California Math Triumphs VOL 5b Functions (CALIFORNIA MATH TRIUMPHS VOL 5B)

Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber Photo Credits Cover Joe McBride/CORBIS; iv (t...

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Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber

Photo Credits Cover Joe McBride/CORBIS; iv (tl bl br) File Photo, (tc tr) The McGraw-Hill Companies, (cl c) Doug Martin, (cr) Aaron Haupt; v (1 2 3 4 6 7 8 9 11 12) The McGraw-Hill Companies; v (5 10 13 14) File Photo; viii Dynamic Graphics Group/ Creatas/Alamy; viii Jeremy Woodhouse/Getty Images; ix Glen Allison/Getty Images; 2–3 Kenneth Eward/Photo Researchers,Inc.; 3 4 Photodisc/Getty Images; 10 Jeffrey L. Rotman/Peter Arnold,Inc.; 13 Photodisc/Getty Images; 16 (b) Kevin Sanchez/Cole Group/Getty Images; 16 (t) The McGraw-Hill Companies, Inc.; 18 CORBIS; 24 Image Source/SuperStock; 29 Manchan/Getty Images; 32 Jeff Maloney/Getty Images; 38 G.K. & Vikki Hart/Getty Images

Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-878212 MHID: 0-07-878212-0 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 055/027 16 15 14 13 12 11 10 09 08 07

California Math Triumphs Volume 5B

California Math Triumphs Volume 1 Place Value and Basic Number Skills 1A Chapter 1 Counting 1A Chapter 2 Place Value 1A Chapter 3 Addition and Subtraction 1B Chapter 4 Multiplication 1B Chapter 5 Division 1B Chapter 6 Integers Volume 2 Fractions and Decimals 2A Chapter 1 Parts of a Whole 2A Chapter 2 Equivalence of Fractions 2B Chapter 3 Operations with Fractions 2B Chapter 4 Positive and Negative Fractions and Decimals

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

Volume 3 Ratios, Rates, and Percents 3A Chapter 1 Ratios and Rates 3A Chapter 2 Percents, Fractions, and Decimals 3B Chapter 3 Using Percents 3B Chapter 4 Rates and Proportional Reasoning Volume 4 The Core Processes of Mathematics 4A Chapter 1 Operations and Equality 4A Chapter 2 Math Fundamentals 4B Chapter 3 Math Expressions 4B Chapter 4 Linear Equations 4B Chapter 5 Inequalities Volume 5 Functions and Equations 5A Chapter 1 Patterns and Relationships 5A Chapter 2 Graphing 5B Chapter 3 Proportional Relationships 5B Chapter 4 The Relationship Between Graphs and Functions Volume 6 Measurement 6A Chapter 1 How Measurements Are Made 6A Chapter 2 Length and Area in the Real World 6B Chapter 3 Exact Measures in Geometry 6B Chapter 4 Angles and Circles iii

Authors and Consultants AUTHORS

Frances Basich Whitney

Kathleen M. Brown

Dixie Dawson

Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, California

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Math Curriculum Leader Long Beach Unified Long Beach, California

Philip Gonsalves

Robyn Silbey

Kathy Vielhaber

Mathematics Coordinator Alameda County Office of Education Hayward, California

Math Specialist Montgomery County Public Schools Gaithersburg, Maryland

Mathematics Consultant St. Louis, Missouri

Viken Hovsepian Professor of Mathematics Rio Hondo College Whittier, California

Dinah Zike Educational Consultant, Dinah-Might Activities, Inc. San Antonio, Texas

CONSULTANTS Assessment Donna M. Kopenski, Ed.D. Math Coordinator K–5 City Heights Educational Collaborative San Diego, California

iv

Instructional Planning and Support

ELL Support and Vocabulary

Beatrice Luchin

ReLeah Cossett Lent

Mathematics Consultant League City, Texas

Author/Educational Consultant Alford, Florida

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

CONTRIBUTING AUTHORS

California Advisory Board CALIFORNIA ADVISORY BOARD

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

Glencoe wishes to thank the following professionals for their invaluable feedback during the development of the program. They reviewed the table of contents, the prototype of the Student Study Guide, the prototype of the Teacher Wraparound Edition, and the professional development plan.

Linda Anderson

Cheryl L. Avalos

Bonnie Awes

Kathleen M. Brown

4th/5th Grade Teacher Oliveira Elementary School, Fremont, California

Mathematics Consultant Retired Teacher Hacienda Heights, California

Teacher, 6th Grade Math Monroe Clark Middle School San Diego, California

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Carol Cronk

Audrey M. Day

Jill Fetters

Grant A. Fraser, Ph.D.

Mathematics Program Specialist San Bernardino City Unified School District San Bernardino, California

Classroom Teacher Rosa Parks Elementary School San Diego, California

Math Teacher Tevis Jr. High School Bakersfield, California

Professor of Mathematics California State University, Los Angeles Los Angeles, California

Eric Kimmel

Donna M. Kopenski, Ed.D.

Michael A. Pease

Chuck Podhorsky, Ph.D.

Mathematics Department Chair Frontier High School Bakersfield, California

Math Coordinator K–5 City Heights Educational Collaborative San Diego, California

Instructional Math Coach Aspire Public Schools Oakland, California

Math Director City Heights Educational Collaborative San Diego, California

Arthur K. Wayman, Ph.D.

Frances Basich Whitney

Mario Borrayo

Melissa Bray

Professor Emeritus California State University, Long Beach Long Beach, California

Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, CA

Teacher Rosa Parks Elementary San Diego, California

K–8 Math Resource Teacher Modesto City Schools Modesto, California

v

California Reviewers CALIFORNIA REVIEWERS Each California Reviewer reviewed at least two chapters of the Student Study Guides, providing feedback and suggestions for improving the effectiveness of the mathematics instruction. Melody McGuire

Math Teacher California College Preparatory Academy Oakland, California

6th and 7th Grade Math Teacher McKinleyville Middle School McKinleyville, California

Eppie Leamy Chung

Monica S. Patterson

Teacher Modesto City Schools Modesto, California

Educator Aspire Public Schools Modesto, California

Judy Descoteaux

Rechelle Pearlman

Mathematics Teacher Thornton Junior High School Fremont, California

4th Grade Teacher Wanda Hirsch Elementary School Tracy, California

Paul J. Fogarty

Armida Picon

Mathematics Lead Aspire Public Schools Modesto, California

5th Grade Teacher Mineral King School Visalia, California

Lisa Majarian

Anthony J. Solina

Classroom Teacher Cottonwood Creek Elementary Visalia, California

Lead Educator Aspire Public Schools Stockton, California

vi

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

Bobbi Anne Barnowsky

Volume 5A

Functions and Equations

Chapter

Patterns and Relationships

1

1-1 Sort and Classify ...............................................................4. KAF1.1, 1SDAP1.1

1-2 Patterns .............................................................................13 1SDAP2.1, 2SDAP2.1

Progress Check 1 .............................................................20 1-3 Number Relationships ...................................................21 2SDAP2.1, 3AF2.2,

1-4 Solve Equations ...............................................................27 3AF2.1, 4AF1.5

Progress Check 2 .............................................................33 Assessment Study Guide .....................................................................34 Chapter Test .....................................................................38

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

Standards Practice ...................................................40 Dana Meadows near Yosemite National Park

Chapters 1 and 2 are contained in Volume 5A. Chapters 3 and 4 are contained in Volume 5B.

Standards Addressed in This Chapter KAF1.1 Identify, sort, and classify objects by attribute and identify objects that do not belong to a particular group (e.g., all these balls are green, those are red). 1SDAP1.1 Sort objects and data by common attributes and describe the categories. 1SDAP2.1 Describe, extend, and explain ways to get to a next element in simple repeating patterns (e.g., rhythmic, numeric, color, and shape). 2SDAP2.1 Recognize, describe, and extend patterns and determine a next term in linear patterns (e.g., 4, 8, 12 . . . , the number of ears on one horse, two horses, four horses). 3AF2.1 Solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the cost per unit). 3AF2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses may be calculated by counting by 4s or by multiplying the number of horses by 4). 4AF1.5 Understand that an equation such as y = 3x + 5 is a prescription for determining a second number when a first number is given.

vii

Contents Chapter

Graphing

2

Standards Addressed in This Chapter 2-1 Bar Graphs and Picture Graphs ...................................44 1SDAP1.2, 2SDAP1.1, 2SDAP1.2

2-2 Line Plots ..........................................................................53 3SDAP1.3, 2SDAP1.1, 2SDAP1.2

Progress Check 1 .............................................................60 2-3 Ordered Pairs ...................................................................61 5SDAP1.5, 4MG2.0, 5SDAP1.4

2-4 Coordinate Grids ............................................................ 67 4MG2.1, 5SDAP1.4, 5SDAP1.5

Progress Check 2 .............................................................75 Assessment Study Guide .....................................................................76 Chapter Test .....................................................................80 Standards Practice ...................................................82

2SDAP1.1 Record numerical data in systematic ways, keeping track of what has been counted. 2SDAP1.2 Represent the same data set in more than one way (e.g., bar graphs and charts with tallies). 3SDAP1.3 Summarize and display the results of probability experiments in a clear and organized way (e.g., use a bar graph or a line plot). 4MG2.0 Students use twodimensional coordinate grids to represent points and graph lines and simple figures. 4MG2.1 Draw the points corresponding to linear relationships on graph paper (e.g., draw 10 points on the graph of the equation y = 3x and connect them by using a straight line). 5SDAP1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph. 5SDAP1.5 Know how to write ordered pairs correctly; for example, (x, y).

viii

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

Golden Gate Bridge, San Francisco

1SDAP1.2 Represent and compare data (e.g., largest, smallest, most often, least often) by using pictures, bar graphs, tally charts, and picture graphs.

Contents Chapter

Proportional Relationships

3

3-1 Linear Patterns ..................................................................4 3AF2.1, 3AF2.2

3-2 Ratios and Rates ..............................................................11 3AF2.1, 6AF2.1

Progress Check 1 .............................................................18 3-3 Proportional Reasoning..................................................19 3AF2.1, 6NS1.3

Assessment Study Guide .....................................................................26 Chapter Test .....................................................................28 Standards Practice ...................................................30

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

Wind turbines, Altamont

Chapters 1 and 2 are contained in Volume 5A. Chapters 3 and 4 are contained in Volume 5B.

Standards Addressed in This Chapter 3AF2.1 Solve simple problems involving a functional relationship between two quantities (e.g., find the total cost of multiple items given the cost per unit). 3AF2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses may be calculated by counting by 4s or by multiplying the number of horses by 4). 6NS1.3 Use proportions to solve problems (e.g., determine the value of N N 4 if __ = ___, find the length of a side of a 7 21 polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. 6AF2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches.

ix

Contents Chapter

The Relationship Between Graphs and Functions

4

4-1 Introduction to Functions ..............................................34

Standards Addressed in This Chapter

5AF1.5

4-2 Graph Linear and Nonlinear Equations......................41 7AF3.0, 7AF3.1

Progress Check 1.............................................................50 4-3 Direct Variation ...............................................................51 7AF3.3, 7AF3.4

4-4 Slope .................................................................................59 7AF3.3, 7AF3.4

Progress Check 2.............................................................66 Assessment Study Guide ....................................................................67 Chapter Test ....................................................................72 Standards Practice...................................................74

x

7AF3.0 Students graph and interpret linear and some nonlinear functions. 7AF3.1 Graph functions of the y = nx2 and y = nx3 and use in solving problems. 7AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. 7AF3.4 Plot the values of quantities whose ratios are always the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the ratio of the quantities.

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

Redwood National Park

5AF1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.

R E G N E V A SC HUNT Let’s Get Started Use the Scavenger Hunt below to learn where things are located in each chapter. 1 What is the title of Chapter 4?

The Relationship Between Graphs and Functions 2

What is the Key Concept of Lesson 3-1? Linear Patterns

3

According to Lesson 4-2, what is the definition of a function table? a table of ordered pairs that is based on a rule

4

What are the vocabulary words for Lesson 3-3?

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

ratio, proportion, cross multiply 5

How many Examples are presented in the Chapter 3 Study Guide? 4

6

What are the California Standards covered in Lesson 4-3?

7AF3.3, 7AF3.4 7

According to page 29, how many gallons can the aquarium hold? 29 gallons

8

What do you think is the purpose of the Chapter Test on pages 28–29? It allows you to practice problems that are

similar to test items. 9

On what pages will you find the Study Guide for Chapter 4?

pages 67–71 10

In Chapter 4, find the logo and Internet address that tells you where you can take the Online Readiness Quiz. It is

found on page 33. The URL is ca.mathtriumphs.com. 1

Chapter

3

Proportional Relationships These cells are shown larger than they are in real life. This photograph shows red blood cells about 1,700 times larger than the actual cells. To find the actual measurement of these cells, you can set up and solve a proportion.

Copyright © by The McGraw-Hill Companies, Inc.

2

Chapter 3 Proportional Relationships

Kenneth Eward/Photo Researchers, Inc.

STEP

STEP

1 Quiz

2 Preview

Are you ready for Chapter 3? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out. Get ready for Chapter 3. Review these skills and compare them with what you’ll learn in this chapter.

What You Know You know how to add and follow patterns. Example: Your backpack can hold 4 textbooks. If your friend has a backpack just like yours, how many textbooks could you both carry in your backpacks?

8

"EEUFYUCPPLTGPSFBDI BEEJUJPOBMCBDLQBDL

What You Will Learn Lesson 3-1 Patterns follow rules. The rule is “each backpack can hold 4 textbooks.” 1 backpack ⇒ 4 textbooks 2 backpacks ⇒ 8 textbooks 3 backpacks ⇒ 12 textbooks 4 backpacks ⇒ 16 textbooks

Add 4 textbooks for each additional backpack.

Copyright © by The McGraw-Hill Companies, Inc.

You know how to add the prices of objects together to get a total. Example:

Lesson 3-2 To find the price for 5 caps at $15 each, you can think about it in two ways: $15 + $15 + $15 + $15 + $15 = $75 or $15 × 5 = $75

3 Photodisc/Getty Images

Lesson

3-1 Linear Patterns KEY Concept Patterns follow a rule . A rule describes the relationship that one element of a sequence has with the next element of the sequence. A rule can also describe the relationship an element has with its position in the sequence.

+

+

+

+

3AF2.1 Solve simple problems involving a functional relationship between two quantities. 3AF2.2 Extend and recognize a linear pattern by its rules.

VOCABULARY pattern a sequence of numbers, figures, or symbols that follows a rule or design rule tells how the numbers, figures, or symbols in a pattern are related to each other

A table can help you identify and extend a pattern by its rules. +1

+1

+1

+1

+1

Number of Ducks

1

2

3

4

5

6

Number of Feet

2

4

6

8

10

12

+2

+2

+2

+2

For each additional duck, the number of feet increases by 2. Another way to describe the pattern is the number of ducks multiplied by 2 is the number of feet.

When the value of the second variable is determined by the value of the first, the relationship is a functional relationship.

4

Chapter 3 Proportional Relationships

Photodisc/Getty Images

Copyright © by The McGraw-Hill Companies, Inc.

+2

Example 1

YOUR TURN!

Margo rides her bike at 6 miles per hour. How many miles can she travel in 4 hours?

Dawn’s Diner serves 9 cheesesticks in 1 order. How many cheesesticks are in 6 orders?

1. Make a table. 2. For every 1 hour traveled, Margo rides 6 miles. So the number of miles traveled increases by 6 for each additional hour. The rule is add 6 for each hour traveled. 3. Add 6 to a term to obtain the next term in the pattern. +1 +1 +1 Number of Hours

1

Number of Miles Traveled

6

2

3

12

1. Make a table. 2. Each order has 9 cheesesticks. The number of cheesesticks increases by 9 for each additional order. The rule is add 9 for each order. 3. Add 9 to a term to obtain the next term in the pattern. +1 +1 +1 +1 +1

4

18

24

Number of Orders

1

Number of Cheesesticks +6

+6

3

4

5

6

9 18 27 36 45 54

+6 +9 +9 +9 +9 +9

Margo can ride 24 miles in 4 hours. There are

Copyright © by The McGraw-Hill Companies, Inc.

2

54

cheesesticks in 6 orders.

Example 2

YOUR TURN!

For every block, there are 5 apartment buildings. How many apartment buildings are in 8 blocks?

For every car, there are 4 wheels. How many wheels do 10 cars have?

1. For every 1 block, there are 5 apartment buildings. The number of buildings increases by 5 for each additional block. The rule is multiply by 5. 2. Multiply the number of blocks by the number of apartment buildings in each block. 8 × 5 = 40 There are 40 apartment buildings in 8 blocks.

1. For every 1 car, there are 4 wheels. The number of wheels increases by 4 for each additional car. multiply by 4 The rule is . 2. Multiply the number of cars by the number of wheels on each car.

10 number of cars There are

4

×

=

40

number of wheels

40

wheels on 10 cars. GO ON Lesson 3-1 Linear Patterns

5

Who is Correct? How many fingers (including the thumb) do 3 people have?

Lamar

Mayo

Gilda

10 × 1 = 10

3 × 8 = 24

10 × 3 = 30

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Write a possible situation for each rule. 1

Add 2 for each additional person.

Possible answer: The number of ears people have. 2

Multiply by 4 for each additional animal.

Possible answer: The number of legs on a dog or horse. 3

John jogs 2 miles in 20 minutes. How long will it take John to jog 8 miles? The rule is add

20

+2

+2

2

4

6

8

minutes to jog 8 miles.

Number of Minutes

20

40

60

80

2

+20 +20 +20 Step by Step Practice 4

There are 3 feet in a yard. How many feet are there in 5 yards? Step 1 Each yard has 3 feet. The number of feet increases by multiply by 3 additional yard. The rule is .

3

Step 2 Multiply the number of yards by the number of feet in each yard.

5

×

There are

6

3

=

15

feet in 5 yards.

Chapter 3 Proportional Relationships

15

for each

Copyright © by The McGraw-Hill Companies, Inc.

miles.

Number of Miles

for every

80

It will take John

+2

5

Francisco reads at a rate of 30 pages for every 2 hours. How many pages did Francisco read in 4 hours? Francisco read

6

60

pages in 4 hours.

Liz swam at a rate of 18 laps per hour. How many laps will Liz swim in 3 hours? Liz will swim

54

laps in 3 hours.

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Use logical reasoning.

Solve. 7

Understand

Read the problem. Write what you know.

6

Copyright © by The McGraw-Hill Companies, Inc.

Larisa sold

tickets each hour.

Plan

Pick a strategy. One strategy is to use logical reasoning. Multiply the number of tickets Larisa sold each hour by the number of hours.

Solve

Multiply.

6

5

×

number of tickets Larisa sold

number of hours

30

30

=

total tickets

tickets in 5 hours.

Skip count by 6 five times.

Check

6

8

Guess and check. Act it out. Solve a simpler problem. Work backward.

FUND-RAISING Larisa sold raffle tickets for a fund-raiser. She sold the tickets at a rate of 6 tickets each hour. How many tickets did Larisa sell in 5 hours?

+

6

+

6

+

6

+

6

=

30

NATURE How many minutes will it take for Lupe to climb out of a hole that is 27 inches deep? 9 Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

NATURE Lupe the Ladybug is able to climb 9 inches every 3 minutes.

GO ON Lesson 3-1 Linear Patterns

Ladybugs

7

9

FOOD Drew and five friends went to the ice cream parlor. They each ordered two scoops of ice cream. How many scoops of ice cream were served to the group altogether?

12 10

Explain two ways to find the number of ears on 5 students.

The rule is to add 2: 2 + 2 + 2 + 2 + 2 = 10 or the rule is to multiply by 2: 5 × 2 = 10. Five students have 10 ears.

Skills, Concepts, and Problem Solving Write a possible situation for each rule. 11

Add 6 for each additional desk.

Possible answer: the number of pencils in each desk 12

Multiply by 9 for each additional tree.

Possible answer: the number of apples on each tree 13

The rule is add

6

Jeremy walked

18

1

for every

laps in 3 hours. +1

+1

+1

Number of Hours

1

2

3

4

Number of Laps

6

12

18

24

+6 +6 +6

8

Chapter 3 Proportional Relationships

hour(s).

Copyright © by The McGraw-Hill Companies, Inc.

TRACK Jeremy walked at a rate of 6 laps per hour on the track. How many laps did Jeremy walk in 3 hours?

14

TRAVEL Careta traveled at a rate of 70 miles for every 2 hours. How many miles did Careta travel in 6 hours? 210 miles

+2 + 2 + 2 Number of Hours

2

Number of Miles

70 140 210 280

4

6

8

+70 +70 +70 Solve. 15

FOOD Sherita is selling lemonade for $1 per glass. In the first hour of business, she sold 10 glasses. How much money did Sherita make?

$10 16

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GEOMETRY Abner had to cut 7 triangles out of a sheet of construction paper. How many sides did Abner cut out of the construction paper?

21

Copyright © by The McGraw-Hill Companies, Inc.

17

FOOD The chicken-finger appetizer at the Chick Pantry comes with 4 pieces of chicken. The cooks had 6 orders at the same time. How many pieces of chicken did they have to prepare?

24 pieces 18

SLEEP It is recommended that a person get at least 8 hours of sleep each night. What is the least number of hours a person should sleep in one week?

56 hours Vocabulary Check Write the vocabulary word that completes each sentence.

rule

19

A(n)

tells how numbers are related to each other.

20

A(n) pattern is a sequence of numbers, figures, or symbols that follows a rule or design. GO ON Lesson 3-1 Linear Patterns

9

21

Writing in Math Explain how to find the number of legs that 6 octopuses have.

Multiply the number of octopuses by the number of legs. Each octopus has: 8 × 6 = 48. Six octopuses have 48 legs. An octopus has 8 legs.

Spiral Review Solve. 22

(Lesson 2-4, p. 67)

KITTENS A kitten gained 3 ounces every week. How many weeks did it take the kitten to gain 15 ounces? Complete the input/output table.

Input, x

Output, y

1

3 6 9 12 15

2 3

5

4 5

3(1) = 3 3(2) = 6 3(3) = 9 3(4) = 12 3(5) = 15

The line plot shows the results of spinning a spinner 50 times. Use the line plot to answer each question. (Lesson 2-2, p. 53) 4QJOOFS3FTVMUT

CMVF

SFE

SFE

HSFFO

ZFMMPX

23

Which color did the spinner land on the least number of times?

24

How many times did the spinner land on blue?

25

Which color did the spinner land on 16 times?

green

26

How many times did the spinner land on red?

11

10

Chapter 3 Proportional Relationships

Jeffrey L. Rotman/Peter Arnold, Inc.

14

yellow

Copyright © by The McGraw-Hill Companies, Inc.

Lesson

3-2 Ratios and Rates 3AF2.1 Solve simple problems involving a functional relationship between two quantities. 6AF2.1 Convert one unit of measurement to another.

KEY Concept A ratio can be used to compare two quantities. If there are 12 apples and 4 people who want to eat the apples, 12 apples then the ratio of apples to people is _________. 4 people In the example above, since apples and people are different, 12 apples 3 12 = __ , the ratio of _________ is an example of a rate . Since ___ 4 1 4 people the unit rate is 3 apples to 1 person.

APPLESINEACHBAG #AGOFAPPLES

The unit price is the price of a single bag of apples. The unit price of the apples is $6 for 1 bag.

VOCABULARY ratio a comparison of two quantities by division rate a ratio of two measurements or amounts made with different units unit price the price of a single piece or item

Every time Mr. Smith buys another bag of apples, it costs an additional $6, and he gets 12 more apples.

unit rate a rate in which the second amount is 1

There are two ways to find the price for 5 bags of apples: $6 + $6 + $6 + $6 + $6 = $30

or

5 × $6 = $30

Copyright © by The McGraw-Hill Companies, Inc.

There are two ways to find the number of apples in 5 bags: 12 + 12 + 12 + 12 + 12 = 60

or

5 × 12 = 60

The cost for 5 bags of apples is $30, and the number of apples in 5 bags is 60. :PVDBOVTFSFQFBUFE BEEJUJPOPSTLJQDPVOUJOH JOTUFBEPG NVMUJQMZJOH

To find a total price, multiply the unit price by the number of items. To find a unit cost, divide the total price by the number of items.

GO ON Lesson 3-2 Ratio and Rates

11

Example 1

YOUR TURN!

Find the unit price.

Find the unit price.

Randy spent $46 to buy 4 DVDs. Each DVD had the same price. Find the unit price for a DVD.

Fina spent $18 to buy 8 notebooks. Find the unit price for a notebook.

1. The price was $46 for 4 DVDs. 2. Divide the total price by the number of DVDs. 46 ÷ 4 = $11.50 The unit price for a DVD was $11.50.

1. The price was $ 18

8

for

notebooks.

2. Divide the total price by the number of notebooks.

18

÷

8

= $2.25

The unit price for a notebook was $2.25 .

Example 2

YOUR TURN!

Find the total price.

Find the total price.

Anessa bought 6 pairs of shorts for soccer practice. Each pair of shorts cost $8. How much did Anessa spend?

Vadim bought 5 pounds of nuts. Each pound of nuts cost $2. How much did Vadim spend?

1. The unit price was $8 for 1 pair of shorts.

1. The unit price is $ of nuts.

8 + 8 + 8 + 8 + 8 + 8 = 48 Anessa spent $48 for 6 pairs of shorts.

2. Add

2

for 1 pound

, or skip-count by

5

times.

2

+

2

2

=

10

+

2

+

2 2

,

+

Vadim spent $10 for 5 pounds of nuts.

12

Chapter 3 Proportional Relationships

Copyright © by The McGraw-Hill Companies, Inc.

2. Add 8, or skip-count by 8, six times.

2

Example 3

YOUR TURN!

Convert.

Convert.

One inch is equal to 2.54 centimeters. How many centimeters are in 36 inches?

There are 5,280 feet in one mile. How many miles are in 26,400 feet?

1. The rate of centimeters to inches is 2.54 cm to 1 in.

1. The rate of feet to 1 mi . to

2. Multiply the centimeters in an inch by the number of inches.

2. Divide the

2.54 × 36 = 91.44 There are 91.44 centimeters in 36 inches.

feet

miles

feet

is 5,280 ft

by the number of

in a mile.

26,400 ÷ 5 There are

5,280

=

5

miles in 26,400 feet.

Who is Correct? The unit price for a box of cereal is $4. What is the cost of 3 boxes of cereal?

Melissa

Alvin

Mateo

$4 + 3 = $7

$4 + $4 + $4 = $12

$3 + $3 + $3 = $9

Copyright © by The McGraw-Hill Companies, Inc.

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Write each ratio. 1

What is the ratio of baseballs to footballs?

2:3 (or 2/3; 2 to 3) 2

What is the ratio of squares to circles?

3:5 (or 3/5, 3 to 5) 3

The rate of feet to inches is 1 ft

to 12 in. .

GO ON Lesson 3-2 Ratio and Rates

Photodisc/Getty Images

13

Find each unit price. 4

Juan bought 10 pencils for $2.50.

5

The unit price is $ 0.25 for 1 pencil.

Beng bought 12 picture frames for $60. The unit price is $

5

for 1 frame.

To find the unit price, divide the total price by the number of pencils.

Step by Step Practice Find the total price. 6

Hinto bought 7 video games. Each video game cost $9. How much did Hinto spend? Step 1 Find the unit price. The unit price is $

9

for 1 video game.

Step 2 Multiply the number of video games by the cost per video game.

7

×

9

=

63

Hinto spent a total of $ 63 .

7

Copyright © by The McGraw-Hill Companies, Inc.

Find each total price. Carpet was on sale at The Rug Factory. Keisha bought 20 square yards of carpet. Each square yard of carpet cost $15. How much did Keisha spend? The unit price is $ 15 for 1 square yard of carpet. Multiply the number of square yards by the cost per square yard.

20

×

15

= 300

Keisha spent $ 300 for 20 square yards of carpet. 8

Pedro bought 5 gallons of gasoline. How much did Pedro spend? The unit price is $ 2 for 1 gallon of gasoline. Multiply the number of gallons by the cost per gallon.

5

×

2

=

Pedro spent $ 10 14

10 for 5 gallons of gasoline.

Chapter 3 Proportional Relationships

QFS HBMMPO

9

FITNESS Tora jogs 3 miles each day. How many miles will Tora jog in 9 days? Tora will jog

10

27

miles in 9 days.

The city where Carlota lives averages 4 centimeters of rain each month. How much rain does Carlota’s city get in 6 months? Carlota’s city gets

24

centimeters of rain in 6 months.

Convert each unit of measure. 11

1 km = 0.62 mi; 12 km = 7.44 mi

13

1 gal = 4 qt; 6 gal =

24

qt

1 ft = 12 in.; 3 ft =

14

1 kg = 2.20 lb; 10 kg =

Step by Step Problem-Solving Practice

Copyright © by The McGraw-Hill Companies, Inc.

6

for one T-shirt.

Pick a strategy. One strategy is to look for a pattern.

$6

more.

Multiply the number of shirts by the unit price.

8

×

$6

= $48

Fedele spent $ 48 Check

Guess and check. Solve a simpler problem. Work backward.

Read the problem. Write what you know.

Each additional shirt cost Solve

lb

Draw a diagram.

The unit price was $ Plan

22

✓ Look for a pattern.

SHOPPING Fedele bought T-shirts for his volleyball team. He bought 8 shirts for $6 each. How much did Fedele spend on volleyball shirts? Understand

in.

Problem-Solving Strategies

Solve. 15

36

12

for 8 T-shirts.

Skip-count by 6 eight times.

6 6

+ +

6 6

+ +

6 6

+ =

6 48

+

6

+

GO ON Lesson 3-2 Ratio and Rates

15

16

CARS Refer to the photo caption at the right. How many cars can 3 parking lots hold? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

Three parking lots can hold 450 cars. 17

can hold 150 cars.

Mr. Jenkins is ordering pizza for a party. He estimates each guest will eat 3 slices of pizza. There will be 18 guests. Each pizza will have 8 slices. How many pizzas should Mr. Jenkins order? He needs

54

He should order 18

CARS One parking lot

slices of pizza.

7

pizzas for 18 guests.

The unit price of a basketball is $18. Explain the meaning of the unit price.

The unit price means that each basketball costs $18.

Skills, Concepts, and Problem Solving 19

Presta bought 7 envelopes for a total of $0.84.

David bought 25 gumballs for $2.

20

The unit price is $ 0.12 for 1 envelope.

The unit price is $ 0.08 for 1 gumball.

Find each total. Use the table to answer Exercises 21–24. 21

How much do 16 loaves of wheat bread cost?

22

How much do 8 loaves of rye bread cost?

23

How much do 5 loaves of white bread cost?

24

$24 $5

How much do 5 loaves of white bread, 2 loaves of rye bread, and 6 loaves of wheat bread cost? Explain how you found your answer.

See TWE margin. 16

$32

Chapter 3 Proportional Relationships

(t)John Flournoy, (b)Kevin Sanchez/Cole Group/Getty Images

#AKERY#READ4ALE 5YPEOF#READ 8HITE#READ 8HEAT#READ 3YE#READ

$OSTPER-OAF

Copyright © by The McGraw-Hill Companies, Inc.

Find each unit price.

Convert each unit of measure. 25

1 in. = 2.54 cm; 5 in. = 12.7 cm

27

1 lb = 16 oz; 3 lb =

29

1 km = 0.62 mi; 4 km = 2.48 mi

48

oz

26

1 m = 100 cm; 350 cm = 3.5 m

28

1 c = 8 fl oz; 10 c =

30

1 kg = 2.20 lb; 5 kg =

80

fl oz

11

lb

Vocabulary Check Write the vocabulary word that completes each sentence.

unit rate

31

A(n) is 1.

32

A

33

Writing in Math Marlon bought 16 pens for $20. Explain how to find the unit price.

unit price

is a rate in which the second amount is the price of a single piece or item.

Divide the cost by 16: 20 ÷ 16 = 1.25. Each pen cost $1.25.

Spiral Review

Copyright © by The McGraw-Hill Companies, Inc.

34

GAMES Carlos and Roza are playing tic-tac-toe. It took 8 seconds each time they marked a square on the board. There were 8 marks made altogether. How many seconds did it take to finish the game?

64 seconds

(Lesson 3-1, p. 4)

For Exercises 35–37, use the graph shown at the right. Name the ordered pair for each point. (Lesson 2-3, p. 61) 35

X

(-8, 4)

36

Y

(5, -6)

37

Z

(-6, -7)

Y

0

[

Solve each equation when x = 1, 5, and –8. 38

Z

Y Z

(Lesson 1-4, p. 27)

y = –2x + 4 x=1 y=

x=5

2

y=

x = –8

-6

y=

20 Lesson 3-2 Ratio and Rates

17

Chapter

3

Progress Check 1

(Lessons 3-1 and 3-2)

Find each unit price. 3AF2.1, 3AF2.2 1

Samson bought 7 books for $112. The unit price is $ 1 book.

2

McKenzie bought 5 dolls for $50. The unit price is $ 1 doll.

16

for

10

for

Solve. 3AF2.1, 3AF2.2 3

HEALTH Sari sleeps 9 hours each night. How many hours does she sleep in one week? Sari sleeps

4

63

hours in one week.

MEASUREMENT One bucket can hold 8 gallons of water. How many gallons of water can 7 buckets hold?

56

Seven buckets can hold 5

FITNESS Steve exercises 0.75 hour each day. How many hours will he exercise in 8 days? Steve will exercise

6

gallons of water.

6

hours in 8 days.

SHARKS About how many inches does a great white shark grow in 7 years?

7

8

SHARKS Great white sharks grow inches

about 10 inches per year.

TRAVEL Mr. Kim drove 320 miles on 16 gallons of gasoline. How many miles did he drive on 1 gallon of gasoline? 20 miles

GROCERIES Kelsey bought Box A. Anita bought Box B. Who bought the box of granola bars that is Anita less expensive per ounce?

PVO

DFT

PVODF

T

#PY"

#PY#

Convert each unit of measure. 6AF2.1

216

9

1 yd = 36 in.; 6 yd =

10

1 day = 24 h; 7 days =

18

Chapter 3 Proportional Relationships

Corbis

168

in. h

11

1 gal = 128 fl oz; 5 gal =

12

1 L = 1,000 mL; 7.5 L =

640 7,500

fl oz mL

Copyright © by The McGraw-Hill Companies, Inc.

70

A great white shark grows about in 7 years.

Lesson

3-3 Proportional Reasoning

3AF2.1 Solve simple problems involving a functional relationship between two quantities. 6NS1.3 Use proportions to solve problems. Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.

KEY Concept 2 are equivalent or equal. 4 and __ The ratios __ 6 3 This equation is an example of a proportion .

2 4 = __ __ 6

3

VOCABULARY

The cross products of a proportion are equal. Cross multiply to solve proportions when one value in the proportion is not known. 5 · 6 is one cross product. 5 __

=

3

6

5·6=3·x 30 = 3x 10 = x

(Lesson 3-2, p. 11)

proportion an equation stating that two ratios or rates are equivalent

3x is the other cross product.

x __

ratio a comparison of two quantities by division

Cross multiply. Simplify. Divide each side of the equation by 3.

cross multiply find the product of the numerator of one fraction and the denominator of the other fraction

A proportion is a comparison of ratios. It can be written two ways. Both are read “5 is to 3 as x is to 6.”

Copyright © by The McGraw-Hill Companies, Inc.

Proportions always have equal signs.

Example 1

YOUR TURN!

Find the value of y.

Find the value of c. 6 c = __ ___

8 4 ___ = __ 12

8 · y = 12 · 4 8y = 48 8y ___ 48 ___ = 8

8

y=6

5

10

y

Cross multiply. Simplify. Divide each side of the equation by 8. Simplify.

c

·

5

= 10 ·

5c = 60 5c _ 5

=

60 _ 5

c = 12

6

Cross multiply. Simplify. Divide each side of the equation by 5 . Simplify.

GO ON Lesson 3-3 Proportional Reasoning

19

Example 2 Yolanda rode her bike 15 miles in 2 hours. How many miles can she ride in 8 hours? Set up a proportion. Let x represent the number of miles Yolanda can ride in 8 hours. 15 miles x miles ________ = _______ 2 hours

8 hours

15 · 8 = 2 · x 120 = 2x 120 ___ ____ = 2x 2 2 x = 60

Cross multiply. Simplify. Divide each side of the equation by 2. Simplify.

Yolanda can ride 60 miles in 8 hours. YOUR TURN! A farmer claims that 10 cows can be raised on 4 acres of land. How many acres of land would the farmer need to raise 15 cows? Set up a proportion. Let x represent the number of acres of land needed to raise 15 cows. 10 cows _________ ________ = 15 cows

x acres

4 acres

10

·

x = 10x = 10

15

Divide each side of the equation by

10 6

x=

6

Cross multiply. Simplify.

60 _

=

The farmer needs

·

Simplify. acres of land to raise 15 cows.

Who is Correct? Find the value of m.

3 4 __ = ___ m

12

Nancy

Saber

Mariah

3m = 4 · 12 3m = 48 m = 16

3 · 4 = m · 12 12 = 12m 12 12 1 m=

3 · 12 = m · 4 36 = 4m 4 4 m=9

_ _

Circle correct answer(s). Cross out incorrect answer(s). 20

Chapter 3 Proportional Relationships

_ _

10 .

Copyright © by The McGraw-Hill Companies, Inc.

10x _

4 60

Guided Practice Find the value of each variable. n 7 1 __ = __ n = 14 8 4

2

6 __ __ =h 3

9

h=

18

Step by Step Practice 3

Berto made a scale model of his bedroom. His bedroom is 10 feet long. What is the width of Berto’s bedroom?

8JEUIJODIFT

Step 1 Set up a proportion. Let x represent the width of Berto’s bedroom.

12 inches 8 inches _________ ________ =

model length actual length

x feet

10 feet

-FOHUIJODIFT

model width actual width

.Y#EDROOM

Step 2 Solve the proportion.

8

·

x = 10 · 8x = 120 8x 120 = 8 8 x = 15

_

Copyright © by The McGraw-Hill Companies, Inc.

Step 3 Berto’s bedroom is

4

12

_

15

Cross multiply. Simplify. Divide each side of the equation by 8 . Simplify.

feet wide.

GAMES A game board has 9 blue sections for every 15 red sections. How many blue sections are there if the game board has 45 red sections? Let x represent the number of blue sections.

x blue 9 blue _______ _______ = 15 red

9

·

45

45 red = 15 ·

x

405 = 15x 405 _ 15

=

x= There are

27

15x _ 15

27

blue sections for a game board with 45 red sections.

GO ON

Lesson 3-3 Proportional Reasoning

21

5

CHESS There are 5 boys to every 2 girls in the chess club. This year there are 20 boys in the chess club. How many girls are in the club? There are

6

8

girls in the chess club.

PARADES Jessy is in charge of buying paint for the float that will be in the Fall Harvest Parade. Two cans of red paint and 3 cans of yellow make 5 cans of orange paint. She has 9 cans of yellow paint. How many cans of red paint does Jessy need to make orange paint? Jessy needs

6

cans of red paint.

Step by Step Problem-Solving Practice

Problem-Solving Strategies

Solve. 7

SOFTBALL A 4-team softball league has 32 players. If the softball league grows to 9 teams, how many players will there be in the league? Understand

Draw a diagram. Look for a pattern. ✓ Write an equation. Act it out. Work backward.

Read the problem. Write what you know. There are 4 teams with 32 players. You need to find out how many players there would be with 9 teams.

Plan

4 teams 9 teams __________ = ______________

x

32 players

players

Cross multiply and solve.

4

·

x 4x

32 · = 288 4x = 288 4 4 x = 72

_

=

9

_

A softball league with 9 teams has Check

players.

Compare the ratios. In simplest form, both should be equal. 4 teams 9 teams __________ __________ 32 players

22

72

72 players

Chapter 3 Proportional Relationships

Are the ratios equal? yes

Copyright © by The McGraw-Hill Companies, Inc.

Solve

Pick a strategy. One strategy is to write an equation. Let x represent the number of players in the league if the softball league grows to 9 teams. Set up a proportion.

8

9

PACKAGING A small package has 3 bars with nuts and 5 bars with raisins. Keishon bought a large box that has 9 bars with nuts. If the ratio of bars with nuts to bars with raisins remains the same, how many bars with raisins are in the large box? 15 Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

GROCERIES A bag of 5 pounds of potatoes costs $3. Gage needs 20 pounds of potatoes. How much will it cost Gage to buy 20 pounds of potatoes? $12 Is 28 the solution to the proportion below? Explain.

10

40 10 ___ ___ = 7

x

Yes; 10 × x = 7 × 40; 10x = 280; x = 28.

Skills, Concepts, and Problem Solving

Copyright © by The McGraw-Hill Companies, Inc.

Find the value of each variable. 11

k = ___ 12 __

k=

6

12

5 ___ __ = d

d=

50

13

13 ___ ___ = z

z=

39

14

18 2 = ___ __

j=

4

15

6 12 = __ ___

g=

7

16

8 r = __ ___

r=

32

3

5

14

6

15

g

7

j

16

70

36

4

GO ON Lesson 3-3 Proportional Reasoning

23

Find the value of each variable. 17

9 1 = __ __

h=

27

19

5 m = __ ___

m=

21

3 __c __ =

3

h

63

9

8

4

18

3 ___ __ = t

t=

15

35

20

56 ___ ___ = 14

z=

6

c = 1.5

22

2 = __ 1 __

n = 2.5

4

20

z

24

5

n

Solve. 23

ARCHITECTURE The distance from Ginger’s family room to the laundry room is 24 feet. How many centimeters is this on Ginger’s drawing?

9 cm

ARCHITECTURE

24

SHOPPING Lora purchased decorations online. The decoration company charges $2 for shipping on every $15 spent. How much will Lora’s total be after the shipping charge is added?

$10 25

One gallon of milk costs $2.99. How much do three gallons cost?

$8.97 26

One dozen eggs costs $0.99. How much do four dozen eggs cost?

$3.96 27

One dozen eggs costs $0.99. What is the unit price of one egg to the nearest cent?

$0.08 24

Chapter 3 Proportional Relationships

Image Source/SuperStock

Copyright © by The McGraw-Hill Companies, Inc.

On Ginger’s drawing, 3 centimeters equal 8 feet.

Vocabulary Check Write the vocabulary word that completes each sentence. 28

proportion A(n) rates are equivalent.

29

cross multiply , find the product of the numerator of one To fraction and the denominator of the other fraction.

30

is an equation stating that two ratios or

Writing in Math Explain how to solve for a variable in a proportion.

Cross multiply to set up an equation. Simplify both sides of the equation. Divide both sides of the equation by the coefficient of the variable.

Spiral Review Solve. 31

(Lesson 3-2, p. 11)

SHOPPING One dress costs $25. How much do 5 dresses cost?

125

Five dresses cost $ 32

FITNESS Jairo swims 4 laps each day. How many laps will Jairo swim in 12 days? Jairo will swim

Copyright © by The McGraw-Hill Companies, Inc.

.

48

laps in 12 days.

The line plot shows the results of rolling a number cube 40 times. Use the line plot to answer each question. (Lesson 2-2, p. 53)

33

Which number did the number cube land on the least number of times?

3 34

How many times did the number cube land on 6?

9 Lesson 3-3 Proportional Reasoning

25

Chapter

Study Guide

3

Vocabulary and Concept Check cross multiply, p. 19

Write the vocabulary word that completes each sentence.

pattern, p. 4

1

proportion, p. 19 rate, p. 11

2

ratio, p. 11 rule, p. 4 unit price, p. 11

rule A(n) tells how the numbers, figures, or symbols in a pattern are related to each other. A(n) or item.

unit price

is the price of a single piece

3

pattern A(n) is a sequence of numbers, figures, or symbols that follows a rule or design.

4

rate A(n) is a ratio of two measurements or amounts made with different units.

unit rate, p. 11

Label each diagram below. Write the correct vocabulary term in each blank.

proportion

5

8 2 = __ __ y

3

cross multiply

6

5 x ; 30 = 3x __ __ 3

6

ratio

7

3 tennis balls :1 can

Lesson Review Linear Patterns (pp. 4–10)

Solve. 8

A spider has 8 legs. How many legs do 4 spiders have?

Janice works at her father’s business for $8 per hour. How much money will Janice earn if she works for 5 hours? Make a table.

Joselyn can run 1 mile in 7 minutes. If she keeps a constant pace, how many miles can she run in 35 minutes?

For every 1 hour worked, Janice earns $8. So the amount of money earned increases by $8 for each additional hour. The rule is add 8 for each hour worked. +1 +1 +1 +1

32

9

5

10

Ross and his five friends each bought three comic books. How many comic books did they buy in all?

18

26

Example 1

Chapter 3 Study Guide

Number of Hours

1

2

3

4

5

Amount Earned

8

16 24 32 40

+8 +8 +8 +8 Janice will earn $40 for working 5 hours.

Copyright © by The McGraw-Hill Companies, Inc.

3-1

3-2

Constant Rate (pp. 11–17)

Find the total price. 11

Rebecca bought 8 gallons of gasoline. Each gallon of gasoline cost $3. How much did Rebecca spend?

Find the unit price. Karina spent $19.96 to purchase 4 T-shirts that each had the same price. Find the unit price for a T-shirt.

$24 Find each unit price. 12

Example 2

Malik spent $7.50 to buy 3 packages of trading cards that each had the same price. Find the unit price for a package of trading cards.

The price was $19.96 for 4 T-shirts. Divide the total price by the number of T-shirts. $19.96 ÷ 4 = $4.99 The unit price for a T-shirt is $4.99.

The unit price for a package of trading $2.50 . cards is Convert. 13

Example 3

One mile is equal to 1,760 yards. How many yards are in 4 miles?

7,040 yd

Copyright © by The McGraw-Hill Companies, Inc.

14

One cup is equal to 8 fluid ounces. How many ounces are in 9 cups?

72 fl oz

Convert. How many ounces are in 6 pounds? The rate of ounces to pounds is 16 ounces to 1 pound. Multiply the ounces in a pound by the number of pounds. 16 × 6 = 96 There are 96 ounces in 6 pounds.

3-3 15

Proportional Reasoning (pp. 19–25)

Find the value of d. 12 d = ___ __ 60

5

d= 16

1

Find the value of h. 4 12 = __ ___ 15

h=

h

5

Example 4 Find the value of t. 6 __t __ =

9 3 6·3=9·t 18 = 9t 9t 18 __ ___ = 9 9

Cross multiply. Simplify. Divide each side of the equation by 9.

t=2 Chapter 3 Study Guide

27

Chapter

Chapter Test

3

State the rule. Then find the solution for each of the following. 3AF2.1, 3AF2.2 1

How many tires do four “18-wheeler” tractor trailer trucks have?

See TWE margin. 2

How many legs do 7 cats have?

See TWE margin. Find the total price. 3AF2.1 3

Stacey bought 5 bottles of hairspray. Each bottle of hairspray cost $2.00. How much did Stacey spend?

$10.00

4

I took my friend to lunch. We each ordered the special for $6.00. What was the total bill?

5

The unit price is

6

Alma bought 6 of the same granola bar for $4.50. The unit price is $0.75 for 1 granola bar.

/P

/P

/P

/P

/P

/P

/P

P /P

/P /P

/P /

for 1 pencil. 1FODJMT

$0.15

/P

Find each unit price. 3AF2.1

/P

$12.00

7

9

1 ft = 12 in.; 108 9 ft =

8

in.

1 pint = 16 ounces; 48 3 pints =

10

ounces

Find the value of each variable. 6AF1.3 f 3 11 ___ = ___ 17 51 f=

1

12

1 week = 7 days; 42 6 weeks =

days

1 kilogram = 1,000 grams; 7,000 grams 7 kilograms =

5 20 __ ___ = y

y=

9

36 GO ON

28

Chapter 3 Test

Copyright © by The McGraw-Hill Companies, Inc.

Convert each unit of measure. 6AF1.2

Solve. 3AF2.1, 3AF2.2, 6AF1.3 13

FITNESS Eric jogs 1.5 miles each day. How many miles does he jog in 2 weeks?

21 miles

Eric jogs 14

FISH

How many gallons of water can 11 of these aquariums hold?

Eleven aquariums hold 15

miles in 2 weeks.

319

gallons of water.

CONSTRUCTION Ruben made a scale model of his deck. The scale measures 18 inches long and 12 inches wide. Ruben’s actual deck is 27 feet long. What is the actual width of Ruben’s deck?

FISH One aquarium can hold 29 gallons of water.

18 feet wide 16

ENTERTAINMENT Mireya and four of her friends went to the movie theater. They each purchased a ticket for $7. What was the total cost of all of their tickets?

$35 17

FOOD Mireya and her four friends each bought three items at the concession stand. How many items did the group purchase in all?

15 items

Copyright © by The McGraw-Hill Companies, Inc.

18

POPULATION The population of Scottville grows by approximately 300 citizens every 4 years. By about how many citizens will the population of Scottville grow in 12 years?

approximately 900 Correct the mistakes. Sheila and Sam are making a casserole. The recipe calls for 6 ounces of chopped almonds. Sheila and Sam want to double the recipe. At the supermarket, almonds only come in 8-oz packages for $2. Sheila says, “The unit cost of the almonds is $0.25 per ounce. Six ounces of almonds will only cost $1.50.” 19

What two mistakes did Sheila make?

See TWE margin. 20

How many ounces of almonds should Sheila and Sam buy? How much will they cost?

See TWE margin.

Chapter 3 Test Manchan/Getty Images

29

Chapter

Standards Practice

3

Choose the best answer and fill in the corresponding circle on the sheet at right. 1

2

Input (x)

Output (___)

17

11

33

27

49

43

A x-5

C x+8

B x-6

D x÷2

5

What is the missing output term for this function table? 3AF2.2 Input (x)

Output (x ÷ 3)

246

82

201

?

153

51

Luis is playing a video game. He earns 530 points for every level he completes. If he has made it through 21 levels, how many points has Luis earned? 6AF1.3 F 551 points

H 10,670 points

G 1,590 points

J 11,130 points

Brandon can travel 270 miles on 9 gallons of gas. At this rate, how many miles can he travel on 5 gallons of gas? A 30 miles

C 150 miles

B 120 miles

D 270 miles

6AF1.3, 3AF2.1 6

Music CDs are on sale this week, 3 for $29.97. How much will 7 CDs cost during this sale? 6AF1.2, 3AF2.1 F $44.96

F 31

H 102

G 67

J 133

G $59.94 H $64.93 J $69.93

Paige’s family is gathered in the living room for family game night. She has a large family and a few dogs. If there is a total of 24 legs in this room, how many humans and how many dogs are there? A 6 humans, 3 dogs

3AF2.1

7

SudzMax is priced at $14.99 for 200 ounces. What is this cleaner’s unit price? 3AF2.1 A $0.75 per ounce B 7.5¢ per ounce

B 4 humans, 2 dogs

C 15¢ per ounce

C 3 humans, 6 dogs

D $7.50 per ounce

D 5 humans, 4 dogs 30

4

Chapter 3 Standards Practice

GO ON

Copyright © by The McGraw-Hill Companies, Inc.

3

What is the output rule for this function table? 3AF2.2

8

Blanca read a 240-page book in 8 days. She read the same number of pages each day. How many pages did she read each day? 4NS3.4

10

Which ordered pair does not fall on the line for the equation y = x + 1? 4MG2.1 G Y

F 28 G 30 H 32 J 34

Y

9

Paco is reporting the results of the class presidential election. Which sentence can he include in his report? 1SDAP1.2

,RJ?BOäLCä4LQBP

Copyright © by The McGraw-Hill Companies, Inc.

!ERIF

.OFV>

H (4, 6)

G (3, 4)

J (2, 3)

ANSWER SHEET Directions: Fill in the circle of each correct answer.

#IB@QFLKä0BPRIQP

F (8, 9)

S>

!>KAFA>QBP

A Chuli and Ava received more than half the total votes. B Ava received the fewest number of votes.

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C Chuli received half as many votes as Priya. D A total of 178 students voted in this election.

Success Strategy Watch for answer choices that are very close. One of them is probably correct. Be careful of calculation errors. Chapter 3 Standards Practice

31

Chapter

4

The Relationship Between Graphs and Functions You can use functions to help you answer important questions. Let’s say you want to make some money this summer. You plan to work 15 hours a week at the library. You will be paid $10 per hour. How much money will you make after two weeks? After four weeks? By the end of the summer?

Copyright © by The McGraw-Hill Companies, Inc.

32

Chapter 4 The Relationship Between Graphs and Functions

Jeff Maloney/Getty Images

STEP

STEP

1 Quiz

Are you ready for Chapter 4? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.

2 Preview

Get ready for Chapter 4. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know how to substitute a value for a variable to evaluate an expression. Example:

4x + 5 = 4(2) + 5 = 8 + 5 = 13

x=2

TRY IT! Evaluate each expression when x = 2. 1

x+2-1= 3

2

3x - 2 = 4

Copyright © by The McGraw-Hill Companies, Inc.

You know how to graph ordered pairs on a coordinate grid. Example: Graph the ordered pair (5, -3).

Z

o

o o o 0 o o o

Y

TRY IT! Graph each ordered pair. 3

(0, 4)

4

(-1, 5)

5

(3, 6)

6

(2, 7)

o

Lesson 4-1 You can make a table to show the values of x and y (the solution). For the expression 4x + 5, you can substitute 1, 2, 3, 4, and 5 for x. You will get 9, 13, 17, 21, and 25 for y. x y

1 9

2 13

3 17

4 21

5 25

Lesson 4-2 If you graph the ordered pairs listed below and connect the points, the graph is a smooth curve. This represents a nonlinear function. (1, 1)

(4, 16)

(2, 4)

(5, 25)

Z

(3, 9)

Y

33

Lesson

4-1 Introduction to Functions KEY Concept A function is a set of ordered pairs that are related to each other by a rule. The function assigns exactly one output value to each input value using a rule. A function table uses the function, or rule, to create ordered pairs. Function Table Function Input, x 2x Ouput, y y = 2x 0 0 0·2 1

1·2

2

-1

-1 · 2

-2

2

2·2

4

-2

-2 · 2

-4

5AF1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.

VOCABULARY function a relationship in which one quantity depends upon another quantity (for every x-value, there is exactly one y-value) function table a table of ordered pairs that is based on a rule linear function a function whose graph is a line

x is the input value. y is the output value.

The graph of a linear function is a straight line. Graph: Linear Function

Z

Z Y

Y

Given a function, you can create a function table. Use the ordered pairs in the function table to graph points. Then connect the points to make the graph of a function.

34

Chapter 4 The Relationship Between Graphs and Functions

Copyright © by The McGraw-Hill Companies, Inc.

Example 1 Write a function and make a function table. AGES

Jodi is 3 years older than Max.

1. The function to describe the situation is: y = x + 3 Where x = Max’s age and y = Jodi’s age. 2. Make a function table using the rule y = x + 3. Max’s Age, x

1

2

3

4

5

6

7

x+3

1+3

2+3

3+3

4+3

5+3

6+3

7+3

Jodi’s Age, y

4

5

6

7

8

9

10

3. How old will Max be when Jodi is 10? Max will be 7 when Jodi is 10. YOUR TURN! Write a function and make a function table. TRIANGLES Every triangle has 3 sides. 1. Write a function to describe the total number of sides in a collection of triangles.

Copyright © by The McGraw-Hill Companies, Inc.

Let x = the number of triangles and y = the number of sides. y=

3x

2. Make a function table using the rule y =

3x .

Number of Triangles, x

1

2

3

4

5

6

3x

3·1

3·2

3·3

3·4

3·5

3·6

Number of Sides, y

3

6

9

12

15

18

3. How many sides do 6 triangles have? Six triangles have

18

sides.

GO ON Lesson 4-1 Introduction to Functions

35

Example 2 Complete the table of values for the equation y = 2x + 1. Then write each pair of values as ordered pairs. Graph the ordered pairs. Ordered Pairs

x

2x + 1

y

-2

2(-2) + 1

-3

(-2, -3)

-1

2(-1) + 1

-1

(-1, -1)

0

2(0) + 1

1

1

2(1) + 1

3

2

2(2) + 1

5

(0, 1)

o o o 0 o o o

(1, 3)

Z

Y

(2, 5)

YOUR TURN! Complete the table of values for the equation y = x + 2. Then write each pair of values as ordered pairs. Graph the ordered pairs. x

x+2

y

-2

-2 + 2

-1

-1 + 2

0 1 2 3 4

0

0+2

1

1+2

2

2+2

Ordered Pairs

(-2, 0) (-1, 1) (0, 2) (1, 3) (2, 4)

Z

o o o 0 o o o

Y

Graph ordered pairs for the equation y = 2x - 1.

Mathew

Z

Y

Z

Y

Circle correct answer(s). Cross out incorrect answer(s). 36

Liza

Dean

Chapter 4 The Relationship Between Graphs and Functions

Z

Y

"

Copyright © by The McGraw-Hill Companies, Inc.

Who is Correct?

Guided Practice Write a function to represent each situation. 1

Mike walks 2 miles less than Darrell every day.

2

Jarrod feeds 4 times as many fish as Patricia.

y=x-2 y = 4x

Step by Step Practice Write a function and make a function table. 3

MUSIC A music club charges $3 for membership and $2 for every downloaded song. How much will Alonso pay for joining the music club and downloading 6 songs? Step 1

Write a function to describe the cost of membership and x downloads. Let x = the number of downloaded songs and y = cost. y=

Step 2

2x + 3

Make a function table using the rule y =

Number of Downloads, x

2d + 3 Copyright © by The McGraw-Hill Companies, Inc.

Total Cost, y Step 3

1

2

3

2x + 3 . 4

5

6

2(1) + 3 2(2) + 3 2(3) + 3 2(4) + 3 2(5) + 3 2(6) + 3 5 7 9 11 13 15

Alonso will pay $ 15 downloading 6 songs.

for joining the music club and

Write a function and make a function table. 4

SCHOOL Each book Cari carries weighs 3 pounds. Her backpack weighs 1 pound. Write a function to show the weight of her backpack if she is carrying b books. w=

3b + 1

Number of Books, b Weight, w

1

2

3

4

4

7

10

13

If Cari carries 3 books, her backpack weighs

10

pounds. GO ON Lesson 4-1 Introduction to Functions

37

Step by Step Problem-Solving Practice

Problem-Solving Strategies Draw a diagram.

Write a function and graph the ordered pairs. 5

SCIENCE Martin is learning about insects, which have exactly 6 legs. He found a picture of four insects. How many legs would Martin have counted? Understand

6

legs.

Plan

Pick a strategy. One strategy is to look for a pattern. Write a function to describe the situation.

Solve

Let x = the number of insects and y = the number of legs.

SCIENCE Martin is learning about insects.

6x

y=

Make a function table using the rule y = Number of insects, x

6x Legs, y

1

2

3

4

6(1) 6

6(2) 12

6(3) 18

6(4) 24

Graph the ordered pairs.

y= =

6

(

4

insect legs.

Z

Y

for x in the function. )

24

CELEBRATIONS A high-school reunion is celebrated every 5 years. Keanu was 18 years old when he graduated. How old will he be at his third high-school reunion? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

y= 38

4

Substitute

24

6x .

18 + 5x

Chapter 4 The Relationship Between Graphs and Functions

G.K. & Vikki Hart/Getty Images

Copyright © by The McGraw-Hill Companies, Inc.

Martin counted Check

Guess and check. Act it out. Work backward.

Read the problem. Write what you know. Insects have

6

✓ Look for a pattern.

Reunion, x

1

18 + 5x

18 + 5(1) Keanu’s age, y 23 Keanu will be

33

2

3

4

18 + 5(2) 28

18 + 5(3) 33

18 + 5(4) 38

years old.

Does the function y = 4x – 3 match the data in the function table? Explain.

7

-2

x

4x - 3 y

-1

4(-2) - 3 4(-1) - 3 -11 -7

0

1

2

4(0) - 3 -3

4(1) - 3 1

4(2) - 3 5

Yes; substitute each x value in the table for x in the function y = 4x - 3 to verify.

Skills, Concepts, and Problem Solving Write a function to represent each situation. 8

Linda writes 3 fewer e-mails than Suja each day.

y=x-3

9

Bruce works 5 times as many hours as Ronnie.

y = 5x

Copyright © by The McGraw-Hill Companies, Inc.

Write a function and make a function table. 10

TRAVEL Kala drove from Detroit to New York City. She drove 45 miles every hour. If it took Kala 15 hours to drive from Detroit to New York City, how many miles did Kala drive to get to New York City? y=

45x Number of Hours, x

3

6

9

12

15

45x

45(3)

45(6)

45(9)

45(12)

45(15)

Number of Miles, y

135

270

405

540

675

Kala drove

675

miles to get to New York City. GO ON Lesson 4-1 Introduction to Functions

39

11

CHEMISTRY In a lab experiment, a scientist used 2.5 milliliters of water for every liter of solution. If the scientist had to mix 8 liters of solution, how many milliliters of water will the scientist need? y=

2.5x

Milliliters of Water, x

2

4

6

8

2.5x

2.5(2)

2.5(4)

2.5(6)

2.5(8)

Liters of Solution, y

5

10

15

20

The scientist needs

20

.JMMJMJUFSTPG8BUFS

Write a function, make a function table, and graph the ordered pairs.

Z

Y -JUFSTPG4PMVUJPO

parts of Solution B.

Vocabulary Check Write the vocabulary word that completes each sentence. 12

function table A(n) is based on a rule.

13

function A(n) is a relationship in which one quantity depends upon another quantity.

14

Writing in Math Explain how to graph a function.

is a table of ordered pairs that

ordered pairs. Connect the consecutive ordered pairs with line segments.

Spiral Review 15

GROCERIES Two pounds of grapes cost $5. Courtney bought 7 pounds of grapes. How much did Courtney pay for the grapes? (Lesson 3-2, p. 11)

$17.50

40

Chapter 4 The Relationship Between Graphs and Functions

Copyright © by The McGraw-Hill Companies, Inc.

Make a function table using the rule of a function. Then graph the

Lesson

4-2 Graph Linear and Nonlinear Functions

7AF3.0 Students graph and interpret linear and some nonlinear functions. 7AF3.1 Graph functions of the form y = nx2 and y = nx3 and use in solving problems.

KEY Concept A nonlinear function is a set of ordered pairs that are related to each other by a non constant rate. A function table can be used to create ordered pairs. The function y = x + 3 is a straight line when graphed. The distance between each point along the x-axis is constant (the same). The distance between each point along the y-axis is constant.

Z

Z Y

Y

Copyright © by The McGraw-Hill Companies, Inc.

0

function table a table of ordered pairs that is based on a rule (Lesson 4-1, p. 34)

(Lesson 4-1, p. 34)

Linear functions show a constant rate of change when graphed. So, y = x + 3 is a linear function.

nonlinear function a function whose graph is not a straight line

function a relationship in which one quantity depends upon another quantity

Linear Function

VOCABULARY

Z

Z Y

Y

The function y = x3 is not a line when graphed. The distance between each point along the x-axis is constant. The distance between each corresponding point along the y-axis is non constant.

linear function a function whose graph is a straight line (Lesson 4-1, p. 34)

constant rate of change the rate of change remains the same between points when a linear function in graphed non constant rate of change the rate of change varies between points when a nonlinear function is graphed

Nonlinear Function

Nonlinear functions show a non constant rate of change when graphed. So, y = x3 is a nonlinear function.

The graph of a nonlinear function is not a straight line.

GO ON Lesson 4-2 Graph Linear and Nonlinear Functions

41

Example 1 Make a function table and a graph for the function y = x 2. Is the function linear or nonlinear? 1. Make a function table using the rule y = x2. x

1

2

3

4

5

x2

12

22

32

42

52

y

1

4

9

16

25

Z

ZY

Y

2. Graph the ordered pairs. 3. Determine if the ordered pairs show a constant or non constant rate of change. If the rate of change is constant, draw a line. If the rate of change is non constant, draw a smooth curve to connect the points. The function is nonlinear. The graph should show a smooth curve. YOUR TURN! Make a function table and a graph for the function y = 2x 2. Is the function linear or nonlinear? 1. Make a function table using the rule y =

2x 2 .

1

2

3

4

5

6

2x2

2 · 12

2 · 22

2 · 32

2 · 42

2 · 52

2 · 62

y

2

8

18

32

50

72

2. Graph the ordered pairs. 3. The rate of change is non constant. The function is nonlinear. The graph should show a smooth curve.

42

Chapter 4 The Relationship Between Graphs and Functions

Z Z Y

Y

Copyright © by The McGraw-Hill Companies, Inc.

x

Example 2 Match y = -3x2 - 1 with its function table and its graph. -2 13

x

A.

y I.

-1 4

1

2

0

4

13

B. II.

Z

0

ZY

0 Y

x

-2

-1

0

1

2

y

-13

-4

-1

-4

-13

Z 0 Y ZY

1. Use a table to check the output values. 0 -2 -1 -3(-2)2 - 1 -3(-1)2 - 1 -3(0)2 - 1 -13 -4 -1

x 2

-3x -1 y

1

2

-3(1)2 - 1 -4

-3(2)2 - 1 -13

2. Table B is the function table for y = -3x 2 - 1. The graph of y = -3x 2 - 1 is Graph II. YOUR TURN! Match y = 4x + 3 with its function table and its graph.

Copyright © by The McGraw-Hill Companies, Inc.

A.

x

-2

-1

0

1

2

y

-5

-1

3

7

11

I.

B.

x

-2

-1

0

1

2

y

-11

-7

3

7

11

II.

Z

0 Y

Z

Z Y 0 Y

1. Use a table to check the output values. x 4x + 3 y

-2

-1

4(-2) + 3 4(-1) + 3 -5 -1

0

1

2

4(0) + 3 3

4(1) + 3 7

4(2) + 3 11

A 2. Table is the function table for y = 4x + 3. The graph of y = 4x + 3 is Graph II .

GO ON

Lesson 4-2 Graph Linear and Nonlinear Functions

43

Who is Correct? Make a function table for y = -2x 3.

Mindy x y

RJ

0

1

2

0

-2

16

0

x y

0

Santos 2

1

-2 -12

x

0

y

0

1

2

-2 -16

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Complete each function table. 1

y = x3 - 5 x y

3

2

-2

-1

-13 -6

y = 5x + 10

0

1

2

x

-2

-1

0

1

2

-5

-4

3

y

0

5

10

15

20

y = x2 x

-2

-1

0

1

2

y

4

1

0

1

4

4

Make a function table and a graph for the function y = 100x 2. Is the function linear or nonlinear? Step 1 Make a function table using the rule y =

100x 2 .

x

1

2

3

4

y

100

400

900

1,600

Step 2 Graph the ordered pairs. Step 3 The rate of change is

non constant

.

nonlinear . Step 4 The function is smooth curve The graph should show a 44

Chapter 4 The Relationship Between Graphs and Functions

.

Z

ZY

Y

Copyright © by The McGraw-Hill Companies, Inc.

Step by Step Practice

Match each function with its function table and its graph. 5

Make a function table and a graph for y = 3x. Is the function linear or nonlinear? x

2

4

6

8

10

y

6

12

18

24

30

linear

.

The function is 6

y = -x 2 function table I graph

8

y = 5x - 2 function table graph II

A.

Copyright © by The McGraw-Hill Companies, Inc.

C.

I.

C

D -1 3

0

1

2

y

-2 9

1

3

9

x

-2

-1

0

1

2

y

-4

-1

0

-1

-4

x

y = x3 function table graph IV

B

9

y = 2x 2 + 1 function table graph III

A

B.

D.

Y

x

-2

-1

0

1

2

y

-8

-1

0

1

8

x

-2

-1

0

1

2

y

-12

-7

-2

3

8

III.

Y

7

ZY

ZY

II.

Z

Z

Z ZY

0 Y

IV.

Z

ZY

0 Y

Z

ZY Y

GO ON

Lesson 4-2 Graph Linear and Nonlinear Functions

45

Step by Step Problem-Solving Practice 10

Problem-Solving Strategies ✓ Make a table.

INVESTING Jada invested in a new company. The amount of money she earned is less than the cube of the number of years she has invested. How much money did Jada earn in the third year? Understand

Look for a pattern. Guess and check. Act it out. Solve a simpler problem.

Read the problem. Write what you know. Write a function to describe the situation. Let x = the number of years and y = the number of dollars Jada earned.

Plan

Pick a strategy. One strategy is to make a table and draw a graph.

Solve

Make a function table using the rule y = 1

2

3

Number of Dollars, y

0

7

26

Graph the ordered pairs. Determine if they show a constant or non constant rate of change. Connect the points.

/VNCFSPG%PMMBST

Number of Years, x

x3- 1 .

In the third year, Jada earned $26 . Substitute

Check

3

Z

ZY

Y /VNCFSPG:FBST

for x in the function. y = ( 3 y = 26

)3 - 1 Copyright © by The McGraw-Hill Companies, Inc.

Make a function table and a graph for each function. SAVING Fay earns $3 per hour. Check off each step. y =

✔

Understand

✔

Plan

✔

Solve

✔

Check

Number of Hours, x

1

2

3

4

5

Amount Earned, y

3

6

9

12

15

Fay earns $ 15 46

after 5 hours of work.

Chapter 4 The Relationship Between Graphs and Functions

3x

"NPVOU&BSOFE

11

Z

ZY

Y /VNCFSPG)PVST

Does the function y = 3x 3 match the data in the function table? Explain.

12

X

-2

-1

0

1

2

y

-24

-3

0

3

9

No; substitute each x value in the table for x in the function y = 3x 3: 3(-2) 3 = -24; 3(-1) 3 = -3; 3(0) 3 = 0; 3(1) 3 = 3; 3(2) 3 = 24; 24 ≠ 9.

Skills, Concepts, and Problem Solving Complete each function table. 13

y=x+7

14

y = 4x 2

x

-2

-1

0

1

2

x

-2

-1

0

1

2

y

5

6

7

8

9

y

16

4

0

4

16

Make a function table and a graph for each function. Is the function linear or nonlinear?

Copyright © by The McGraw-Hill Companies, Inc.

15

y = x(x + 8)

16

y = x 3 - 5.

x

1

2

3

4

5

x

2

3

4

5

y

9

20

33

48

65

y

3

22

59

120

Z

Z Y Y

Y

Z

The function is nonlinear .

ZY

Y

The function is nonlinear .

GO ON Lesson 4-2 Graph Linear and Nonlinear Functions

47

Match each function with its function table and its graph. 17

y = -x function table graph III

19

y = 3x 2 - 2 function table graph IV

A.

x y

C.

x y

I.

A

C

-2 2

-1 1

0

1

2

0

-1

-2

-2 10

-1 1

0

1

2

-2

1

10

18

y = -2x 3 - 1 function table I graph

D

20

y = –x 2 + 1 function table graph II

B

B.

D.

ZY

Y

0

2

y

1

0

-3

x

-2 15

-1 1

0

1

2

-1

-3

-17

Z

ZY

Y

IV.

Z

ZY Y

48

1

Chapter 4 The Relationship Between Graphs and Functions

0

Z

ZY Y

Copyright © by The McGraw-Hill Companies, Inc.

0

-3

-1 0

0

III.

-2

y

II.

Z

x

Vocabulary Check Write the vocabulary word that completes each sentence. 21

A(n) nonlinear function is a function whose graph is not a straight line.

22

function A(n) is a relationship in which one quantity depends upon another quantity.

23

Writing in Math Explain how to graph a nonlinear function.

Make a function table using the rule of a function. Then graph the ordered pairs. Draw a smooth curve to connect the ordered pairs.

Spiral Review 24

SHOPPING Valerie bought a watch that was 4 times as much as the hat Bea bought. Bea spent $22. How much did Valerie spend? (Lesson 3-1, p. 4)

$88

Solve each equation when x = 5.

Copyright © by The McGraw-Hill Companies, Inc.

25

y = -3x + 10

(Lesson 1-4, p. 13)

-5

26

45 y = ___ x -8

1

What is the pattern? Write the next term in the pattern. (Lesson 1-2, p. 13)

27

A, B, a, b, A, B, a, b, A The repeating terms are The next term is

B

A

,

B

,

a

,

b

□ ,

.

.

28

The repeating terms are The next term is

□ ,

□ ,

□ .

Lesson 4-2 Graph Linear and Nonlinear Functions

49

Chapter

Progress Check 1

4

(Lessons 3-1 and 3-2)

Write a function to represent each situation. 7AF3.0, 7AF3.1, 5AF1.5

y=x+8

1

Hannah earns $8 an hour more than Russ.

2

Enrico earns $4 less than the cube of the number of dollars he invested.

y = x3 - 4 3

A quilting club charges $50 for membership and $100 for every y = 50 + 100x quilt purchased.

4

A Spanish club charges $10 for membership and the square of the y = 10 + x 2 number of activities planned.

Write a function, make a function table, and make a graph. Is the function linear or nonlinear? 7AF3.0, 5AF1.5 CRAFTS Carlita makes bracelets at Jade’s Jewelry. She makes approximately 5 new bracelets to sell each day. y=

5x Number of Days, x

1

2

3

4

5

6

Number of Bracelets, y

5

10

15

20

25

30

In 6 days, about how many bracelets would Carlita make to sell at the jewelry store? Carlita would make about 30 The function is linear .

ZY

bracelets in 6 days.

Jae made a graph of how many cups of water drip from a broken sink faucet each hour. y=

2x Hours Passed, x

1

2

3

4

Cups of Water, y

2

4

6

8

How many cups of water have dripped at 4 hours?

8 cups of water have dripped. At 4 hours, linear The function is . 50

Z

Y /VNCFSPG%BZT

Chapter 4 The Relationship Between Graphs and Functions

$VQTPG8BUFS

6

Z

ZY

Y )PVST1BTTFE

Copyright © by The McGraw-Hill Companies, Inc.

/VNCFSPG#SBDFMFUT

5

Lesson

4-3 Direct Variation

7AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of the graph. 7AF3.4 Plot the values of quantities whose ratios are always the same. Fit a line to the plot and understand that the slope of the line equals the ratio of the quantities.

KEY Concept A ratio shows a constant rate of change between two quantities.

Z

&

Point

A

B

C

D

x

1

2

3

4

5

y

7

14

21

28

35

%

E

VOCABULARY

$

#

ratio a relationship that compares two quantities

"

Y

y Use the ratio __ x for each point on the graph to determine if the

(Lesson 3-2, p. 11)

constant of variation the rate of change in a direct variation

rate of change is constant. 28 ___ 35 7 = ___ 14 = ___ 21 = ___ Since __ = , the rate of change is constant. 2 3 5 1 4 7 or 7. So, the constant of variation can be described as __ 1

direct variation a special type of linear equation that describes a rate of change; a relationship such that as x changes in value, y increases or decreases at a constant rate

The value of y increases or decreases at a constant rate, as the value of x changes. The graph shows a direct variation .

MONEY The graph shows the ratio of the number of gallons of gasoline, y, to the number of dollars Angelica spends, x. Use the graph to find the constant of variation. Interpret the constant of variation.

Point

A

B

C

D

E

Dollars Spent, x

1

2

3

4

5

Gallons of Gasoline, y

2

4

6

8

10

%PMMBST4QFOU

Copyright © by The McGraw-Hill Companies, Inc.

Example 1

1. Use the points on the line to make a function table.

2. Find the ratio of the amount of gasoline Angelica buys to the amount of money Angelica spends, y or __ x . The ratio is constant. 2 or 2. The constant of variation is __ 1 3. The constant of variation shows that for every 1 gallon of gasoline she purchases, she spends 2 dollars.

Point A Point B Point C Point D Point E

Z

& % $ # " (BMMPOTPG(BTPMJOF

Y

_y = _2 = 2 x 1 _y = _4 = _2 = 2 x 2 1 y_ _ 6 _ 2 x=3=1=2 _y = _8 = _2 = 2 x 4 1 y_ _ 10 _ 2 = =2 = x

5

1

Lesson 4-3 Direct Variation

51

YOUR TURN!

5FTU4DPSF

TESTS The graph shows the ratio of the test score, y, to the number of correct answers, x. Use the graph to find the constant of variation. Interpret the constant of variation. 1. Use the points on the line to make a function table. Point Number of Correct Answers, x Test Score, y

A

B

C

1 3

2 6

3 9

y 2. Find the ratio of the test score to correct answers, or __ x. Point A

_3 1 _9

_y

x =

_y

3

=

Point B

_3

_y

x =

= = 3 x = 1 3 The ratio is constant. The constant of variation is

Point C

3.The constant of variation was worth 3 points.

_3 1

Point B Point C Point D Point E

C 9 3

D 16 4

3

or

"

Y /VNCFSPG$PSSFDU"OTXFST

=

3

.

_y = _1 = 1 x x 1 _y = _2 = _1 x 4 2 _y = _3 = _1 x 9 3 4 _y = _ _1 x 16 = 4 5 _1 _y = _ =

Do the ordered pairs in the table represent a linear or nonlinear function? Explain. Point x y

E 25 5

x 25 5 2. This function is nonlinear because the ratios between different pairs of points are not constant. 52

1

1

#

A 1 3

B 2 6

C 3 9

D 4 12

E 5 15

y 1. Compare the ratios __ x. y Point A x = 3 = 3 1 y 6 3 Point B x = = = 3 2 1 y 9 3 Point C x = = = 3 3 1 y 12 3 = =3 Point D x = 1 4 y 15 3 = =3 Point E x = 5 1

_ _ _ _ _

_ _ _ _ _

2. This function is ratios are

Chapter 4 The Relationship Between Graphs and Functions

_ _ _ _

linear because the constant .

Copyright © by The McGraw-Hill Companies, Inc.

B 4 2

y 1. Compare the ratios __. Point A

2

_3

_3

=

$

YOUR TURN!

Do the ordered pairs in the table represent a linear or nonlinear function? Explain. A 1 1

_6

Z

means each question

Example 2

Point x y

Who is Correct? Interpret the ratio of the number of wheels, y, to the number 4 of cars, x. __ 1

Lori

Carter

_4 means that there are 1

_4 means that 4 wheels

4 wheels on 1 car.

1

are on 4 cars.

Sonja

_4 means that for every

1 s. 1 car, there are 4 wheel

Circle correct answer(s). Cross out incorrect answer(s).

Guided Practice Interpret each ratio. 1

2. The ratio of the number of eyes, y, to the number of people, x, is __ 1

_

The ratio 2 means that there are 2 eyes for each 1 person. 1

2

8 The ratio of the number of legs, y, to the number of spiders, x, is __. 1

_

Copyright © by The McGraw-Hill Companies, Inc.

The ratio 8 means that 8 legs are on 1 spider. 1

Do the ordered pairs in the table represent a linear or nonlinear function? Explain. 3

Point x y

A 2 3

B 4 6

C 6 9

This function is linear points are constant . 4

Point x y

A 1 3

This function is pairs of points are

B 2 8

C 3 12

D 8 12

E 10 15

because the ratios between pairs of

D 4 16

E 5 25

nonlinear not constant

because the ratios between .

GO ON

Lesson 4-3 Direct Variation

53

Step by Step Practice TRAVEL The graph shows the ratio of the number of steps Miguel went north, y, to the number of steps he went east, x. Use the graph to find the constant of variation. Interpret the constant of variation.

Point

A

B

C

D

Steps East, x

1

2

3

4

Steps North, y

4

8

12

16

4UFQT/PSUI

5

Step 1 Use the points on the line to make a function table.

Step 2 Find the ratio of the number of steps north to the y number of steps east, or __ x. y y 4 __ 4 = Point B = Point A __ x x= 1

_ 12 _

% $ # " Y 4UFQT&BTU

_8 2 16 _

_

y y 4 Point C __ = = 4 Point D __ x= x= 1 3 y The ratio __ x is constant. The constant of variation is 4 or 4 .

_

Z

4

= =

_4 1 _4 1

=

4

=

4

_4

1

Step 3 Interpret the direct variation. The constant of 1 means that for every 4 steps Miguel took north, he took 1 step east.

Point

A

B

C

D

Number of Lunches, x

1

2

3

4

Cost, y

5

10

15

20

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LUNCHES The graph shows the ratio of the cost, y, to the number of lunches, x. Use the graph to find the constant of variation. Interpret the constant of variation.

y The ratio __ x is constant. The constant of variation is The constant of variation $ 5 for 1 lunch.

54

_5

_5 1 1

or

5

.

means that it cost

Chapter 4 The Relationship Between Graphs and Functions

Z

% $ # "

Y /VNCFSPG-VODIFT

Copyright © by The McGraw-Hill Companies, Inc.

6

Step by Step Problem-Solving Practice

Problem-Solving Strategies Draw a diagram.

Solve. GROCERIES The graph shows the ratio of cost, y, to the number of pounds of peanuts, x. Use the graph to find the constant of variation. Interpret the constant of variation. Understand

Read the problem. Write what you know. y cost The ratio __ to x is

pounds of peanuts Plan Solve

Act it out. Solve a simpler problem. Work backward.

.

Pick a strategy. One strategy is to make a table.

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7

✓ Make a table.

Use the points to make a function table. Point

A

B

C

D

Pounds of Peanuts, x

1

2

3

4

Cost, y

2.5

5

7.5

10

Z

% $ # "

Y 1PVOETPG1FBOVUT

y Find the ratio of the cost to pounds of peanuts, or __ x. Point A

y __ x=

Copyright © by The McGraw-Hill Companies, Inc.

y Point B __ x= y Point C __ x=

2.5 _ 1 _5 2 7.5 _ 3 10 _

= 2.5 = =

2.5 _ 1 2.5 _ 1 2.5 _

= 2.5 = 2.5

y Point D __ = = 2.5 x= 4 1 y The ratio __ x is constant. The constant of variation is

2.5 _ 1

or 2.5 .

2.5 _

The constant of variation means that for every 1 2.50 1 $ , Mora can buy pound of peanuts. Check

Verify that the points in the table are on the graph.

Lesson 4-3 Direct Variation

55

TRAVEL Colby rented a bicycle. He was charged $0.50 for every mile. The graph shows the ratio of the cost, y, to the number of miles, x. Use the graph to find the constant of variation. Interpret the constant of variation. Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

Z & % $ # " Y /VNCFSPG.JMFT

Point

A

B

C

D

E

Number of Miles, x

2

4

6

8

10

Cost, y

1

2

3

4

5

y The ratio __ x is constant. The constant of variation is The constant of variation $ 0.50 for Colby to drive

0.5 _

0.5 _ 1

or 0.5 .

means that it cost

1 1 mile.

What is the constant of variation of the graph? Explain.

9

$PTU

8

_3; find the ratio of each point using _y x

1

$

#

"

Y

Skills, Concepts, and Problem Solving ENTERTAINMENT The graph shows the ratio of the cost, y, to the number of pony rides, x. Use the graph to find the constant of variation. Interpret the constant of variation. Point

A

B

C

x

1

2

3

y

3

6

9

y __ = x

y __ = x

y __ = x

56

_3 1 _6 2 _9 3

= = =

$PTU

10

3

_3 1 _3 1

=

3

=

3

Z $

#

"

Y /VNCFSPG3JEFT

Chapter 4 The Relationship Between Graphs and Functions

1ONY3IDES

Copyright © by The McGraw-Hill Companies, Inc.

Z

The ratio is constant. The constant of variation is

_3

or

1

3

.

means that it cost

The constant of variation 1 $ 3 for 1 pony ride.

BOOKS The graph shows the ratio of the cost, y, to the number of books Diego buys, x. Use the graph to find the constant of variation. Interpret the direct variation. Point

A

B

C

Number of Books, x

1

2

3

Cost, y

4

8

12

_4 1 _8 2 12 _

y __ = x

y __ = x

y __ =

3

x

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11

_3

= = =

4

_4 1 _4 1

=

4

=

4

y The ratio __ x is constant. The constant of variation is 4 . The constant of variation 1 book.

_4 1

means that it cost

_4

or

$4

for

1

$

Z #

"

Y /VNCFSPG#PPLT

Copyright © by The McGraw-Hill Companies, Inc.

Do the ordered pairs in the table represent a linear or nonlinear function? Explain. 12

Point x y

A 2 5

B 4 10

C 6 15

E 10 25

linear

This function is

constant

13

D 8 20

because the ratios are

.

Point

A

B

C

D

E

x

3

6

9

18

21

y

2

4

8

16

14

This function is

not constant

nonlinear

because the ratios are

. GO ON Lesson 4-3 Direct Variation

57

Vocabulary Check Write the vocabulary word that completes each sentence. 14

ratio A(n) is a relationship between two quantities in which the first measures a certain number of units and the second measures another number of units. Constant of variation

15

is the rate of change in a direct

variation. 16

10 Writing in Math Interpret the ratio ___ where y represents the 1 number of ten-dollar bills, and x represents the number of hundred-dollar bills.

_

The ratio 10 means that there are 10 ten-dollar bills in 1 1 one-hundred-dollar bill.

Spiral Review Write a function and make a function table. 17

MONEY Ferdinand pays $15 every month for a daily newspaper subscription. How much money does Ferdinand spend each year to receive a newspaper every day? (Lesson 4-1, p. 34) y=

15x 3

6

Cost ($), y

45

90

9

12

135 180

Ferdinand will spend $ 180 each year to receive a newspaper every day. Find each unit price. 18

(Lesson 3-2, p. 11)

Ricardo bought 15 erasers for $1.50.

19

The unit price is $ 0.10 for 1 eraser. 20

LaShonda bought 6 juice boxes for $4.50.

Julie bought 5 mugs for $60. The unit price is $ 12

21

The unit price is $ 0.75 for 1 juice box.

Dah-Chou bought 8 picture frames for $120. The unit price is $ 15 1 picture frame.

58

Chapter 4 The Relationship Between Graphs and Functions

for 1 mug.

for

Copyright © by The McGraw-Hill Companies, Inc.

Number of Months, x

Lesson

4-4 Slope KEY Concept The slope of a line illustrates the ratio of the number of units of rise to the number of units of run for a linear function. You can find the slope of a line from the graph of that line or by using the slope formula. change in y Δy Slope = m = ___ = ___________ Δx change in x =

"MPXFSDBTFN SFQSFTFOUTTMPQF

y2 - y1 m = ______ x2 - x1 , where x2 ≠ x1

Copyright © by The McGraw-Hill Companies, Inc.

Look at the graph at the right. Choose two points on the line, such as (-3, -2) and (0, 0).

5IFCMVF BSSPXTTIPX UIFSJTF VOJUT

7AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of the graph. 7AF3.4 Plot the values of quantities whose ratios are always the same. Fit a line to the plot and understand that the slope of the line equals the ratio of the quantities.

VOCABULARY slope the ratio of the change in the y-value to the corresponding change in the x-value in a linear function

Z

• Start at point (-3, -2). 0 • Move up parallel to the y-axis (rise) 2 units. • Move to the right parallel to the x-axis (run) 3 units. • You are at another point on the line, (0, 0).

Y

5IFSFE BSSPXTTIPX UIFSVO VOJUT

• From the point (0, 0) move up (rise) 2 units. • Move to the right (run) 3 units. You are at another point on the line, (3, 2). +2 +2 You can continue to move up 2 units and right 3 units because the slope of the line is constant.

y

-2

0

2

x

-3

0

3

+3 Another way to find the slope is to use the points (0, 0) and (3, 2) in the formula. y2 - y1 0_____ - 2 __ 2 m = _______ x2 - x1 = 0 - 3 = 3

+3

GO ON Lesson 4-4 Slope

59

Example 1

YOUR TURN!

Graph the function y = 2x + 3 and determine its slope.

Graph the function y = 1 x - 8 and 2 determine its slope.

_

1. Complete a function table for the equation. +1

+1

+1

+1

x

-2

-1

0

1

2

y

-1

1

3

5

7

+2

+2

+2

+2

1. Complete a function table for the equation.

+2

0

Z

+2

-2

0

2

4

6

y

-9 -8

-7

-6

-5

-1

-1

-1

-1

2. Graph the ordered pairs on the graph.

ZY Y

3. From the points on the graph, determine rise ____ the slope run . Find the change in

( )

To move from (-2, -1) to (-1, 1), you move up 2 units and right 1 unit. The change in x-values and the change in y-values are constant from one point to another. So, the rise is 2 units, and the run is 1 unit. rise __ 2 Slope = ____ run = 1 or 2

0

Z

Y

@@ ZY

3. From the points on the graph, determine rise ____ the slope run . Find the change in

( )

x-values and the change in y-values. To move from the point ( -2 , -9 ) to the point ( 0 , -8 ), you will move up 1 unit(s) and right 2 unit(s). The change in x-values and the change in y-values are constant from one point to another. So, the rise is 1 2 units. rise ___ 1 Slope = ____ run = 2

Chapter 4 The Relationship Between Graphs and Functions

unit, and the run is

Copyright © by The McGraw-Hill Companies, Inc.

x-values and the change in y-values.

60

+2

x

2. Graph the ordered pairs on the graph.

+2

Example 2

YOUR TURN! Determine the slope of the graph.

Determine the slope of the graph. 0

@@ ZY $

0

Y

#

1. Complete a function table for the graph. Point x y

A 2 2

B 0 5

_ _

_

"

Point

A

B

C

x

5 -5

0 -2

-5 1

y

y2 - y1 5-8 3 m= x -x = =0 (-2) 2 2 1 3. 3. The slope of the line is -__ 2

Y

1. Complete a function table for the graph.

C -2 8

2. Substitute the x and y values in the slope formula to find the slope of the line.

Copyright © by The McGraw-Hill Companies, Inc.

Z @@ ZY

Z

2. Substitute the x and y values in the slope formula to find the slope of the line.

-5 - ( -2 ) _ __

y2 - y1

m= x -x = 2 1

5

= -

_3 5

3. The slope of the line is -

Who is Correct?

_ _

run Slope = rise 7 = 2

Landon

_ _

rise Slope = run 2 = 7

_3 . 5

@@ ZY

What is the slope of the line on the graph?

Dale

0

-

Jena

_ _

rise Slope = run -2 = 7

0

Z

Y

Circle correct answer(s). Cross out incorrect answer(s). GO ON Lesson 4-4 Slope

61

Guided Practice Use the graph to answer each question. 1 2

$

(4, -2)

What is the location of Point A?

@@ ZY Y

0 #

(0, -1)

What is the location of Point B?

Z

"

Step by Step Practice 3

Determine the slope of the function y = 3x - 9. Step 1 Complete a function table for the equation. x

0

1

2

3

y

-9

-6

-3

0

Step 2 Plot the points on the graph. Step 3 From the points on the graph, determine the slope rise ____ run . Find the change in x-values and the change in y-values.

( )

To move from ( 0 , -9 ) to ( 1 you move up 3 unit(s) and right

,

1

0

-6 ),

Z ZY

Y

unit(s).

So, the rise is 3 units , and the run is 1 unit . rise ___ 3 3 Slope = ____ run = 1 or

Graph each function and determine its slope. 3 Z 4 y = - __x - 3 4

_

The slope is

-3 4

.

5

@@ ZY

0

Y

62

Chapter 4 The Relationship Between Graphs and Functions

y = 6x + 2 The slope is

6

.

0

Z

ZY Y

Copyright © by The McGraw-Hill Companies, Inc.

The change in x-values and the change in y-values are constant from one point to another.

Step by Step Problem-Solving Practice

Problem-Solving Strategies Draw a diagram. Look for a pattern. Guess and check. ✓ Write an equation. Work backward.

Solve. 6

PRICES Gasoline costs $2 per gallon. Graph an equation to represent the cost of purchasing x gallons of gas. Understand

Read the problem. Write what you know.

$2 a gallon

Gasoline costs Plan

.

Pick a strategy. One strategy is to write an equation. Let x represent the number of gallons and y represent the cost.

y = 2x

Use the equation to make a function table.

+1

+1

+1

Number of Gallons, x

0

1

2

3

Cost, y

0

2

4

6

$PTU JO

Solve

+2

+2

+2

Z

ZY

Y /VNCFSPG(BMMPOT

Copyright © by The McGraw-Hill Companies, Inc.

Graph the ordered pairs in the table on the graph.

2

To find the slope, move up right 1 unit(s).

unit(s) and to the

The change in x-values and the change in y-values constant are from one point to another. rise ___ 2 or slope = ____ run =

1

Check

2

.

Use the formula for slope to check your answer. y2 - y1 2______ - (0) m = ______ = = x2 - x1 1-0

_2 1

GO ON Lesson 4-4 Slope

63

MONEY Jay earns $12 per hour. Graph an equation to represent the total amount Jay earns if he works x hours. Include at least three points on the graph. Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

%PMMBST&BSOFE

7

Write an equation. Let x represent the hours and y represent the dollars.

y = 12x 12 .

The slope is 8

ZY

Y /VNCFSPG)PVST

Draw two lines on the graph to the right. One line 1 . The second line should should have a slope of __ 3 1 __ have a slope of - . What do you notice about direction of 3 the two lines?

Z

@@ ZY

_

0

_

Skills, Concepts, and Problem Solving Graph each function and determine its slope. 9

y = - 6x + 4 0

10

Z

Y

The slope is - 6 . 64

y = 4x - 5

Chapter 4 The Relationship Between Graphs and Functions

0

Z

Y

The slope is

4

.

Y

@@ ZY

Copyright © by The McGraw-Hill Companies, Inc.

The line with the positive slope of 1 moves up as 3 you travel from left to right along the x-axis. The line with the negative slope of - 1 moves down 3 as you travel from left to right along the x-axis.

Z

SWIMMING Ines can swim 7 laps in 20 minutes. Graph an equation to represent the number of minutes it takes Ines to swim x laps.

.JOVUFT

11

Write an equation. Let x represent the laps that Ines can swim and y represent the minutes.

y = 7x The slope is 12

7

.

FITNESS It costs $40 a month to belong to Rosemill Athletic Club. Graph an equation to represent the total cost of belonging to the club for x months. .POUIT

Write an equation. Let x represent the cost and y represent the months.

y = 40x The slope is 40 .

Z

ZY

Y -BQT Z

ZY

Y $PTU

Vocabulary Check Write the vocabulary word that completes each sentence.

Slope

13 14

is the ratio of the rise over the run.

Writing in Math Explain how to find the slope for the equation y = -2x + 7.

To find the slope, first determine different points on the line. It helps to the change in x-values and the change in y-values to determine the slope. the change in y-values The slope is the rise or . run the change in x-values

_ __

Spiral Review 15

MONEY The graph shows the ratio of the cost, y, to the number of tickets sold, x. Use the graph to find the constant of variation of the cost to the number of tickets sold. Then interpret the constant of variation. (Lesson 4-3, p. 49) Tickets Sold, x

1

2

3

4

5

Cost, y

10

20

30

40

50

10 _

The ratio is is

1

or

constant 10 .

The constant of variation of

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Copyright © by The McGraw-Hill Companies, Inc.

make a table. Next, plot the points on a coordinate graph. Finally, find

, so the constant of variation

10 _ 1

means it cost $ 10

for

1

Z

ZY

Y 5JDLFUT4PME

ticket. Lesson 4-4 Slope

65

Chapter

4

Progress Check 2

(Lessons 4-3 and 4-4)

Interpret each ratio. 7AF3.4 1

2. The ratio of the number of ears, y, to the number of people, x, is __ 1

_

The ratio 2 means that there are 2 ears for each person. 1

2

The ratio of the number of legs, y, to the number of centipedes, x, 100 is ____. 1

_

The ratio 100 means that there are 100 legs on each centipede. 1 #

Determine the slope of the graph. 7AF3.3 3

What is the “rise” of the line?

3

-2 3 What is the slope of the line? 2 What is the “run” of the line?

Z Zĕ Y "

0

_

Y

Graph each equation. 7AF3.3 4

Graph the equation y = 5x - 1.

0

Graph the equation y = 3x

Z

Y

66

Chapter 4 The Relationship Between Graphs and Functions

Z

Y

Copyright © by The McGraw-Hill Companies, Inc.

5

Chapter

Study Guide

4

Vocabulary and Concept Check constant of variation, p. 51 function, p. 34

Write the vocabulary word that completes each sentence. 1

function table, p. 34 linear function, p. 34

Slope

is the ratio between the change in the y-value and change in the x-value.

2

function A(n) is a relationship in which one quantity depends upon another quantity.

3

The constant of variation is the rate of change in a direct variation.

4

function table A(n) that is based on a rule.

is a table of ordered pairs

5

linear function A(n) graph is a line.

is a function whose

nonlinear function, p. 41 slope, p. 59

Label each diagram below. Write the correct vocabulary term in each blank.

linear function

Copyright © by The McGraw-Hill Companies, Inc.

6

Z

7

nonlinear function ZY

ZY

Y

0

Z

Y

Chapter 4 Study Guide

67

Lesson Review Introduction to Functions

(pp. 34–40)

Write a function, make a function table, and graph the ordered pairs. 8

Mary is 3 years younger than RaeAnne. Let x = RaeAnne’s age and y = Mary’s age. x-3 y= RaeAnne’s 14 15 16 17 18 19 Age, x Mary’s 11 12 13 14 15 16 Age, y

.BSZT"HF

Graph the ordered pairs.

QUADRILATERALS Every quadrilateral has 4 sides. What is the total number of sides in a collection of x quadrilaterals? The function to describe the situation is y = 4x where x = the number of quadrilaterals and y = the number of sides. Make a function table using the rule y = 4x. Number of Quadrilaterals, x Number of Sides, y

1

2

3

4

5

4

8

12 16 20

Z

Graph the ordered pairs.

ZY

Y 3FB"OOFT"HF

How old will RaeAnne be when Mary is 16?

RaeAnne will be 19 years old when Mary is 16 years old.

68

Write a function, make a function table, and graph the ordered pairs.

Chapter 4 Study Guide

Z

ZY

Y /VNCFSPG2VBESJMBUFSBMT

How many sides do 5 quadrilaterals have? Five quadrilaterals have a total of 20 sides.

Copyright © by The McGraw-Hill Companies, Inc.

Example 1

/VNCFSPG4JEFT

4-1

Graph Linear and Nonlinear Functions

Write a function, make a function table, and make a graph. Is the function linear or nonlinear? 9

VIDEOS It will cost Peyton $15 to join the video game club. Then the club will charge him $4 for each video game rental. y = 4x + 15

"NPVOUJO%PMMBST

Number of Video Games, x Amount (in dollars), y

Copyright © by The McGraw-Hill Companies, Inc.

1

2

3

4

19 23 27 31

(pp. 41–49)

Example 2 Write a function, make a function table, and make a graph. Is the function linear or nonlinear? GRADES For every A Jase earns on his report card, he receives three times the square of the number of A’s in quarters from his grandfather. The function to describe this situation is y = 3x2 where x = the number of A’s on Jase’s report card and y = the amount of quarters received. Make a function table using the rule y = 3x2.

Z

ZY

Number of A’s, x Amount (in quarters), y

1

2

3

4

5

6

3

12

27

48

75 108

Graph the ordered pairs. Draw a smooth curve to connect the ordered pairs. Y /VNCFSPG7JEFP(BNFT

How much does Peyton pay if he joins the club and then rents 3 games?

27 for a Peyton pays $ membership plus $4 per video game rental. The function is linear .

"NPVOU JORVBSUFST

4-2

Z

ZY

Y /VNCFSPG"T

How many quarters does Jase receive if he earns 6 A’s on his report card? Jase will receive 108 quarters, or $27, from his grandfather if he earns 6 A’s on his report card. The function is nonlinear.

Chapter 4 Study Guide

69

4-3

Example 3

(pp. 51–58)

MONEY The graph shows the ratio of money earned, y, to the number of hours worked, x. Use the graph to find the constant of variation. Interpret the constant of variation.

Z

& % $ # " Y )PVST8PSLFE

A

B

C

D

E

x

1 8

2 16

3 24

4 32

5 40

y Point A __ x= y __ x=

Point C

y __ x=

y Point D __ x= y Point E __ x=

5

= = = = =

1

y The ratio __ x is constant. The constant of variation is

=

8

=

8

=

8

=

8

_8 or 8 1

The constant of variation means for every 1 hour worked, $ 8 was earned.

70

Chapter 4 Study Guide

& % $ # " Y (BMMPOTPG5SFBUFE8BUFS

Use the points on the line to make a function table.

8

_8 1 _8 1 _8 1 _8

Z

Point

A

B

C

D

E

x

1

2

3

4

5

y

6

12

18

24

30

Find the ratio of the cost in dollars to the y number of gallons of treated water, or __ x. y 6 Point A x = = 6 1 y 6 12 Point B x = = =6 2 1 y 6 18 Point C x = = =6 3 1 y 6 24 Point D x = = =6 4 1 y 30 6 Point E x = = =6 5 1 y The ratio __ x is constant. The constant of 6 variation is __ or 6. 1 The constant of variation means that it costs 6 dollars for 1 gallon of treated water.

_ _ _ _ _

.

_ _ _ _ _

_ _ _ _

Copyright © by The McGraw-Hill Companies, Inc.

Point B

_8 1 16 _ 2 24 _ 3 32 _ 4 40 _

Point y

SHAPES The graph shows the ratio of the cost in dollars, y, to the number of gallons of treated water, x. Use the graph to find the constant of variation. Interpret the constant of variation.

$PTUJO%PMMBST

.POFZ&BSOFE

10

Direct Variation

4-4 11

Slope

(pp. 59–65)

Determine the slope of the function y = 3x - 1.

Example 4 Determine the slope of the function 1 y = x + 1. 4 1. Complete a function table for the equation.

1. Complete a function table for the equation.

+1

+1

x

-2

-1

y

-7 -4 +3

+1

_

+1

0

1

2

-1

2

5

+3

+3

+3

2. Plot the points on the graph.

+4

+4

+4

+4

x

-4

0

4

8

12

y

0

1

2

3

4

+1

+1

+1

+1

2. Plot the points on the graph. 0

Copyright © by The McGraw-Hill Companies, Inc.

Z

Y ZY

0

Z

@@ ZY

Y

3. From the points on the graph, determine ⎛ rise ⎞ ⎥. Use the graph and find the slope ⎪____ ⎝ run ⎠ the change in x-values and the change in y-values. To move from the point ( -2 , -7 ) to the point ( -1 , -4 ), you will 1 move up 3 units and right unit.

3. From the points on the graph, determine ⎛ rise ⎞ ⎥. Use the graph and find the the slope ⎪____ ⎝ run ⎠ change in x-values and the change in y-values. To move from the point (-4, 0) to the point (0, 1), you will move up 1 unit and right 4 units.

The change in x-values and the change constant in y-values are from one point to another.

The change in x-values and the change in y-values are constant from one point to another.

The slope is a rise of 3 units , run of

The slope is a rise of 1 unit, run of 4 units, 1. or __ 4

1 unit ,

_3 1

or

3

.

Chapter 4 Study Guide

71

Chapter

Chapter Test

4

Write a function to represent each situation. 5AF1.5

y=x+4

1

Tanner is 4 years older than Justice.

2

Tonya spent $10 more than twice the amount Virginia spent.

y = 2x + 10

Write a function, make a table function, and make a graph. Is the function linear or nonlinear? 5AF1.5, 7AF3.0 3

CRAFTS Tyler is making key chains for the upcoming craft show. Each day when he gets home from school he makes approximately 6 key chains. y = 6x Number of Days, x

1

2

3

4

5

6

Number of Key Chains, y

6

12

18

24

30

36

Determine the slope of the graph. 7AF3.3, 7AF3.4

_1 2

Z

0 "

Z

Y /VNCFSPG%BZT Copyright © by The McGraw-Hill Companies, Inc.

4

/VNCFSPG,FZ$IBJOT

About how many key chains will Tyler make in 6 days? Tyler will make about 36 key chains in 6 days. The function is linear .

Y $ #

Graph each equation. 7AF3.3 5

y=x+2

6

Z

0

72

Chapter 4 Test

y = –x

Y

0

Z

Y

GO ON

Write a function, make a function table, and graph the ordered pairs.

5AF1.5, 7AF3.1, 7AF3.3

COOKING Diana is baking some cookies for her annual cookie exchange. She is going to make 4 batches of her cookies, and each batch requires 3 large eggs. How many eggs does she need to make x batches of cookies? y = 3x Number of Cookie Batches, x

1

2

3

4

Number of Eggs, y

3

6

9

12

/VNCFSPG&HHT

7

Z

Y /VNCFSPG$PPLJF#BUDIFT

How many eggs would Diana use to make all 4 batches of her cookies? 12

Make a function table and graph for each function. Is the function linear or nonlinear? 5AF1.5, 7AF3.0 8

y = x3 + 6 x

1

2

3

4

y

6

14

33

70

The function is

nonlinear

.

Y

Solve. 5AF1.5, 7AF3.0, 7AF3.3, 7AF3.4 9

PACKAGING Each box at the Cardboard Warehouse can hold 6 glass jars. Graph an equation to represent the total number of jars contained in x boxes. The equation is The slope is

y = 6x 6

/VNCFSPG+BST

Copyright © by The McGraw-Hill Companies, Inc.

Z

.

Z

Y /VNCFSPG#PYFT

Correct the mistakes. 5AF1.5, 7AF3.3 10

On Nikki’s math quiz, the problem stated: “For every mile you bicycle, you burn 35 Calories. Fill in the blanks.” Let x =

; let y =

.

Nikki x = Calories y = miles

What is the mistake Nikki made?

x = miles, y = total Calories burned Chapter 4 Test

73

Chapter

4

Standards Practice

Choose the best answer and fill in the corresponding circle on the sheet at right. 1

Which ordered pair falls on the line of the equation y = x - 2? 7AF3.3

4

A (-2, -4) B (0, -4)

Molly is selling T-shirts for a school fund-raiser. Use her chart to make a line graph that shows the data. If her goal is to raise $60, how many T-shirts does Molly need to sell? 7AF3.3

C (-3, -2) D (3, 2)

0

2

$18

3

$27

7AF3.0

Z

Y

Z

Y 54IJSUT4PME

F y=x-4

F 4 T-shirts

G y = 2x - 1

G 5 T-shirts

H y=x+2

H 6 T-shirts

J y = 2x + 1

J 7 T-shirts

Which ordered pair falls on the line of the equation y = 3x + 4?

5

What is the slope of the equation y = -6x - 5? 7AF3.3, 7AF3.4

A (-1, -1)

A -6

B (3, 4)

B -5

C (-3, -5)

C 5

D (5, 9)

D 6

GO ON 74

Chapter 4 Standards Practice

Copyright © by The McGraw-Hill Companies, Inc.

3

Money Raised, y $9

Which equation is graphed below?

.POFZ3BJTFE JOEPMMBST

2

T-Shirts Sold, x 1

6

Which equation could have been used to create this function table? 5AF1.5 Input (x) 2

Output (y) 3

5

24

8

63

9

Mila’s parents spent $52.50 on 5 children’s tickets to the amusement park. What was the price for one child’s ticket? 3AF2.1 A $1.50

C $10.50

B $9.70

D $15.00

ANSWER SHEET F y = x2 - 1

Directions: Fill in the circle of each correct answer.

G y=x+1

7

1

A

B

C

D

J y = 4x - 1

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

Which point on the grid corresponds to the coordinate pair (4, 6)? 5SDAP1.5

Copyright © by The McGraw-Hill Companies, Inc.

H y = 2x

Z # $

" %

Y

A A Success Strategy

B B

Read each question completely. Then think of the answer in your head before looking at the answer choices. You will be better able to spot the answers that are trying to trick you.

C C D D

8

Jorge read 90 pages in his book over 5 days. At this rate, how many days will it take him to read 144 pages? 5AF1.5 F 28 days

H 8 days

G 18 days

J 6 days Chapter 4 Standards Practice

75

Index A Algebra and Functions, 4, 11, 19, 34, 41, 51, 59

K Key Concept, 4, 11, 19, 34, 41, 51, 59

Answer sheet, 31, 75

C California Mathematics Content Standards, 4, 11, 19, 34, 41, 51, 59 Chapter Preview, 3, 33 Chapter Test, 28–29, 72–73 constant of variation, 51–58 coordinate grid, 17, 33, 34, 36, 38–48, 50–52, 54–56, 57, 59–75 Correct the Mistakes, 29, 73 cross multiply, 19–25

L line plot, 10, 25 linear function, 34–40, 41–50

M Mathematical Reasoning, see Step-by-Step Problem Solving multi-step equation, 10–25

N nonlinear function, 41–50

D

number pattern, 4–10 Number Sense, 19

direct variation, 51–58

P pattern, 4–10

equation, 19–25, 51–58 equivalent fractions, 11–17, 19–25 equivalent ratios, 11–17, 19–25

F fraction equivalent, 11–17, 19–25 ratios, 11–17 simplest form, 11–17 function, 34–40, 41–50 function table, 34–40, 41–50

G graphs, 34–40, 41–50, 51–58, 59–66

76

Index

Problem-Solving, see Step-byStep Problem Solving product, 19–25 Progress Check, 18, 50, 66 proportion, 19–25 rate 11–17 ratio 11–17, 19–25, 51–58, 59–66

Real-World Applications architecture, 24 books, 57 cars, 16 celebrations, 38 chemistry, 40 chess, 22 construction, 29 cooking, 73 crafts, 50, 72 entertainment, 29, 56 fish, 29 fitness, 15, 18, 25, 29, 65 food, 8, 9, 29 fund-raising, 7 games, 17, 21 geometry, 9 grades, 69 groceries, 18, 23, 40, 55 health, 18 kittens, 10 lunches, 54 measurement, 18 money, 51, 58, 65, 70, 73 music, 37 packaging, 23, 73 parades, 22 population, 29 prices, 63 quadrilaterals, 68 saving, 46 school, 37 science, 38 shapes, 70 sharks, 18 shopping, 15, 24, 25, 49 sleep, 9 softball, 22 swimming, 65 tests, 52 track, 8 travel, 9, 18, 39, 53, 56 videos, 69 Reflect, 8, 16, 23, 39, 47, 56, 64 rule pattern, 4–10

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

E

R

S slope, 59–65 Spiral Review, 10, 17, 25, 40, 49, 58, 65 Standards Practice, 30–31, 74–75 Step-by-Step Practice, 6, 14, 21, 37, 44, 54, 62 Step-by-Step Problem Solving Practice, 7–8, 15–16, 22–23, 38–39, 46–47, 55–56, 63–64 Look for a pattern, 15, 38 Make a table, 46, 55 Use logical reasoning, 7 Write an equation, 22, 63 Study Guide, 26–27, 67–71 Success Strategy, 31, 75

U unit price, 11–17 unit rate, 11–17

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

V variable, 19–25 Vocabulary, 4, 11, 19, 34, 41, 51, 59 Vocabulary and Concept Check, 26, 67 Vocabulary Check, 9, 17, 25, 40, 49, 58, 65

W Who is Correct?, 6, 13, 20, 36, 44, 53, 61 Writing in Math, 10, 17, 25, 40, 49, 58, 65

Index

77