rational and radical...

Rational and Radical Functions

Section 8A Section 8BRational Functions Radical Functions

8-1 Algebra Lab Model Inverse Variation

8-1 Variation Functions

8-2 Multiplying and Dividing Rational Expressions

8-3 Adding and Subtracting Rational Expressions

8-4 Technology Lab Explore Holes in Graphs

8-4 Rational Functions

8-5 Solving Rational Equations and Inequalities

8-6 Radical Expressions and Rational Exponents

Connecting Algebra to Geometry Area and Volume Relationships

8-7 Radical Functions

8-8 Solving Radical Equations and Inequalities

Pacing Guide for 45-Minute Classes Calendar Planner

Chapter 8 Countdown Weeks 17, 18 DAY 1 DAY 2 DAY 3 DAY 4 DAY 5

8-1 Algebra Lab 8-1 Lesson 8-2 Lesson 8-3 Lesson 8-4 Technology Lab8-4 Lesson

DAY 6 DAY 7 DAY 8 DAY 9 DAY 10

8-4 Lesson 8-5 Lesson Multi-Step Test PrepReady to Go On?

8-6 Lesson Connecting Algebra to Geometry8-7 Lesson

DAY 11 DAY 12 DAY 13 DAY 14

8-7 Lesson 8-8 Lesson Multi-Step Test PrepReady to Go On?

Chapter 8 Test

Pacing Guide for 90-Minute Classes Calendar Planner

Chapter 8DAY 1 DAY 2 DAY 3 DAY 4 DAY 5

8-1 Algebra Lab8-1 Lesson

8-2 Lesson8-3 Lesson

8-4 Technology Lab8-4 Lesson

8-5 LessonMulti-Step Test PrepReady to Go On?

8-6 LessonConnecting Algebra to Geometry8-7 Lesson

DAY 6 DAY 7

8-7 Lesson8-8 Lesson

Multi-Step Test PrepReady to Go On?Chapter 8 Test

564A Chapter 8

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Before Chapter 8 TestingDiagnose mastery of concepts in chapter.

Ready to Go On? SE pp. 609, 637Multi-Step Test Prep SE pp. 608, 636Section Quizzes ARTest and Practice Generator

Prescribe intervention.

Ready to Go On? InterventionScaffolding Questions TE pp. 608, 636Reteach CRB

Lesson Tutorial Videos

Before High Stakes TestingDiagnose mastery of benchmark concepts.

College Entrance Exam Practice SE p. 643Standardized Test Prep SE pp. 646647


College Entrance Exam Practice

Before Every LessonDiagnose readiness for the lesson.

Warm Up TEPrescribe intervention.

Reteach CRB

During Every LessonDiagnose understanding of lesson concepts.

Check It Out! SEQuestioning Strategies TEThink and Discuss SEWrite About It SEJournal TE


Reading Strategies CRBSuccess for Every Learner


After Every LessonDiagnose mastery of lesson concepts.

Lesson Quiz TETest Prep SE

Test and Practice Generator


Reteach CRBTest Prep Doctor TE

Homework Help Online

DIAGNOSE PRESCRIBE

KEY: SE = Student Edition TE = Teachers Edition CRB = Chapter Resource Book AR = Assessment Resources Available online Available on CD- or DVD-ROM

AssessPrior

Knowledge

FormativeAssessment

SummativeAssessment

Before Chapter 8Diagnose readiness for the chapter.

Are You Ready? SE p. 565Prescribe intervention.

Are You Ready? Intervention

After Chapter 8Check mastery of chapter concepts.

Multiple-Choice Tests (Forms A, B, C)Free-Response Tests (Forms A, B, C)Performance Assessment AR

Cumulative Test AR

Test and Practice Generator


Reteach CRB


564B

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Power Presentations Teacher One Stop Premier Online EditionDynamic presentations to engage students. Complete PowerPoint presentations for every lesson in Chapter 8.

Easy access to Chapter 8 resources and assessments. Includes lesson planning, test generation, and puzzle creation software.

Chapter 8 includes Tutorial Videos, Lesson Activities, Lesson Quizzes, Homework Help, and Chapter Project.

Lesson Resources

Technology Highlights for the Teacher

KEY: SE = Student Edition TE = Teachers Edition English Language Learners Spanish version available Available online Available on CD- or DVD-ROM

Before the Lesson

Prepare Teacher One Stop Editable lesson plans Calendar Planner Easy access to all chapter resources

Lesson Transparencies Teacher Tools

Teach the Lesson

Introduce Alternate Openers: Explorations Lesson Transparencies

Warm Up Problem of the Day

Teach Lesson Transparencies Teaching Transparencies

Know-It Notebook Vocabulary Key Concepts

Power Presentations Lesson Tutorial Videos Interactive Online Edition

Lesson Activities Lesson Tutorial Videos

Lab ActivitiesLab Resources Online Online Interactivities TechKeys

Practice the Lesson

Practice Chapter Resources Practice A, B, C

Practice and Problem Solving Workbook IDEA Works! Modifi ed Worksheets and TestsExamView Test and Practice Generator

Homework Help Online Online Interactivities Interactive Online Edition

Homework Help

Apply Chapter Resources Problem Solving

Practice and Problem Solving Workbook Interactive Online Edition

Chapter ProjectProject Teacher Support

After the Lesson

Reteach Chapter Resources Reteach Reading Strategies

Success for Every Learner

Review Interactive Answers and Solutions Solutions KeyKnow-It Notebook

Big Ideas Chapter Review

Extend Chapter Resources Challenge

564C Chapter 8

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Lesson Tutorial Videos Multilingual Glossary Online InteractivitiesStarring Holt authors Ed Burger and Freddie Renfro! Live tutorials to support every lesson in Chapter 8.

Searchable glossary includes defi nitions in English, Spanish, Vietnamese, Chinese, Hmong, Korean, and 4 other languages.

Interactive tutorials provide visually engaging alternative opportunities to learn concepts and master skills.

Technology Highlights for Reaching All Learners

KEY: SE = Student Edition TE = Teachers Edition CRB = Chapter Resource Book Spanish version available Available online Available on CD- or DVD-ROM

Reaching All Learners

All LearnersLab ActivitiesPractice and Problem Solving Workbook Know-It Notebook

Special Needs StudentsPractice A .............................................................................CRBReteach.................................................................................CRBReading Strategies ...............................................................CRBAre You Ready? ............................................................SE p. 565Inclusion .....TE pp. 575, 580, 584, 589, 598, 611, 615, 620, 625IDEA Works! Modifi ed Worksheets and TestsReady to Go On? Intervention Know-It NotebookOnline Interactivities Lesson Tutorial Videos

Developing LearnersPractice A .............................................................................CRBReteach.................................................................................CRBReading Strategies ...............................................................CRBAre You Ready? ............................................................SE p. 565Vocabulary Connections ..............................................SE p. 566Questioning Strategies ...........................................................TEReady to Go On? Intervention Know-It NotebookHomework Help Online Online Interactivities Lesson Tutorial Videos

On-Level LearnersPractice B .............................................................................CRBProblem Solving ..................................................................CRBVocabulary Connections ..............................................SE p. 566Questioning Strategies ...........................................................TEReady to Go On? Intervention Know-It NotebookHomework Help Online Online Interactivities

Advanced LearnersPractice C .............................................................................CRBChallenge .............................................................................CRBChallenge Exercises ...............................................................SEReading and Writing Math Extend ..............................TE p. 567Critical Thinking ................................TE pp. 587, 596, 612, 630

Are You Ready? Enrichment

Ready To Go On? Enrichment

English Language LearnersReading Strategies ...............................................................CRBAre You Ready? Vocabulary ........................................SE p. 565Vocabulary Connections ..............................................SE p. 566Vocabulary Review ......................................................SE p. 638English Language Learners ................TE pp. 570, 585, 611, 649Success for Every LearnerKnow-It NotebookSpanish Study Guide (Resumen y Repaso)Multilingual Glossary Lesson Tutorial Videos

Teaching tips to help all learners appear throughout the chapter. A few that target specific students are included in the lists below.

ENGLISH LANGUAGE LEARNERS

564D

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Are You Ready? Ready to Go On? Test and Practice GeneratorAutomatically assess readiness and prescribe intervention for Chapter 8 prerequisite skills.

Automatically assess understanding of and prescribe intervention for Sections 8A and 8B.

Use Chapter 8 problem banks to create assessments and worksheets to print out or deliver online. Includes dynamic problems.

Ongoing Assessment

Technology Highlights for Assessment

Assessing Prior KnowledgeDetermine whether students have the prerequisite concepts and skills for success in Chapter 8.

Are You Ready? ............................SE p. 565Warm Up .................................................................TE

Test PreparationProvide review and practice for Chapter 8 and standardized tests.

Multi-Step Test Prep ..........................................SE pp. 608, 636Study Guide: Review .........................................SE pp. 638641Test Tackler ........................................................SE pp. 644645Standardized Test Prep ......................................SE pp. 646647College Entrance Exam Practice ..................................SE p. 643Countdown to Testing ........................ SE pp. C4C27IDEA Works! Modifi ed Worksheets and Tests

Alternative AssessmentAssess students understanding of Chapter 8 concepts and combined problem-solving skills.

Chapter 8 Project .........................................................SE p. 564Alternative Assessment ..........................................................TEPerformance Assessment ..................................................... ARPortfolio Assessment ............................................................ AR

Lesson AssessmentProvide formative assessment for each lesson of Chapter 8.

Questioning Strategies ...........................................................TEThink and Discuss ...................................................................SECheck It Out! Exercises ...........................................................SEWrite About It .........................................................................SEJournal ....................................................................................TELesson Quiz .............................................................TEAlternative Assessment ..........................................................TEIDEA Works! Modifi ed Worksheets and Tests

Weekly AssessmentProvide formative assessment for each section of Chapter 8.

Multi-Step Test Prep ..........................................SE pp. 608, 636Ready to Go On? .................SE pp. 609, 637Section Quizzes ..................................................................... ARTest and Practice Generator ................. Teacher One Stop

Chapter AssessmentProvide summative assessment of Chapter 8 mastery.

Chapter 8 Test ..............................................................SE p. 642Chapter Test (Levels A, B, C) ................................................ AR

Multiple Choice Free Response

Cumulative Test .................................................................... ARTest and Practice Generator ................. Teacher One StopIDEA Works! Modifi ed Worksheets and Tests

KEY: SE = Student Edition TE = Teachers Edition AR = Assessment Resources Spanish version available Available online Available on CD- or DVD-ROM

564E Chapter 8

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Formal Assessment

Three levels (A, B, C) of multiple-choice and free-response chapter tests, along with a performance assessment, are available in the Assessment Resources.

A Chapter 8 Test

C Chapter 8 Test

A Chapter 8 Test

C Chapter 8 Test

B Chapter 8 Test

B Chapter 8 Test (continued)

MULTIPLE CHOICE

B Chapter 8 Test

B Chapter 8 Test (continued)

FREE RESPONSE

Chapter 8 Test

Chapter 8 Test (continued)

PERFORMANCE ASSESSMENT

564F

Name ________________________________________ Date ___________________ Class __________________

Performance Assessment Rational and Radical Functions

A rational function, R(x) has the following characteristics: a vertical asymptote at x = 3, a horizontal asymptote at y = 2, and a hole at(2, 2). Sketch the function and determine what it could be.

1. Put in the factor that would account for the vertical asymptote at x = 3.

_________________________________________________________________________________________

2. Add in the factors that would account for a hole at x = 2.

_________________________________________________________________________________________

3. Determine what must be true about the numerator and denominator for there to be a horizontal asymptote at y = 2.

_________________________________________________________________________________________

4. Add the factors that would account for the horizontal asymptote at y = 2.

_________________________________________________________________________________________

5. Describe what you must do in order for the hole to appear at (2, 2).

_________________________________________________________________________________________

6. Find a.

_________________________________________________________________________________________

7. Write the completed function.

_________________________________________________________________________________________

CHAPTER

8

Name ________________________________________ Date ___________________ Class __________________

Performance Assessment Teacher Support Rational and Radical Functions

Purpose:To assess student understanding of analyzing and graphing rational functions.

Time:2030 minutes

Grouping:Individuals or partners

Preparation Hints: Review what creates asymptotes and holes in rational functions.

Introduce the Task: Students are presented with a description of a rational function. By working backward, students will be able to re-create the function that fits the description.

Performance Indicators:

______ Accounts for the vertical asymptote.

______ Accounts for the x-coordinate of the hole.

______ Accounts for the horizontal asymptote.

______ Accounts for the y-coordinate of the hole.

______ Writes the completed function.

Scoring Rubric:

Level 4: Student solves problems correctly and gives good explanations. Level 3: Student solves problems but does not give satisfactory

explanations.Level 2: Student solves some problems but does not give satisfactory

explanations.

Level 1: Student is not able to solve any of the problems.

CHAPTER

8

Create and customize Chapter 8 Tests. Instantly

generate multiple test versions, answer keys, and

practice versions of test items.

Test & Practice Generator

Modified chapter tests that address special learning needs are available in IDEA Works!Modified Worksheets and Tests.

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KEYWORD: MB7 ChProj

564 Chapter 8

Rational and Radical Functions

You can use rational expressions and functions to determine a bicyclists average speed in a race with multiple stages.

8A Rational Functions Lab Model Inverse Variation



8-3 Adding and Subtracting Rational Expressions

Lab Explore Holes in Graphs

8-4 Rational Functions

8-5 Solving Rational Equations and Inequalities

8B Radical Functions 8-6 Radical Expressions and

Rational Exponents

8-7 Radical Functions

8-8 Solving Radical Equations and Inequalities

Apply algebraic reasoning to solve problems with rational and radical expressions.

Make connections among multiple representations of rational and radical functions.

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564 Chapter 8

S E C T I O N 8ARational Functions

On page 608, stu-dents write rational functions and solve rational equations

to solve problems involving the Indianapolis 500.

Exercises designed to prepare students for success on the Multi-Step Test Prep can be found on pages 575, 581, 589, 598, and 606.

S E C T I O N 8BRadical Functions

On page 636, students apply radical func-tions and

equations to model a real-world situation involving clockpendulums.

Exercises designed to prepare students for success on the Multi-Step Test Prep can be found on pages 616, 626, and 634.

About the ProjectIn the Chapter Project, students investigate how the speeds of bicyclists change as they ride up and down hills. They use rational functions to calculate bicyclists average speed during a race with multiple stages by using a weighted harmonic mean.

Race to the Finish

Project ResourcesAll project resources for teachers and students are provided online.

Materials: graphing calculator

KEYWORD: MB7 ProjectTS

Interactivities Online

Lessons 8-5 and 8-6

Lesson Tutorials Online

Lesson Tutorial Videos are available for EVERY example.

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VocabularyMatch each term on the left with a definition on the right.

1. asymptote

2. rational number

3. reflection

4. translation

5. zero of a function

A. any number that can be expressed as a quotient of two integers, where the denominator is not zero

B. a transformation that flips a figure across a line

C. any number x such that f (x) = 0

D. a line that a curve approaches as the value of x or ybecomes very large or very small

E. a whole number or its opposite

F. a transformation that moves each point in a figure the same distance in the same direction

Properties of ExponentsSimplify each expression. Assume that all variables are nonzero.

6. x 11y 5_x 4y 7

7. (3x 2y_z )4

8. (x 3)-2

9. (3x3y)(6x y 5) 10. (2x-4)3 11. 12 x0

Combine Like TermsSimplify each expression.

12. 5 x 2 + 10x - 4x + 6 13. 3x + 12 - 10x 14. x 2 + x + 3 x2 - 4x

Greatest Common FactorFind the greatest common factor of each pair of expressions.

15. 3 a 2 and 12a 16. c 2 d and c d 2 17. 16 x 4 and 40 x 3

Factor TrinomialsFactor each trinomial.

18. x 2 - 4x - 5 19. x2 + 2x - 24 20. x 2 + 12x + 32

21. x 2 + 9x + 18 22. x 2 - 6x + 9 23. x 2 - 8x - 20

Solve Quadratic EquationsSolve.

24. 5 x 2 = 45 25. 4 x 2 - 7 = 93 26. 2 (x - 2)2 = 32

Rational and Radical Functions 565

D

AB

F

C

x 7_y 2

81x 8y 4_

z 4 1_

x 6

5x 2 + 6x + 6 -7x + 12 4x 2 - 3x

3a cd 8x 3

3 5 -2, 6

18x4y6 8_x12

12

(x + 3)(x + 6)

19. (x - 4)(x + 6)18. (x - 5)(x + 1)(x + 4)(x + 8)

(x - 3) 2 (x - 10)(x + 2)

Are You Ready? 565

NOINTERVENE

YESENRICHDiagnose and Prescribe

ARE YOU READY? Intervention, Chapter 8

Prerequisite Skill Worksheets CD-ROM Online

Properties of Exponents Skill 59 Activity 59

Diagnose and Prescribe Online

Combine Like Terms Skill 57 Activity 57

Greatest Common Factor Skill 4 Activity 4

Factor Trinomials Skill 67 Activity 67

Solve Quadratic Equations Skill 81 Activity 81

ARE YOU READY? Enrichment, Chapter 8

WorksheetsCD-ROMOnline

OrganizerObjective: Assess students understanding of prerequisite skills.

Prerequisite Skills

Properties of Exponents

Combine Like Terms

Greatest Common Factor

Factor Trinomials

Solve Quadratic Equations

Assessing Prior Knowledge

INTERVENTIONDiagnose and Prescribe

Use this page to determine whether intervention is necessary or whether enrichment is appropriate.

ResourcesAre You Ready? Intervention and Enrichment Worksheets

Are You Ready? CD-ROM

Are You Ready? Online

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566 Chapter 8

KeyVocabulary/Vocabulariocomplex fraction fraccin compleja

constant of variation constante de variacin

continuous function funcin continua

direct variation variacin directa

discontinuous function funcin discontinua

extraneous solutions soluciones extraas

hole (in a graph) hoyo (en una grfica)

inverse variation variacin inversa

radical equation ecuacin radical

radical function funcin radical

rational equation ecuacin racional

rational exponent exponente racional

rational function funcin racional

Vocabulary Connections

To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like.

1. The word extraneous contains the word extra. What does extra mean? What do you think an extraneous solution is?

2. The graph of a continuous function has no gaps or breaks. How do you think a discontinuous function differs from a continuous function?

3. Do you think a hole could occur in the graph of a continuous function or a discontinuous function? Why?

4. A rational number can be written as a ratio of two integers. What do you think a rational exponent is?

Previously, you solved problems with linear

functions.

simplified polynomial expressions.

graphed functions with asymptotes.

solved quadratic equations and inequalities.

You will study solving problems with

variation functions.

simplifying rational and radical expressions.

graphing rational and radical functions.

solving rational and radical equations and inequalities.

You can use the skills in this chapter

in future math classes, including Precalculus.

to solve problems in other classes, such as Chemistry, Physics, and Biology.

outside of school to make predictions involving time, money, or speed.

566 Chapter 8

Study Guide: Preview

OrganizerObjective: Help students organize the new concepts they will learn in Chapter 8.

Online EditionMultilingual Glossary

Resources

Multilingual Glossary Online

KEYWORD: MB7 Glossary

Answers toVocabulary ConnectionsPossible answers:

1. Extra means more than is needed. An extraneous solution might be a solution that is not needed.

2. The graph of a discontinuous function has at least 1 gap or break.

3. Discontinuous; a hole is probably a type of break.

4. an exponent that can be written as a ratio of 2 integers.

C H A P T E R

8

PuzzleView


Study Strategy: Make Flash CardsYou can use flash cards to help you remember a sequence of steps, the definitions of vocabulary words, or important formulas and properties.

Use these hints to make useful flash cards:

Write a vocabulary word or the name of a formula or property on one side of a card and the meaning on the other.

When memorizing a sequence of steps, make a flash card for each step.

Use examples or diagrams if needed.

Label each card with a lesson number in case you need to look back at your textbook for more information.

Try This

Make flash cards that can help you remember each piece of information.

1. The Product of Powers Property states that to multiply powers with the same base, add the exponents. (See Lesson 1-5.)

2. The quadratic formula, x = -b b 2 - 4ac____________

2a, can be used to find the roots of an

equation with the form a x 2 + bx + c = 0 (a 0) . (See Lesson 5-6.)

Rational and Radical Functions 567

logb m ?

Front

Lesson 74

Quotient Property of Logarithms

n

Back

logb m logbm logbn n

example:

log2 (16) log216 log222

For any positive numbers m, n, and b(b 1

),

WORDS NUMBERS ALGEBRAThe logarithm of a quotient is the logarithm of the dividend minusthe logarithm of the divisor.

logb ( 16_2 ) = logb16 - logb 2 logb m_n = logb m - logb n

Quotient Property of Logarithms

From Lesson

7-4

Sample Flash Card

Reading and Writing Math 567

OrganizerObjective: Help students apply strategies to understand and retain key concepts.

Online Edition

Study Strategy: Make Flash CardsDiscuss Ask students to describe when and how they have used flash cards before. Elicit that flash cards can be a useful study tool because they can provide a quick check or review of what you have learned.Extend Have students work in pairs to create a set of flash cards for the chapter, and ask them to develop a game using the cards that will help them review the chapter content.

Answers to Try This 1. Possible answer:

2. See p. A36.

C H A P T E R

8

Reading Connection

The Square Root of 2by David FlanneryMore than a discourse on the most famous radical of all, 2 , and, by extension, of irrational numbers, this book is a celebration of the many modes of mathematical inquiry. Written in the form of a dialogue between a master teacher and a student, the book provides insights into how mathematicians solve problems.

Activity After students have studied proofs of the irrationality of 2 in the book, ask them to demonstrate one for the class.


Rational Functions

One-Minute Section PlannerLesson Lab Resources Materials

8-1 Algebra Lab Model Inverse Variation Explore an inverse variation relationship. SAT-10 NAEP ACT SAT SAT Subject Tests

Algebra Lab Activities8-1 Lab Recording Sheet

Requiredtape, ruler (MK), pennies, graph paper and/or graphing calculator

Lesson 8-1 Variation Functions Solve problems involving direct, inverse, joint, and combined variation. SAT-10 NAEP ACT SAT SAT Subject Tests

Technology Lab Activities8-1 Technology Lab

Requiredgraph paper and/or graphing calculator

Optionalscience textbook

Lesson 8-2 Multiplying and Dividing Rational Expressions Simplify rational expressions. Multiply and divide rational expressions. SAT-10 NAEP ACT SAT SAT Subject Tests


Optionalhighlighters or colored pencils (MK), flash cards

Lesson 8-3 Adding and Subtracting Rational Expressions Add and subtract rational expressions. Simplify complex fractions. SAT-10 NAEP ACT SAT SAT Subject Tests


Optionalposter board

8-4 Technology Lab Explore Holes in Graphs Identify breaks, or holes, in graphs. SAT-10 NAEP ACT SAT SAT Subject Tests

Technology Lab Activities8-4 Lab Recording Sheet

Requiredgraphing calculator

Lesson 8-4 Rational Functions Graph rational functions. Transform rational functions by changing parameters. SAT-10 NAEP ACT SAT SAT Subject Tests

Algebra Lab Activities8-4 Algebra Lab


Lesson 8-5 Solving Rational Equations and Inequalities Solve rational equations and inequalities. SAT-10 NAEP ACT SAT SAT Subject Tests

Requiredgraphing calculator

Optionalgraph paper

MK = Manipulatives Kit

SECTION

8A

568A Chapter 8

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568B

INVERSE VARIATIONLesson 8-1This section deals with rational expressions and ratio-nal functions. The simplest rational functions are given by inverse variations. Specifically, an inverse variation is a function of the form y = k __ x , where k is a nonzero constant.

The graph of an inverse variation is a hyperbola. The x- and y-axes are asymptotes. The absolute value of k determines the shape of the hyperbola. The smaller the absolute value of k, the closer the curve is to the origin. This is illustrated in the graphs of y = k __ x for k = 1, k = 6, and k = 20.

RATIONAL EXPRESSIONSLessons 8-2, 8-3A rational expression is a quotient of two polynomi-als. Students should realize, however, that a rational expression may be written in a form in which it is not immediately obvious that the expression is a quotient of polynomials. Consider the following expression.

(x - 1) 2

_ x (x + 2) -2

Using the Negative Exponent Property, the expression may be rewritten as

(x - 1) 2 (x + 2) 2

__ x .

Squaring both binomials in the numerator and then multiplying shows that the expression is indeed a rational expression.

Operations on rational expressions are similar to opera-tions on fractions. For example, if a (x) , b (x) , c (x) , and d (x) are polynomials, then

a (x)

_ b (x)

+ c (x)

_ d (x)

= a (x) d (x) + b (x) c (x)

__ b (x) d (x)

except where b (x) = 0 and d (x) = 0. As with frac-tions, it is generally best to simplify rational expressions before adding, subtracting, multiplying, or dividing.

RATIONAL FUNCTIONSLesson 8-4A rational function has the form f (x) =

p (x) ____ q (x) , where

p (x) and q (x) are polynomials. Note that q (x) may be a constant, in which case the rational function simplifies to a polynomial. In this text, however, q (x) is gener-ally taken to be a polynomial of degree greater than or equal to 1 in order to restrict the discussion to rational functions that are not polynomials.

The graph of a rational function has a vertical asymp-tote at each real value of x for which q (x) = 0 and p (x) 0. However, not every rational function has vertical asymptotes. For example, the rational function f (x) = 1 _____

x 2 + 1 has a denominator that is never 0, and the

graph of this function is a continuous curve.

It is also interesting to note that rational functions may have asymptotes that are neither horizontal nor vertical. Example 3 on page 594 shows the graph of f (x) = x

2 - 2x - 3 _________ x - 2 . Using long division, this function

may be written as f (x) = x - 3 ____ x - 2 . For very large values of x , 3 ____ x - 2 is close to 0, so the value of f (x) is close to that of the function g (x) = x. Thus, the line y = x is an asymptote.

Math Background

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Use with Lesson 8-1

568 Chapter 8 Rational and Radical Functions

Model Inverse VariationIn this activity, you will explore the relationship between the mass of an object and the objects distance from the pivot point, or fulcrum, of a balanced lever.

8-1

Activity

1 Secure a pencil to a tabletop with tape. The pencil will be the fulcrum.

2 Draw an arrow on a piece of tape, and use the arrow to mark the midpoint of a ruler. Then tape a penny to the end of the ruler.

3 Place the midpoint of the ruler on top of the pencil. The ruler is the lever.

4 Place one penny on the lever opposite the taped penny. If needed, move the untaped penny to a position that makes the lever balanced. Find the distance from the untaped penny to the fulcrum, and record the distance in a table like the one below. (Measure from the center of the penny.) Repeat this step with stacks of two to seven pennies.

Let x be the number of pennies and y be the distance from the fulcrum. Plot the points from your table on a graph. Then draw a smooth curve through the points.

Try This

1. Multiply the corresponding x- and y-values together. What do you notice?

2. Use your answer to Problem 1 to write an equation relating distance from the fulcrum to the number of pennies.

3. Would it be possible to balance a stack of 20 pennies on the lever? Use your equation from Problem 2 to justify your answer.

4. Make a Conjecture The relationship between the mass of an object on a balanced lever and the objects distance from the fulcrum can be modeled by an inverse variation function. Based on your data and graph, how are the variables in an inverse variation related?

Number of Pennies 1 2 3 4 5 6 7

Distance from Fulcrum (cm)

15 7.5 5 3.8 3 2.5 2.1Possible answer: The products are all approximately equal.

Possible answer: xy = 15

Possible answers:

Organizer

Pacing: Traditional 1 dayBlock 1 __ 2 day

Objective: Explore an inverse variation relationship.

Materials: pencil, tape, ruler, 8 pennies

Online Edition

Countdown Week 17

ResourcesAlgebra Lab Activities

8-1 Lab Recording Sheet

TeachDiscussAsk students to predict whether they should move the pennies closer to or farther from the fulcrum as they increase the number of pennies.

CloseKey ConceptIn an inverse variation, one quantity increases as another decreases, and the product of the quantities is constant.

AssessmentJournal Have students describe the relationship they observe between the number of pennies in the stack and the distance in centimeters from the fulcrum.

Use with Lesson 8-1

568 Chapter 8

Answer to Activity

Answers to Try This 3. Possible answer: Substituting 20 for

x in the equation yields 20y = 15, or y = 0.75. According to the equation, it is possible to balance a stack of 20 pennies by placing them 0.75 cm from the fulcrum. However, this would probably not be possible because of the size of the pennies.

4. Possible answer: An increase in 1 variable causes a decrease in the other variable, and a decrease in 1 variable causes an increase in the other variable.

KEYWORD: MB7 Resources

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Why learn this?You can use variation functions to determine how many people are needed to complete a task, such as building a home, in a given time. (See Example 5.)

8- 1 Variation Functions 569

In Chapter 2, you studied many types of linear functions. One special type of linear function is called direct variation. A direct variation is a relationship between two variables x and y that can be written in the form y = kx, where k 0. In this relationship, k is the constant of variation . For the equation y = kx, y varies directly as x.

A direct variation equation is a linear equation in the form y = mx + b, where b = 0 and the constant of variation k is the slope. Because b = 0, the graph of a direct variation always passes through the origin.

1E X A M P L E Writing and Graphing Direct VariationGiven: y varies directly as x, and y = 14 when x = 3.5. Write and graph the direct variation function.

y = kx y varies directly as x.

14 = k(3.5) Substitute 14 for y and 3.5 for x.

4 = k Solve for the constant of variation k.

y = 4x Write the variation function by using the value of k.

Graph the direct variation function.

The y-intercept is 0, and the slope is 4.

Check Substitute the original values of x and y into the equation.

y = 4x 14 4 (3.5)

14 14

1. Given: y varies directly as x, and y = 6.5 when x = 13. Write and graph the direct variation function.

When you want to find specific values in a direct variation problem, you can solve for k and then use substitution or you can use the proportion derived below.

y 1 = kx1 y1_x1

= k and y2 = kx2 y2_x2

= k so, y1_x1

= y2_x2

.


ObjectiveSolve problems involving direct, inverse, joint, and combined variation.

Vocabularydirect variationconstant of variationjoint variationinverse variationcombined variation

If k is positive in a direct variation, the value of y increases as the value of xincreases.

y = 0.5x

Lesson 8-1 569

Introduce1

Explorations and answers are provided in Alternate Openers: Explorations Transparencies.

8-1 OrganizerPacing: Traditional 1 day

Block 1 __ 2 dayObjective: Solve problems involving direct, inverse, joint, and combined variation.

Technology LabIn Technology Lab Activities

Online EditionTutorial Videos

Countdown Week 17

Warm UpSolve each equation.

1. 2.4 _ x = 2 _ 9 10.8

2. 1.6x = 1.8 (24.8) 27.9

Determine whether each data set could represent a linear function.

3. x 2 4 6 8

y 12 6 4 3

no

4. x -2 -1 0 1

y -6 -2 2 6

yes

Also available on transparency

Q: Whats reverse variation?

A: The relationship between the length of a driveway and the chance of backing into a mailbox.

MotivateGive students a few examples of quantities that vary directly or indirectly, such as number of songs downloaded and total cost of songs, or number of workers and time needed to build a highway. Ask students to describe how an increase in one quantity would affect the other. Explain to students that relationships such as these can be described mathematically by using variation functions.


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2E X A M P L E Solving Direct Variation Problems The circumference of a circle C varies directly as the radius r, and C = 7 ft

when r = 3.5 ft. Find r when C = 4.5 ft.

Method 1 Find k. Method 2 Use a proportion.

C 1_r1

= C2_r2

7

_3.5

= 4.5 _r Substitute. 7r = 15.75 Find the cross

products.

r = 2.25 Solve for r.

C = kr

7 = k(3.5) Substitute.

2 = k Solve for k.

Write the variation function.

C = (2) r Use 2 for k.

4.5 = (2) r Substitute 4.5 for C. 2.25 = r Solve for r.

The radius r is 2.25 ft.

2. The perimeter P of a regular dodecagon varies directly as the side length s, and P = 18 in. when s = 1.5 in. Find s when P = 75 in.

A joint variation is a relationship among three variables that can be written in the form y = kxz, where k is the constant of variation. For the equation y = kxz,y varies jointly as x and z.

3E X A M P L E Solving Joint Variation Problems The area A of a triangle varies jointly as the base b and the height h, and

A = 12 m2 when b = 6 m and h = 4 m. Find b when A = 36 m 2 and h = 8 m.

Step 1 Find k. Step 2 Use the variation function.

A = kbh Joint variation A = 1_2

bh Use 1_2

for k.

12 = k(6)(4) Substitute. 36 = 1_2

b(8) Substitute.

1_2

= k Solve for k. 9 = b Solve for b.

The base b is 9 m.

3. The lateral surface area L of a cone varies jointly as the base radius r and the slant height , and L = 63 m 2 when r = 3.5 m and = 18 m. Find r to the nearest tenth when L = 8 m 2 and = 5 m.

A third type of variation describes a situation in which one quantity increases and the other decreases. For example, the table shows that the time needed to drive 600 miles decreases as speed increases.

This type of variation is an inverse variation. An inverse variation is a relationship between two variables x and y that can be written in the form y = k__x , where k 0. For the equation y =

k__x ,

y varies inversely as x.

Speed(mi/h)

Time(h)

Distance(mi)

30 20 600

40 15 600

50 12 600

The phrases yvaries directly as xand y is directly proportional to x have the same meaning.

6.25 in.

1.6 m

570 Chapter 8

Example 1

Given: y varies directly as x, and y = 27 when x = 6. Write and graph the direct variation function. y = 4.5x

Example 2

The cost of an item in euros e varies directly as the cost of the item in dollars d, and e = 3.85 euros when d = $5.00. Find d when e = 10.00 euros. $12.99

Example 3

The volume V of a cone varies jointly as the area of the base B and the height h, and V = 12 ft 3 when B = 9 ft 2 and h = 4 ft. Find B when V = 24 ft 3 and h = 9 ft. 8 ft 2

Additional Examples

INTERVENTION Questioning Strategies

EXAMPLE 1

How do you determine the con-stant of variation?

How can you determine the slope and y-intercept of a direct variation function before you graph it?

EXAMPLE 2

How do you set up a proportion to solve a direct variation problem?

EXAMPLE 3

How do you know how to write the joint variation equation from the information given in the problem?

Reading Math Emphasize that the phrase y varies jointly as x and z indicates

that y varies directly as the product of x and z, rather than as the sum of x and z.

Guided InstructionAs you work through the direct-variation examples, point out that one quantity increases as the other increases, but that this alone does not make a direct variation. The relationship is a direct variation only if the ratio of the x- and y-values is constant. Likewise, a relationship in which one quan-tity increases as the other decreases is only an inverse variation if the product of the x- and y-values is constant. Introduce stu-dents to joint and combined variation by using what they have learned about direct and inverse variation.

Teach2

Through Cognitive Strategies

Work with students to develop mnemonic devices that they can use to remember the properties of direct and inverse variations. For example, to find the constant of varia-tion of a direct variation, you must divide y by x, and both direct and divide begin with d. Likewise, the graph of an indirect varia-tion is always nonlinear, and both indirect and nonlinear contain the letter n.

ENGLISH LANGUAGE LEARNERS



4E X A M P L E Writing and Graphing Inverse VariationGiven: y varies inversely as x, and y = 3 when x = 8. Write and graph the inverse variation function.

y = k_x y varies inversely as x.

3 = k_8

Substitute 3 for y and 8 for x.

k = 24 Solve for k.

y = 24_x Write the variation function.

To graph, make a table of values for both positive and negative values of x. Plot the points, and connect them with two smooth curves. Because division by 0 is undefined, the function is undefined when x = 0.

x y

-3 -8

-4 -6

-8 -3

-12 -2

x y

3 8

4 6

8 3

12 2

4. Given: y varies inversely as x, and y = 4 when x = 10. Write and graph the inverse variation function.

When you want to find specific values in an inverse variation problem, you can solve for k and then use substitution or you can use the equation derived below.

y 1 = k_x1 y1x1 = k and y2 =

k_x2 y2x2 = k so, y1x1 = y2x2.

5E X A M P L E Community Service ApplicationThe time t that it takes for a group of volunteers to construct a house varies inversely as the number of volunteers v. If 20 volunteers can build a house in 62.5 working hours, how many volunteers would be needed to build a house in 50 working hours?

Method 1 Find k. Method 2 Use t1v1 = t2v2 .

t = k_v t 1v1 = t2v2

62.5 = k_20

Substitute. 62.5(20) = 50v Substitute.

1250 = k Solve for k. 1250 = 50v Simplify.

t = 1250_v Use 1250 for k. 25 = v Solve for v.

50 = 1250_v Substitute 50 for t.

v = 25 Solve for v.

So 25 volunteers would be needed to build a home in 50 working hours.

5. What if...? How many working hours would it take 15 volunteers to build a house?

When graphing an inverse variation function, use values of x that are factors of k so that the y-values will be integers.

83 1_3 working hours

y = 40_x

Lesson 8-1 571

Students might determine the con-stant of variation and then forget to use it to solve the original problem. Remind students that variation prob-lems usually involve more than one step. Often, the first step is to write the variation function by finding the constant of variation. The second step is to use the variation function to solve the problem.

Example 4

Given: y varies inversely as x, and y = 4 when x = 5. Write and graph the inverse variation function. y = 20 _ x

Example 5

The time t needed to complete a certain race varies inversely as the runners average speed s. If a runner with an average speed of 8.82 mi/h completes the race in 2.97 h, what is the average speed of a runner who completes the race in 3.5 h? 7.48 mi/h

Additional Examples


EXAMPLE 4

How do you use the given informa-tion to find the value of k?

Does an inverse variation function have a y-intercept? Explain.

EXAMPLE 5

How can you solve the problem without finding the value of k?

How can you check your answer to an inverse variation problem?

Math Background The constant of variation is also called the constant of proportionality.

Geometry Review with students common formulas for perimeter, area, and volume. Discuss with

students the types of variation in these formulas. For example, the perimeter P of a square varies directly as its side length s, and the volume V of a rectangular prism varies jointly as its length , width w, and height h.



You can use algebra to rewrite variation functions in terms of k.

Direct Variation

Constant ratio

Inverse Variation

Constant product

Notice that in direct variation, the ratio of the two quantities is constant. In inverse variation, the product of the two quantities is constant.

6E X A M P L E Identifying Direct and Inverse VariationDetermine whether each data set represents a direct variation, an inverse variation, or neither.

A x 3 8 10y 9 24 30

B x 4.5 12 2y 8 3 18

In each case, y__x = 3.

The ratio is constant, so this represents a direct variation.

In each case, xy = 36.The product is constant, so this represents an inverse variation.

Determine whether each data set represents a direct variation, an inverse variation, or neither.

6a. x 3.75 15 5

y 12 3 9

6b. x 1 40 26

y 0.2 8 5.2

A combined variation is a relationship that contains both direct and inverse variation. Quantities that vary directly appear in the numerator, and quantities that vary inversely appear in the denominator.

7E X A M P L E Chemistry ApplicationThe volume V of a gas varies inversely as the pressure P and directly as the temperature T. A certain gas has a volume of 10 liters (L), a temperature of 300 kelvins (K), and a pressure of 1.5 atmospheres (atm). If the gas is compressed to a volume of 7.5 L and is heated to 350 K, what will the new pressure be?

Step 1 Find k. Step 2 Use the variation function.

V = kT_P

Combinedvariation

V = 0.05T_P

Use 0.05 for k.

10 = k(300)_

1.5 Substitute. 7.5 =

0.05(350)_P

Substitute.

0.05 = k Solve for k. P = 2.

3 Solve for P.

The new pressure will be 2.

3 , or 2 1_3

, atm.

7. If the gas is heated to 400 K and has a pressure of 1 atm, what is its volume?

A kelvin (K) is a unit of temperature that is often used by chemists.

0C = 273.15 K100C = 373.15 K

inverse direct

20 L

572 Chapter 8

Example 6


A. x 6.5 13 104

y 8 4 0.5

inverse variation

B. x 5 8 12

y 30 48 72

direct variation

C. x 3 6 8

y 5 14 21

neither

Example 7

The change in temperature T of an aluminum wire varies inversely as its mass m and directly as the amount of heat energy E trans-ferred. The temperature of an alu-minum wire with a mass of 0.1 kg rises 5C when 450 joules (J) of heat energy are transferred to it. How much heat energy must be transferred to an aluminum wire with a mass of 0.2 kg to raise its temperature 20C? 3600 J

Additional Examples


EXAMPLE 6

What does a constant ratio y _ x

indicate about a data set? a con-stant product xy?

EXAMPLE 7

How do you know which quanti-ties to write in the numerator and which to write in the denominator?

Assess After the Lesson8-1 Lesson Quiz, TE p. 576

Alternative Assessment, TE p. 576

Monitor During the LessonCheck It Out! Exercises, SE pp. 569572Questioning Strategies, TE pp. 570572

Diagnose Before the Lesson8-1 Warm Up, TE p. 569

and INTERVENTIONSummarizeAsk students to describe the type of varia-tion that each function represents, assum-ing that d is the dependent variable.

d = 26 _ m d varies inversely as m.

d = (7.3s) t d varies jointly as s and t.

d = _ 4 p d varies directly as p.

d = 8g

_ 3h

d varies directly as g and inversely as h.

Close3


8-1 ExercisesExercises


THINK AND DISCUSS 1. Explain why the graph of a direct variation is a line.

2. Describe the type of variation between the length and the width of a rectangular room with an area of 400 ft 2 .

3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the general variation equation, draw a graph, or give an example.

GUIDED PRACTICE 1. Vocabulary A variation function in which k is positive and one quantity decreases

when the other increases is a(n) ? . (direct variation or indirect variation)

SEE EXAMPLE 1 p. 569

Given: y varies directly as x. Write and graph each direct variation function.

2. y = 6 when x = 3 3. y = 45 when x = -5 4. y = 54 when x = 4.5


5. Physics The wavelength of a wave of a certain frequency varies directly as the velocity v of the wave, and = 60 ft when v = 15 ft/s. Find when v = 3 ft/s.

6. Work The dollar amount d that Julia earns varies directly as the number of hours tthat she works, and d = $116.25 when t = 15 h. Find t when d = $178.25.


7. Geometry The volume V of a rectangular prism of a particular height varies jointly as the length and the width w, and V = 224 ft 3 when = 8 ft and w = 4 ft. Find when V = 210 ft 3 and w = 5 ft.

8. Economics The total cost C of electricity for a particular light bulb varies jointly as the time t that the light bulb is used and the cost k per kilowatt-hour, and C = 12 when t = 50 h and k = 6 per kilowatt-hour. Find C to the nearest cent when t = 30 h and k = 8 per kilowatt-hour.


Given: y varies inversely as x. Write and graph each inverse variation function.

9. y = 2 when x = 7 10. y = 8 when x = 4 11. y = 1_2

when x = -10


12. Travel The time t that it takes for a salesman to drive a certain distance d varies inversely as the average speed r. It takes the salesman 4.75 h to travel between two cities at 60 mi/h. How long would the drive take at 50 mi/h?



13. x 2 5 9

y 3 6 4

14. x 6 4 1

y 2 3 12

15. x 24 4 12

y 30 5 15

KEYWORD: MB7 Parent

KEYWORD: MB7 8-1


indirect variation

y = 2x y = -9x y = 12x

12 ft

23 h

6 ft

10

y = 14_x y = 32_x y = - 5_x5.7 h

neither inverse direct

Lesson 8-1 573


Answers to Think and DiscussPossible answers:

1. A direct variation equation is in the form y = mx + b, with m = k and b = 0.

2. The length varies inversely as the width, with a constant of varia-tion of 400.

3. See p. A9.

Assignment Guide

Assign Guided Practice exercises as necessary.

If you finished Examples 13 Basic 1723, 39, 40 Average 1723, 39, 40, 50 Advanced 1723, 39, 40, 4950

If you finished Examples 17 Basic 1742, 4448, 5256 Average 1748, 50, 5256 Advanced 1756

Homework Quick CheckQuickly check key concepts.Exercises: 18, 20, 22, 24, 27, 28,

31, 39

Science Link In Exercise 5, the symbol is the Greek letter lambda. This symbol

is traditionally used by physicists to represent wavelength.

Answers 2.

3.

4.

9.

10.

11.

a207te_c08_l1_569-576.indd 573a207te_c08_l1_569-576.indd 573 12/20/05 3:31:01 PM12/20/05 3:31:01 PM



16. Cars The power P that must be delivered by a car engine varies directly as the distance d that the car moves and inversely as the time t required to move that distance. To move the car 500 m in 50 s, the engine must deliver 147 kilowatts (kW) of power. How many kilowatts must the engine deliver to move the car 700 m in 30 s?

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

1719 1 2021 2 2223 3 2426 4 27 5 2830 6 31 7

Independent Practice Given: y varies directly as x. Write and graph each direct variation function.

17. y = 4 when x = 8 18. y = 12 when x = 2 19. y = -15 when x = 5

20. Medicine The dosage d of a drug that a physician prescribes varies directly as the patients mass m, and d = 100 mg when m = 55 kg. Find d to the nearest milligram when m = 70 kg.

21. Nutrition The number of Calories C in a horned melon varies directly as its weight w, and C = 25 Calwhen w = 3.5 oz. How many Calories are in the horned melon shown on the scale? Round to the nearest Calorie.

22. Agriculture The number of bags of soybean seeds N that a farmer needs varies jointly as the number of acres a to be planted and the pounds of seed needed per acre p, and N = 980 when a = 700 acres and p = 70 lb/acre. Find N when a = 1000 acres and p = 75 lb/acre.

23. Physics The heat Q required to raise the temperature of water varies jointly as the mass m of the water and the amount of temperature change T, and Q = 20,930 joules (J) when m = 1 kg and T = 5C. Find m when Q = 8372 J and T = 10C.

Given: y varies inversely as x. Write and graph the inverse variation function.

24. y = 1 when x = 0.8 25. y = 1.75 when x = 6 26. y = -2 when x = 3

12.35 oz


28. x 5 6.25 10

y 5 4 2.5

29. x 5 7 9

y 3 5 7

30. x 8 14 24

y 12 21 36

31. Chemistry The volume V of a gas varies inversely as the pressure P and directly as the temperature T. A certain gas has a volume of 20 L, a temperature of 320 K, and a pressure of 1 atm. If the gas is compressed to a volume of 15 L and is heated to 330 K, what will the new pressure be?

Tell whether each statement is sometimes, always, or never true.

32. Direct variation is a linear function.

33. A linear function is a direct variation.

34. An inverse variation is a linear function.

35. In a direct variation, x = 0 when y = 0.

36. The graph of an inverse variation passes through the origin.

Skills Practice p. S18Application Practice p. S39

Extra Practice

Broadway shows are plays or musicals presented in larger theaters near New York Citys Times Square. In 2004, 11.3 million tickets were sold to Broadway shows for a total of almost $750 million.

Entertainment

27. Entertainment The number of days it takes a theater crew to set up a stage for a muscial varies inversely as the number of workers. If the stage can be set up in 3 days by 20 workers, how many days would it take if only 12 workers were available?

343 kilowatts

y = 1_2

xy = 6x

y = -3x

127 mg

88 Cal

1500

0.2 kg

5 days

inverse neither direct

1.375 atm

alwayssometimes

never

always

never

574 Chapter 8

Answers 17.

18.

19.

24. y = 4 _ 5x

8-1 RETEACH8-1 READING STRATEGIES

8-1 PRACTICE C

8-1 PRACTICE B

8-1 PRACTICE A



38. Data Collection Use a graphing calculator, a motion detector, and a light detector to measure the intensity of light as distance from the light source increases. Position the detectors next to each other. Place a flashlight in front of the detectors, and then pull the flashlight away from them. Find an appropriate model for the intensity of the light as a function of the square of the distance from the light source.

39. Multi-Step Interest earned on a certificate of deposit (CD) at a certain rate varies jointly as the principal in dollars and the time in years.

a. Diane purchased a CD for $2500 that earned $12.50 simple interest in 3 months. Write a variation function for this data.

b. At which bank did Diane buy her CD?

c. How much interest would Diane earn in 6 months on a $3000 CD bought from the same bank?

Complete each table.

40. y varies jointly as x and z. 41. y varies directly as x and inversely as z.

x y z

2 4

5 52.5 7

1.5 -36

1.38 23

x y z

25 13.75 4

1 11

17 18.7

10 5

42. Estimation Shane swims 42 laps in 26 min 19 s. Without using a calculator, estimate how many minutes it would take Shane to swim 15 laps at the same average speed.

43. Critical Thinking Explain why only one point (x, y) is needed to write a direct variation function whose graph passes through this point.

44. Write About It Explain how to identify the type of variation from a list of ordered pairs.

45. Which of the following would best be represented by an inverse variation function?

The distance traveled as a function of speed

The total cost as a function of the number of items purchased

The area of a circular swimming pool as a function of its radius

The number of posts in a 20-ft fence as a function of distance between posts

37. This problem will prepare you for the Multi-Step Test Prep on page 608.

In an auto race, a car with an average speed of 200 mi/h takes an average of 31.5 s to complete one lap of the track.

a. Write an inverse variation function that gives the average speed s of a car in miles per hour as a function of the time t in seconds needed to complete one lap.

b. How many seconds does it take the car to complete one lap at an average speed of 210 mi/h?

s = 6300_t

30 s

38. Answers will vary. Possible answer:

I(d) = 1150____d2

, where

I is intensity in milliwatts per square centimeter and d is distance from thelight source in centimeters

I = 0.02Pt

39b. True Federal Bank$30

12

9 min

43. Possible answer: 2 points are needed to determine a line. Since direct variations always include (0, 0) , only 1 other point is needed to write the equation.

Possible answer: If the ratios of the coordinates in each ordered pair are the same, the variation is direct. If the products of the coordi-nates in each ordered pair are the same, the variation is inverse.

0.04

-16

5

4.4

2

Lesson 8-1 575

8-1 PROBLEM SOLVING 8-1 CHALLENGE

Students may have difficulty graph-ing the inverse variation functions in Exercises 2426. Remind them that the graphs will have two unconnected branches and that neither branch will cross the x- or y-axis.

Exercise 37 involves writing and using an inverse varia-

tion function that gives the average speed of a car. This exercise prepares students for the Multi-Step Test Prep on page 608.

Data Collection To help students com-plete Exercise 38, see

Technology Lab Activities.

Inclusion To complete the tables in Exercises 4041,students should start by

using the completed row to find the constant of variation. Then they can use the variation functions to com-plete the rest of the table.

Answers

25. y = 10.5_x

26. y = - 6_x



46. Which statement is best represented by the graph?

y varies directly as x2.

y varies inversely as x.

y varies directly as x.

x varies inversely as y.

47. Which equation is best represented by the following statement: y varies directly as the square root of x?

y = k_ x

y = k_x2

k = xy y = k x

48. Gridded Response The cost per student of a ski trip varies inversely as the number of students who attend. It will cost each student $250 if 24 students attend. How many students would have to attend to get the cost down to $200?

CHALLENGE AND EXTEND 49. Given: y varies jointly as x and the square of z, and y = 189 when x = 7 and z = 9.

Find y when x = 2 and z = 6.

50. Government The number of U.S. Representatives that each state receives can be approximated with a direct variation function where the number of representatives (rounded to the nearest whole number) varies directly with the states population.

a. Given that Pennsylvania has 19 representatives, find k to eight decimal places and write the direct variation function.

b. Find the number of representatives for each state shown.

c. Given that Texas had 32 U.S. representatives in the year 2000, estimate the states population in that year.

51. Estimation Given: y is inversely proportional to x, directly proportional to z2 , and the constant of variation is 7. Estimate the value of y when x = 12 and z = 2.

SPIRAL REVIEW 52. Architecture Brad stands next to the Eiffel Tower. He is 6 ft 8 in. tall and casts

a shadow of 9 ft 4 in. The Eiffel Tower is 985 ft tall. How long is the Eiffel Towers shadow, in feet? (Lesson 2-2)

Write an equation of the line that includes the points in the table. (Lesson 2-4)

53. x 1 3 5 7 9

y 1.5 4 6.5 9 11.5

54. x -4 -2 0 2 4

y 10 9 8 7 6

Make a table of values, and graph the exponential function. Describe the asymptote. Tell how the graph is transformed from the graph of f (x) = 4 x. (Lesson 7-7)

55. g(x) = 1_2

( 4 x) - 2 56. h(x) = ( 4 x - 1) + 1

Florida15,982,378

Illinois12,419,293

Pennsylvania12,281,054

Michigan9,938,444

State Populationsfrom the 2000 U.S. Census

30

24

50a. k 0.00000155; r 0.00000155p

FL: 25; IL: 19; MI: 15

y 7

1379 ft

53. y = 5_4

x + 1_4

54. y = - 1_2

x + 8

21,000,000

576 Chapter 8

In Exercise 48, stu-dents who answered 19 or 20 may have

solved the problem using direct vari-ation instead of inverse variation.

JournalHave students compare and contrast direct and inverse variation func-tions, including characteristics of the functions graphs.

Have students choose formulas from a science textbook that exem-plify each type of variation function covered in this lesson (direct, joint, inverse, combined). For each formu-la, ask them to identify the type of variation and the constant of varia-tion and to describe the relationship between the quantities involved.

8-1

1. The volume V of a pyramid varies jointly as the area of the base B and the height h, and V = 24 ft 3 when B = 12 f t 2 and h = 6 ft. Find B when V = 54 ft 3 and h = 9 ft. 18 f t 2

2. The cost per person c of chartering a tour bus varies inversely as the number of passengers n. If it costs $22.50 per person to charter a bus for 20 passengers, how much will it cost per person to charter a bus for 36 passengers? $12.50

3. The pressure P of a gas varies inversely as its volume V and directly as the temperature T. A certain gas has a pressure of 2.7 atm, a volume of 3.6 L, and a temperature of 324 K. If the volume of the gas is kept constant and the temperature is increased to 396 K, what will the new pressure be? 3.3 atm


Answers 55.

x -2 -1 0 1 2

g (x) - 63 _ 32

- 15 _ 8

- 3 _ 2

0 6

asymptote: y = -2; vertically com-pressed by a factor of 1 __ 2 and translated 2 units down

56. x -2 -1 0 1 2

h (x) 65 _ 64 17 _ 16

5 _ 4

2 5

asymptote: y = 1; translated 1 unit right and 1 unit up


8- 2 Multiplying and Dividing Rational Expressions 577

In Lesson 8-1, you worked with inverse variation functions such as y = 5__x . The expression on the right side of this equation is a rational expression. A rational expression is a quotient of two polynomials. Other examples of rational expressions include the following:

x2 - 4_

x + 2 10_

x2 - 6 x + 3_

x - 7

Because rational expressions are ratios of polynomials, you can simplify them the same way as you simplify fractions. Recall that to write a fraction in simplest form, you can divide out common factors in the numerator and denominator.

1E X A M P L E Simplifying Rational ExpressionsSimplify. Identify any x-values for which the expression is undefined.

A 3x7_

2x4

3_2

x7 - 4 = 3_2

x3 Quotient of Powers Property

The expression is undefined at x = 0 because this value of x makes 2x4 equal 0.

B x2

- 2x - 3__x2 + 5x + 4

(x - 3)(x + 1)__(x + 1)(x + 4)

= (x - 3)_(x + 4)

Factor; then divide out common factors.

The expression is undefined at x = -1 and x = -4 because these values of x make the factors (x + 1) and (x + 4) equal 0.

Check Substitute x = -1 and x = -4 into the original expression.

(

-1) 2 - 2(-1) - 3__(

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

0_0

(

-4) 2 - 2(-4) - 3__(

-4)2 + 5 (-4) + 4 =

21_0

Both values of x result in division by 0, which is undefined.

Simplify. Identify any x-values for which the expression is undefined.

1a. 16x11_

8x2 1b. 3x + 4__

3x2 + x - 4 1c. 6x

2 + 7x + 2__6x2 - 5x - 6


ObjectivesSimplify rational expressions.

Multiply and divide rational expressions.

Vocabularyrational expression

Why learn this?You can simplify rational expressions to determine the probability of hitting an archery target. (See Exercise 35.)

9_24

= 3 3_8 3

= 3_8

When identifying values for which a rational expression is undefined, identify the values of the variable that make the original denominator equal to 0.

1a. 2 x9 ; x 0

1b. 1_x - 1

; x - 4_3

and x 1

1c. (2x + 1)_(2x - 3)

; x - 2_3 and x 3_

2

Lesson 8-2 577

Introduce1

Explorations and answers are provided in Alternate Openers: Explorations Transparencies.

8-2 OrganizerPacing: Traditional 1 day

Block 1 __ 2 dayObjectives: Simplify rational expressions.

Multiply and divide rational expressions.

Online EditionTutorial Videos, TechKeys

Countdown Week 17

Warm UpSimplify each expression. Assume all variables are nonzero.

1. x 5 x 2 x 7 2. y 3 y 3 y 6

3. x 6 _

x 2 x 4 4.

y 2 _

y 5 1 _

y 3

Factor each expression.

5. x 2 - 2x - 8 (x - 4) (x + 2)

6. x 2 - 5x x (x - 5)

7. x 5 - 9 x 3 x 3 (x - 3) (x + 3)


Q: Why did the doctor send the expression to a psychiatrist?

A: Because it wasnt rational.

MotivateAsk students to explain the steps needed to multiply fractions such as 5 __ 8 and 1 __ 5 . Then have students discuss how they would multiply the fractions if the number 5 in each fraction were replaced by the expression x + 5. Explain that in this lesson students will learn to multiply and divide fractions that have algebraic expres-sions in the numerator and denominator.




2E X A M P L E Simplifying by Factoring -1Simplify 2x - x

2_______x2 - x - 2

. Identify any x-values for which the expression

is undefined.

-1(x2 - 2x)__x2 - x - 2

Factor out 1 in the numerator so that x2

is positive, and reorder the terms.

-1(x)(x - 2)__(x - 2)(x + 1)

Factor the numerator and denominator.

Divide out common factors.

-x_x + 1

Simplify.

The expression is undefined at x = 2 and x = -1.

Check The calculator screens suggest that 2x - x

2________x2 - x - 2

= -x____x + 1 except

when x = 2 or x = -1.


2a. 10 - 2x_x - 5

2b. - x2 + 3x__

2x2 - 7x + 3

You can multiply rational expressions the same way that you multiply fractions.

Multiplying Rational Expressions

1. Factor all numerators and denominators completely.

2. Divide out common factors of the numerators and denominators.

3. Multiply numerators. Then multiply denominators.

4. Be sure the numerator and denominator have no common factors other than 1.

3E X A M P L E Multiplying Rational ExpressionsMultiply. Assume that all expressions are defined.

A 2x4y5_

3x2 15x

2_8x3y2

B x + 2_3x + 12

x + 4_x2 - 4

2x4 1 y5 3_3x2

515x2_

48x3y2 x + 2_

3(x + 4) x + 4__

(x + 2)(x - 2)

5xy3_4

1_3(x - 2)

or 1_3x - 6

Multiply. Assume that all expressions are defined.

3a. x_15

x7_

2x 20_

x4 3b. 10x - 40__

x2 - 6x + 8 x + 3_

5x + 15

-2; x 5

-x_2x - 1

; x 3

and x 1_2

2x3_3 2_

x - 2

578 Chapter 8

Example 1


A. 10 x 8 _

6 x 4 5 _

3 x 4 ; x 0

B. x 2 + x - 2

__ x 2 + 2x - 3

x + 2

_ x + 3

; x -3, 1

Example 2

Simplify 4x - x 2 _________

x 2 - 2x - 8 . Identify

any x-values for which the expression is undefined. -x _

x + 2 ;

x -2, 4

Example 3


A. 3 x 5 y 3

_ 2 x 3 y 7

10 x 3 y 4

_ 9 x 2 y 5

5 x 3 _

3 y 5

B. x - 3 _ 4x + 20

x + 5

_ x 2 - 9

1 _ 4 (x + 3)

Additional Examples


EXAMPLE 1

How do you know which values of the variable make the expression undefined?

EXAMPLE 2

When might you want to factor out -1?

EXAMPLE 3

Why should you factor the numera-tors and the denominators before you multiply?

Technology In Example 2, remind students that they can access the table

feature of their graphing calculators by pressing TABLE. Point out that the entry ERROR appears in the table for values of x that make the functions Y1 or Y2 undefined.

Guided InstructionBefore introducing operations with rational expressions, make sure that students can simplify rational expressions by dividing out common factors. Lead students to make connections between the processes of multiplying and dividing numerical frac-tions and the processes of multiplying and dividing rational expressions.

Teach2

Through Visual Cues

When students simplify rational expres-sions, suggest that they highlight each different factor in the numerator and denominator with a different color by using highlighters or colored pencils. Colored pencils can be found in the Manipulatives Kit (MK). This process can help them keep track of common factors, as shown below.

x 2 (x + 1) (x - 2) 2

__ x (x + 3) (x + 1)

= x (x - 2) 2

_ (x + 3)



You can also divide rational expressions. Recall that to divide by a fraction, you multiply by its reciprocal.

4E X A M P L E Dividing Rational ExpressionsDivide. Assume that all expressions are defined.

A 4x3_

9x2y 16_

9y5 B x

5- 4x3_

x2 - x - 2 x

5- x4 - 2x3__

x2 - 1

4x3_9x2y

9y5_16

Rewrite as

multiplicationby the

reciprocal.

x5 - 4x3_

x2 - x - 2 x

2 - 1__x5 - x4 - 2x3

4x3 1

_9x2y

9y5

4_164

x3(x2 - 4)_x2 - x - 2

x2 - 1__

x3(x2 - x - 2)

x y 4_4

x3(x - 2)(x + 2)__

(x - 2)(x + 1)

(x - 1)(x + 1)__x3(x - 2)(x + 1)

(x + 2)(x - 1)__(x + 1)(x - 2)

or x2 + x - 2_

x2 - x - 2

Divide. Assume that all expressions are defined.

4a. x2_

4

x4y_12y2

4b. 2x2 - 7x - 4__x2 - 9

4x2 - 1__

8x2 - 28x + 12

5E X A M P L E Solving Simple Rational EquationsSolve. Check your solution.

A x2

- 9_x + 3

= 7 B x2

+ 3x - 4__x - 1

= 5

(x - 3)(x + 3)__x + 3

= 7 Note thatx -3.

(x - 1)(x + 4)__

x - 1= 5 Note that

x 1. x - 3 = 7 x + 4 = 5

x = 10 x = 1

Check

x2 - 9_

x + 3 = 7 Because the left side of the

original equation is undefined when x = 1, there is no solution.

(10)2 - 9_10 + 3

7

91_13

7

7 7

Solve. Check your solution.

5a. x2 + x - 12__

x + 4 = -7 5b. 4x

2 - 9_(2x + 3)

= 5

1_2

3_4

= 1_2

4_3

2

= 2_3

Check A graphing calculator shows that 1 is not a solution.

As you simplify a rational expression, take note of values that must be excluded. The excluded values are those that make the rational expression undefined.

3y_x2

4(x - 4)_x + 3

no solution4

Lesson 8-2 579

Students sometimes forget that when all of the factors in the numer-ator (or denominator) divide out, as in Example 3B, the numerator (or denominator) becomes 1, not 0. A numerical example may help them to realize this. For instance, they know that 6__12 simplifies to

1__2 , not

to 0__2 .

Example 4


A. 5 x 4 _

8 x 2 y 2 15 _

8 y 5

x 2 y 3 _

3

B. x 4 - 9 x 2 __

x 2 - 4x + 3

x 4 + 2 x 3 - 8 x 2 __

x 2 - 16

(x + 3) (x - 4)

__ (x - 1) (x - 2)

Example 5


A. x 2 - 25 _ x - 5

= 14 x = 9

B. x 2 + 3x - 10

__ x - 2

= 7

no solution

Additional Examples


EXAMPLE 4

How do you find the reciprocal of a rational expression?

EXAMPLE 5

Why must you check your solutions when you solve rational equations?

Reading Math In Examples 3 and 4, discuss with students the mean-

ing of the statement Assume that all expressions are defined. Explain that when this direction line is given, students do not need to worry about identifying values of the variables that make the expressions unde-fined.

Assess After the Lesson8-2 Lesson Quiz, TE p. 582

Alternative Assessment, TE p. 582

Monitor During the LessonCheck It Out! Exercises, SE pp. 577579Questioning Strategies, TE pp. 578579

Diagnose Before the Lesson8-2 Warm Up, TE p. 577

and INTERVENTIONSummarizeAsk students to list the steps they would use to multiply two rational expressions and the steps they would use to divide two rational expressions. Possible answer: To multiply, factor the expressions, divide out common factors, multiply numerators, and multiply denominators. To divide, multiply by the reciprocal of the divisor. Suggest that students list these steps on flash cards and that they make additional flash cards as they learn new operations.

Close3




GUIDED PRACTICE 1. Vocabulary How can you tell if an algebraic expression is a rational expression?



2. 4x6_

2x - 6 3. 6x

2 + 13x - 5__6x2 - 23x + 7

4. x + 4__3x2 + 11x - 4


5. -x - 4__x2 - x - 20

6. 6x2 + 7x - 3__

-3x2 + x7. 6x

3 + 6x_x2 + 1



8. x - 2_2x - 3

4x - 6_x2 - 4

9. x - 2_x - 3

2x - 6_x + 5

10. x2 - 16__

x2 - 4x + 4 x - 2__

x2 + 6x + 8



11. x5y4_3xy

1_x3y

12. x + 3__x2 - 2x + 1

x + 3_x - 1

13. x2 - 25__

2x2 + 5x - 12 x

2 - 3x - 10__x2 + 9x + 20

14. x2 + 2x + 1__

x2 - 3x - 18 x

2 - 1__x2 - 7x + 6



15. 16x2 - 9_

4x + 3 = -6 16. 2x

2 + 7x - 15__2x - 3

= 10 17. x2 - 4_

x - 2 = 1

PRACTICE AND PROBLEM SOLVINGSimplify. Identify any x-values for which the expression is undefined.

18. 4x - 8_x2 - 2x

19. 8x - 4__2x2 + 9x - 5

20. x2 - 36__

x2 - 12x + 36

21. 3x + 18__24 - 2x - x2

22. -2x2 - 9x_

4x2 - 81 23. 4x + 20_

-5 - x

THINK AND DISCUSS 1. Explain how you find undefined values for a rational expression.

2. Explain why it is important to check solutions to rational equations.

3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write a worked-out example.

8-2

KEYWORD: MB7 Parent

KEYWORD: MB7 8-2ExercisesExercises

Possible answer: A rational expression is the quotient of 2 polynomials.

2x6_x - 3

; x 3

2_x + 2

2(x - 2)_(x + 5)

x 7y4_3 1_

x - 1

no solution

5 -1

-3_x - 4

; x -6 and x 4

-x_2x - 9

; x - 9_2 and x 9_

2

-4; x -5

x + 1_x + 3

10. x - 4__(x + 2)(x - 2)

20. x + 6_x - 6

; x 6

580 Chapter 8


Answers to Think and DiscussPossible answers:

1. Set the denominator of the unsimplified expression equal to 0, and solve.

2. Solving a rational equation may produce extraneous solutions.

3. See p. A9.

Assignment Guide

Assign Guided Practice exercises as necessary.

If you finished Examples 13 Basic 1827, 3637 Average 1827, 3637 Advanced 1827, 3637, 5051

If you finished Examples 15 Basic 1841, 4349, 5459 Average 1849, 52, 5459 Advanced 1844, 4659

Homework Quick CheckQuickly check key concepts.Exercises: 18, 22, 24, 28, 32, 43

Inclusion For Exercises 2123, encourage students to check the sign of the

leading coefficient in the numera-tor and denominator. If the leading coefficient is negative, suggest that students factor out -1 before they factor the expression further.

8-2 RETEACH8-2 READING STRATEGIES


Archery was practiced in ancient times on every inhabited continent except Australia. The painting in the photo above dates from about 1400 B.C.E. and shows archers from ancient Egypt.

Archery


For See Exercises Example

1820 1 2123 2 2427 3 2831 4 3234 5

Independent Practice Multiply. Assume that all expressions are defined.

24.x2y_4xy

x_6

3y5_x4

25. x - 4_x - 3

2x - 1_x + 4

26. x2 - 2x - 8__9x2 - 16

3x2 + 10x + 8__x2 - 16

27. 4x2 - 20x + 25__

x2 - 4x 3x - 12_

2x - 5


28. 4x2 + 15x + 9__

8x2 + 10x + 3 x

2 + 4x_2x + 1

29. x2 - 4x - 5__

x2 - 3x + 2 x

2 - 3x - 10__x2 - 4

30. x + 2_x - 4

1_3x - 12

31. x2 - 2x - 3__x2 - x - 2

x2 + 2x - 15__x2 + x - 6


32. 3x2 + 10x + 8__

-x - 2= -2 33. x

2 - 9_x - 3

= 5 34. x2 + 3x - 28__

(x + 7)(x - 4)= -11

35. Archery An archery target consists of an inner circle and four concentric rings. The width of each ring is equal to the radius r of the inner circle. Write a rational expression in terms of r that represents the probability that an arrow hitting the target at random will land in the inner circle. Then simplify the expression.

Multiply or divide. Assume that all expressions are defined.

36. 2x_3

x3_

6x - 8 37. 4x

2 - 3x_4x2 - 1

2x + 1_x

38. 1_25x2 - 49

x_10x - 14

39. 2xy 2x2_

y

y2_2x

40. 14x4_

xy x

3_6y3

5x2_

12y5 41. (y + 4)

4x + 4 + xy + y__3

42. Critical Thinking What polynomial completes the equation x - 5_x - 2

_x - 5

= x + 1?

43. Geometry Use the table to determine the following.

a. For each figure, find the ratio of the volume to the area of the base.

b. For each figure, find the ratio of the surface area to the volume.

c. What if...? If the radius and the height of a cylinder were doubled, what effect would this have on the ratio of the cylinders surface area to its volume?

Square Prism Cylinder

Area of Base s2 r2

Volume s2h r2h

Surface Area 2s2 + 4sh 2r2 + 2rh

Skills Practice p. S18Application Practice p. S39

Extra Practice

44. This problem will prepare you for the Multi-Step Test Prep on page 608.

For a car moving with initial speed v0 and acceleration a, the distance d that the car travels in time t is given by d = v0t +

1__2

a t 2 .

a. Write a rational expression in terms of t for the average speed of the car during a period of acceleration. Simplify the expression.

b. During a race, a driver accelerates for 3 s at a rate of 10 ft/ s 2 in order to pass another car. The drivers initial speed was 264 ft/s. What was the drivers average speed during the acceleration?

y 5_8x 2

(x - 4)(2x - 1)__(x - 3)(x + 4)

x + 3_x (x + 4)

x + 1_x - 1

-

2_3

2no solution

x 4_3(3x - 4)

4x - 3_2x - 1

3(2x - 5)_x

3x + 6x + 3_x + 5

2x2y2

28x4y_5 3_

x + 1

43a. square prism: h_1 ;

cylinder: h_1

43b. square prism: 2s + 4h_

sh;

cylinder: 2r + 2h_rh The ratio would be reduced by a factor of

1__2 .

279 ft/s

r 2_(5r)2

; 1_25

2_x (5x + 7)

Lesson 8-2 581

Students often list only the values that make the simplest form of a rational expression undefined. In Exercises 1823, stress the impor-tance of recording all of the values for which the original expression is undefined.

Probability In Exercise 35, remind students how to determine probability

by using geometric figures. Point out that the probability that the arrow will hit the inner circle is the ratio of the area of the inner circle to the area of the entire target.

Exercise 44 involves simplifying and evaluating a rational

expression for the average speed of a car during acceleration. This exercise prepares students for the Multi-Step Test Prep on page 608.

Answers 35. See p. A36.

6. 2x + 3_-x ; x 0 and x

1_3

7. 6x; defined for all real values of x

13. (x + 5) 2 __

(2x - 3) (x + 2)

18. 4_x ; x 0 and x 2 19. 4_

x + 5 ; x 1_

2 and x -5

26. (x + 2) 2 __

(3x - 4) (x + 4)

42. x2- x - 2

44a. v 0 t +

1 __ 2 a t 2 _t

= v 0 +1_2

at

8-2 PRACTICE C

8-2 PRACTICE B

8-2 PRACTICE A

8-2 PROBLEM SOLVING 8-2 CHALLENGE

a207te_c08_l2_577-582.indd 581a207te_c08_l2_577-582.indd

rational and radical...

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