

d to

hav

e a

fun

ctio

n—

a se

t ca

lled

th

e do

mai

n,

a se

t ca

lled

th

e ra

nge

, an

d a

rule

th

at m

atch

es e

ach

ele

men

t in

th

e do

mai

n w

ith

on

ly o

ne

elem

ent

in t

he

ran

ge. H

ere

is a

n e

xam

ple.

Ru

le: f

(x)

= 2

x +

1

3 5-

5

f(x)

x 1 2-

3

f(x)

= 2

x +

1

f(1)

= 2

(1)

+ 1

= 2

+ 1

= 3

f(2)

= 2

(2)

+ 1

= 4

+ 1

= 5

f(

-3)

= 2

(-3)

+ 1

= 2

6 +

1 =

-5

Su

ppos

e w

e h

ave

thre

e se

ts A

, B, a

nd

C a

nd

two

fun

ctio

ns

desc

ribe

d as

sh

own

bel

ow.

Ru

le: f

(x)

= 2

x +

1

Ru

le: g

(y)

= 3

y -

4A

BC

f(x) 3

5

x 1

g[f(x

)]

g(

y) =

3y

- 4

g(3)

= 3

(3)

- 4

= 5

Let

’s f

ind

a ru

le t

hat

wil

l m

atch

ele

men

ts o

f se

t A

wit

h e

lem

ents

of

set

C w

ith

out

fin

din

g an

y el

emen

ts i

n s

et B

. In

oth

er w

ords

, let

’s f

ind

a ru

le f

or t

he

com

pos

ite

fun

ctio

n g

[f(x

)].

Sin

ce f

(x)

= 2

x +

1, g

[f(x

)] =

g(2

x +

1).

Sin

ce g

(y)

= 3

y -

4, g

(2x

+ 1

) =

3(2

x +

1)

- 4

, or

6x -

1.

Th

eref

ore,

g[f

(x)]

= 6

x -

1.

Fin

d a

ru

le f

or t

he

com

pos

ite

fun

ctio

n g

[f(x

)].

1. f

(x)

= 3

x an

d g(

y) =

2y

+ 1

2.

f(x

) =

x2

+ 1

an

d g(

y) =

4y

3. f

(x)

= -

2x a

nd

g(y)

= y

2 -

3y

4. f

(x)

=

1 −

x -

3 a

nd

g(y)

= y

-1

5. I

s it

alw

ays

the

case

th

at g

[f(x

)] =

f[g

(x)]

? Ju

stif

y yo

ur

answ

er.

g[f

(x)]

= 6

x +

1

g[f

(x)]

= 4

x2

+ 4

g[f

(x)]

= 4

x2

+ 6

x

g[f

(x)]

= x

- 3

N

o. F

or

exam

ple

, in

Exe

rcis

e 1,

f[g

(x)]

= f

(2x +

1)

= 3

(2x +

1)

= 6

x +

3, n

ot

6x +

1.

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

1713

/12/

10

3:53

PM



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

18

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

Rate

of

Ch

an

ge a

nd

Slo

pe

3-3

Rat

e o

f C

han

ge

Th

e ra

te o

f ch

ange

tel

ls, o

n a

vera

ge, h

ow a

qu

anti

ty i

s ch

angi

ng

over

tim

e.

PO

PULA

TIO

N T

he

grap

h s

how

s th

e p

opu

lati

on g

row

th i

n C

hin

a.

a.

Fin

d t

he

rate

s of

ch

ange

for

195

0–19

75 a

nd

for

20

00–2

025.

1950

–197

5: ch

ange

in

pop

ula

tion

−

−

chan

ge i

n t

ime

=

0.93

- 0

.55

−

1975

- 1

950

=

0.38

−

25

or 0

.015

2

2000

–202

5: ch

ange

in

pop

ula

tion

−

−

chan

ge i

n t

ime

= 1.

45 -

1.

27

−

2025

-

20

00

=

0.18

−

25

or 0

.007

2

b

. Exp

lain

th

e m

ean

ing

of t

he

rate

of

chan

ge i

n e

ach

cas

e.F

rom

195

0–19

75, t

he

grow

th w

as 0

.015

2 bi

llio

n p

er y

ear,

or 1

5.2

mil

lion

per

yea

r.F

rom

200

0–20

25, t

he

grow

th i

s ex

pect

ed t

o be

0.0

072

bill

ion

per

yea

r, or

7.2

mil

lion

pe

r ye

ar.

c.

How

are

th

e d

iffe

ren

t ra

tes

of c

han

ge s

how

n o

n t

he

grap

h?

Th

ere

is a

gre

ater

ver

tica

l ch

ange

for

195

0–19

75 t

han

for

200

0–20

25. T

her

efor

e, t

he

sect

ion

of

the

grap

h f

or 1

950–

1975

has

a s

teep

er s

lope

.

Exer

cise

s1.

LO

NG

EVIT

Y T

he g

raph

sho

ws

the

pred

icte

d li

feex

pect

ancy

for

men

and

wom

en b

orn

in a

giv

en y

ear.

a.

Fin

d th

e ra

tes

of c

han

ge f

or w

omen

fro

m 2

000–

2025

an

d 20

25–2

050.

0.1

6/yr

, 0.1

2/yr

b

. Fin

d th

e ra

tes

of c

han

ge f

or m

en f

rom

200

0–20

25 a

nd

2025

–205

0. 0

.16/

yr, 0

.12/

yr

c. E

xpla

in t

he

mea

nin

g of

you

r re

sult

s in

Exe

rcis

es 1

an

d 2.

Bo

th m

en a

nd

wo

men

incr

ease

d t

hei

r lif

e ex

pec

tan

cy a

t th

e sa

me

rate

s.d

. Wh

at p

atte

rn d

o yo

u s

ee i

n t

he

incr

ease

wit

h e

ach

25

-yea

r pe

riod

? W

hile

life

exp

ecta

ncy

in

crea

ses,

it d

oes

no

t in

crea

se a

t a

con

stan

t ra

te.

e.

Mak

e a

pred

icti

on f

or t

he

life

exp

ecta

ncy

for

205

0–20

75. E

xpla

in h

ow y

ou a

rriv

ed a

t yo

ur

pred

icti

on.

Sam

ple

an

swer

: 89

for

wo

men

an

d 8

3 fo

r m

en; t

he

dec

reas

e in

rat

e fr

om

200

0–20

25 t

o 2

025–

2050

is 0

.04/

yr. I

f th

e d

ecre

ase

in t

he

rate

rem

ain

s th

e sa

me,

th

e ch

ang

e o

f ra

te f

or

2050

–207

5 m

igh

t b

e 0.

08/y

r an

d 2

5(0.

08)

= 2

yea

rs o

f in

crea

se o

ver

the

25-y

ear

span

.

Pre

dic

tin

g L

ife E

xp

ect

an

cy

Year

Bo

rn

Age

2000

Wo

men

Men

100 95 90 85 80 75 70 65

2050

*20

25*

*Est

imat

ed

Sour

ce: U

SA T

ODA

Y8084

8178

74

87

Po

pu

lati

on

Gro

wth

in

Ch

ina

Year

People (billions)

1950

1975

2000

2.0

1.5

1.0

0.5 0

2025

*

*Est

imat

ed

Sour

ce: U

nite

d Na

tions

Pop

ulat

ion

Divis

ion

0.55

0.93

1.27

1.45

Exam

ple

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

1812

/15/

10

11:3

6 P

M

Lesson 3-3


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

19

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Rate

of

Ch

an

ge a

nd

Slo

pe

3-3

Fin

d S

lop

e T

he

slop

e of

a l

ine

is t

he

rati

o of

ch

ange

in

th

e y-

coor

din

ates

(ri

se)

to t

he

chan

ge i

n t

he

x- co

ordi

nat

es (

run

) as

you

mov

e in

th

e po

siti

ve d

irec

tion

.

Slo

pe

of

a L

ine

m =

rise

−

run o

r m

= y 2

- y

1

−

x 2 -

x 1 , w

he

re (

x 1,

y 1)

an

d (

x 2,

y 2)

are

th

e c

oo

rdin

ate

s

of

any t

wo

po

ints

on

a n

onve

rtic

al lin

e

F

ind

th

e sl

ope

of t

he

lin

e th

at p

asse

s th

rou

gh (

-3,

5)

and

(4

, -2)

.

Let

(-

3, 5

) =

(x 1,

y 1) a

nd

(4, -

2) =

(x 2,

y 2).

m =

y 2 -

y1

−

x 2 -

x1

Slo

pe form

ula

=

-

2 -

5

−

4 -

(-

3)

y 2 =

-2,

y 1 =

5,

x 2 =

4,

x 1 =

-3

=

-7

−

7 S

implif

y.

=

-1

Exer

cise

s F

ind

th

e sl

ope

of t

he

lin

e th

at p

asse

s th

rou

gh e

ach

pai

r of

poi

nts

.

1. (

4, 9

), (1

, 6)

1 2.

(-

4, -

1), (

-2,

-5)

- 2

3.

(-

4, -

1), (

-4,

-5)

u

nd

efi n

ed 4

. (2,

1),

(8, 9

) 4 −

3 5.

(14

, -8)

, (7,

-6)

-

2 −

7 6.

(4,

-3)

, (8,

-3)

0

7. (

1, -

2), (

6, 2

) 4 −

5 8.

(2,

5),

(6, 2

) -

3 −

4 9.

(4,

3.5

), (-

4, 3

.5)

0

Fin

d t

he

valu

e of

r s

o th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts h

as t

he

give

n s

lop

e.

10. (

6, 8

), (r

, -2)

, m =

1 -

4 11

. (-

1, -

3), (

7, r

), m

= 3 −

4 3 12

. (2,

8),

(r, -

4) m

= -

3 6

13. (

7, -

5), (

6, r

), m

= 0

-5

14. (

r, 4

), (7

, 1),

m =

3 −

4 11

15. (

7, 5

), (r

, 9),

m =

6 23

−

3

F

ind

th

e va

lue

of r

so

that

th

e li

ne

thro

ugh

(10

, r)

and

(3,

4)

has

a

slop

e of

- 2 −

7 .

m

= y

2 -

y1

−

x 2 -

x1

S

lope form

ula

-

2 −

7 =

4 -

r

−

3 -

10

m

= -

2

−

7 ,

y 2 =

4,

y 1 =

r,

x 2 =

3,

x 1 =

10

-

2 −

7 = 4

- r

−

-7

Sim

plif

y.

-2(

-7)

= 7

(4 -

r)

Cro

ss m

ultip

ly.

14

= 2

8 -

7r

Dis

trib

utive

Pro

pert

y

-

14 =

-7r

S

ubtr

act

28 f

rom

each s

ide.

2

= r

D

ivid

e e

ach s

ide b

y -

7.

Exam

ple

1Ex

amp

le 2

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

1913

/12/

10

3:53

PM



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

20

Gle

ncoe

Alg

ebra

1

Skill

s Pr

acti

ceR

ate

of

Ch

an

ge a

nd

Slo

pe

3-3

Fin

d t

he

slop

e of

th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts.

1.

( 0, 1

)

( 2, 5

)

x

y O

2.

( 0, 0

)( 3, 1

)

x

y

O

3.

( 0, 1

) ( 1, -

2)x

y

O

2

1 −

3

-3

4. (

2, 5

), (3

, 6)

1 5.

(6,

1),

(-6,

1)

0

6. (

4, 6

), (4

, 8)

un

defi

ned

7.

(5,

2),

(5, -

2) u

nd

efi n

ed

8. (

2, 5

), (-

3, -

5) 2

9.

(9,

8),

(7, -

8) 8

10. (

-5,

-8)

, (-

8, 1

) -

3 11

. (-

3, 1

0), (

-3,

7)

un

defi

ned

12. (

17, 1

8), (

18, 1

7) -

1 13

. (-

6, -

4), (

4, 1

) 1 −

2

14. (

10, 0

), (-

2, 4

) -

1 −

3 15

. (2,

-1)

, (-

8, -

2)

1 −

10

16. (

5, -

9), (

3, -

2) -

7 −

2 17

. (12

, 6),

(3, -

5) 11

−

9

18. (

-4,

5),

(-8,

-5)

5 −

2 19

. (-

5, 6

), (7

, -8)

- 7 −

6

Fin

d t

he

valu

e of

r s

o th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts

has

th

e gi

ven

slo

pe.

20. (

r, 3

), (5

, 9),

m =

2 2

21

. (5,

9),

(r, -

3), m

= -

4 8

22. (

r, 2

), (6

, 3),

m =

1 −

2 4

23. (

r, 4

), (7

, 1),

m =

3 −

4 11

24. (

5, 3

), (r

, -5)

, m =

4 3

25

. (7,

r),

(4, 6

), m

= 0

6

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

2013

/12/

10

3:53

PM

Lesson 3-3


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

21

Gle

ncoe

Alg

ebra

1

Prac

tice

Rate

of

Ch

an

ge a

nd

Slo

pe

3-3

Fin

d t

he

slop

e of

th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts.

1.

( –1,

0)

( –2,

3)

x

y

O

2.

( 3, 1

)

( –2,

–3)

x

y

O

3.

( –

2, 3

)( 3

, 3)

x

y O

-

3 4

−

5 0

4. (

6, 3

), (7

, -4)

-7

5. (

-9,

-3)

, (-

7, -

5) -

1

6. (

6, -

2), (

5, -

4) 2

7.

(7,

-4)

, (4,

8)

-4

8. (

-7,

8),

(-7,

5)

un

defi

ned

9.

(5,

9),

(3, 9

) 0

10. (

15, 2

), (-

6, 5

) -

1 −

7 11

. (3,

9),

(-2,

8)

1 −

5

12. (

-2,

-5)

, (7,

8)

13

−

9 13

. (12

, 10)

, (12

, 5)

un

defi

ned

14. (

0.2,

-0.

9), (

0.5,

-0.

9) 0

15

. ( 7 −

3 , 4 −

3 ) , (

- 1 −

3 , 2 −

3 ) 1 −

4

Fin

d t

he

valu

e of

r s

o th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts h

as t

he

give

n s

lop

e.

16. (

-2,

r),

(6, 7

), m

= 1 −

2 3

17. (

-4,

3),

(r, 5

), m

= 1 −

4 4

18. (

-3,

-4)

, (-

5, r

), m

= -

9 −

2 5

19. (

-5,

r),

(1, 3

), m

= 7 −

6 -4

20. (

1, 4

), (r

, 5),

m u

nde

fin

ed 1

21

. (-

7, 2

), (-

8, r

), m

= -

5 7

22. (

r, 7

), (1

1, 8

), m

= -

1 −

5 16

23. (

r, 2

), (5

, r),

m =

0 2

24. R

OO

FIN

G T

he

pitc

h o

f a

roof

is

the

nu

mbe

r of

fee

t th

e ro

of r

ises

for

eac

h 1

2 fe

et

hor

izon

tall

y. I

f a

roof

has

a p

itch

of

8, w

hat

is

its

slop

e ex

pres

sed

as a

pos

itiv

e n

um

ber?

2 −

3

25. S

ALE

S A

dai

ly n

ewsp

aper

had

12,

125

subs

crib

ers

wh

en i

t be

gan

pu

blic

atio

n. F

ive

year

s la

ter

it h

ad 1

0,10

0 su

bscr

iber

s. W

hat

is

the

aver

age

year

ly r

ate

of c

han

ge i

n t

he

nu

mbe

r of

su

bscr

iber

s fo

r th

e fi

ve-y

ear

peri

od?

-40

5 su

bsc

rib

ers

per

yea

r

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

2113

/12/

10

3:53

PM



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

22

Gle

ncoe

Alg

ebra

1

Wor

d Pr

oble

m P

ract

ice

Rate

of

Ch

an

ge a

nd

Slo

pe

3-3

1.H

IGH

WA

YS

Roa

dway

sig

ns

such

as

the

one

belo

w a

re u

sed

to w

arn

dri

vers

of

an

upc

omin

g st

eep

dow

n g

rade

th

at c

ould

le

ad t

o a

dan

gero

us

situ

atio

n. W

hat

is

the

grad

e, o

r sl

ope,

of

the

hil

l de

scri

bed

on t

he

sign

?2

− 25

2. A

MU

SEM

ENT

PAR

KS

Th

e S

hei

Kra

ro

ller

coa

ster

at

Bu

sch

Gar

den

s in

T

ampa

, Flo

rida

, fea

ture

s a

138-

foot

ve

rtic

al d

rop.

Wh

at i

s th

e sl

ope

of t

he

coas

ter

trac

k at

th

is p

art

of t

he

ride

? E

xpla

in.

T

he

slo

pe

is u

nd

efi n

ed b

ecau

se

the

dro

p is

ver

tica

l.

3. C

ENSU

S T

he

tabl

e sh

ows

the

popu

lati

on

den

sity

for

th

e st

ate

of T

exas

in

var

iou

s ye

ars.

Fin

d th

e av

erag

e an

nu

al r

ate

of

chan

ge i

n t

he

popu

lati

on d

ensi

ty f

rom

20

00 t

o 20

09.

in

crea

sed

ab

ou

t 1.

9 p

eop

le p

er

squ

are

mile

4.R

EAL

ESTA

TE A

rea

ltor

est

imat

es t

he

med

ian

pri

ce o

f an

exi

stin

g si

ngl

e-fa

mil

y h

ome

in C

edar

Rid

ge i

s $2

21,9

00. T

wo

year

s ag

o, t

he

med

ian

pri

ce w

as

$195

,200

. Fin

d th

e av

erag

e an

nu

al r

ate

of c

han

ge i

n m

edia

n h

ome

pric

e in

th

ese

year

s. $1

3,35

0

5. C

OA

L EX

POR

TS T

he

grap

h s

how

s th

e an

nu

al c

oal

expo

rts

from

U.S

. min

es i

n

mil

lion

s of

sh

ort

ton

s.

S

ourc

e: E

nerg

y In

form

atio

n As

soci

atio

n

a. W

hat

was

th

e ra

te o

f ch

ange

in

coa

l ex

port

s be

twee

n 2

001

and

2002

?

-9

mill

ion

to

ns

per

yea

r o

r -

9 −

1

b.

How

doe

s th

e ra

te o

f ch

ange

in

coa

l ex

port

s fr

om 2

005

to 2

006

com

pare

to

that

of

2001

to

2002

?

In 2

005

–20

06, t

he

rate

was

0 co

mp

ared

to

-

9 −

1 i

n

2001

–20

02.

c. E

xpla

in t

he

mea

nin

g of

th

e pa

rt o

f th

e gr

aph

wit

h a

slo

pe o

f ze

ro.

Th

e sl

op

e in

dic

ates

th

at t

her

e w

as n

o c

han

ge

in t

he

amo

un

t o

f co

al e

xpo

rted

bet

wee

n 2

005

an

d 2

006

.

Million Short Tons

4050 30 060708090100

2004

2003

2000

2001

2002

2006

2005

Tota

l Exp

orts

Po

pu

lati

on

Den

sity

Year

Peo

ple

Per

Sq

uar

e M

ile

19

30

22

.1

19

60

36

.4

19

80

54

.3

20

00

79

.6

20

09

96

.7

Sour

ce: B

urea

u of

the

Cen

sus,

U.S

. Dep

t. of

Com

mer

ce

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

2213

/12/

10

3:53

PM

Lesson 3-3


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

23

Gle

ncoe

Alg

ebra

1

3-3

Enri

chm

ent

Treasu

re H

un

t w

ith

Slo

pes

Usi

ng

the

def

init

ion

of

slop

e, d

raw

seg

men

ts w

ith

th

e sl

opes

lis

ted

b

elow

in

ord

er. A

cor

rect

sol

uti

on w

ill

trac

e th

e ro

ute

to

the

trea

sure

.

1. 3

2.

1 −

4 3.

- 2 −

5 4.

0

5. 1

6.

-1

7. n

o sl

ope

8. 2

−

7

9. 3

−

2 10

. 1 −

3 11

. - 3 −

4 12

. 3

Star

t Her

e

Trea

sure

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

2313

/12/

10

3:53

PM



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

24

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

Dir

ect

Vari

ati

on

3-4

Dir

ect

Var

iati

on

Eq

uat

ion

s A

dir

ect

vari

atio

n i

s de

scri

bed

by a

n e

quat

ion

of

the

form

y =

kx,

wh

ere

k ≠

0. W

e sa

y th

at y

var

ies

dir

ectl

y as

x. I

n t

he

equ

atio

n y

= k

x, k

is

the

con

stan

t of

var

iati

on.

N

ame

the

con

stan

t of

va

riat

ion

for

th

e eq

uat

ion

. Th

en f

ind

th

e sl

ope

of t

he

lin

e th

at p

asse

s th

rou

gh t

he

pai

r of

poi

nts

.

For

y =

1 −

2 x, t

he

con

stan

t of

var

iati

on i

s 1 −

2 .

m

= y

2 -

y 1

−

x 2 -

x 1

S

lope form

ula

=

1 -

0

−

2 -

0

(x1,

y 1)

= (

0,

0),

(x 2

, y 2

) =

(2,

1)

=

1 −

2 S

implif

y.

Th

e sl

ope

is 1 −

2 .

S

up

pos

e y

vari

es d

irec

tly

as x

, an

d y

= 3

0 w

hen

x =

5.

a. W

rite

a d

irec

t va

riat

ion

eq

uat

ion

th

at r

elat

es x

an

d y

.F

ind

the

valu

e of

k.

y =

kx

Direct

vari

ation e

quation

30 =

k(5

) R

epla

ce y

with 3

0 a

nd x

with 5

.

6 =

k

Div

ide e

ach s

ide b

y 5

.

Th

eref

ore,

th

e eq

uat

ion

is

y =

6x.

b.

Use

th

e d

irec

t va

riat

ion

eq

uat

ion

to

fin

d x

wh

en y

= 1

8.

y =

6x

Direct

vari

ation e

quation

18 =

6x

Repla

ce y

with 1

8.

3 =

x

Div

ide e

ach s

ide b

y 6

.

Th

eref

ore,

x =

3 w

hen

y =

18.

Exer

cise

sN

ame

the

con

stan

t of

var

iati

on f

or e

ach

eq

uat

ion

. Th

en d

eter

min

e th

e sl

ope

of t

he

lin

e th

at p

asse

s th

rou

gh e

ach

pai

r of

poi

nts

.

1.

2.

( 1, 3

)

( 0, 0

)x

y

O

y =

3x

3.

-

2; -

2 3

; 3

3 −

2 ; 3 −

2

Su

pp

ose

y va

ries

dir

ectl

y as

x. W

rite

a d

irec

t va

riat

ion

eq

uat

ion

th

at r

elat

es x

to

y.

Th

en s

olve

.

4. I

f y

= 4

wh

en x

= 2

, fin

d y

wh

en x

= 1

6. y

= 2

x;

325.

If

y =

9 w

hen

x =

-3,

fin

d x

wh

en y

= 6

. y =

-3x

; -

26.

If

y =

-4.

8 w

hen

x =

-1.

6, f

ind

x w

hen

y =

-24

. y =

3x;

-8

7. I

f y

= 1 −

4 wh

en x

= 1 −

8 , fi

nd

x w

hen

y =

3 −

16 .

y =

2x;

3 −

32

( –2,

–3)

( 0, 0

)x

y O

y =

3 2x( –

1, 2

)( 0

, 0)

x

y

O

y =

–2x

( 2, 1

)( 0

, 0)

x

y

Oy

= 1 2x

Exam

ple

1Ex

amp

le 2

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

2413

/12/

10

3:53

PM

Lesson 3-4


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

25

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Dir

ect

Vari

ati

on

3-4

Dir

ect

Var

iati

on

Pro

ble

ms

Th

e d

ista

nce

for

mu

la d

= r

t is

a d

irec

t va

riat

ion

eq

uat

ion

. In

th

e fo

rmu

la, d

ista

nce

d v

arie

s di

rect

ly a

s ti

me

t, an

d th

e ra

te r

is

the

con

stan

t of

var

iati

on.

TR

AV

EL A

fam

ily

dro

ve t

hei

r ca

r 22

5 m

iles

in

5 h

ours

.

a.

Wri

te a

dir

ect

vari

atio

n e

qu

atio

n t

o fi

nd

th

e d

ista

nce

tra

vele

d f

or a

ny

nu

mb

er

of h

ours

. U

se g

iven

val

ues

for

d a

nd

t to

fin

d r.

d

= r

t O

rigin

al equation

22

5 =

r(5

) d

= 2

25 a

nd t

= 5

45

= r

D

ivid

e e

ach s

ide b

y 5

.

Th

eref

ore,

th

e di

rect

var

iati

on e

quat

ion

is

d =

45t

. A

uto

mo

bil

e T

rip

s

Tim

e (h

ou

rs)

Distance (miles)

10

23

45

67

8t

d

360

270

180 90

( 1, 4

5)

( 5, 2

25)

d =

45t

b.

Gra

ph

th

e eq

uat

ion

.

Th

e gr

aph

of

d =

45t

pas

ses

thro

ugh

th

e or

igin

wit

h

slop

e 45

.

m =

45

−

1

ri

se

−

run

�C

HEC

K (

5, 2

25)

lies

on

th

e gr

aph

.

c.

Est

imat

e h

ow m

any

hou

rs i

t w

ould

tak

e th

e fa

mil

y to

dri

ve 3

60 m

iles

.

d

= 4

5t

Ori

gin

al equation

36

0 =

45t

R

epla

ce d

with 3

60.

t

= 8

D

ivid

e e

ach s

ide b

y 4

5.

Th

eref

ore,

it

wil

l ta

ke 8

hou

rs t

o dr

ive

360

mil

es.

Exer

cise

s 1

. RET

AIL

Th

e to

tal

cost

C o

f bu

lk je

lly

bean

s is

$4.4

9 ti

mes

th

e n

um

ber

of p

oun

ds p

.

a. W

rite

a d

irec

t va

riat

ion

equ

atio

n t

hat

rel

ates

th

e va

riab

les.

C =

4.4

9pb

. G

raph

th

e eq

uat

ion

on

th

e gr

id a

t th

e ri

ght.

c. F

ind

the

cost

of

3 −

4 pou

nd

of je

lly

bean

s. $

3.37

2. C

HEM

ISTR

Y C

har

les’

s L

aw s

tate

s th

at, a

t a

con

stan

t

pres

sure

, vol

um

e of

a g

as V

var

ies

dire

ctly

as

its

tem

pera

ture

T. A

vol

um

e of

4 c

ubi

c fe

et o

f a

cert

ain

ga

s h

as a

tem

pera

ture

of

200

degr

ees

Kel

vin

.

a. W

rite

a d

irec

t va

riat

ion

equ

atio

n t

hat

rel

ates

th

e va

riab

les.

V

= 0

.02T

b.

Gra

ph t

he

equ

atio

n o

n t

he

grid

at

the

righ

t.

c. F

ind

the

volu

me

of t

he

sam

e ga

s at

250

deg

rees

Kel

vin

. 5

ft3

Co

st o

f Je

lly B

ean

s

Wei

gh

t (p

ou

nd

s)

Cost (dollars)

20

4p

C

18.0

0

13.5

0

9.00

4.50

Ch

arl

es’

s La

w

Tem

per

atu

re (

K)

Volume (cubic feet)

100

020

0T

V 4 3 2 1

Exam

ple

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

2513

/12/

10

3:53

PM



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

26

Gle

ncoe

Alg

ebra

1

Skill

s Pr

acti

ceD

irect

Vari

ati

on

3-4

Nam

e th

e co

nst

ant

of v

aria

tion

for

eac

h e

qu

atio

n. T

hen

det

erm

ine

the

slop

e of

th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts.

1.

( 3, 1

)( 0

, 0)

x

y O

y =

1 3 x

1 −

3 ; 1 −

3 2.

-2;

-2

3.

- 3 −

2 ;

Gra

ph

eac

h e

qu

atio

n.

4. y

= 3

x

5. y

= -

3 −

4 x

6. y

= 2 −

5 x

Su

pp

ose

y va

ries

dir

ectl

y as

x. W

rite

a d

irec

t va

riat

ion

eq

uat

ion

th

at r

elat

es

x an

d y

. Th

en s

olve

.

7. I

f y

= -

8 w

hen

x =

-2,

fin

d x

8.

If

y =

45

wh

en x

= 1

5, f

ind

x w

hen

y =

32.

y =

4x;

8

wh

en y

= 1

5. y

= 3

x;

59.

If

y =

-4

wh

en x

= 2

, fin

d y

10

. If

y =

-9

wh

en x

= 3

, fin

d y

wh

en x

= -

6. y

= -

2x;

12

w

hen

x =

-5.

y =

-3x

; 15

11. I

f y

= 4

wh

en x

= 1

6, f

ind

y

12. I

f y

= 1

2 w

hen

x =

18,

fin

d x

w

hen

x =

6.

y =

1 −

4 x;

3 −

2

wh

en y

= -

16.

y =

2 −

3 x;

-24

Wri

te a

dir

ect

vari

atio

n e

qu

atio

n t

hat

rel

ates

th

e va

riab

les.

T

hen

gra

ph

th

e eq

uat

ion

.

13. T

RA

VEL

Th

e to

tal

cost

C o

f ga

soli

ne

14

. SH

IPPI

NG

Th

e n

um

ber

of d

eliv

ered

toy

s T

is $

3.00

tim

es t

he

nu

mbe

r of

gal

lon

s g.

is 3

tim

es t

he

tota

l n

um

ber

of c

rate

s c.

Gaso

lin

e C

ost

Gal

lon

s

Cost ($)

20

46

71

35

g

C 28 24 20 16 12 8 4

C =

3.0

0g

Toys

Sh

ipp

ed

Cra

tes

Toys

20

46

71

35

c

T

21 18 15 12 9 6 3

T

= 3

c

x

y

Ox

y

Ox

y

O

( –2,

3)

( 0, 0

)x

y

O

y =

– 3 2

x

( -1,

2)

( 0, 0

)x

y

O

y =

-2x

- 3 −

2

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

2613

/12/

10

3:53

PM

Lesson 3-4


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

27

Gle

ncoe

Alg

ebra

1

3-4

Prac

tice

Dir

ect

Vari

ati

on

Nam

e th

e co

nst

ant

of v

aria

tion

for

eac

h e

qu

atio

n. T

hen

det

erm

ine

the

slop

e of

th

e li

ne

that

pas

ses

thro

ugh

eac

h p

air

of p

oin

ts.

1.

( 4, 3

)

( 0, 0

)x

y O

y =

3 x4

2.

( 3

, 4)

( 0, 0

)x

y O

y =

4 3x

3.

( –2,

5)

( 0, 0

)x

y

O

y =

- 5 2

x

Gra

ph

eac

h e

qu

atio

n.

4. y

= -

2x

5. y

= 6 −

5 x

6. y

= -

5 −

2 x

Su

pp

ose

y va

ries

dir

ectl

y as

x. W

rite

a d

irec

t va

riat

ion

eq

uat

ion

th

at r

elat

es

x an

d y

. Th

en s

olve

.

7. I

f y

= 7

.5 w

hen

x =

0.5

, fin

d y

wh

en x

= -

0.3.

y =

15x

; -

4.5

8. I

f y

= 8

0 w

hen

x =

32,

fin

d x

wh

en y

= 1

00.

y =

2.5

x;

40

9. I

f y

= 3 −

4 wh

en x

= 2

4, f

ind

y w

hen

x =

12.

y =

1

−

32 x

; 3 −

8

Wri

te a

dir

ect

vari

atio

n e

qu

atio

n t

hat

rel

ates

th

e va

riab

les.

Th

en g

rap

h t

he

equ

atio

n.

10. M

EASU

RE

Th

e w

idth

W o

f a

11

. TIC

KET

S T

he

tota

l co

st C

of

tick

ets

isre

ctan

gle

is t

wo

thir

ds o

f th

e le

ngt

h �

. $

4.50

tim

es t

he

nu

mbe

r of

tic

kets

t.

W

= 2 −

3 �

C =

4.5

0t

12. P

RO

DU

CE

Th

e co

st o

f ba

nan

as v

arie

s di

rect

ly w

ith

th

eir

wei

ght.

Mig

uel

bou

ght

3 1 −

2 pou

nds

of

ban

anas

for

$1.

12. W

rite

an

equ

atio

n t

hat

rel

ates

th

e co

st o

f th

e ba

nan

as

to t

hei

r w

eigh

t. T

hen

fin

d th

e co

st o

f 4

1 −

4 pou

nds

of

ban

anas

. C

= 0

.32p

; $1

.36

x

y

O

x

y

Ox

y

O

Co

st o

f Ti

ckets

Tick

ets

Cost ($)

20

46

13

5t

C 25 20 15 10 5

Rect

an

gle

Dim

en

sio

ns

Len

gth

Width

40

812

26

10�

W 10 8 6 4 2

4 −

3 ; 4 −

3 -

5 −

2 ; - 5 −

2 3

−

4 ; 3 −

4

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

2713

/12/

10

3:53

PM



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

28

Gle

ncoe

Alg

ebra

1

Wor

d Pr

oble

m P

ract

ice

Dir

ect

Vari

ati

on

3-4

1. E

NG

INES

Th

e en

gin

e of

a c

hai

nsa

w

requ

ires

a m

ixtu

re o

f en

gin

e oi

l an

d ga

soli

ne.

Acc

ordi

ng

to t

he

dire

ctio

ns,

oil

an

d ga

soli

ne

shou

ld b

e m

ixed

as

show

n

in t

he

grap

h b

elow

. Wh

at i

s th

e co

nst

ant

of v

aria

tion

for

th

e li

ne

grap

hed

? 2.

5

2. R

AC

ING

In

a r

ecen

t ye

ar, E

ngl

ish

dri

ver

Lew

is H

amil

ton

won

th

e U

nit

ed S

tate

s G

ran

d P

rix

at t

he

Indi

anap

olis

Mot

or

Spe

edw

ay. H

is s

peed

du

rin

g th

e ra

ce

aver

aged

125

.145

mil

es p

er h

our.

Wri

te a

di

rect

var

iati

on e

quat

ion

for

th

e di

stan

ce

d t

hat

Ham

ilto

n d

rove

in

h h

ours

at

that

sp

eed.

3. C

UR

REN

CY

Th

e ex

chan

ge r

ate

from

on

e cu

rren

cy t

o an

oth

er v

arie

s ev

ery

day.

R

ecen

tly

the

exch

ange

rat

e fr

om U

.S.

doll

ars

to B

riti

sh p

oun

d st

erli

ng

(£) w

as

$1.5

8 to

£1.

Wri

te a

nd

solv

e a

dire

ct

vari

atio

n e

quat

ion

to

dete

rmin

e h

ow

man

y po

un

ds s

terl

ing

you

wou

ld r

ecei

ve

in e

xch

ange

for

$90

of

U.S

. cu

rren

cy.

4. S

ALA

RY H

enry

sta

rted

a n

ew jo

b in

w

hic

h h

e is

pai

d $9

.50

an h

our.

Wri

te

and

solv

e an

equ

atio

n t

o de

term

ine

Hen

ry’s

gro

ss s

alar

y fo

r a

40-h

our

wor

k w

eek.

5. S

ALE

S TA

X A

mel

ia r

ecei

ved

a gi

ft c

ard

to a

loc

al m

usi

c sh

op f

or h

er b

irth

day.

S

he

plan

s to

use

th

e gi

ft c

ard

to b

uy

som

e n

ew C

Ds.

a. A

mel

ia c

hos

e 3

CD

s th

at e

ach

cos

t $1

6. T

he

sale

s ta

x on

th

e th

ree

CD

s is

$3.

96. W

rite

a d

irec

t va

riat

ion

eq

uat

ion

rel

atin

g sa

les

tax

to t

he

pric

e.

b.

Gra

ph t

he

equ

atio

n y

ou w

rote

in

pa

rt a

.

c. W

hat

is

the

sale

s ta

x ra

te t

hat

Am

elia

is

pay

ing

on t

he

CD

s?

Oil (fl oz)

34 2 15910 8 7 6

Gas

olin

e (g

al)

32

10

54

x

y

6

Pric

e ($

)30

2010

5040

9080

7010

0P

60

Sales Tax ($)

34 2 15T 8 7 6

d =

125

.145

h

u =

1.5

8b;

abo

ut

£56

.96

p =

9.

5h;

$380

T =

0.0

825P

8.25

%

015_

028_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

2813

/12/

10

3:53

PM

Lesson 3-4


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

29

Gle

ncoe

Alg

ebra

1

Enri

chm

ent

3-4

nth

Po

wer

Vari

ati

on

An

equ

atio

n o

f th

e fo

rm y

= k

xn, w

her

e k

≠ 0

, des

crib

es a

n n

th p

ower

var

iati

on. T

he

vari

able

n c

an b

e re

plac

ed b

y 2

to i

ndi

cate

th

e se

con

d po

wer

of

x (t

he

squ

are

of x

) or

by

3 to

in

dica

te t

he

thir

d po

wer

of

x (t

he

cube

of

x).

Ass

um

e th

at t

he

wei

ght

of a

per

son

of

aver

age

buil

d va

ries

dir

ectl

y as

th

e cu

be o

f th

at

pers

on’s

hei

ght.

Th

e eq

uat

ion

of

vari

atio

n h

as t

he

form

w =

kh

3 .T

he w

eigh

t th

at a

per

son’

s le

gs w

ill s

uppo

rt is

pro

port

iona

l to

the

cros

s-se

ctio

nal a

rea

of t

he

leg

bone

s. T

his

area

var

ies

dire

ctly

as

the

squa

re o

f th

e pe

rson

’s h

eigh

t. T

he e

quat

ion

of

vari

atio

n ha

s th

e fo

rm s

= k

h2 .

An

swer

eac

h q

ues

tion

.

1. F

or a

per

son

6 f

eet

tall

wh

o w

eigh

s 20

0 po

un

ds, f

ind

a va

lue

for

k in

th

e eq

uat

ion

w =

kh

3 .

k =

0.9

3 2

. Use

you

r an

swer

fro

m E

xerc

ise

1 to

pre

dict

th

e w

eigh

t of

a p

erso

n w

ho

is 5

fee

t ta

ll.

a

bo

ut

116

po

un

ds

3. F

ind

the

valu

e fo

r k

in t

he

equ

atio

n w

= k

h3

for

a ba

by w

ho

is 2

0 in

ches

lo

ng

and

wei

ghs

6 po

un

ds.

k

= 1

.296

fo

r h

= 5 −

3 ft

4. H

ow d

oes

you

r an

swer

to

Exe

rcis

e 3

dem

onst

rate

th

at a

bab

y is

si

gnif

ican

tly

fatt

er i

n p

ropo

rtio

n t

o it

s h

eigh

t th

an a

n a

dult

?

k

has

a g

reat

er v

alu

e. 5

. For

a p

erso

n 6

fee

t ta

ll w

ho

wei

ghs

200

pou

nds

, fin

d a

valu

e fo

r k

in t

he

equ

atio

n s

= k

h2 .

k

= 5

.56

6. F

or a

bab

y w

ho

is 2

0 in

ches

lon

g an

d w

eigh

s 6

pou

nds

, fin

d an

“in

fan

t va

lue”

for

k i

n t

he

equ

atio

n s

= k

h2 .

k

= 2

.16

for

h =

5 −

3 ft

7. A

ccor

din

g to

th

e ad

ult

equ

atio

n y

ou f

oun

d (E

xerc

ise

1), h

ow m

uch

w

ould

an

im

agin

ary

gian

t 20

fee

t ta

ll w

eigh

?

7

440

po

un

ds

8. A

ccor

din

g to

th

e ad

ult

equ

atio

n f

or w

eigh

t su

ppor

ted

(Exe

rcis

e 5)

, how

m

uch

wei

ght

cou

ld a

20-

foot

tal

l gi

ant’s

leg

s ac

tual

ly s

upp

ort?

o

nly

222

4 p

ou

nd

s 9

. Wh

at c

an y

ou c

oncl

ude

fro

m E

xerc

ises

7 a

nd

8?

A

nsw

ers

will

var

y. F

or

exam

ple

, bo

ne

stre

ng

th li

mit

s th

e si

ze h

um

ans

can

att

ain

.

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

2913

/12/

10

3:53

PM



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

30

Gle

ncoe

Alg

ebra

1

Rec

og

niz

e A

rith

met

ic S

equ

ence

s A

seq

uen

ce i

s a

set

of n

um

bers

in

a s

peci

fic

orde

r. If

th

e di

ffer

ence

bet

wee

n s

ucc

essi

ve t

erm

s is

con

stan

t, t

hen

th

e se

quen

ce i

s ca

lled

an

ar

ith

met

ic s

equ

ence

.

W

rite

an

eq

uat

ion

for

th

e n

th t

erm

of

the

seq

uen

ce

12, 1

5, 1

8, 2

1, .

. . .

In t

his

seq

uen

ce, a

1 is

12.

Fin

d th

e co

mm

on

diff

eren

ce.

12

+ 3

+ 3

+ 3

1518

21

Th

e co

mm

on d

iffe

ren

ce i

s 3.

Use

th

e fo

rmu

la f

or t

he

nth

ter

m t

o w

rite

an

equ

atio

n.

a n =

a

1 + (n

-

1)

d

Form

ula

for

the n

th t

erm

a n =

12

+

(n

-

1)

3 a 1

= 12,

d =

3

a n =

12

+

3n

-

3

Dis

trib

utive

Pro

pert

y

a n =

3n

+

9

Sim

plif

y.

Th

e eq

uat

ion

for

th

e n

th t

erm

is

a n =

3n

+

9.

Exer

cise

sD

eter

min

e w

het

her

eac

h s

equ

ence

is

an a

rith

met

ic s

equ

ence

. Wri

te y

es o

r n

o.

Exp

lain

.

1. 1

, 5, 9

, 13,

17,

. . .

2.

8, 4

, 0, -

4, -

8, .

. .

3. 1

, 3, 9

, 27,

81,

. . .

Fin

d t

he

nex

t th

ree

term

s of

eac

h a

rith

met

ic s

equ

ence

.

4. 9

, 13,

17,

21,

25,

. . .

5.

4, 0

, -4,

-8,

-12

, . .

. 6.

29,

35,

41,

47,

. . .

Wri

te a

n e

qu

atio

n f

or t

he

nth

ter

m o

f ea

ch a

rith

met

ic s

equ

ence

. Th

en g

rap

h t

he

firs

t fi

ve t

erm

s of

th

e se

qu

ence

.

7. 1

, 3, 5

, 7, .

. .

8. -

1, -

4, -

7, -

10, .

. .

9. -

4, -

9, -

14, -

19, .

. .

D

eter

min

e w

het

her

th

e se

qu

ence

1, 3

, 5, 7

, 9, 1

1, .

. . i

s an

ar

ith

met

ic s

equ

ence

. Ju

stif

y yo

ur

answ

er.

If p

ossi

ble,

fin

d th

e co

mm

on d

iffe

ren

ce

betw

een

th

e te

rms.

Sin

ce 3

-

1

=

2,

5

- 3

= 2,

an

d so

on

, th

e co

mm

on

diff

eren

ce i

s 2.

Sin

ce t

he

diff

eren

ce b

etw

een

th

e te

rms

of

1, 3

, 5, 7

, 9, 1

1, .

. . i

s co

nst

ant,

th

is i

s an

ar

ith

met

ic s

equ

ence

.

3-5

Stud

y G

uide

and

Inte

rven

tion

Ari

thm

eti

c S

eq

uen

ces a

s L

inear

Fu

ncti

on

s

Ari

thm

etic

Seq

uen

cea

nu

me

rica

l p

atte

rn t

ha

t in

cre

ase

s o

r d

ecre

ase

s a

t a

co

nsta

nt

rate

or

valu

e c

alle

d t

he

co

mm

on

diff

eren

ce

Term

s o

f an

Ari

thm

etic

Seq

uen

ce If

a1 is t

he

fi r

st

term

of

an

ari

thm

etic s

eq

ue

nce

with

co

mm

on

diff

ere

nce

d,

the

n t

he

se

qu

en

ce

is a

1,

a 1 +

d

, a 1

+

2d

, a 1

+

3d

, . . . .

nth

Ter

m o

f an

Ari

thm

etic

Seq

uen

cea n

= a 1

+

(n

-

1

)d

Exam

ple

1Ex

amp

le 2

2

9, 3

3, 3

7 -

16, -

20, -

24

53,

59,

65

y

es;

d =

4

yes

; d

= -

4 n

o;

no

co

mm

on

d

iffer

ence

a

n =

2n

-

1

a n =

-

3n +

2

a n =

-

5n +

1

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

3013

/12/

10

3:53

PM

Lesson 3-5


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

31

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Ari

thm

eti

c S

eq

uen

ces a

s L

inear

Fu

ncti

on

s

Ari

thm

etic

Seq

uen

ces

and

Fu

nct

ion

s A

n a

rith

met

ic s

equ

ence

is

a li

nea

r fu

nct

ion

in

wh

ich

n i

s th

e in

depe

nde

nt

vari

able

, a n i

s th

e de

pen

den

t va

riab

le, a

nd

the

com

mon

dif

fere

nce

d i

s th

e sl

ope.

Th

e fo

rmu

la c

an b

e re

wri

tten

as

the

fun

ctio

n

a n =

a 1

+ (

n -

1)

d, w

her

e n

is

a co

un

tin

g n

um

ber.

SE

ATI

NG

T

her

e ar

e 20

sea

ts i

n t

he

firs

t ro

w o

f th

e b

alco

ny

of t

he

aud

itor

ium

. Th

ere

are

22 s

eats

in

th

e se

con

d r

ow, a

nd

24

seat

s in

th

e th

ird

row

.

a. W

rite

a f

un

ctio

n t

o re

pre

sen

t th

is s

equ

ence

. T

he

firs

t te

rm a

1 is

20.

Fin

d th

e co

mm

on d

iffe

ren

ce.

20

+ 2

+ 2

2224

Th

e co

mm

on d

iffe

ren

ce i

s 2.

a

n =

a 1

+ (

n -

1)d

= 2

0 +

(n

- 1

)2

=

20

+

2n

-

2

= 1

8 +

2n

Th

e fu

nct

ion

is

a n =

18

+ 2

n.

b. G

rap

h t

he

fun

ctio

n.

Th

e ra

te o

f ch

ange

is

2. M

ake

a ta

ble

and

plot

poi

nts

.

n a

n

12

0

22

2

32

4

42

6

nO

1

2022242628

23

4

a n

Exer

cise

s 1

. KN

ITTI

NG

Sar

ah l

earn

s to

kn

it f

rom

her

gra

ndm

oth

er. T

wo

days

ago

, sh

e m

easu

red

the

len

gth

of

the

scar

f sh

e is

kn

itti

ng

to b

e 13

in

ches

. Yes

terd

ay, s

he

mea

sure

d th

e le

ngt

h o

f th

e sc

arf

to b

e 15

.5 i

nch

es. T

oday

it

mea

sure

s 18

in

ches

. Wri

te a

fu

nct

ion

to

repr

esen

t th

e ar

ith

met

ic s

equ

ence

.

2. R

EFR

ESH

MEN

TS Y

ou a

gree

to

pou

r w

ater

in

to t

he

cups

for

th

e B

oost

er C

lub

at a

foo

tbal

l ga

me.

Th

e pi

tch

er c

onta

ins

64 o

un

ces

of

wat

er w

hen

you

beg

in. A

fter

you

hav

e fi

lled

8 c

ups

, th

e pi

tch

er i

s em

pty

and

mu

st b

e re

fill

ed.

a. W

rite

a f

un

ctio

n t

o re

pres

ent

the

arit

hm

etic

seq

uen

ce.

an =

-

8n

b.

Gra

ph t

he

fun

ctio

n.

3-5

Form

ula

for

the n

th term

a 1 =

20 a

nd d

= 2

Dis

trib

utive

Pro

pert

y

Sim

plif

y.

a n

nO

18

24

6

816324864 24405672

35

7

Exam

ple

an =

13

+ 2

.5n

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

3113

/12/

10

3:54

PM



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

32

Gle

ncoe

Alg

ebra

1

Skill

s Pr

acti

ceA

rith

meti

c S

eq

uen

ces a

s L

inear

Fu

ncti

on

s

Det

erm

ine

wh

eth

er e

ach

seq

uen

ce i

s an

ari

thm

etic

seq

uen

ce. W

rite

yes

or

no.

E

xpla

in.

1. 4

, 7, 9

, 12,

. . .

2.

15,

13,

11,

9, .

. .

3. 7

, 10,

13,

16,

. . .

4.

-6,

-5,

-3,

-1,

. . .

5. -

5, -

3, -

1, 1

, . .

. 6.

-9,

-12

, -15

, -18

, . .

.

7. 1

0, 1

5, 2

5, 4

0, .

. . n

o

8. -

10, -

5, 0

, 5, .

. .

yes

; 5

Fin

d t

he

nex

t th

ree

term

s of

eac

h a

rith

met

ic s

equ

ence

.

9. 3

, 7, 1

1, 1

5, .

. .

10. 2

2, 2

0, 1

8, 1

6, .

. .

11. -

13, -

11, -

9, -

7 . .

. 12

. -2,

-

5, -

8, -

11, .

. .

13. 1

9, 2

4, 2

9, 3

4, .

. .

14. 1

6, 7

, -2,

-11

, . .

.

15. 2

.5, 5

, 7.5

, 10,

. . .

12.

5, 1

5, 1

7.5

16. 3

.1, 4

.1, 5

.1, 6

.1, .

. .

7.1,

8.1

, 9.1

Wri

te a

n e

qu

atio

n f

or t

he

nth

ter

m o

f ea

ch a

rith

met

ic s

equ

ence

. Th

en g

rap

h t

he

firs

t fi

ve t

erm

s of

th

e se

qu

ence

.

17. 7

, 13,

19,

25,

. . .

18

. 30,

26,

22,

18,

. . .

19

. -7,

-4,

-1,

2, .

. .

20. V

IDEO

DO

WN

LOA

DIN

G B

rian

is

dow

nlo

adin

g ep

isod

es o

f h

is f

avor

ite

TV

sh

ow t

o pl

ay

on h

is p

erso

nal

med

ia d

evic

e. T

he

cost

to

dow

nlo

ad 1

epi

sode

is

$1.9

9. T

he

cost

to

dow

nlo

ad 2

epi

sode

s is

$3.

98. T

he

cost

to

dow

nlo

ad 3

epi

sode

s is

$5.

97. W

rite

a f

un

ctio

n

to r

epre

sen

t th

e ar

ith

met

ic s

equ

ence

.

a

n =

1.9

9n

n

a n

24

6

4

-4

-8O

n

a n

O2

46

30 20 10

n

a n

O2

46

30 20 10

3-5

no

yes;

-2

yes;

3n

o

yes;

2ye

s; -

3

-5,

-3,

-1

-14

, -17

, -20

39, 4

4, 4

9-

20, -

29, -

38

a

n =

6n

+

1

a n =

-

4n +

34

a

n =

3n

-

10

19, 2

3, 2

714

, 12,

10

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

3213

/12/

10

3:54

PM

Lesson 3-5


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

33

Gle

ncoe

Alg

ebra

1

Prac

tice

Ari

thm

eti

c S

eq

uen

ces a

s L

inear

Fu

ncti

on

s

Det

erm

ine

wh

eth

er e

ach

seq

uen

ce i

s an

ari

thm

etic

seq

uen

ce. W

rite

yes

or

no.

E

xpla

in.

1. 2

1, 1

3, 5

, -3,

. . .

2.

-5,

12,

29,

46,

. . .

3.

-2.

2, -

1.1,

0.1

, 1.3

, . .

.

4. 1

, 4, 9

, 16,

. . .

5. 9

, 16,

23,

30,

. . .

6. -

1.2,

0.6

, 1.8

, 3.0

, . .

.

Fin

d t

he

nex

t th

ree

term

s of

eac

h a

rith

met

ic s

equ

ence

.

7. 8

2, 7

6, 7

0, 6

4, .

. .

8. -

49, -

35, -

21, -

7, .

. .

9. 3 −

4 , 1 −

2 , 1 −

4 , 0,

. . .

10. -

10, -

3, 4

, 11

. . .

11

. 12,

10,

8, 6

, . .

. 12

. 12,

7, 2

, -3,

. . .

Wri

te a

n e

qu

atio

n f

or t

he

nth

ter

m o

f ea

ch a

rith

met

ic s

equ

ence

. Th

en g

rap

h t

he

firs

t fi

ve t

erm

s of

th

e se

qu

ence

.

13. 9

, 13,

17,

21,

. . .

14

. -5,

-2,

1, 4

, . .

. 15

. 19,

31,

43,

55,

. . .

16. B

AN

KIN

G C

hem

dep

osit

ed $

115.

00 i

n a

sav

ings

acc

oun

t. E

ach

wee

k th

erea

fter

, he

depo

sits

$35

.00

into

th

e ac

cou

nt.

a. W

rite

a f

un

ctio

n t

o re

pres

ent

the

tota

l am

oun

t C

hem

has

dep

osit

ed f

or a

ny

part

icu

lar

nu

mbe

r of

wee

ks a

fter

his

in

itia

l de

posi

t.

b.

How

mu

ch h

as C

hem

dep

osit

ed 3

0 w

eeks

aft

er h

is i

nit

ial

depo

sit?

17. S

TOR

E D

ISPL

AY

S T

amik

a is

sta

ckin

g bo

xes

of t

issu

e fo

r a

stor

e di

spla

y. E

ach

row

of

tiss

ues

has

2

fe

wer

box

es t

han

th

e ro

w b

elow

. Th

e fi

rst

row

has

23

boxe

s of

tis

sues

.

a. W

rite

a f

un

ctio

n t

o re

pres

ent

the

arit

hm

etic

seq

uen

ce.

b.

How

man

y bo

xes

wil

l th

ere

be i

n t

he

ten

th r

ow?

n

a n

O60 40 20

24

6

n

a n

24

6

8 4

-4O

n

a n

O2

46

30 20 10

3-5

y

es;

d =

-8

yes

; d

= 1

7 n

o;

no

co

mm

on

d

iffer

ence

5

8, 5

2, 4

6 7

, 21,

35

- 1 −

4 , -

1 −

2 , -

3 −

4

a

n =

4n

+

5

a n =

3n

-

8

a n =

12

n +

7

a n =

35

n +

11

5

$116

5

5

1

8, 2

5, 3

2 4

, 2, 0

-

8, -

13, -

18

a n =

-

2n +

25

n

o;

no

co

mm

on

y

es;

d =

7

no

; n

o c

om

mo

n

diff

eren

ce

d

iffer

ence

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

3313

/12/

10

3:54

PM



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

34

Gle

ncoe

Alg

ebra

1

1.PO

STA

GE

Th

e pr

ice

to s

end

a la

rge

enve

lope

fir

st c

lass

mai

l is

88

cen

ts f

or

the

firs

t ou

nce

an

d 17

cen

ts f

or e

ach

ad

diti

onal

ou

nce

. Th

e ta

ble

belo

w s

how

s th

e co

st f

or w

eigh

ts u

p to

5ou

nce

s.

So

urc

e:U

nite

d S

tate

s P

osta

l S

erv

cie

H

ow m

uch

did

a l

arge

en

velo

pe w

eigh

th

at c

ost

$2.0

7 to

sen

d?

2. S

POR

TS W

anda

is

the

man

ager

for

th

e so

ccer

tea

m. O

ne

of h

er d

uti

es i

s to

han

d ou

t cu

ps o

f w

ater

at

prac

tice

. Eac

h c

up

of

wat

er i

s 4

oun

ces.

Sh

e be

gin

s pr

acti

ce

wit

h a

128

-ou

nce

coo

ler

of w

ater

. How

m

uch

wat

er i

s re

mai

nin

g af

ter

she

han

ds

out

the

14th

cu

p?

3. T

HEA

TER

A t

hea

ter

has

20

seat

s in

th

e fi

rst

row

, 22

in t

he

seco

nd

row

, 24

in t

he

thir

d ro

w, a

nd

so o

n f

or 2

5 ro

ws.

How

m

any

seat

s ar

e in

th

e la

st r

ow?

4.N

UM

BER

TH

EORY

On

e of

th

e m

ost

fam

ous

sequ

ence

s in

mat

hem

atic

s is

th

e F

ibon

acci

seq

uen

ce. I

t is

nam

ed a

fter

L

eon

ardo

de

Pis

a (1

170–

1250

) or

Fil

ius

Bon

acci

, ali

as L

eon

ardo

Fib

onac

ci. T

he

firs

t se

vera

l n

um

bers

in

th

e F

ibon

acci

se

quen

ce a

re:

1, 1

, 2, 3

, 5, 8

, 13,

21,

34,

55,

89,

. . .

Doe

s th

is r

epre

sen

t an

ari

thm

etic

se

quen

ce?

Wh

y or

wh

y n

ot?

5. S

AV

ING

S In

ga’s

gra

ndf

ath

er d

ecid

es t

o st

art

a fu

nd

for

her

col

lege

edu

cati

on. H

e m

akes

an

in

itia

l co

ntr

ibu

tion

of

$300

0 an

d ea

ch m

onth

dep

osit

s an

add

itio

nal

$5

00. A

fter

on

e m

onth

he

wil

l h

ave

con

trib

ute

d $3

500.

a. W

rite

an

equ

atio

n f

or t

he

nth

ter

m o

f th

e se

quen

ce.

b.

How

mu

ch m

oney

wil

l In

ga’s

gr

andf

ath

er h

ave

con

trib

ute

d af

ter

24 m

onth

s?$1

5,0

00

Wo

rd P

rob

lem

Pra

ctic

eA

rith

meti

c S

eq

uen

ces a

s L

inear

Fu

ncti

on

s

3-5 W

eig

ht

(ou

nce

s)1

23

45

Po

stag

e (d

olla

rs)

0.8

81.

05

1.2

21.

39

1.5

6

8 o

un

ces

72 o

un

ces

68 s

eats

No

, bec

ause

th

e d

iffer

ence

b

etw

een

ter

ms

is n

ot

con

stan

t.

an =

30

00

+ 50

0n

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

3413

/12/

10

3:54

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 3-5

Cha

pte

r 3

35

Gle

ncoe

Alg

ebra

1

Ari

thm

eti

c S

eri

es

An

arit

hmet

ic s

erie

s is

a s

erie

s in

whi

ch e

ach

term

aft

er t

he f

irst

may

be

foun

d by

add

ing

the

sam

e nu

mbe

r to

the

pre

cedi

ng t

erm

. Let

S s

tand

for

the

fol

low

ing

seri

es i

n w

hich

eac

h te

rm

is 3

mor

e th

an t

he p

rece

ding

one

.S

=

2

+ 5

+ 8

+ 11

+

14

+

17

+

20

Th

e se

ries

rem

ain

s th

e sa

me

if w

e

S =

2

+

5

+

8

+

11

+

14

+

17

+

20

reve

rse

the

orde

r of

all

th

e te

rms.

So

S

=

20

+

17

+

14

+

11

+

8

+

5

+

2

let

us

reve

rse

the

orde

r of

th

e te

rms

2S

=

22

+

22

+

22

+

22

+

22

+

22

+

22

and

add

one

seri

es t

o th

e ot

her

, ter

m

2S =

7(

22)

by t

erm

. Th

is i

s sh

own

at

the

righ

t.

S =

7(

22)

−

2 =

7(

11) =

77

Let

a r

epre

sen

t th

e fi

rst

term

of

the

seri

es.

Let

� r

epre

sen

t th

e la

st t

erm

of

the

seri

es.

Let

n r

epre

sen

t th

e n

um

ber

of t

erm

s in

th

e se

ries

.In

th

e pr

eced

ing

exam

ple,

a =

2,

� =

20

, an

d n

=

7.

Not

ice

that

wh

en y

ou a

dd t

he

two

seri

es,

term

by

term

, th

e su

m o

f ea

ch p

air

of t

erm

s is

22.

Th

at s

um

can

be

fou

nd

by a

ddin

g th

e fi

rst

and

last

ter

ms,

2

+ 20

or

a +

�. N

otic

e al

so t

hat

th

ere

are

7, o

r n

, su

ch s

um

s. T

her

efor

e, t

he

valu

e of

2S

is

7(22

), or

n(a

+

�)

in t

he

gen

eral

cas

e. S

ince

th

is i

s tw

ice

the

sum

of

th

e se

ries

, you

can

use

th

e fo

rmu

la

S =

n

(a +

�)

−

2

to f

ind

the

sum

of

any

arit

hm

etic

ser

ies.

F

ind

th

e su

m: 1

+

2

+ 3

+ 4

+ 5

+ 6

+ 7

+ 8

+ 9.

a =

1,

� =

9,

n =

9,

so

S =

9(1

+ 9)

−

2 =

9

. 10

−

2 =

45

F

ind

th

e su

m: -

9 +

(-

5) +

(-

1) +

3

+ 7

+ 11

+

15

.

a =

29

, � =

15

, n =

7,

so

S =

7(

-9

+ 15

) −

2 =

7.

6 −

2 =

21

Exer

cise

sF

ind

th

e su

m o

f ea

ch a

rith

met

ic s

erie

s.

1.

3 +

6

+ 9

+ 12

+

15

+

18

+

21

+

24

2. 1

0 +

15

+

20

+

25

+

30

+

35

+

40

+

45

+

50

3. -

21 +

(-

16) +

(-

11) +

(-

6) +

(-

1) +

4

+ 9

+ 14

4. e

ven

wh

ole

nu

mbe

rs f

rom

2 t

hro

ugh

100

5. o

dd w

hol

e n

um

bers

bet

wee

n 0

an

d 10

0

Enri

chm

ent

3-5

Exam

ple

1

Exam

ple

2

108

270

-28

2550

250

0

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

3513

/12/

10

3:54

PM



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

36

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

Pro

po

rtio

nal

an

d N

on

pro

po

rtio

nal

Rela

tio

nsh

ips

Pro

po

rtio

nal

Rel

atio

nsh

ips

If t

he

rela

tion

ship

bet

wee

n t

he

dom

ain

an

d ra

nge

of

a re

lati

on i

s li

nea

r, th

e re

lati

onsh

ip c

an b

e de

scri

bed

by a

lin

ear

equ

atio

n. I

f th

e eq

uat

ion

pa

sses

th

rou

gh (

0, 0

) an

d is

of

the

form

y =

kx,

th

en t

he

rela

tion

ship

is

prop

orti

onal

.

CO

MPA

CT

DIS

CS

Su

pp

ose

you

pu

rch

ased

a n

um

ber

of

pac

kag

es o

f b

lan

k c

omp

act

dis

cs. I

f ea

ch p

ack

age

con

tain

s 3

com

pac

t d

iscs

, you

cou

ld m

ake

a ch

art

to s

how

th

e re

lati

onsh

ip b

etw

een

th

e n

um

ber

of

pac

kag

es o

f co

mp

act

dis

cs

and

th

e n

um

ber

of

dis

cs p

urc

has

ed. U

se x

for

th

e n

um

ber

of

pac

kag

es a

nd

y

for

the

nu

mb

er o

f co

mp

act

dis

cs.

Mak

e a

tabl

e of

ord

ered

pai

rs f

or s

ever

al p

oin

ts o

f th

e gr

aph

.

N

um

ber

of

Pac

kag

es1

23

45

Nu

mb

er o

f C

Ds

36

912

15

Th

e di

ffer

ence

in

th

e x

valu

es i

s 1,

an

d th

e di

ffer

ence

in

th

e y

valu

es i

s 3.

Th

is p

atte

rn

show

s th

at y

is

alw

ays

thre

e ti

mes

x. T

his

su

gges

ts t

he

rela

tion

y =

3x.

Sin

ce t

he

rela

tion

is

als

o a

fun

ctio

n, w

e ca

n w

rite

th

e eq

uat

ion

in

fu

nct

ion

not

atio

n a

s f(

x) =

3x.

Th

e re

lati

on i

ncl

ude

s th

e po

int

(0, 0

) be

cau

se i

f yo

u b

uy

0 pa

ckag

es o

f co

mpa

ct d

iscs

, you

w

ill

not

hav

e an

y co

mpa

ct d

iscs

. Th

eref

ore,

th

e re

lati

onsh

ip i

s pr

opor

tion

al.

Exer

cise

s 1

. NA

TUR

AL

GA

S N

atu

ral

gas

use

is

ofte

n m

easu

red

in “

ther

ms.

” T

he

tota

l am

oun

t a

gas

com

pan

y w

ill

char

ge f

or n

atu

ral

gas

use

is

base

d on

how

mu

ch n

atu

ral

gas

a h

ouse

hol

d u

ses.

Th

e ta

ble

show

s th

e re

lati

onsh

ip b

etw

een

nat

ura

l ga

s u

se a

nd

the

tota

l co

st.

G

as U

sed

(th

erm

s)1

23

4

Tota

l Co

st (

$)$

1.3

0$

2.6

0$

3.9

0$

5.2

0

a. G

raph

th

e da

ta. W

hat

can

you

ded

uce

fro

m t

he

patt

ern

ab

out

the

rela

tion

ship

bet

wee

n t

he

nu

mbe

r of

th

erm

s u

sed

and

the

tota

l co

st?

Th

e re

lati

on

ship

is p

rop

ort

ion

al.

b.

Wri

te a

n e

quat

ion

to

desc

ribe

th

is r

elat

ion

ship

. y =

1.3

0x

c. U

se t

his

equ

atio

n t

o pr

edic

t h

ow m

uch

it

wil

l co

st i

f a

hou

seh

old

use

s 40

th

erm

s.$5

2.0

0

3-6

Exam

ple

Total Cost ($) 23 1

0

456y

Gas

Used

(the

rms)

32

1x

4

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

3613

/12/

10

3:54

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 3-6

Cha

pte

r 3

37

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Pro

po

rtio

nal

an

d N

on

pro

po

rtio

nal

Rela

tio

nsh

ips

No

np

rop

ort

ion

al R

elat

ion

ship

s If

th

e ra

tio

of t

he

valu

e of

x t

o th

e va

lue

of y

is

diff

eren

t fo

r se

lect

ord

ered

pai

rs o

n t

he

lin

e, t

he

equ

atio

n i

s n

onpr

opor

tion

al.

W

rite

an

eq

uat

ion

in

fu

nct

ion

al n

otat

ion

for

th

e re

lati

on s

how

n

in t

he

grap

h.

Sel

ect

poin

ts f

rom

th

e gr

aph

an

d pl

ace

them

in

a t

able

.

x

-1

01

23

y4

20

-2

-4

y

x

Th

e di

ffer

ence

bet

wee

n t

he

x–va

lues

is

1, w

hil

e th

e di

ffer

ence

bet

wee

n t

he

y-va

lues

is

–2. T

his

su

gges

ts

that

y =

–2x

.If

x =

1, t

hen

y =

–2(

1) o

r –2

. Bu

t th

e y–

valu

e fo

r x

= 1

is

0.

x

1

2

3

-2x

-2

-4

-6

y

0-

2-

4

y is

alw

ays 2

more

than -

2x

Th

is p

atte

rn s

how

s th

at 2

sh

ould

be

adde

d to

on

e si

de o

f th

e eq

uat

ion

. Th

us,

th

e eq

uat

ion

is

y =

-2x

+ 2

.

Exer

cise

sW

rite

an

eq

uat

ion

in

fu

nct

ion

not

atio

n f

or t

he

rela

tion

sh

own

in

th

e ta

ble

. Th

en c

omp

lete

th

e ta

ble

.

1.

x-

10

12

34

y-

22

610

1418

2

. x

-2

-1

01

23

y10

7

4 1

-2

-5

Wri

te a

n e

qu

atio

n i

n f

un

ctio

n n

otat

ion

for

eac

h r

elat

ion

.

3.

x

y

O

4.

x

y O

3-6

Exam

ple

f(x

) = 4x

+

2

f(x

) = -

3x +

4

f(x

) = -

x +

2

f(x

) = 2x

+

2

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

3712

/15/

10

11:4

5 P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

38

Gle

ncoe

Alg

ebra

1

Skill

s Pr

acti

ceP

rop

ort

ion

al

an

d N

on

pro

po

rtio

nal

Rela

tio

nsh

ips

Wri

te a

n e

qu

atio

n i

n f

un

ctio

n n

otat

ion

for

eac

h r

elat

ion

.

1.

2.

3.

4.

5.

6.

7. G

AM

ESH

OW

S T

he

tabl

e sh

ows

how

man

y po

ints

are

aw

arde

d fo

r an

swer

ing

con

secu

tive

qu

esti

ons

on a

gam

esh

ow.

Qu

esti

on

an

swer

ed1

23

45

Po

ints

aw

ard

ed2

00

40

06

00

80

010

00

a.

Wri

te a

n e

quat

ion

for

th

e da

ta g

iven

. y

= 2

00x

b

. F

ind

the

nu

mbe

r of

poi

nts

aw

arde

d if

9 q

ues

tion

s w

ere

answ

ered

. 18

00

xO

f(x)

xO

f(x)

xO

f(x)

xO

f(x)

xO

f(x)

xO

f(x)

3-6

f

(x) =

-

2x

f(x

) = x

-

2

f

(x) =

1

- x

f(x

) = x

+

6

f

(x) =

5

- x

f(x

) = 2x

-

1

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

3813

/12/

10

4:39

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 3-6

Cha

pte

r 3

39

Gle

ncoe

Alg

ebra

1

Prac

tice

Pro

po

rtio

nal

an

d N

on

pro

po

rtio

nal

Rela

tio

nsh

ips

1.B

IOLO

GY

Mal

e fi

refl

ies

flas

h in

var

ious

pat

tern

s to

sig

nal l

ocat

ion

and

perh

aps

to w

ard

off

pred

ator

s. D

iffe

rent

spe

cies

of

fire

flie

s ha

ve d

iffe

rent

fla

sh c

hara

cter

isti

cs, s

uch

as t

he

inte

nsit

y of

the

fla

sh, i

ts r

ate,

and

its

shap

e. T

he t

able

bel

ow s

how

s th

e ra

te a

t w

hich

a

mal

e fi

refl

y is

fla

shin

g.

Tim

es (

seco

nd

s)1

23

4 5

Nu

mb

er o

f F

lash

es2

46

810

a. W

rite

an

equ

atio

n i

n f

un

ctio

n n

otat

ion

for

th

e re

lati

on.

b.

How

man

y ti

mes

wil

l th

e fi

refl

y fl

ash

in

20

seco

nds

?

2. G

EOM

ETRY

Th

e ta

ble

show

s th

e n

um

ber

of d

iago

nal

s th

at c

an b

e dr

awn

fro

m o

ne

vert

ex i

n a

pol

ygon

. Wri

te

an e

quat

ion

in

fu

nct

ion

not

atio

n f

or t

he

rela

tion

an

d fi

nd

the

nu

mbe

r of

dia

gon

als

that

can

be

draw

n f

rom

on

e ve

rtex

in

a 1

2-si

ded

poly

gon

.

Wri

te a

n e

qu

atio

n i

n f

un

ctio

n n

otat

ion

for

eac

h r

elat

ion

.

3.

4.

5.

For

eac

h a

rith

met

ic s

equ

ence

, det

erm

ine

the

rela

ted

fu

nct

ion

. Th

en d

eter

min

e if

th

e fu

nct

ion

is

pro

por

tion

al

or n

onp

rop

orti

ona

l. E

xpla

in.

6. 1

, 3, 5

, . .

. 7.

2, 7

, 12,

. . .

8. -

3, -

6, -

9, .

. .

a(n

) =

2n

- 1

; a

(n)

= 5

n-

3;

a

(n)

=-

3n;

n

on

pro

po

rtio

nal

; n

on

pro

po

rtio

nal

; p

rop

ort

ion

al;

n

ot

of

form

y=

kx

no

t o

f fo

rm y

=kx

of

form

y=

kx

x

y O

x

y

O

x

y

O

3-6

Sid

es3

45

6

Dia

go

nal

s0

12

3

f

(x) =

-

1 −

2 x

f(x

) = 3x

-

6

f(x

) = 2x

+

4

f(t)

=

2t

, wh

ere

t is

th

e

40

f(s) =

s

-

3,

wh

ere

s is

th

e n

um

ber

of

sid

es

tim

e in

sec

on

ds

and

f(t

) is

th

e n

um

ber

of

fl as

hes

and

f(s

) is

th

e n

um

ber

of

dia

go

nal

s; 9

029_

042_

ALG

1_A

_CR

M_C

03_C

R_6

6138

3.in

dd

3913

/12/

10

3:54

PM



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 3

40

Gle

ncoe

Alg

ebra

1

Wor

d Pr

oble

m P

ract

ice

Pro

po

rtio

nal

an

d N

on

pro

po

rtio

nal

Rela

tio

nsh

ips

1.O

NLI

NE

SHO

PPIN

G R

icar

do i

s bu

yin

g co

mpu

ter

cabl

es f

rom

an

on

lin

e st

ore.

If

he

buys

4 c

able

s, t

he

tota

l co

st w

ill

be

$24.

If

he

buys

5 c

able

s, t

he

tota

l co

st

wil

l be

$29

. If

the

tota

l co

st c

an b

e re

pres

ente

d by

a l

inea

r fu

nct

ion

, wil

l th

e fu

nct

ion

be

prop

orti

onal

or

non

prop

orti

onal

? E

xpla

in.

2. F

OO

D I

t ta

kes

abou

t fo

ur

pou

nds

of

grap

es t

o pr

odu

ce o

ne

pou

nd

of r

aisi

ns.

T

he

grap

h s

how

s th

e re

lati

on f

or t

he

nu

mbe

r of

pou

nds

of

grap

es n

eede

d, x

, to

mak

e y

pou

nds

of

rais

ins.

Wri

te a

n

equ

atio

n i

n f

un

ctio

n n

otat

ion

for

th

e re

lati

on s

how

n.

3. P

AR

KIN

G P

alm

er T

own

ship

rec

entl

y in

stal

led

park

ing

met

ers

in t

hei

r m

un

icip

al l

ot. T

he

cost

to

park

for

h

hou

rs i

s re

pres

ente

d by

th

e eq

uat

ion

C

= 0

.25h

.

a. M

ake

a ta

ble

of v

alu

es t

hat

rep

rese

nts

th

is r

elat

ion

ship

.

b. D

escr

ibe

the

rela

tion

ship

bet

wee

n t

he

tim

e pa

rked

an

d th

e co

st.

4.M

USI

C A

mea

sure

of

mu

sic

con

tain

s th

e sa

me

nu

mbe

r of

bea

ts t

hro

ugh

out

the

son

g. T

he

tabl

e sh

ows

the

rela

tion

for

th

e n

um

ber

of b

eats

cou

nte

d af

ter

a ce

rtai

n

nu

mbe

r of

mea

sure

s h

ave

been

pla

yed

in

the

six-

eigh

t ti

me.

Wri

te a

n e

quat

ion

to

desc

ribe

th

is r

elat

ion

ship

.

Sour

ce: S

heet

Mus

ic U

SA

5. G

EOM

ETRY

A f

ract

al i

s a

patt

ern

co

nta

inin

g pa

rts

wh

ich

are

ide

nti

cal

to

the

over

all

patt

ern

. Th

e fo

llow

ing

geom

etri

c pa

tter

n i

s a

frac

tal.

a.

Com

plet

e th

e ta

ble.

b.

Wh

at a

re t

he

nex

t th

ree

nu

mbe

rs i

n

the

patt

ern

?

c. W

rite

an

equ

atio

n i

n f

un

ctio

n n

otat

ion

fo

r th

e pa

tter

n.

Pou

nd

s o

f G

rap

es2

10

43

f(x

)

x5

67

8

Pounds of Raisins

1.5

2.0

1.0

0.5

2.5

4.0

3.5

3.0

3-6

Mea

sure

sP

laye

d (

x)

12

34

56

Tota

l Nu

mb

ero

f B

eats

(y)

612

18

24

30

36

Term

x1

23

4

Nu

mb

er o

f S

mal

ler

Tria

ng

les

y1

4 9

16

y

=

6x

25, 3

6, 4

9

f(x) =

x

2

no

np

rop

ort

ion

al;

do

es n

ot

pas

s th

rou

gh

(0,

0)

f

(x) =

0.

25x

It c

ost

s 25

cen

ts f

or

each

h

ou

r yo

u p

ark

in t

he

lot.

Tim

eh

12

34

Co

stC

0.25

0.5

0.75

1.0

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NA

ME

DAT

E

P

ER

IOD

Lesson 3-6

Cha

pte

r 3

41

Gle

ncoe

Alg

ebra

1

Enri

chm

ent

Taxic

ab

Gra

ph

sYo

u h

ave

use

d a

rect

angu

lar

coor

din

ate

syst

em t

o gr

aph

x

y

O

y =

x -

1

y =

-2

equ

atio

ns

such

as

y =

x

- 1

on a

coo

rdin

ate

plan

e. I

n a

co

ordi

nat

e pl

ane,

th

e n

um

bers

in

an

ord

ered

pai

r (x

, y)

can

be

an

y tw

o re

al n

um

bers

.A

tax

icab

pla

ne

is d

iffe

ren

t fr

om t

he

usu

al c

oord

inat

e pl

ane.

T

he

only

poi

nts

all

owed

are

th

ose

that

exi

st a

lon

g th

e h

oriz

onta

l an

d ve

rtic

al g

rid

lin

es. Y

ou m

ay t

hin

k of

th

e po

ints

as

tax

icab

s th

at m

ust

sta

y on

th

e st

reet

s.

Th

e ta

xica

b gr

aph

sh

ows

the

equ

atio

ns

y =

-

2 an

d y

= x

- 1.

N

otic

e th

at o

ne

of t

he

grap

hs

is n

o lo

nge

r a

stra

igh

t li

ne.

It

is

now

a c

olle

ctio

n o

f se

para

te p

oin

ts.

Gra

ph

th

ese

equ

atio

ns

on t

he

taxi

cab

pla

ne

at t

he

righ

t.

x

y

O

1.

1.2.

2.

3.3.

4. 4.

1. y

=

x

+ 1

2. y

=

-

2x +

3

3. y

=

2.

5 4.

x =

-

4

Use

you

r gr

aph

s fo

r th

ese

pro

ble

ms.

5. W

hic

h o

f th

e eq

uat

ion

s h

as t

he

sam

e gr

aph

in

bot

h t

he

usu

al c

oord

inat

e pl

ane

and

the

taxi

cab

plan

e?

6. D

escr

ibe

the

form

of

equ

atio

ns

that

hav

e th

e sa

me

grap

h i

n b

oth

th

e u

sual

coo

rdin

ate

plan

e an

d th

e ta

xica

b pl

ane.

In

th

e ta

xica

b p

lan

e, d

ista

nce

s ar

e n

ot m

easu

red

dia

gon

ally

, bu

t al

ong

the

stre

ets.

W

rite

th

e ta

xi-d

ista

nce

bet

wee

n e

ach

pai

r of

poi

nts

.

7. (

0, 0

) an

d (5

, 2)

8. (

0, 0

) an

d (-

3, 2

) 9.

(0,

0)

and

(2, 1

.5)

10. (

1, 2

) an

d (4

, 3)

11. (

2, 4

) an

d (-

1, 3

) 12

. (0,

4)

and

(-2,

0)

Dra

w t

hes

e gr

aph

s on

th

e ta

xica

b g

rid

at

the

righ

t.

13. T

he

set

of p

oin

ts w

hos

e ta

xi-d

ista

nce

fro

m (

0, 0

) is

2

un

its.

14. T

he

set

of p

oin

ts w

hos

e ta

xi-d

ista

nce

fro

m (

2, 1

) is

3 u

nit

s.

x

y

O

3-6

x =

-

4

x =

A

an

d y

=

B

, wh

ere

A a

nd

B a

re in

teg

ers

7 u

nit

s 5

un

its

3.5

un

its

4 u

nit

s 4

un

its

6 u

nit

s

ind

icat

ed b

y ×

ind

icat

ed b

y d

ots

029_

042_

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PM



Chapter 3 Assessment Answer KeyQuiz 1 (Lessons 3-1 and 3-2) Quiz 3 (Lessons 3-4 and 3-5) Mid-Chapter TestPage 45 Page 46 Page 47

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Quiz 2 (Lessons 3-3)

Page 45Quiz 4 (Lesson 3-6)

Page 46

1.

2.

3.

4.

5.

yes; 2x - y = 1

D

about 400 miles

y

xO

x = 15

1.

2.

3.

4.

5.

2 −

9

7 −

4

B

not linear

increases by 5/yr

1.

2.

3.

4.

5. a = 0, b = 33

y = 3x + 3

y = 9x - 2

no

y = 5x + 4

1.

2.

3.

4.

5.

C

B

F

G

B

6.

7.

8.

9.

yes; 2x - y = 3

y

xO

no solution

-2

25

1.

2.

3.

4.

5.

no

21, 25, 29

B

$285


Chapter 3 Assessment Answer KeyVocabulary Test Form 1Page 48 Page 49 Page 50

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c.

An

swer

s

PDF Pass


1.

2.

3.

4.

5.

6.

7.

8.

9. C

F

B

J

D

H

B

J

D 10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

G

H

C

F

B

B

B

F

B

G

J

See students’ work; x: 3, y: -3

B:

root

linear equation

x-intercept

common difference

standard form

term

inductive reasoning

Sample answer: the y-coordinate of the point at which the graph crosses the y-axis

Sample answer: Rate of change is a ratio that describes, on average, how much a quantity changes with respect to a change in another quantity.

arithmetic

sequence

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.


Chapter 3 Assessment Answer KeyForm 2A Form 2BPage 51 Page 52 Page 53 Page 54

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A

C

D

G

J

A

F

C

G

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

H

A

11:08 A.M. B:

H

H

C

F

D

A

J

C

H

12.

13.

14.

15.

16.

17.

18.

19.

20.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. D

F

D

G

C

D

B

G

H

B

G

G

H

A

F

B

G

A

F

B

12.

13.

14.

15.

16.

17.

18.

19.

20.

7:49 A.M.B:


Chapter 3 Assessment Answer KeyForm 2CPage 55 Page 56

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An

swer

s

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1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

yes; 2x + 3y = -4

6

y

xO

no

x = 2

9

3

y

xO

4

1

- 11 −

9

0

36 mi

about - 0.07/yr

y

xO

(3, -1), (-1, -1),

(3, 3)

6

36, 43, 50

yes; 3

an = -7n + 19

Sample answer:

f(d) = 0.04d

y

x

x = 3

14. y

xO

15.

16.

17.

18.

19.

20.

B:


Chapter 3 Assessment Answer KeyForm 2DPage 57 Page 58

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14.

15.

16.

17.

18.

19.

20.

B:

f(d) = 0.05d

an = 9n - 6

-13, -10, -7

yes; 4

y

xO

(-2, 3) (4, 3)

(4, -2)(-2, -2)

(-2, -2)

y

xO

7

y

x

x = 1

7 − 3

39

8 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13. y

xO

y

x

no

yes; 4x - 2y = 0

8

1 −

2 h

about - 1 − 3 %/yr

undefi ned

- 3 − 5

0

A20_A34_ALG1_A_CRM_C03_AN_661383.indd A24A20_A34_ALG1_A_CRM_C03_AN_661383.indd A24 12/29/10 9:27 AM12/29/10 9:27 AM

Chapter 3 Assessment Answer KeyForm 3Page 59 Page 60

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An

swer

s

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1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

yes; 3x - 7y = -6

12

-4.2

3.75

y

xO

4

y

x

x = 1.5

60

undefi ned

- 9 −

13

690.4/yr

12. y

xO

13.

14.

15.

16.

17.

18.

19.

20.

B:

12 yrs

about 10 1 −

2 yrs

v(t) = 2000 - 190t

yes; 1 −

2

c = 359.88y

70

−

13

-13

y = -1

an

n1

2

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

-10-12-14

an = 4n - 19


Chapter 3 Assessment Answer Key Page 61, Extended-Response Assessment Scoring Rubric

Score General Description Specifi c Criteria

4 SuperiorA correct solution that is supported by well-developed, accurate explanations

• Shows thorough understanding of the concepts of points on the coordinate plane, transformations, relations, functions, linear equations, arithmetic sequences, and patterns.

• Uses appropriate strategies to solve problems. • Computations are correct. • Written explanations are exemplary. • Graphs are accurate and appropriate. • Goes beyond requirements of some or all problems.

3 SatisfactoryA generally correct solution, but may contain minor flaws in reasoning or computation

• Shows an understanding of the concepts of points on the coordinate plane, transformations, relations, functions, linear equations, arithmetic sequences, and patterns.

• Uses appropriate strategies to solve problems. • Computations are mostly correct. • Written explanations are effective. • Graphs are mostly accurate and appropriate. • Satisfies all requirements of problems.

2 Nearly SatisfactoryA partially correct interpretation and/or solution to the problem

• Shows an understanding of most of the concepts of points on the coordinate plane, transformations, relations, functions, linear equations, arithmetic sequences, and patterns.

• May not use appropriate strategies to solve problems. • Computations are mostly correct. • Written explanations are satisfactory. • Graphs are mostly accurate. • Satisfies the requirements of most of the problems.

1 Nearly Unsatisfactory A correct solution with no supporting evidence or explanation

• Final computation is correct. • No written explanations or work is shown to

substantiate the final computation. • Graphs may be accurate but lack detail or explanation. • Satisfies minimal requirements of some of the problems.

0 UnsatisfactoryAn incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given

• Shows little or no understanding of most of the concepts of points on the coordinate plane, transformations, relations, functions, linear equations, arithmetic sequences, and patterns.

• Does not use appropriate strategies to solve problems. • Computations are incorrect. • Written explanations are unsatisfactory. • Graphs are inaccurate or inappropriate. • Does not satisfy requirements of problems. • No answer may be given.

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Chapter 3 Assessment Answer Key Page 61, Extended-Response Test Sample Answers

1a. After drawing a graph, students should explain that they can use the two points on the graph to determine the slope. This can be done by counting squares for the rise and run of the line or by using the coordinates of the points in the slope formula.

1b. The slope is the value of m in the slope-intercept form y = mx + b. By substituting the value of m and the coordinates of one of the points for x and y, the value of b can be found and an equation written using m and b. The slope and either ordered pair can be used to write the point-slope form of an equation. The standard form of an equation is an algebraic manipulation of either of the other two forms of equations. See students’ equations for the line drawn.

2a. The student writes an equation in the form C = kf. Sample answer: C = 2f

2b. The student should explain that the constant of variation represents the price per foot of the lumber.

2c. The student explains the new function is in the form C = kf + 10, which is not a

direct variation.

3a. Sample answer: 2, 5, 8, 11, … ; The common difference is 3; a6 = 17

3b. Sample answer: 5, 3, 8, 6, 11, 9, 14, … ; The pattern is to subtract 2 from the first term to find the second term, then add 5 to the second term to find the third term. Repeat the process of subtracting 2 then adding 5.

3c. The sequence 1, 1, 1, 1, … is a set of numbers whose difference between successive terms is the constant number 0. Thus, this sequence is an arithmetic sequence by the definition.

4. The student should draw a line on the coordinate grid and then identify the x- and y-coordinates. The student should write a linear equation in function notation for the line they drew.

In addition to the scoring rubric found on page A26, the following sample answers may be used as guidance in evaluating assessment items.

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Chapter 3 Assessment Answer KeyStandardized Test Practice

Page 62 Page 63

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9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

7 7

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

2 818.

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

8. F G H J

9. A B C D

10. F G H J

11. A B C D

12. F G H J

13. A B C D

14. F G H J

15. A B C D

16. F G H J

17.


Chapter 3 Assessment Answer KeyStandardized Test Practice

Page 64

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19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30a.

30b.

Yes, because each

element of the

domain is paired

with exactly one

element of the range.

Yes, because it can

be represented by an

equation in the form

y = 18x.

1.25

y

xO

not a function

{(-4, 3), (-4, -1),

(-2, 2), (0, -1), (2, 3),

(3, -3), (4, 1)}; domain:

{-4, -2, 0, 2, 3, 4};

range: {-3, -1, 1, 2, 3}

no

-2

$12.75

x divided by 4 minus y

equals negative 2 times

the quotient of x and y.

- 5 −

2

8m + 9p

4 −

5


Chapter 3 Resource Masters - Commack Schools

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Transcript of Chapter 3 Resource Masters - Commack Schools