Rational Expressions and Equations Chapter Overview and Pacing

Transcription

1 Rational Epressions and Equations Chapter verview and Pacing PACING (das) Regular Block Basic/ Basic/ Average Advanced Average Advanced Multipling and Dividing Rational Epressions (pp. 7 78) Simplif rational epressions. Simplif comple fractions. LESSN BJECTIVES Adding and Subtracting Rational Epressions (pp. 79 8) Determine the LCM of polnomials. Add and subtract rational epressions. Graphing Rational Functions (pp. 85 9) 0.5 Determine the vertical asmptotes and the point discontinuit for the graphs of (with 9- rational functions. Follow-Up) Graph rational functions. Follow-Up: Graphing Rational Functions Direct, Joint, and Inverse Variation (pp. 9 98) Recognize and solve direct and joint variation problems. Recognize and solve inverse variation problems. Classes of Functions (pp ) Identif graphs as different tpes of functions. Identif equations as different tpes of functions. Solving Rational Equations and Inequalities (pp ).5 Solve rational equations. (with 9-6 Solve rational inequalities. Follow-Up) Follow-Up: Solving Rational Equations b Graphing Stud Guide and Practice Test (pp. 5 57) Standardized Test Practice (pp ) Chapter Assessment TTAL 6 6 Pacing suggestions for the entire ear can be found on pages T0 T. 70A Chapter 9 Rational Epressions and Equations

2 Timesaving Tools Chapter Resource Manager All-In-ne Planner and Resource Center See pages T T. CHAPTER 9 RESURCE MASTERS Stud Guide and Intervention Practice (Skills and Average) Reading to Learn Mathematics Enrichment Assessment Applications* 5-Minute Check Transparencies Interactive Chalkboard SC AlgePASS: Tutorial Plus (lessons) Materials , 569 GCS 9-9- balance, metric measuring cup, graph paper (Follow-Up: graphing calculator) GCS, 9-9- SC 8, SM string, grid paper (Follow-Up: graphing calculator) , *Ke to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual Chapter 9 Rational Epressions and Equations 70B

3 Mathematical Connections and Background Continuit of Instruction Prior Knowledge Students have simplified rational numbers, written equivalent rational numbers, and found least common denominators. The have graphed functions from tables of values and the have eplored functions whose graphs are lines or other shapes. Also, the have solved linear and polnomial equations. This Chapter Students etend basic arithmetic operations to rational epressions and etend solving equations to rational equations and inequalities. The graph rational functions and identif discontinuities in the graphs. The investigate equations that represent direct, inverse, and joint variation. The look at graphs of various shapes, including continuous curves, discontinuous curves, and lines, and associate each graph with a specific kind of function. Future Connections Students will etend operations on rational epressions to finding powers and roots of rational epressions. The will continue to stud situations involving direct or inverse variation, and the will etend the stud of joint variation to variables with eponents. Also, the will continue to see how the shape of a graph is used to classif the function represented b the graph. Multipling and Dividing Rational Epressions In this lesson students look at some familiar ideas from fractions and appl them to rational epressions. ne idea is that a fraction is undefined if the denominator is zero. To etend that idea, students eamine polnomials that are the denominators of rational epressions and use their skill in factoring to identif values of variables for which the denominator would be zero. Students etend another idea to rational epressions, simplifing fractions, b finding common factors in the numerator and denominator, and replacing the quotient of those factors with. As another etension of simplifing, students identif factors of the form a b and b a in the numerator and denominator of a rational epression, and replace the quotient of those factors with. The ideas of multipling or dividing fractions and simplifing comple fractions have direct etensions. To multipl two rational epressions, students divide the product of the numerators b the product of the denominators; to divide b a rational epression, the multipl b the reciprocal of that epression; and to simplif a comple fraction involving rational epressions, the rewrite it and treat it as a division epression. Adding and Subtracting Rational Epressions Students continue to look at familiar ideas from fractions and etend those ideas to rational epressions. For the idea of a least common multiple, students factor two or more polnomials. The write each factor of either of the given polnomials, with each factor having an eponent that indicates the maimum number of times that factor appears in an one of the given polnomials. Another familiar idea is writing two fractions as equivalent fractions with a common denominator. To etend this idea to rational epressions, students find the LCM of the given denominators and rewrite each rational epression as an equivalent epression whose denominator is that LCM. The ideas of adding and subtracting fractions are etended to rational epressions b writing the rational epressions with a common denominator (again, the common denominator is the LCM of the given denominators) and then adding or subtracting the numerators. 70C Chapter 9 Rational Epressions and Equations

4 Graphing Rational Functions In this lesson (and the net two) students use graphs to eamine properties of functions. This lesson introduces graphs of rational functions, which are functions whose numerator and denominator are both polnomials. To look at values of for which the denominator and thus the function is undefined, students stud two kinds of rational functions. In one kind, the denominator is a factor of the numerator; for eample, f() ( ) 5 ( )( )(5 ) or g(). These ( )(5 ) functions can be simplified, and the graph of the simplified function is a continuous curve. However, there are one or more values of the variable for which the denominator of the original polnomial is zero. These values, called point discontinuities, represent places when the function is undefined. Students eplore these tpes of functions b reducing the function, graphing the reduced function, and identifing the holes in the graph. Another kind of rational epression is one in which the entire denominator is not a factor of the numerator. For these functions, each value of the variable for which the denominator is zero is associated with a vertical asmptote on the graph. Students eplore these functions b identifing all the vertical asmptotes, and then using tables of values and the asmptotes to graph the function. Direct, Joint, and Inverse Variation In this lesson students eamine graphs for two-variable equations that represent two tpes of relationships. For variables and and constant k, the relationship k (also written as k) is called direct variation. The graph of a direct variation is a line; the line goes through the origin (0, 0) and has k slope k. The relationship (or k) is called inverse variation; its graph is a hperbola. Students eplore direct and inverse variation b finding the constant or a missing value of a variable for a given tpe of variation or b stating the tpe of variation for a given graph or set of values. Students also eplore the relationship kz among the variables,, and z and constant k. Students eplore this tpe of variation, called joint variation, b using given values to find the value of the constant or a missing value of a variable. Classes of Functions In this lesson students organize information learned previousl about graphs and functions. Given the graph of a line, the describe that line as the graph of a constant function, a direct variation function, or the identit function; if a line has a hole, the identif it as the graph of one tpe of rational function. Given a graph that is a continuous curve, the describe that curve as the graph of a quadratic function or a square root function. Also, the relate V-shaped graphs to absolute value functions and relate discontinuous graphs to greatest integer functions, inverse variation functions, and rational functions. Solving Rational Equations and Inequalities In this lesson students return to the topics of the chapter s first two lessons and solve equations involving rational epressions. In general, the first step is to multipl both sides of the equation b the least common denominator of all the denominators. The result is to rewrite the original equation as an equation with no denominators; the new equation can be solved using familiar methods for solving linear or polnomial equations. Students appl these methods to several kinds of word problems. ne kind is work problems, in which one complete job is the sum of partial jobs, each partial job being the quotient of some number of time units divided b a per-unit rate. Another kind is rate problems, in which a total duration is the sum of two smaller durations, each one being the quotient of distance divided b rate. Also in this lesson students eplore rational inequalities b finding values that make the denominator equal to 0, solving a related equation, and identifing intervals on the number line. Additional mathematical information and teaching notes are available in Glencoe s Algebra Ke Concepts: Mathematical Background and Teaching Notes, which is available at The lessons appropriate for this chapter are as follows. Simplifing Rational Epressions (Lesson 5) Multipling Rational Epressions (Lesson 6) Dividing Rational Epressions (Lesson 7) Rational Epressions with Unlike Denominators (Lesson 8) Chapter 9 Rational Epressions and Equations 70D

5 and Assessment Tpe Student Edition Teacher Resources Technolog/Internet ASSESSMENT INTERVENTIN ngoing Prerequisite Skills, pp. 7, 78, 8, 90, 98, 50 Practice Quiz, p. 8 Practice Quiz, p. 98 Mied Review Error Analsis Standardized Test Practice pen-ended Assessment Chapter Assessment pp. 78, 8, 90, 98, 50, 5 Find the Error, pp. 8, 509 pp. 7, 76, 78, 8, 90, 98, 50, 50, 5, 57, Writing in Math, pp. 77, 8, 90, 98, 50, 5 pen Ended, pp. 76, 78, 8, 88, 95, 50, 509 Stud Guide, pp Practice Test, p Minute Check Transparencies Quizzes, CRM pp Mid-Chapter Test, CRM p. 569 Stud Guide and Intervention, CRM pp , 5 5, 59 50, 55 56, 5 5, Cumulative Review, CRM p. 570 Find the Error, TWE pp. 8, 509 Unlocking Misconceptions, TWE pp. 7, 86, 9 Tips for New Teachers, TWE pp. 78, 8, 87, 98, 50, 5 TWE p. 7 Standardized Test Practice, CRM pp Modeling: TWE pp. 98, 50 Speaking: TWE p. 78 Writing: TWE pp. 8, 90, 5 pen-ended Assessment, CRM p. 565 Multiple-Choice Tests (Forms, A, B), CRM pp Free-Response Tests (Forms C, D, ), CRM pp Vocabular Test/Review, CRM p. 566 AlgePASS: Tutorial Plus Standardized Test Practice CD-RM standardized_test TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes vocabular_review Ke to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters Additional Intervention Resources The Princeton Review s Cracking the SAT & PSAT The Princeton Review s Cracking the ACT ALEKS TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment fleibilit: Worksheet Builder to make worksheet and tests Student Module to take tests on screen (optional) Management Sstem to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests. 70E Chapter 9 Rational Epressions and Equations

6 Intervention Technolog AlgePASS: Tutorial Plus CD-RM offers a complete, self-paced algebra curriculum. Algebra Lesson AlgePASS Lesson 9-7 Simplifing Rational Epressions 9-8 perations with Rational Functions ALEKS is an online mathematics learning sstem that adapts assessment and tutoring to the student s needs. Subscribe at Intervention at Home Log on for student stud help. For each lesson in the Student Edition, there are Etra Eamples and Self-Check Quizzes. For chapter review, there is vocabular review, test practice, and standardized test practice For more information on Intervention and Assessment, see pp. T8 T. Reading and Writing in Mathematics Glencoe Algebra provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition Foldables Stud rganizer, p. 7 Concept Check questions require students to verbalize and write about what the have learned in the lesson. (pp. 76, 8, 88, 95, 50, 509, 5) Writing in Math questions in ever lesson, pp. 77, 8, 90, 98, 50, 5 WebQuest, p. 50 Teacher Wraparound Edition Foldables Stud rganizer, pp. 7, 5 Stud Notebook suggestions, pp. 76, 8, 88, 95, 50, 509 Modeling activities, pp. 98, 50 Speaking activities, p. 78 Writing activities, pp. 8, 90, 5 ELL Resources, pp. 70, 77, 8, 89, 96, 50, 50, 5 Additional Resources Vocabular Builder worksheets require students to define and give eamples for ke vocabular terms as the progress through the chapter. (Chapter 9 Resource Masters, pp. vii-viii) Reading to Learn Mathematics master for each lesson (Chapter 9 Resource Masters, pp. 5, 57, 5, 59, 55, 55) Vocabular PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabular lists that ou can customize. Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. Reading and Writing in the Mathematics Classroom WebQuest and Project Resources For more information on Reading and Writing in Mathematics, see pp. T6 T7. Chapter 9 Rational Epressions and Equations 70F

7 Notes Rational Epressions and Equations Have students read over the list of objectives and make a list of an words with which the are not familiar. Point out to students that this is onl one of man reasons wh each objective is important. thers are provided in the introduction to each lesson. Lessons 9- and 9- Simplif rational epressions. Lesson 9- Graph rational functions. Lesson 9- Solve direct, joint, and inverse variation problems. Lesson 9-5 Identif graphs and equations as different tpes of functions. Lesson 9-6 Solve rational equations and inequalities. Ke Vocabular rational epression (p. 7) asmptote (p. 85) point discontinuit (p. 85) direct variation (p. 9) inverse variation (p. 9) Rational epressions, functions, and equations can be used to solve problems involving mitures, photograph, electricit, medicine, and travel, to name a few. Direct, joint, and inverse variation are important applications of rational epressions. For eample, scuba divers can use direct variation to determine the amount of pressure at various depths. You will learn how to determine the amount of pressure eerted on the ears of a diver in Lesson 9-. NCTM Local Lesson Standards bjectives 9-, 6, 8, 9, 0 9-, 6, 8, 9, 0 9-, 6, 8, 9, 0 9-, 6, 8 Follow-Up 9-,, 6, 8, 9, 0 9-5, 6, 8, 9, 0 9-6, 6, 8, 9, 0 9-6, 6 Follow-Up Ke to NCTM Standards: =Number & perations, =Algebra, =Geometr, =Measurement, 5=Data Analsis & Probabilit, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 0=Representation 70 Chapter 9 Rational Epressions and Equations Vocabular Builder ELL The Ke Vocabular list introduces students to some of the main vocabular terms included in this chapter. For a more thorough vocabular list with pronunciations of new words, give students the Vocabular Builder worksheets found on pages vii and viii of the Chapter 9 Resource Masters. Encourage them to complete the definition of each term as the progress through the chapter. You ma suggest that the add these sheets to their stud notebooks for future reference when studing for the Chapter 9 test. 70 Chapter 9 Rational Epressions and Equations

8 Prerequisite Skills To be successful in this chapter, ou ll need to master these skills and be able to appl them in problem-solving situations. Review these skills before beginning Chapter 9. For Lesson 9- Solve Equations with Rational Numbers Solve each equation. Write our answer in simplest form. (For review, see Lesson -.) t a m b s r n r 5 6 For Lesson 9- Determine Asmptotes and Graph Equations Draw the asmptotes and graph each hperbola. (For review, see Lesson 8-5.) 0. ( ) ( 5). ( ). ( ) ( ) See margin. For Lesson 9- Solve Proportions Solve each proportion.. r m 5 0 t a b 6 9. v p 8. 5 z 7 This section provides a review of the basic concepts needed before beginning Chapter 9. Page references are included for additional student help. Prerequisite Skills in the Getting Read for the Net Lesson section at the end of each eercise set review a skill needed in the net lesson. For Lesson Prerequisite Skill 9- Solving Equations (p. 78) 9- Graphing Hperbolas (p. 8) 9- Solving Proportions (p. 90) 9-5 Special Functions (p. 98) 9-6 Least Common Multiples of Polnomials (p. 50) Answers 0. Fold Make this Foldable to help ou organize what ou learn about rational epressions and equations. Begin with a sheet of plain 8 " " paper. Cut and Label ( ) ( 5) 9 Fold in half lengthwise leaving a '' margin at the top. Fold again in thirds. pen. Cut along the second folds to make three tabs. Label as shown. Epressions Functions Equations Rational Reading and Writing As ou read and stud the chapter, write notes and eamples for each concept under the tabs.. ( ) Chapter 9 Rational Epressions and Equations 7 For more information about Foldables, see Teaching Mathematics with Foldables. TM rganization of Data with a Concept Map Concept maps are visual stud guides that allow students to view main ideas or ke words and use them to recall and organize what the know and what the have learned. Begin b writing Rational on the base of the Foldable and the words Epressions, Functions, and Equations on the tabs of the concept map. Under the tabs of their Foldable, have students take notes, define terms, record concepts, and write eamples. Students can check their responses and memor b reviewing their notes under the tabs ( ) ( ) 5 6 Chapter 9 Rational Epressions and Equations 7

9 Multipling and Dividing Rational Epressions Lesson Notes Focus 5-Minute Check Transparenc 9- Use as a quiz or review of Chapter 8. Simplif rational epressions. Simplif comple fractions. Vocabular rational epression comple fraction Mathematical Background notes are available for this lesson on p. 70C. are rational epressions used in mitures? Add lb of peanuts. The Goodie Shoppe sells cand and nuts b the pound. ne of their items is a miture of peanuts and cashews. This miture is made with 8 pounds of peanuts and 5 pounds of cashews lb of peanuts 5 lb of cashews lb of nuts 8 Therefore, or of the miture is peanuts. If the store manager adds an additional pounds of peanuts to the are rational epressions used in mitures? Ask students: How can the term rational epression help ou recall what it means? The word rational contains the word ratio. What does it mean to sa that 6 is the GCF of and 0? It is the greatest integer that divides into both and 0 without a remainder. 8 miture, then of the miture will be peanuts. SIMPLIFY RATINAL EXPRESSINS A ratio of two polnomial epressions 8 such as is called a rational epression. Because variables in algebra represent real numbers, operations with rational numbers and rational epressions are similar. To write a fraction in simplest form, ou divide both the numerator and denominator b their greatest common factor (GCF). To simplif a rational epression, ou use similar properties. Eample Simplif a Rational Epression ( 5) ( 5)( ) a. Simplif. Look for common factors. ( 5) 5 ( 5)( ) 5 0 How is this similar to simplifing? 5 Simplif. b. Under what conditions is this epression undefined? Just as with a fraction, a rational epression is undefined if the denominator is equal to 0. To find when this epression is undefined, completel factor the original denominator. ( 5) ( 5) ( 5)( ) ( 5)( )( ) ( )( ) The values that would make the denominator equal to 0 are 5,, or. So the epression is undefined when 5,, or. These numbers are called ecluded values. 7 Chapter 9 Rational Epressions and Equations Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 59 Practice, p. 50 Reading to Learn Mathematics, p. 5 Enrichment, p. 5 School-to-Career Masters, p. 7 Teaching Algebra With Manipulatives Masters, p. 7 Transparencies 5-Minute Check Transparenc 9- Answer Ke Transparencies Technolog AlgePASS: Tutorial Plus, Lesson 7 Interactive Chalkboard

10 Standardized Test Practice Test-Taking Tip Sometimes ou can save time b looking at the possible answers and eliminating choices, rather than actuall evaluating an epression or solving an equation. Eample Multiple-Choice Test Item Use the Process of Elimination For what value(s) of is undefined? 7 A, B C 0 D, Read the Test Item You want to determine which values of make the denominator equal to 0. Solve the Test Item Look at the possible answers. Notice that if equals 0 or a positive number, 7 must be greater than 0. Therefore, ou can eliminate choices C and D. Since both choices A and B contain, determine whether the denominator equals 0 when. 7 () 7() 9 0 Multipl. Simplif. Since the denominator equals 0 when, the answer is A. Sometimes ou can factor out in the numerator or denominator to help simplif rational epressions. Eample Simplif b Factoring ut Simplif z w z z. z w z w z z zw z ( w ) z Factor the numerator and the denominator. ( w) z ()( w) w (w ) or ( w) z ( w) z or z z Simplif. Remember that to multipl two fractions, ou first multipl the numerators and then multipl the denominators. To divide two fractions, ou multipl b the multiplicative inverse, or reciprocal, of the divisor. Multiplication Division or The same procedures are used for multipling and dividing rational epressions. Standardized Test Practice Lesson 9- Multipling and Dividing Rational Epressions 7 Eample Make sure students know to stud onl the denominator to determine the values that make the epression undefined. In this eample, the numerator is irrelevant. Teach SIMPLIFY RATINAL EXPRESSINS In-Class Eamples Power Point ( 7) a. Simplif. ( 7)( 9) 9 b. Under what conditions is this epression undefined? when 7,, or Teaching Tip Ask students to eplain wh division b zero is undefined. For what value(s) of p is p p undefined? B p p 5 A 5 B, 5 C, 5 D 5,, a Simplif b a. a a a b Teaching Tip Point out that rational epressions are usuall used without specificall ecluding those values that make the epression undefined. It is understood that onl those values for which the epression has meaning are included. Interactive Chalkboard PowerPoint Presentations This CD-RM is a customizable Microsoft PowerPoint presentation that includes: Step-b-step, dnamic solutions of each In-Class Eample from the Teacher Wraparound Edition Additional, Your Turn eercises for each eample The 5-Minute Check Transparencies Hot links to Glencoe nline Stud Tools Lesson 9- Multipling and Dividing Rational Epressions 7

11 In-Class Eamples Power Point Simplif each epression. 8 7 a a b. c bc ac b 5a b b 0ps 5ps 5 Simplif. ds c d 6c d Teaching Tip To help students understand wh division is equivalent to multipling b the reciprocal, discuss simple eamples such as this: dividing 8 marbles between people means that each person gets one-half, or 9, of the marbles. Stud Tip Alternative Method When multipling rational epressions, ou can multipl first and then divide b the common factors. For instance, in Eample, a 5b 5 b 6a 6 0ab 8 0a. b Now divide the numerator and denominator b the common factors. b 60 ab b 80 a b a a Multipling Rational Epressions Words To multipl two rational epressions, multipl the numerators and the denominators. a c Smbols For all rational epressions and d, a c ac b d b, if b 0 and d 0. b d Dividing Rational Epressions Words To divide two rational epressions, multipl b the reciprocal of the divisor. a c Smbols For all rational epressions and d, a c a d b d b b c a d, bc if b 0, c 0, and d 0. The following eamples show how these rules are used with rational epressions. Eample Multipl Rational Epressions Simplif each epression. a. a 5b 5b 6a a 5b a 5 b b 5 b 6a 5 b a a a b a a b a Factor. Simplif. Simplif. b. 8 ts 5sr 5r t s 8 ts 5 r t t s 5 s r 5r t ss 5 r r t t t s s r t Factor. Simplif. Rational Epressions Eample 5 Divide Rational Epressions Simplif 5ab. 5 ab 5ab 5ab 5ab 5ab 5 a b b b 5 a a a b b b a a a Multipl b the reciprocal of the divisor. Factor. Simplif. Simplif. 7 Chapter 9 Rational Epressions and Equations Unlocking Misconceptions Simplifing the Quotient of pposites Help students understand wh the quotient of ( ) and ( ) is b pointing out that these two epressions are opposites (or additive inverses) just as are and. Division b Zero B definition, a b c if a bc. If students think 6 0 0, use the definition to show 6 0 if 6 0 0, which is false. 0 7 Chapter 9 Rational Epressions and Equations

12 Stud Tip Factor First As in Eample 6, sometimes ou must factor the numerator and/or the denominator first before ou can simplif a quotient of rational epressions. TEACHING TIP Have students replace a with a value such as. Then simplif both a a a a a 9 and a. The values a are the same. These same steps are followed when the rational epressions contain numerators and denominators that are polnomials. Eample 6 Polnomials in the Numerator and Denominator Simplif each epression. a. 8 8 ( )( ) ( ) Factor. ( )( ) ( ) ( ) Simplif. ( ) Simplif. b. a a a a a 9 a a a a a 9 a a a 9 a Multipl b the reciprocal of the divisor. a (a )(a )(a ) Factor. (a )(a )(a ) a Simplif. a SIMPLIFY CMPLEX FRACTINS A comple fraction is a rational epression whose numerator and/or denominator contains a rational epression. The epressions below are comple fractions. 5 a b t t 5 m 9 8 m p p Remember that a fraction is nothing more than a wa to epress a division problem. For eample, 5 can be epressed as 5. So to simplif an comple fraction, rewrite it as a division epression and use the rules for division. In-Class Eample In-Class Eample 7 Power Point 6 Simplif each epression. k k a. k. k k d 6 d b. d d d d (d ) d SIMPLIFY CMPLEX FRACTINS 9 Simplif or ( ) ( ) Power Point Eample 7 Simplif a Comple Fraction r r 5s Simplif. r 5s r r r 5s r 5s r r r 5s r Epress as a division epression. 5s r r r 5s 5s r r Multipl b the reciprocal of the divisor. r r()(r 5s) (r 5s)(r 5s)r r r or r 5s r 5s Factor. Simplif. Lesson 9- Multipling and Dividing Rational Epressions 75 Lesson 9- Multipling and Dividing Rational Epressions 75

13 Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 9. add the Ke Concepts in this lesson to their notebook, adding their own eamples for each one. add the Test-Taking Tip to their list of test-taking tips for review as the prepare for standardized tests. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Simplif Rational Epressions: 5 Simplif Comple Fractions: 6 dd/even Assignments Eercises and 6 7 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 5 7 odd,, 7 70 Average: 5 odd,, 5, 7 70 Advanced: even, 6, 8 6 (optional: 65 70) Answer. To multipl rational numbers or rational epressions, ou multipl the numerators and multipl the denominators. To divide rational numbers or rational epressions, ou multipl b the reciprocal of the divisor. In either case, ou can reduce our answer b dividing the numerator and the denominator of the results b an common factors. Concept Check. Sample answer: 6, ( ) 6( ) Guided Practice GUIDED PRACTICE KEY Eercises Eamples 6, 7 0 6, 7 Standardized Test Practice indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples, ,, 50 Etra Practice See page ( ) ( ) 76 Chapter 9 Rational Epressions and Equations. PEN ENDED Write two rational epressions that are equivalent.. Eplain how multiplication and division of rational epressions are similar to multiplication and division of rational numbers. See margin.. Determine whether d 5 d 5 is sometimes, alwas, or never true. Eplain. Never; solving the equation using cross products leads to 5 0, which is never true. Simplif each epression.. 5mn 9 m 5. a b n7 n a b a b 5 7. a bc c 5 b 8. 5 c 8a 0b t 6 t 6 7t 7 5t p c d 6p 6 6p a (p ). cd. ( ) p 0 c d ( ) a p 5 p. Identif all of the values of for which the epression is undefined. D A,, 6 B 6,, C, 0, 6 D, 6 Simplif each epression.. 0bc 5c b b mn 5. mn n 7m 6. ( ) 9 7. ( rs) s rs 8. 5 t 5 5 c 5 t 9. t c 0 a a a a a a. z 6 z. ab c bc c 8a 7a. 5 d 9 p 5d f 5. f q p q p 6. z 5 z z 7. 8 a ab b t 8. t t t 9. w w t 0. 9 ( t ) t t. p p 9 p 7p p w w w w 5w 8w 80 w 9w 50 0 w w. r r 8 r r ( r ) 5. a a 5 a (a 5) r r r a (a ) (a ) Differentiated Instruction Intrapersonal Have students think about what aspects of multipling and dividing rational epressions the find most challenging. Have them write a paragraph eplaining wh, and what steps the can take to help their challenges or confusions. 76 Chapter 9 Rational Epressions and Equations

14 p m 6. m n n n q 5 7. p 8. m n m m m n p m n 9 n q 5 Stud Stud Guide and and Intervention Intervention, p. 57 (shown) and p NAME DATE PERID Multipling and Dividing Rational Epressions Simplif Rational Epressions A ratio of two polnomial epressions is a rational epression. To simplif a rational epression, divide both the numerator and the denominator b their greatest common factor (GCF) d( d ). Under what conditions is (d )( d undefined? d,, or ). Under what conditions is a b b a a b undefined? a b or b a c a c ac b, d b d bd Multipling Rational Epressions For all rational epressions and, if b 0 and d 0. a c a ad c b, d b d bc Dividing Rational Epressions For all rational epressions and, if b 0, c 0, and d 0. Eample a a. 5 b (ab) a 5 b (ab) r b. s 5t r s 5t 0t 9r s Simplif each epression. 0t r r s s s 5 t t s s 9r s 5 t t t t r r r s r t t c a a a a a b b a a a a b b b b a b 8 s rt Lesson 9- Basketball After graduating from the U.S. Naval Academ, David Robinson became the NBA Rookie of the Year in 990. He has plaed basketball in different lmpic Games. Source: NBA BASKETBALL For Eercises and 5, use the following information. At the end of the season, David Robinson had made 687 field goals out of,9 attempts during his NBA career.. Write a fraction to represent the ratio of the number of career field goals made to career field goals attempted b David Robinson at the end of the season. 68 7,9 5. Suppose David Robinson attempted a field goals and made m field goals during the season. Write a rational epression to represent the number of career field goals made to the number of career field goals attempted at the end of the season. 687 m, 9 a nline Research Data Update What are the current scoring statistics of our favorite NBA plaer? Visit to learn more. 6. GEMETRY A parallelogram with an area of square units has a base of 5 units. Determine the height of the parallelogram. units 7. GEMETRY Parallelogram L has an area of 0 square meters and a height of meters. Parallelogram M has an area of 0 square meters and a height of meters. Find the area of rectangle N. ( 5) m ( ) m M b ( ) m b N L Eercises Simplif each epression. a b. ( ab ) 0ab m m 5 c c 5 m c 5 m c c. m (m ) 5. 6m c c (m ) m 9m 6 8z 5z 5 6. m 7. m 6m 9 m 9 6p 8. 8p p m 7p 9. p 7p 5 m m 0 p(p ) (p ) 9- ( )( )( ) ( )( )( ) NAME 57 DATE PERID Practice (Average) ( ) 5 5z 5 (m )(m 5) Simplif each epression. 9a. b (m m. n ) n. 7a b c a bc 8m 5 n c c 5 m m 8 Skills Practice, p. 59 and Practice, p. 50 (shown) k k 5 k 5 5 v v 5. k 5. v k 9 v v 0 5u u z 5 u z 5 a n a n 6 a w 0. n 5 5 n w. 5 w n a a ( 5) w 7 w Multipling and Dividing Rational Epressions w 5 w s s 5 9 ( ) (s ) 9 a a a 5a 6 5a ( ). ( ) 8 a w w 5 n 6n n 8 a w n ( ) s 0s 5 s ( ) s (s )(s 5) 8. CRITICAL THINKING Simplif (a 5a 6) ( a ) ( a ) (. a ) a. GEMETRY A right triangle with an area of square units has a leg that measures units. Determine the length of the other leg of the triangle. units. GEMETRY A rectangular pramid has a base area of 0 square centimeters and a height of centimeters. Write a rational epression to describe the 5 6 volume of the rectangular pramid. 5 cm 6 9. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 59A 59D. How are rational epressions used in mitures? Include the following in our answer: an eplanation of how to determine whether the rational epression representing the nut miture is in simplest form, and 8 an eample of a miture problem that could be represented b. Lesson 9- Multipling and Dividing Rational Epressions 77 NAME DATE PERID 9- Enrichment, p. 5 Reading Algebra In mathematics, the term group has a special meaning. The following numbered sentences discuss the idea of group and one interesting eample of a group. 0 To be a group, a set of elements and a binar operation must satisf four conditions: the set must be closed under the operation, the operation must be associative, there must be an identit element, and ever element must have an inverse. 0 The following si functions form a group under the operation of composition of functions: f (), f (), f (), f () ( ), f 5 (), and f ( ) 6 (). ( ) 0 This group is an eample of a noncommutative group. For eample, f f f, but f f f 6. 0 Some eperimentation with this group will show that the identit element is f. er is its own inver for f a ach s the Reading to Learn Mathematics, p NAME 50 DATE PERID Reading to Learn Mathematics ELL Multipling and Dividing Rational Epressions Pre-Activit How are rational epressions used in mitures? Read the introduction to Lesson 9- at the top of page 7 in our tetbook. Suppose that the Goodie Shoppe also sells a cand miture of chocolate mints and caramels. If this miture is made with pounds of chocolate mints and pounds of caramels, then 7 of the miture is mints and 7 of the miture is caramels. If the store manager adds another pounds of mints to the miture, what fraction of the miture will be mints? 7 Reading the Lesson. a. In order to simplif a rational number or rational epression, factor the numerator and denominator greatest common factor and divide both of them b their. b. A rational epression is undefined when its denominator is equal to 0. To find the values that make the epression undefined, completel factor the original denominator and set each factor equal to 0.. a. To multipl two rational epressions, multipl the numerators and multipl the denominators. b. To divide two rational epressions, multipl b the reciprocal of the divisor.. a. Which of the following epressions are comple fractions? ii, iv, v 8 z z r i. ii. r 5 iii. r 5 iv. z v. 5 6 r 5 b. Does a comple fraction epress a multiplication or division problem? division How is multiplication used in simplifing a comple fraction? Sample answer: To divide the numerator of the comple fraction b the denominator, multipl the numerator b the reciprocal of the denominator. Helping You Remember. ne wa to remember something new is to see how it is similar to something ou alread know. How can our knowledge of division of fractions in arithmetic help ou to understand how to divide rational epressions? Sample answer: To divide rational epressions, multipl the first epression b the reciprocal of the second.this is the same invert and multipl process that is used when dividing arithmetic fractions. Lesson 9- Lesson 9- Multipling and Dividing Rational Epressions 77

15 Assess pen-ended Assessment Speaking Have students eplain the procedures and cautions for multipling and dividing rational epressions, demonstrating with eamples. Standardized Test Practice 50. For what value(s) of is the epression undefined? C A, B, 0, C 0, D 0 E, 5. Compare the quantit in Column A and the quantit in Column B. Then determine whether: A A the quantit in Column A is greater, B the quantit in Column B is greater, C the two quantities are equal, or D the relationship cannot be determined from the information given. Column A Column B Intervention Encourage students who are having difficult with these problems to use several steps, writing each one below the previous one, and keeping each line equivalent to the one above. Caution them to make onl one change per step. New Getting Read for Lesson 9- PREREQUISITE SKILL Students will add and subtract rational epressions in Lesson 9-. As with equations containing fractions, students will find common denominators, combine like terms, and simplif equations. Use Eercises to determine our students familiarit with solving equations containing fractions. Answers 5. ( ) Maintain Your Skills Mied Review 5. ( ) ; parabola 55. ( 7) 9 ( ) ; hperbola Getting Read for the Net Lesson Find the eact solution(s) of each sstem of equations. (Lesson 8 7) (, ), (5, ) ( 7, ) Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hperbola. Then graph the equation. (Lesson 8 6) See margin for graphs. Determine whether each graph represents an odd-degree function or an evendegree function. Then state how man real zeros each function has. (Lesson 7 ) 56. f() 57. f() 58. f() a a 0 a a a 6 a even; odd; even; 0 Solve each equation b factoring. (Lesson 6 ) 59. r r, 60. 8u u 6, 6. d 5d 0 0, 5 6. ASTRNMY Earth is an average kilometers from the Sun. If light travels 0 5 kilometers per second, how long does it take sunlight to reach Earth? (Lesson 5 ).99 0 s or about 8 min 9 s Solve each equation. (Lesson ) , 9 6 PREREQUISITE SKILL Solve each equation. (To review solving equations, see Lesson.) Chapter 9 Rational Epressions and Equations ( 7) 9 ( ) 78 Chapter 9 Rational Epressions and Equations

16 Adding and Subtracting Rational Epressions Lesson Notes Determine the LCM of polnomials. Add and subtract rational epressions. is subtraction of rational epressions used in photograph? To take sharp, clear pictures, a photographer must focus the camera precisel. The distance from the object to the lens p and the distance from the lens to the film q must be accuratel calculated to ensure a sharp image. The focal length of the lens is ƒ. The formula p q q ƒ f can be p used to determine how far the film should be placed from the lens to create a perfect photograph. object lens image on film LCM F PLYNMIALS To find 5 6 or ƒ, ou must first find the least p common denominator (LCD). The LCD is the least common multiple (LCM) of the denominators. To find the LCM of two or more numbers or polnomials, factor each number or polnomial. The LCM contains each factor the greatest number of times it appears as a factor. LCM of 6 and LCM of a 6a 9 and a a 6 a 6a 9 (a ) a a (a )(a ) LCM or LCM (a ) ( a ) Focus 5-Minute Check Transparenc 9- Use as a quiz or review of Lesson 9-. Mathematical Background notes are available for this lesson on p. 70C. is subtraction of rational epressions used in photograph? Ask students: If ou drew a bo to represent the camera, which of the variables shown would describe dimensions within the camera? f and q In a camera, what is a tpical value for q? Sample answer: 0.5 in. Eample LCM of Monomials Find the LCM of 8r s 5, r st, and 5s t. 8r s 5 r s 5 Factor the first monomial. r st r s t Factor the second monomial. 5s t 5 s t Factor the third monomial. LCM 5 r s 5 t 60r s 5 t Use each factor the greatest number of times it appears as a factor and simplif. Lesson 9- Adding and Subtracting Rational Epressions 79 Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters Stud Guide and Intervention, pp. 5 5 Skills Practice, p. 55 Practice, p. 56 Reading to Learn Mathematics, p. 57 Enrichment, p. 58 Assessment, p. 567 Transparencies 5-Minute Check Transparenc 9- Answer Ke Transparencies Technolog AlgePASS: Tutorial Plus, Lesson 8 Interactive Chalkboard Lesson - Lesson Title 79

17 Teach LCM F PLYNMIALS In-Class Eamples In-Class Eamples Power Point Find the LCM of 5a bc, 6b 5 c, and 0a c 6. 0a b 5 c 6 Find the LCM of and. ( )( ) ADD AND SUBTRACT RATINAL EXPRESSINS 5a 9 Simplif. 6b a b 5a b 7 a b Power Point Teaching Tip Have students discuss the differences between procedures for adding and multipling fractions. 0 Simplif Stud Tip Common Factors Sometimes when ou simplif the numerator, the polnomial contains a factor common to the denominator. Thus, the rational epression can be further simplified. Eample ADD AND SUBTRACT RATINAL EXPRESSINS As with fractions, to add or subtract rational epressions, ou must have common denominators. Specific Case General Case Find equivalent fractions that a c b a d b d c have a common denominator. c d d c Eample Simplif each numerator and denominator. a d bc c cd d 9 5 Add the numerators. ad bc cd Monomial Denominators 7 Simplif The LCD is 90. Find equivalent fractions that have this denominator Simplif each numerator and denominator. Eample LCM of Polnomials Find the LCM of p 5p 6p and p 6p 9. p 5p 6p p( p )( p ) Factor the first polnomial. p 6p 9 ( p ) Factor the second polnomial. LCM p( p )( p ) 5 90 Add the numerators. Polnomial Denominators Use each factor the greatest number of times it appears as a factor. w w Simplif w 6 w 8 w w w w Factor the denominators. w 6 w 8 (w ) ( w ) w ( w ) ( ) The LCD is (w). (w ) ( w ) ( ) (w ) ()(w ) Subtract the numerators. (w ) w w 8 Distributive Propert ( w ) w Combine like terms. (w ) (w ) or Simplif. (w ) Sometimes simplifing comple fractions involves adding or subtracting rational epressions. ne wa to simplif a comple fraction is to simplif the numerator and the denominator separatel, and then simplif the resulting epressions. 80 Chapter 9 Rational Epressions and Equations 80 Chapter 9 Rational Epressions and Equations

18 TEACHING TIP Point out that can also be simplified b multipling the numerator and denominator b. Stud Tip Check Your Solution You can check our answer be letting p equal an nonzero number, sa. Use the definition of slope to find the slope of the line through the points. Eample 5 Simplif. Eample 6 Simplif Comple Fractions The LCD of the numerator is. The LCD of the denominator is. Simplif the numerator and denominator. Write as a division epression. Multipl b the reciprocal of the divisor. or Simplif. ( ) Use a Comple Fraction to Solve a Problem CRDINATE GEMETRY Find the slope of the line that passes through A, p and B, p. m Definition of slope 6 p p p 6 p p p p,,, and p The LCD of the numerator is p. The LCD of the denominator is p. 6 p p p 6 Write as a division epression. p 6 p p or The slope is. p p 6 In-Class Eamples 5 6 a b Simplif. b a b a b or a( b) a ab Practice/Appl Stud Notebook Power Point CRDINATE GEMETRY Find the slope of the line that passes through P k, and Q, k. Teaching Tip Remind students that the slope of a line is the change in divided b the change in, or the rise over the run. Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 9. include an item(s) that the find helpful in mastering the skills in this lesson. Concept Check. Catalina; ou need a common denominator, not a common numerator, to subtract two rational epressions. FIND THE ERRR Catalina and Yong-Chan are simplifing a b. Catalina - b = b a a - a b a b = b - a ab Who is correct? Eplain our reasoning. Yong-Chan a - b = a - b Lesson 9- Adding and Subtracting Rational Epressions 8 Differentiated Instruction Interpersonal Have students work with a partner, one in the role of coach and the other in the role of athlete. The athlete works the problem, using steps and eplaining the thinking, while the coach listens and watches for errors, correcting as necessar. Then the partners echange roles. FIND THE ERRR ne wa to find the error is to substitute values for the variables. With, a 5, and b, a b becomes 5. Since the first fraction is less than and the second is greater than, the result must be negative, which means that the answer a b or cannot be correct. Lesson 9- Adding and Subtracting Rational Epressions 8

19 About the Eercises rganization b bjective LCM of Polnomials: Add and Subtract Rational Epressions: 9 dd/even Assignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 5 7 odd, 50 6 Average: 5 odd, 9 6 Advanced: even, 8, (optional: 59 6) All: Practice Quiz ( 0) Answers a. Since a, b, and c are factors of abc, abc is alwas a common denominator of a b c. b. If a, b, and c have no common factors, then abc is the LCD of a b c. c. If a and b have no common factors and c is a factor of ab, then ab is the LCD of a b c. d. If a and c are factors of b, then b is the LCD of a b c. e. Since a b c bc ac ab, the sum is a bc a bc a bc alwas bc ac ab. abc. Sample answer: d d, d Guided Practice GUIDED PRACTICE KEY Eercises Eamples 6, 7, 5 6 indicates increased difficult Practice and Appl Homework Help For See Eercises Eamples, 9, Etra Practice See page 87. Application. PEN ENDED Write two polnomials that have a LCM of d d.. Consider a b if a, b, and c are real numbers. Determine whether each c statement is sometimes, alwas, or never true. Eplain our answer. a. abc is a common denominator. alwas a e. See margin for eplanations. b. abc is the LCD. sometimes c. ab is the LCD. sometimes d. b is the LCD. sometimes e. The sum is bc ac ab. alwas abc Find the LCM of each set of polnomials.., ab, 5b a, 0ac 6., 80ab c ( )( ) Simplif each epression. 7. 7a b 8. 5b a 5b 8 ab 90ab 5 9. m 7m m d 6 0. m d d d ( d ) a a 0. a. a 0 a (a 5)(a ). GEMETRY Find the perimeter of the quadrilateral. Epress in simplest form. 9 units ( ) ( ) Find the LCM of each set of polnomials.. 0s, 5s t 70s t 5. 6, 0z 80 z 6. a, 5bc, b 0a b c 7. 9p q, 6pq, p 6p q 8. w, w 6 (w ) 9., ( )( ) 0. t t, t 5t. n 7n, n n 8 (t )(t )(t ) (n )(n )(n ) Simplif each epression. 6. a b 8 a 6 8b 5 7. ab 6 v v v. 5 r 7 5 7r r a b 7a 5b 5b 7a 5a b 8. q 5 q q 0q w 6 5w 0w 90w 7 6 a 0.. a a a a m 5m 6 ( 9). m. m 6 (m ) (m ) 9 ( ) ( ) 8 Chapter 9 Rational Epressions and Equations 8 Chapter 9 Rational Epressions and Equations

20 5. 8d 0 (d )(d )(d ) h 5 6. (h )( h 5) 6 7. ( ) ( ) 9. ( )( ) d d ( 7)( ) d d 8 d h 9h 0 h 7. 0h m n m n m n n m m 0 9. n ( ) b b b b b b 0 b ( ). Write s s s s in simplest form. s s Stud Stud Guide and and Intervention Intervention, p. 5 (shown) and p NAME DATE PERID Adding and Subtracting Rational Epressions LCM of Polnomials To find the least common multiple of two or more polnomials, factor each epression. The LCM contains each factor the greatest number of times it appears as a factor. Eample Find the LCM of 6p q Eample r, Find the LCM of 0pq r, and 5p r. m m 6 and m m 0. 6p q r p q r m m 6 (m )(m ) 0pq r 5 p q r m m 0 (m )(m 5) 5p r 5 p r LCM (m )(m )(m 5) LCM 5 p q r 0p q r Eercises Find the LCM of each set of polnomials.. ab,bc,8a c. 8cdf,8c f,5d f 6a b c 80c d f. 65,0,6. mn 5,8m n,0mn 0 980m n 5 Lesson 9-. What is the simplest form of 5 a a 0 7? a 7 a 5. 5a b,50a b,0b 8 6. p 7 q,0p q,5pq 600a b 8 60p 7 q 7. 9b c,5b c,c 8.,,0 56b c stv,s v,70t v 0., s t v 5( )( ) ELECTRICITY For Eercises and 5, use the the following information. In an electrical circuit, if two resistors with resistance R and R are connected in parallel as shown, the relationship between these resistances and the resulting combination resistance R is R. R R. If R is ohms and R is ohms less than twice ohms, write an epression for. R ( ) 5. Find the effective resistance of a 0-ohm resistor and a 0-ohm resistor that are connected in parallel. ohms R R. 9, , 6 ( ) ( ) ( )( )( 5) , , 5 5 ( 5)( 6)( ) 5( 5)( 5)(5 ) , , 5 5 6( ) ( ) 5(5 )( )( ) 7., 8. 5, 6 ( )( )( )( ) 6( )( )( ) 9- NAME 5 DATE PERID Practice (Average) Skills Practice, p. 55 and Practice, p. 56 (shown) Adding and Subtracting Rational Epressions Find the LCM of each set of polnomials..,. a b c, abc., a b c ( )( ). g, g g 5. r, r r, r 6., w, w (g )(g ) r(r ) 6(w )(w ) 7. 8, 8. 6, d 6d 9, (d 9) ( )( ) ( )( )( ) (d )(d ) Biccling The Tour de France is the most popular biccle road race. It lasts das and covers 500 miles. Source: World Book Encclopedia BICYCLING For Eercises 6 8, use the following information. Jalisa is competing in a 8-mile biccle race. She travels half the distance at one rate. The rest of the distance, she travels miles per hour slower. 6. If represents the faster pace in miles per hour, write an epression that represents the time spent at that pace. h 7. Write an epression for the amount of time spent at the slower pace. h 8. Write an epression for the amount of time Jalisa needed to complete the race. 8( ) h ( ) 9. MAGNETS For a bar magnet, the magnetic field strength H at a point P along m m the ais of the magnet is H L(d L). L(d L) Write a simpler epression for H. md md (d L ) (d L) or (d L ) 50. CRITICAL THINKING Find two rational epressions whose sum is. ( ) ( ) Sample answer:, L d L Lesson 9- Adding and Subtracting Rational Epressions 8 9- Enrichment, p. 58 Superellipses The circle and the ellipse are members of an interesting famil of curves that were first studied b the French phsicist and mathematician Gabriel Lamé ( ). The general equation for the famil is a n b d NAME DATE PERID n, with a 0, b 0, and n 0. For even values of n greater than, the curves are called superellipses.. Consider two curves that are not superellipses. Graph each equation on the grid at the right. State the tpe of curve produced each time. circle a. d L Point P b. ellipse Simplif each epression ab 8a 5 6c d b ab m 8 mn a ( n) n 6 5m m m 9 9 m p p 5p 6 p 9 5n 5 ( ) 5 60 ( )( ) 7 9m m 9 p p (p )(p )(p ) d 9c c d a 7 (a )(a 5) ( )( ) (6 5n) 0n 9 a 5 r 6 r r a a 6... r a a a 9 r r r r a r GEMETRY The epressions,, and represent the lengths of the sides of a triangle. Write a simplified epression for the perimeter of the triangle. 5( 6) ( )( ) 6. KAYAKING Mai is kaaking on a river that has a current of miles per hour. If r represents her rate in calm water, then r represents her rate with the current, and r represents her rate against the current. Mai kaaks miles downstream and then d back to her starting point. Use the formula for time, t, where d is the distance, to r write a simplified epression for the total time it takes Mai to complete the trip. r h (r )(r ) Reading to Learn Mathematics, p NAME 56 DATE PERID Reading to Learn Mathematics cd Adding and Subtracting Rational Epressions 7 0n ELL Pre-Activit How is subtraction of rational epressions used in photograph? Read the introduction to Lesson 9- at the top of page 79 in our tetbook. A person is standing 5 feet from a camera that has a lens with a focal length of feet. Write an equation that ou could solve to find how far the film should be from the lens to get a perfectl focused photograph. q 5 Reading the Lesson. a. In work with rational epressions, LCD stands for least common denominator and LCM stands for least common multiple. The LCD is the LCM of the denominators. b. To find the LCM of two or more numbers or polnomials, factor each number or polnomial. The LCM contains each factor the greatest number of times it appears as a factor.. To add and, ou should first factor the denominator of 5 6 each fraction. Then use the factorizations to find the LCM of 5 6 and LCD for the two fractions.. This is the. When ou add or subtract fractions, ou often need to rewrite the fractions as equivalent fractions. You do this so that the resulting equivalent fractions will each have a denominator equal to the LCD of the original fractions.. To add or subtract two fractions that have the same denominator, ou add or subtract their numerators and keep the same denominator. 5. The sum or difference of two rational epressions should be written as a polnomial or as a fraction in simplest form. Helping You Remember 6. Some students have trouble remembering whether a common denominator is needed to add and subtract rational epressions or to multipl and divide them. How can our knowledge of working with fractions in arithmetic help ou remember this? Sample answer: In arithmetic, a common denominator is needed to add and subtract fractions, but not to multipl and divide them. The situation is the same for rational epressions. Lesson 9- Lesson 9- Adding and Subtracting Rational Epressions 8

21 Assess pen-ended Assessment Writing Have students write their own problems of the tpes in this lesson b beginning with an answer and working backward to create a problem. Intervention The skills for combining and simplifing done in this lesson are used etensivel in algebra. Take time to clear up student errors and misconceptions before proceeding. New Getting Read for Lesson 9- PREREQUISITE SKILL Students will graph rational functions using asmptotes in Lesson 9-. In previous course material, students graphed hperbolas b using asmptotes and will appl these skills to graphing rational functions. Use Eercises 59 6 to determine our students familiarit with graphing hperbolas. Assessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons 9- and 9-. Lesson numbers are given to the right of eercises or instruction lines so students can review concepts not et mastered. Quiz (Lessons 9- and 9-) is available on p. 567 of the Chapter 9 Resource Masters. Standardized Test Practice Maintain Your Skills Mied Review Getting Read for the Net Lesson P ractice Quiz 8 Chapter 9 Rational Epressions and Equations 5. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 59A 59D. How is subtraction of rational epressions used in photograph? Include the following in our answer: an eplanation of how to subtract rational epressions, and an equation that could be used to find the distance between the lens and the film if the focal length of the lens is 0 centimeters and the distance between the lens and the object is 60 centimeters. t 5. For all t 5, 5 B t 5 t 5. t 5. t 5. t 5. t 5 A B C D E. t 5. What is the sum of and? C A B C D E 0 Simplif each epression. (Lesson 9-) 9 5. (5 z) ( ) 0 5 z a 0 a a( a ) a 0a 0 a Solve each sstem of inequalities b graphing ( ) 6 Simplif each epression. (Lesson 9-) t t 6. t t. ab 6t 9 t 8a ac c b 9b 6 b a 9 7 6a w w w w (w )(w ) Simplif each epression. (Lesson 9-) 7. a a 8. a b b a a b 5 ab a b 6a 0b 5a b 5 n n 6 n (n 6)(n ) 6 (Lesson 8-7) See pp. 59A 59D. 58. GARDENS Helene Jonson has a rectangular garden 5 feet b 50 feet. She wants to increase the garden on all sides b an equal amount. If the area of the garden is to be increased b 00 square feet, b how much should each dimension be increased? (Lesson 6-).5 ft PREREQUISITE SKILL Draw the asmptotes and graph each hperbola. (To review graphing hperbolas, see Lesson 8-5.) See pp. 59A 59D ( ) ( 5) Lessons 9- and 9-8 Chapter 9 Rational Epressions and Equations

22 Graphing Rational Functions Lesson Notes Vocabular rational function continuit asmptote point discontinuit Stud Tip Look Back To review asmptotes, see Lesson 8-5. Determine the vertical asmptotes and the point discontinuit for the graphs of rational functions. Graph rational functions. can rational functions be used when buing a group gift? A group of students want to get their favorite teacher, Mr. Salgado, a retirement gift. The plan to get him a gift certificate for a weekend package at a lodge in a state park. The certificate costs $50. If c represents the cost for each student and s represents the number of students, then c 5 0. s Cost (dollars) C student could pa $50. 5 students could each pa $6. 50 students could each pa $ s Students VERTICAL ASYMPTTES AND PINT DISCNTINUITY The function 0 is an eample of a rational function. A rational function is an equation of c 5 s the form ƒ() p ( ), where p() and q() are polnomial functions and q() 0. Here q( ) are other eamples of rational functions. 5 ƒ() g() h() 6 ( )( ) No denominator in a rational function can be zero because division b zero is not defined. In the eamples above, the functions are not defined at, 6, and and, respectivel. The graphs of rational functions ma have breaks in continuit. This means that, unlike polnomial functions, which can be traced with a pencil never leaving the paper, not all rational functions are traceable. Breaks in continuit can appear as a vertical asmptote or as a point discontinuit. Recall that an asmptote is a line that the graph of the function approaches, but never crosses. Point discontinuit is like a hole in a graph. Vertical Asmptotes Propert Words Eample Model Vertical If the rational For f(), f() Asmptote epression of a is a vertical function is written in asmptote. f() simplest form and the function is undefined for a, then a is a vertical asmptote. Focus 5-Minute Check Transparenc 9- Use as a quiz or review of Lesson 9-. Mathematical Background notes are available for this lesson on p. 70D. Building on Prior Knowledge In Chapter 7, students learned to graph polnomial equations. In this lesson, the will appl the same skills to graphing rational functions. can rational functions be used when buing a group gift? Ask students: What does the cost for one student depend on? the number of students who participate What happens to the value of c as the value of s increases? It decreases. Lesson 9- Graphing Rational Functions 85 Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 5 Practice, p. 5 Reading to Learn Mathematics, p. 5 Enrichment, p. 5 Assessment, pp. 567, 569 Graphing Calculator and Spreadsheet Masters, p. Teaching Algebra With Manipulatives Masters, p. 7 Transparencies 5-Minute Check Transparenc 9- Answer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 85

23 Teach VERTICAL ASYMPTTES AND PINT DISCNTINUITY In-Class Eample In-Class Eample Power Point Determine the equations of an vertical asmptotes and the values of for an holes in the graph of f(). 5 6 is a vertical asmptote and represents a hole in the graph. GRAPH RATINAL FUNCTINS Graph f(). f() Power Point Teaching Tip Suggest that students choose a large unit on their grid paper and estimate points to the nearest tenth. Point out that the will probabl not be able to see the shape of the graph as a whole unless the use a graphing calculator or computer program. Stud Tip Graphing Rational Functions Finding the - and - intercepts is often useful when graphing rational functions. Propert Words Eample Model Point Discontinuit Eample GRAPH RATINAL FUNCTINS You can use what ou know about vertical asmptotes and point discontinuit to graph rational functions. Eample Vertical Asmptotes and Point Discontinuit Determine the equations of an vertical asmptotes and the values of for an holes in the graph of ƒ(). 6 5 First factor the numerator and denominator of the rational epression. 6 ( ) ( ) 5 ( ) ( 5) ( )( ) The function is undefined for and 5. Since, ( )( 5) 5 5 is a vertical asmptote, and represents a hole in the graph. Graph with a Vertical Asmptote Graph ƒ(). The function is undefined for. Since is in simplest form, is a vertical asmptote. Draw the vertical asmptote. Make a table of values. Plot the points and draw the graph. f() If the original function ( )( ) f() is undefined for a but the rational can be simplified epression of the to ƒ(). function in simplest So, form is defined for represents a hole a, then there is in the graph. a hole in the graph at a. f() As increases, it appears that the values of the function get closer and closer to. The line with the equation ƒ() is a horizontal asmptote of the function. f() Point Discontinuit f() ( )( ) f() 86 Chapter 9 Rational Epressions and Equations Unlocking Misconceptions Asmptotes Students should understand that a graph continues to approach an asmptote and gets closer and closer to that value, but never reaches it. This is an abstract mathematical idea that cannot be represented accuratel with an form of visual illustration. 86 Chapter 9 Rational Epressions and Equations

24 As ou have learned, graphs of rational functions ma have point discontinuit rather than vertical asmptotes. The graphs of these functions appear to have holes. These holes are usuall shown as circles on graphs. Eample Graph with Point Discontinuit Graph ƒ() 9. Notice that 9 ( )( ) or. Therefore, the graph of ƒ() 9 is the graph of ƒ() with a hole at. f() f() 9 In-Class Eamples Graph f(). f() Power Point Man real-life situations can be described b using rational functions. Eample Rational Functions The densit of a material can be epressed as D m, where m is the mass V of the material in grams and V is the volume in cubic centimeters. B finding the volume and densit of 00 grams of each liquid, ou can draw a graph of the function D 00. V Collect the Data Use a balance and metric measuring cups to find the volumes of 00 grams of different liquids such as water, cooking oil, isopropl alcohol, sugar water, and salt water. Use D m to find the densit of each liquid. V Analze the Data. Graph the data b plotting the points (volume, densit) on a graph. Connect the points. See pp. 59A 59D.. From the graph, find the asmptotes. 0, 0 In the real world, sometimes values on the graph of a rational function are not meaningful. Use Graphs of Rational Functions TRANSPRTATIN A train travels at one velocit V for a given amount of time t and then another velocit V for a different amount of time t. The average velocit is given b V V t V t. t t a. Let t be the independent variable and let V be the dependent variable. Draw the graph if V 60 miles per hour, V 0 miles per hour, and t 8 hours. The function is V 60t t 0(8) 8 or V 60t 0 t 8. The vertical asmptote is t 8. Graph the vertical asmptote and the function. Notice that the horizontal asmptote is V 60. Algebra Activit V 60t 0 t Lesson 9- Graphing Rational Functions V Materials: balance, metric measuring cups, different liquids, graph paper Choose liquids that are quite different in densit. If ou make sugar or salt water, dissolve as much of the substance as ou can in it. Differences in volume for each 00 grams will be easier to read if ou use the smallest measuring cup that holds the amount. t Teaching Tip Since the discontinuit is onl one point (and a mathematical point has no dimensions), suggest that students draw a circle on their graphs to indicate the discontinuit. TRANSPRTATIN Use the situation and formula given in Eample. a. Draw the graph if V 50 miles per hour, V 0 miles per hour, and t hour t V 60 b. What is the V-intercept of the graph? 0 c. What values of t and V are meaningful in the contet of the problem? Positive values of t and values of V between 0 and 50 are meaningful. Intervention To make sure the situation in Eample is meaningful, ask a student to eplain the situation as a stor without using letter names for variables. For eample, the stor might begin A train travels 0 miles per hour as it goes through towns. Eight hours of its total trip are spent going through towns. New Lesson 9- Graphing Rational Functions 87

25 Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 9. include an other item(s) that the find helpful in mastering the skills in this lesson. About the Eercises rganization b bjective Vertical Asmptotes and Point Discontinuit: 6 Graph Rational Functions: 5, 7 50 dd/even Assignments Eercises 6 9 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 7 9 odd, 6, 5 66 Average: 7 9 odd, 0, 6 66 Advanced: 6 8 even, 0 6 (optional: 6 66) Answer. Each of the graphs is a straight line passing through (5, 0) and (0, 5). However, the graph of f() ( )( 5) has a hole at (, 6), and the graph of g() 5 does not have a hole. Concept Check. Sample answer: ƒ() ( 5) ( ) Guided Practice GUIDED PRACTICE KEY Eercises Eamples, 5 6, 5 Application 88 Chapter 9 Rational Epressions and Equations b. What is the V-intercept of the graph? The V-intercept is 0. c. What values of t and V are meaningful in the contet of the problem? In the problem contet, time and velocit are positive values. Therefore, onl values of t greater than 0 and values of V between 0 and 60 are meaningful.. PEN ENDED Write a function whose graph has two vertical asmptotes located at 5 and.. Compare and contrast the graphs of ƒ() ( )( 5) and g() 5. See margin.. Describe the graph at the right. Include the equations of an asmptotes, the values of an holes, and the - and -intercepts. and 0 are asmptotes of the graph. The -intercept is 0.5 and there is no -intercept because 0 is an asmptote. Determine the equations of an vertical asmptotes and the values of for an holes in the graph of each rational function.. ƒ() asmptote: 5. ƒ() 5 asmptote: 5; hole: Graph each rational function. 6. See pp. 59A 59D ƒ() 7. ƒ() ( ) ( ) 8. ƒ() ƒ() 5 0. ƒ() ( ). ƒ() 6 MEDICINE For Eercises 5, use the following information. For certain medicines, health care professionals ma use Young s Rule, C D, to estimate the proper dosage for a child when the adult dosage is known. In this equation, C represents the child s dose, D represents the adult dose, and represents the child s age in ears.. Use Young s Rule to estimate the dosage of amoicillin for an eight-ear-old child if the adult dosage is 50 milligrams. 00 mg. Graph C. See pp. 59A 59D.. Give the equations of an asmptotes and - and C-intercepts of the graph., C ; 0; 0 5. What values of and C are meaningful in the contet of the problem? 0 and 0 C Differentiated Instruction Visual/Spatial Have students graph one of the eamples from the lesson with colors on a large sheet of posterboard, to clearl show how a graph approaches but never reaches an asmptote or how a graph ma have a hole in it for a certain value of the variable. Displa the results in the classroom. ( ) 88 Chapter 9 Rational Epressions and Equations

26 indicates increased difficult Practice and Appl Homework Help For Eercises See Eamples 6 9, 0 50 Etra Practice See page asmptote: ; hole: 5 Histor Mathematician Maria Gaetana Agnesi was one of the greatest scholars of all time. Born in Milan, Ital, in 78, she mastered Greek, Hebrew, and several modern languages b the age of. Source: A Histor of Mathematics Determine the equations of an vertical asmptotes and the values of for an holes in the graph of each rational function. 6. ƒ() asmptotes: 7. ƒ() 5 6 asmptotes:, 8, 8. ƒ() 7 0. ƒ() asmptote: ; hole: hole: 5 9. ƒ() 5. ƒ() Graph each rational function. 9. See pp. 59A 59D. hole:. ƒ(). ƒ(). ƒ() ƒ() 6. ƒ() 7. ƒ() 8. ƒ() ( ) 9. ƒ() ( ) 0. ƒ(). ƒ(). ƒ() 6. ƒ() 6. ƒ() 5. ƒ() 6. ƒ() ( ) ( 5) ( )( ) 6 7. ƒ() 8. ƒ() ( 6) 9. ƒ() ( ) HISTRY For Eercises 0, use the following information. In Maria Gaetana Agnesi s book Analtical Institutions, Agnesi discussed the characteristics of the equation a (a ), whose graph is called the curve a of Agnesi. This equation can be epressed as. a a 0. Graph ƒ() a if a. See pp. 59A 59D.. Describe the graph. a. Make a conjecture about the shape of the graph of f() a if a. Eplain our reasoning. See pp. 59A 59D.. The graph is bell-shaped with a horizontal asmptote at f() 0. AUT SAFETY For Eercises 5, use the following information. When a car has a front-end collision, the objects in the car (including passengers) keep moving forward until the impact occurs. After impact, objects are repelled. Seat belts and airbags limit how far ou are jolted forward. The formula for the velocit at which ou are thrown backward is V ƒ (m m)v, i where m m m and m are masses of the two objects meeting and v i is the initial velocit.. See pp. 59A 59D.. Let m be the independent variable, and let V ƒ be the dependent variable. Graph the function if m 7 kilograms and v i 5 meters per second.. Give the equation of the vertical asmptote and the m - and V ƒ -intercepts of the graph. m 7; 7; 5 5. Find the value of V ƒ when the value of m is 5 kilograms. about 0.8 m/s ( ) 5( ) 6. Sample answers: f(), f(), f() ( )( ) ( ) ( ) ( ) ( ) 6. CRITICAL THINKING Write three rational functions that have a vertical asmptote at and a hole at. Lesson 9- Graphing Rational Functions 89 Stud Stud Guide and and Intervention Intervention, p. 59 (shown) and p NAME DATE PERID Graphing Rational Functions Vertical Asmptotes and Point Discontinuit p() Rational Function an equation of the form f(), where p() and q() are polnomial epressions and q() q() 0 Vertical Asmptote An asmptote is a line that the graph of a function approaches, but never crosses. of the Graph of a If the simplified form of the related rational epression is undefined for a, Rational Function then a is a vertical asmptote. Point Discontinuit Point discontinuit is like a hole in a graph. If the original related epression is undefined of the Graph of a for a but the simplified epression is defined for a, then there is a hole in the Rational Function graph at a. Eample Determine the equations of an vertical asmptotes and the values of for an holes in the graph of f(). First factor the numerator and the denominator of the rational epression. f() ( )( ) ( )( ) The function is undefined for and. ( )( ) Since, is a vertical asmptote. The simplified epression is ( )( ) defined for, so this value represents a hole in the graph. Eercises Determine the equations of an vertical asmptotes and the values of for an holes in the graph of each rational function.. f(). f() 0. f() asmptotes:, hole: asmptote: 0; 5 hole. f() 5. f() f() asmptote: ; asmptotes:, asmptote: hole: 7 7. f() 8. f() 9. f() asmptotes:, asmptote: ; holes:, 5 hole: 9- NAME 59 DATE PERID Practice (Average) Skills Practice, p. 5 and Practice, p. 5 (shown) Determine the equations of an vertical asmptotes and the values of for an holes in the graph of each rational function f(). f(). f() 0 0 asmptotes:, asmptote: ; asmptote: 5 hole: 7. f() f() 6. f() hole: 0 hole: 6 hole: 5 Graph each rational function. 7. f() 8. f() 9. f() Graphing Rational Functions f() 0. PAINTING Working alone, Tawa can give the shed a coat of paint in 6 hours. It takes her father hours working alone to give the 6 shed a coat of paint. The equation f() describes the 6 portion of the job Tawa and her father working together can 6 complete in hour. Graph f() for 0, 0. If Tawa s 6 father can complete the job in hours alone, what portion of the job can the complete together in hour? 5. LIGHT The relationship between the illumination an object receives from a light source of I foot-candles and the square of the distance d in feet of the object from the source can be d d modeled b I(d). Graph the function I(d) for 0 I 80 and 0 d 80. What is the illumination in foot-candles that the object receives at a distance of 0 feet from the light source?.5 foot-candles Reading to Learn Mathematics, p f() Illumination Distance (ft) NAME 5 DATE PERID Reading to Learn Mathematics Graphing Rational Functions Illumination (foot-candles) ( ) Pre-Activit How can rational functions be used when buing a group gift? Read the introduction to Lesson 9- at the top of page 85 in our tetbook. If 5 students contribute to the gift, how much would each of them pa? $0 If each student pas $5, how man students contributed? 0 students Reading the Lesson. Which of the following are rational functions? A and C 5 A. f() B. g() C. h() a. Graphs of rational functions ma have breaks in continuit. These ma occur as vertical asmptotes or as point discontinuities. b. The graphs of two rational functions are shown below. I f() f() ELL d Lesson 9- I. II. NAME DATE PERID 9- Enrichment, p. 5 Graphing with Addition of -Coordinates Equations of parabolas, ellipses, and hperbolas that are tipped with respect to the - and -aes are more difficult to graph than the equations ou have been studing. ften, however, ou can use the graphs of two simpler equations to graph a more complicated equation. For eample, the graph of the ellipse in the diagram at the right is obtained b adding the -coordinate of each point on the circle and the -coordinate of the corresponding point of the line. 6 Graph each equation. State the tpe of curve for each graph. 6 ellipse. parabola B B A 6 A Graph I has a point discontinuit at. Graph II has a vertical asmptote at. Match each function with its graph above. f() II g() I Helping You Remember. ne wa to remember something new is to see how it is related to something ou alread know. How can knowing that division b zero is undefined help ou to remember how to find the places where a rational function has a point discontinuit or an asmptote? Sample answer: A point discontinuit or vertical asmptote occurs where the function is undefined, that is, where the denominator of the related rational epression is equal to 0. Therefore, set the denominator equal to zero and solve for the variable. Lesson 9- Lesson 9- Graphing Rational Functions 89

27 Assess pen-ended Assessment Writing Have students write their own eamples of rational functions and graph them, showing discontinuities. Getting Read for Lesson 9- BASIC SKILL Students will write and solve direct, joint, and inverse variation problems in Lesson 9-. This will include students writing and solving proportions that relate the values in the variation. Use Eercises 6 66 to determine our students familiarit with solving proportions. Assessment ptions Quiz (Lesson 9-) is available on p. 567 of the Chapter 9 Resource Masters. Mid-Chapter Test (Lessons 9- through 9-) is available on p. 569 of the Chapter 9 Resource Masters. Answers 7. P() 8 P() A rational function can be used to determine how much each person owes if the cost of the gift is known and the number of people sharing the cost is s. Answers should include the following. c 00 c s s s c 0 8. the part in the first quadrant 9. It represents her original free-throw percentage of 60%. 50. ; this represents 00% which she cannot achieve because she has alread missed free throws. Standardized Test Practice Maintain Your Skills Mied Review 5. m m n ( )( ) (w ) (w ) Getting Read for the Net Lesson 90 Chapter 9 Rational Epressions and Equations nl the portion in the first quadrant is significant in the real world because there cannot be a negative number of people nor a negative amount of mone owed for the gift. BASKETBALL For Eercises 7 50, use the following information. Zonta plas basketball for Centerville High School. So far this season, she has made 6 out of 0 free throws. She is determined to improve her free-throw percentage. If she can make consecutive free throws, her free-throw percentage can be 6 determined using P() Graph the function. See margin. 8. What part of the graph is meaningful in the contet of the problem? 9. Describe the meaning of the -intercept. 50. What is the equation of the horizontal asmptote? Eplain its meaning with respect to Zonta s shooting percentage. 5. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How can rational functions be used when buing a group gift? Include the following in our answer: a complete graph of the function c 5 0 with asmptotes, and s an eplanation of wh onl part of the graph is meaningful in the contet of the problem. 5. Which set is the domain of the function graphed at the right? A A { 0, } B {, 0} C { } D { } 5. Which set is the range of the function 8? B A B { } C { 0} D { 0} Simplif each epression. (Lessons 9- and 9-) 5. m m n m n 56. w w w 6 5 Find the coordinates of the center and the radius of the circle with the given equation. Then graph the circle. (Lesson 8-) See margin for graphs. 57. ( 6) ( ) 5 (6, ); (, 0); 59. ART Joce Jackson purchases works of art for an art galler. Two ears ago, she bought a painting for $0,000, and last ear, she bought one for $5,000. If paintings appreciate % per ear, how much are the two paintings worth now? (Lesson 7-) $65,89 Solve each equation b completing the square. (Lesson 6-) i, 0 7 BASIC SKILL Solve each proportion v 9 5 a s b ( 6) ( ) Chapter 9 Rational Epressions and Equations

28 Graphing Rational Functions A TI-8 Plus graphing calculator can be used to eplore the graphs of rational functions. These graphs have some features that never appear in the graphs of polnomial functions. Eample Graph 8 z 5 in the standard viewing window. Find the equations of an asmptotes. Enter the equation in the Y= list. KEYSTRKES: ( 8 X,T,,n 5 ) ( X,T,,n ) ZM 6 B looking at the equation, we can determine that if 0, the function is undefined. The equation of the vertical asmptote is 0. Notice what happens to the values as grows larger and as gets smaller. The values approach. So, the equation for the horizontal asmptote is =. Eample Graph 6 in the window [5,.] b [0, ] with scale factors of. Because the function is not continuous, put the calculator in dot mode. KEYSTRKES: MDE A Follow-Up of Lesson 9- Eercises 6. See pp. 59A 59D for graphs. Use a graphing calculator to graph each function. Be sure to show a complete graph. Draw the graph on a sheet of paper. Write the -coordinates of an points of discontinuit and/or the equations of an asmptotes..,., 5.,. ƒ(), 0. ƒ() 0, 0. ƒ(). ƒ() 5. ƒ() 6 ENTER This graph looks like a line with a break in continuit at. This happens because the denominator is 0 when. Therefore, the function is undefined when. If ou TRACE along the graph, when ou come to, ou will see that there is no corresponding value. 7. Which graph(s) has point discontinuit? 6 8. Describe functions that have point discontinuit. See margin. [0, 0] scl: b [0, 0] scl: Notice the hole at. [5,.] scl: b [0, ] scl: 6. ƒ() 9 point discontinuit at Graphing Calculator Investigation Graphing Rational Functions 9 Graphing Calculator Investigation A Follow-Up of Lesson 9- Getting Started Graphing Window For the eamples, students should use the settings shown below the diagrams. For all the eercises, a good window is [0, 0] scl: b [0, 0] scl:. Graph Stle Students ma find it instructive to eperiment with the graph stle. The can begin b using the usual line stle. This is the stle when the icon to the left of the equation on the Y= list is a backslash. The best alternate stle to use is path stle. The icon for this stle is a small numeral 0 with a short minus sign attached to the left side of the 0. Teach Suggest that students tr graphing Eample in Connected mode as well as Dot mode. Ask them which wa makes it easier to see the discontinuit. Assess Ask: How can ou use the graphing calculator to check the eact value where a discontinuit occurs? Use TRACE to find where there is no value. Answer 8. rational functions where a value of the function is not defined, but the rational epression in simplest form is defined for that value Graphing Calculator Investigation Graphing Rational Functions 9

29 Lesson Notes Direct, Joint, and Inverse Variation Focus 5-Minute Check Transparenc 9- Use as a quiz or review of Lesson 9-. Mathematical Background notes are available for this lesson on p. 70D. is variation used to find the total cost given the unit cost? Ask students: If the number of students increases, what happens to the value of the total spending? It increases. If the number of students decreases, what happens to the value of the total spending? It decreases. Vocabular direct variation constant of variation joint variation inverse variation Recognize and solve direct and joint variation problems. Recognize and solve inverse variation problems. is variation used to find the total cost given the unit cost? The total high-tech spending t of an average public college can be found b using the equation t = 9s, where s is the number of students. USA TDAY Snapshots College high-tech spending Spending for computer hardware and software b colleges was about $.8 billion this school ear. Spending per student: $8 Source: Market Data Retrieval DIRECT VARIATIN AND JINT VARIATIN The relationship given b t 9s is an eample of direct variation. A direct variation can be epressed in the form k. The k in this equation is a constant and is called the constant of variation. Notice that the graph of t 9s is a straight line through the origin. An equation of a direct variation is a special case of an equation written in slope-intercept form, t 600 m b. When m k and b 0, m b becomes k. So the slope of a direct variation equation is its constant of variation. To epress a direct variation, we sa that varies t 9s directl as. In other words, as increases, increases or decreases at a constant rate. 6 s $9 Public schools varies directl as if there is some nonzero constant k such that k. k is called the constant of variation. Private schools B Marc E. Mullins, USA TDAY Direct Variation If ou know that varies directl as and one set of values, ou can use a proportion to find the other set of corresponding values. k and k Therefore,. 9 Chapter 9 Rational Epressions and Equations k k Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 57 Practice, p. 58 Reading to Learn Mathematics, p. 59 Enrichment, p. 50 Graphing Calculator and Spreadsheet Masters, p. School-to-Career Masters, p. 8 Science and Mathematics Lab Manual, pp. 6 Transparencies 5-Minute Check Transparenc 9- Real-World Transparenc 9 Answer Ke Transparencies Technolog Interactive Chalkboard

30 Using the properties of equalit, ou can find man other proportions that relate these same and values. Eample Direct Variation If varies directl as and when, find when 6. Use a proportion that relates the values. Direct proportion 6,, and 6 6() ( ) Cross multipl. 9 Simplif. 6 Divide each side b. When 6, the value of is 6. Another tpe of variation is joint variation. Joint variation occurs when one quantit varies directl as the product of two or more other quantities. Joint Variation varies jointl as and z if there is some number k such that kz, where k 0, 0, and z 0. Teach DIRECT VARIATIN AND JINT VARIATIN In-Class Eamples Power Point If varies directl as and 5 when 5, find when. 9 Suppose varies jointl as and z. Find when 0 and z 5, if when z 8 and. 5 Teaching Tip Discuss with students what happens to the constant of variation in the proportions used to solve these eamples. If ou know varies jointl as and z and one set of values, ou can use a proportion to find the other set of corresponding values. k z and k z z k z k Therefore,. Eample Joint Variation Suppose varies jointl as and z. Find when 8 and z, if 6 when z and 5. Use a proportion that relates the values. Joint variation z 6 5 ( ) z 8 z z () 6, 5, z, 8, and z 8()(6) 5()( ) Cross multipl. 8 0 Simplif. 8. Divide each side b 0. When 8 and z, the value of is 8.. INVERSE VARIATIN Another tpe of variation is inverse variation. For two quantities with inverse variation, as one quantit increases, the other quantit decreases. For eample, speed and time for a fied distance var inversel with each other. When ou travel to a particular location, as our speed increases, the time it takes to arrive at that location decreases. Lesson 9- Direct, Joint, and Inverse Variation 9 nline Lesson Plans USA TDAY Education s nline site offers resources and interactive features connected to each da s newspaper. Eperience TDAY, USA TDAY s dail lesson plan, is available on the site and delivered dail to subscribers. This plan provides instruction for integrating USA TDAY graphics and ke editorial features into our mathematics classroom. Log on to Lesson 9- Direct, Joint, and Inverse Variation 9

31 INVERSE VARIATIN In-Class Eamples Power Point If a varies inversel as b and a 6 when b, find a when b 7. 7 SPACE The net closest planet to the Sun after Mercur is Venus, which is about 67 million miles awa. How much larger would the diameter of the Sun appear on Venus than on Earth? about.9 times as large as it appears from Earth Teaching Tip To understand the situation in the problem, some students ma find it useful to make a sketch showing the relative distances from the Sun to Earth, Mercur, and Venus. TEACHING TIP In Eample, students ma wish to solve the problem b using the equation r t r t. varies inversel as if there is some nonzero constant k such that k or k. Suppose varies inversel as such that 6 or 6. The graph of this equation is shown at the right. Note that in this case, k is a positive value 6, so as the values of increase, the values of decrease. Just as with direct variation and joint variation, a proportion can be used with inverse variation to solve problems where some quantities are known. The following proportion is onl one of several that can be formed. Eample k and k Substitution Propert of Equalit Divide each side b. Inverse Variation Inverse Variation If r varies inversel as t and r 8 when t, find r when t. Use a proportion that relates the values. 6 or 6 r r t t 8 r 8() (r ) Inverse variation r 8, t, and t Cross multipl. Space Mercur is about 6 million miles from the Sun, making it the closest planet to the Sun. Its proimit to the Sun causes its temperature to be as high as 800 F. Source: World Book Encclopedia 5 r Simplif. 0 Divide each side b. 0 When t, the value of r is. Eample Use Inverse Variation SPACE The apparent length of an object is inversel proportional to one s distance from the object. Earth is about 9 million miles from the Sun. Use the information at the left to find how much larger the diameter of the Sun would appear on Mercur than on Earth. Eplore Plan You know that the apparent diameter of the Sun varies inversel with the distance from the Sun. You also know Mercur s distance from the Sun and Earth s distance from the Sun. You want to determine how much larger the diameter of the Sun appears on Mercur than on Earth. Let the apparent diameter of the Sun from Earth equal unit and the apparent diameter of the Sun from Mercur equal m. Then use a proportion that relates the values. 9 Chapter 9 Rational Epressions and Equations Unlocking Misconceptions Direct and Inverse Variation Help students understand the difference between the two tpes of variation b using the eample of gas in the tank of a car, distance, and driving time. The amount of distance increases as the driving time increases (direct). The amount of gas decreases as the driving time increases (inverse). 9 Chapter 9 Rational Epressions and Equations

32 Concept Check Solve distance from Mercur apparent diameter from Earth 6 mi llio u n miles nit 9 m illion miles m units Inverse variation Substitution (6 million miles)(m units) (9 million miles)( unit) Cross multipl. Eamine m distance from Earth apparent diameter from Mercur (9 million miles)( unit) 6 million miles m.58 units Divide each side b 6 million miles. Simplif. Since distance between the Sun and Earth is between and times the distance between the Sun and Mercur, the answer seems reasonable. From Mercur, the diameter of the Sun will appear about.58 times as large as it appears from Earth.. Determine whether each graph represents a direct or an inverse variation. a. inverse b. direct Practice/Appl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 9. write the names and some eamples from their lives for direct, joint, and inverse variation. include an other item(s) that the find helpful in mastering the skills in this lesson.. Both are eamples of direct variation. For 5, increases as increases. For 5, decreases as increases. Guided Practice GUIDED PRACTICE KEY Eercises Eamples 9 0. about 50 ft Application. Compare and contrast 5 and 5.. PEN ENDED Describe two quantities in real life that var directl with each other and two quantities that var inversel with each other. See margin. State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.. ab 0 inverse; direct; A bh joint; Find each value. 7. If varies directl as and 8 when 5, find when Suppose varies jointl as and z. Find when 9 and z 5, if 90 when z 5 and If varies inversel as and when, find when. 8 SWIMMING For Eercises 0, use the following information. When a person swims underwater, the pressure in his or her ears varies directl with the depth at which he. pounds or she is swimming. per square inch (psi) 0. Write an equation of direct variation that represents this situation. P 0.d. Find the pressure at 60 feet. 5.8 psi. It is unsafe for amateur divers to swim where the water pressure is more than 65 pounds per square inch. How deep can an amateur diver safel swim?. Make a table showing the number of pounds of pressure at various depths of water. Use the data to draw a graph of pressure versus depth. See margin. 0 ft Lesson 9- Direct, Joint, and Inverse Variation 95 Answers. Sample answers: wages and hours worked, total cost and number of pounds of apples purchased; distances traveled and amount of gas remaining in the tank, distance of an object and the size it appears. Depth (ft) Pressure (psi) P P 0.d d Differentiated Instruction Auditor/Musical Have students find various kinds of variation in the sounds made b musical instruments. Suggest that the investigate the length and size of guitar strings relative to their vibrations, and the length and diameter of the columns of air used in wind and brass instruments for various notes. Lesson 9- Direct, Joint, and Inverse Variation 95

33 Stud Stud Guide and and Intervention Intervention, p. 55 (shown) and p NAME DATE PERID Direct, Joint, and Inverse Variation Direct Variation and Joint Variation varies directl as if there is some nonzero constant k such that k. k is called the Direct Variation constant of variation. Joint Variation varies jointl as and z if there is some number k such that kz, where 0 and z 0. Eample Find each value. a. If varies directl as and 6 when, find when 0. Direct proportion 6 0 6,, and 0 6 (0)() 5 Cross multipl. Simplif. The value of is 5 when is 0. Eercises b. If varies jointl as and z and 0 when and z, find when and z. Joint variation z z 0 0,, z,, and z 0 8 Simplif. 5 Divide each side b 8. The value of is 5 when and z. Find each value.. If varies directl as and 9 when. If varies directl as and 6 when 6, find when 8. 6, find when 5.. If varies directl as and 5. If varies directl as and when when 5, find when 9. 7, find when. 8 indicates increased difficult Practice and Appl Homework Help Eercises Eamples Etra Practice See page 88. State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation.. direct;.5 6. inverse; 8. m n.5 5. a 5bc joint; 5 6. vw 8 7. b a direct; 8. p V q Bh..5 s t inverse; direct; 7 joint; inverse;.5. CHEMISTRY Bole s Law states that when a sample of gas is kept at a constant temperature, the volume varies inversel with the pressure eerted on it. Write an equation for Bole s Law that epresses the variation in volume V as a function of pressure P. V k P 5. Suppose varies jointl as and z. 6. Suppose varies jointl as and z. Find Find when 5 and z, if 8 when 6 and z 8, if 6 when when and z. 5 and z Suppose varies jointl as and z. 8. Suppose varies jointl as and z. Find Find when and z, if 60 when 5 and z, if 8 when when and z and z If varies directl as and 0. If varies directl as and 00 when when 5, find when..8 50, find when If varies directl as and 9. If varies directl as and 60 when when 5, find when , find when..6. Suppose varies jointl as and z.. Suppose varies jointl as and z. Find Find when 6 and z, if when 5 and z 0, if when 0 when 5 and z. 8 and z 6..5 Lesson 9-. CHEMISTRY Charles Law states that when a sample of gas is kept at a constant pressure, its volume V will increase as the temperature t increases. Write an equation for Charles Law that epresses volume as a function. V kt. GEMETRY How does the circumference of a circle var with respect to its radius? What is the constant of variation? directl; 5. Suppose varies jointl as and z. 6. Suppose varies jointl as and z. Find Find when 7 and z 8, if when 5 and z 7, if 80 when 5 when 6 and z and z NAME 55 DATE PERID 9- Skills Practice Practice, p. 57 and (Average) Practice, Direct, Joint, p. and 58 Inverse (shown) Variation State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. 5 k. u 8wz joint; 8. p s direct;. L inverse; 5..5 inverse;.5 C.5 d g d mn 7. h 8. direct; joint; inverse;.5 inverse; Find each value. 9. If varies directl as and 8 when, find when If varies directl as and 6 when 6, find when..5. If varies directl as and when, find when If varies directl as and 7 when.5, find when.. If varies jointl as and z and when and z, find when and z. 88. If varies jointl as and z and 60 when and z, find when 6 and z If varies jointl as and z and when and z, find when and z If varies inversel as and 6 when, find when. 7. If varies inversel as and when 5, find when If varies inversel as and 8 when 6, find when If varies directl as and 5 when 0., find when GASES The volume V of a gas varies inversel as its pressure P. If V 80 cubic centimeters when P 000 millimeters of mercur, find V when P 0 millimeters of mercur. 500 cm. SPRINGS The length S that a spring will stretch varies directl with the weight F that is attached to the spring. If a spring stretches 0 inches with 5 pounds attached, how far will it stretch with 5 pounds attached? in.. GEMETRY The area A of a trapezoid varies jointl as its height and the sum of its bases. If the area is 80 square meters when the height is 0 meters and the bases are 8 meters and 0 meters, what is the area of a trapezoid when its height is 8 meters and its bases are 0 meters and 5 meters? 00 m Reading to Learn Mathematics, p NAME 58 DATE PERID Reading to Learn Mathematics Direct, Joint, and Inverse Variation Pre-Activit How is variation used to find the total cost given the unit cost? Read the introduction to Lesson 9- at the top of page 9 in our tetbook. For each additional student who enrolls in a public college, the total high-tech spending will increase (increase/decrease) b $9. For each decrease in enrollment of 00 students in a public college, the total high-tech spending will decrease (increase/decrease) b $,900. Reading the Lesson. Write an equation to represent each of the following variation statements. Use k as the constant of variation. k a. m varies inversel as n. m n b. s varies directl as r. s kr c. t varies jointl as p and q. t kpq ELL. Which tpe of variation, direct or inverse, is represented b each graph? a. inverse b. direct More About... Travel Agent Travel agents give advice and make arrangements for transportation, accommodations, and recreation. For international travel, the also provide information on customs and currenc echange. nline Research For information about a career as a travel agent, visit: careers 96 Chapter 9 Rational Epressions and Equations NAME DATE PERID 9- Enrichment, p TRAVEL A map is scaled so that centimeters represents 5 kilometers. How far apart are two towns if the are 7.9 centimeters apart on the map? 8.5 km Find each value. 6. If varies directl as and 5 when, find when If varies directl as and 8 when 6, find when Suppose varies jointl as and z. Find when and z 7, if 9 when 8 and z If varies jointl as and z and 80 when 5 and z 8, find when 6 and z If varies inversel as and 5 when 0, find when. 5. If varies inversel as and 6 when 5, find when 0.. If varies inversel as and when 5, find when If varies inversel as and when, find when If varies directl as and 9 when is 5, find when If varies directl as and 6 when 0.5, find when Suppose varies jointl as and z. Find when and z 6, if 5 when 6 and z If varies jointl as and z and 8 when and z, find when 6 and z WRK Paul drove from his house to work at an average speed of 0 miles per hour. The drive took him 5 minutes. If the drive home took him 0 minutes and he used the same route in reverse, what was his average speed going home? 0 mph 9. WATER SUPPLY Man areas of Northern California depend on the snowpack of the Sierra Nevada Mountains for their water suppl. If 50 cubic centimeters of snow will melt to 8 cubic centimeters of water, how much water does 900 cubic centimeters of snow produce? 00.8 cm Epansions of Rational Epressions Helping You Remember. How can our knowledge of the equation of the slope-intercept form of the equation of a line help ou remember the equation for direct variation? Sample answer: The graph of an equation epressing direct variation is a line. The slope-intercept form of the equation of a line is m b. In direct variation, if one of the quantities is 0, the other quantit is also 0, so b 0 and the line goes through the origin. The equation of a line through the origin is m, where m is the slope. This is the same as the equation for direct variation with k m. 96 Chapter 9 Rational Epressions and Equations Man rational epressions can be transformed into power series. A power series is an infinite series of the form A B C D. The rational epression and the power series normall can be said to have the same values onl for certain values of. For eample, the following equation holds onl for values of such that. for Eample Epand in ascending powers of. Assume that the epression equals a series of the form A B C D. Then multipl both sides of the equation b the denominator. A B C D ( )(A B C D ) A B C D B C

34 Biolog In order to sustain itself in its cold habitat, a Siberian tiger requires 0 pounds of meat per da. Source: Wildlife Fact File ; C 0.0 P P d 5. Sample answer: If the average student spends $.50 for lunch in the school cafeteria, write an equation to represent the amount s students will spend for lunch in d das. How much will 0 students spend in a week? a.50sd; $75 0. RESEARCH According to Johannes Kepler s third law of planetar motion, the ratio of the square of a planet s period of revolution around the Sun to the cube of its mean distance from the Sun is constant for all planets. Verif that this is true for at least three planets. See students work. BILGY For Eercises, use the information at the left.. Write an equation to represent the amount of meat needed to sustain s Siberian tigers for d das. m 0sd. Is our equation in Eercise a direct, joint, or inverse variation? joint. How much meat do three Siberian tigers need for the month of Januar? 860 lb LAUGHTER For Eercises 6, use the following information. According to The Columbus Dispatch, the average American laughs 5 times per da.. Write an equation to represent the average number of laughs produced b m household members during a period of d das. 5md 5. Is our equation in Eercise a direct, joint, or inverse variation? joint 6. Assume that members of our household laugh the same number of times each da as the average American. How man times would the members of our household laugh in a week? See students work. ARCHITECTURE For Eercises 7 9, use the following information. When designing buildings such as theaters, auditoriums, or museums architects have to consider how sound travels. Sound intensit I is inversel proportional to the square of the distance from the sound source d. 7. Write an equation that represents this situation. I d k 8. If d is the independent variable and I is the dependent variable, graph the equation from Eercise 7 when k 6. See margin. 9. The sound will be 9. If a person in a theater moves to a seat twice as far from the speakers, compare heard the new sound intensit to that of the original. as intensel. TELECMMUNICATINS For Eercises 50 5, use the following information. It has been found that the average number of dail phone calls C between Cit Population two cities is directl proportional to the product of the populations P and P of two cities and inversel proportional to Tampa Indianapolis the square of the distance d between the,96,000 cities. That is, C kp,607,000 P. d 50. The distance between Nashville and Charlotte Charlotte is about 5 miles. If the Nashville,99,000 average number of dail phone calls,,000 between the cities is 0,000, find the value of k and write the equation of variation. Round to the nearest hundredth. Source: U.S. Census Bureau 5. Nashville is about 680 miles from Tampa. Find the average number of dail phone calls between them. about 7,57 calls 5. The average dail phone calls between Indianapolis and Charlotte is,80. Find the distance between Indianapolis and Charlotte. about 60 mi 5. Could ou use this formula to find the populations or the average number of phone calls between two adjoining cities? Eplain. no; d 0 5. CRITICAL THINKING Write a real-world problem that involves a joint variation. Solve the problem. Lesson 9- Direct, Joint, and Inverse Variation 97 About the Eercises rganization b bjective Direct Variation and Joint Variation: 9, 7, 9 6 Inverse Variation:, 0, 8, 7 9 dd/even Assignments Eercises 9 are structured so that students practice the same concepts whether the are assigned odd or even problems. Alert! Eercise 0 requires reference materials for planetar data. Assignment Guide Basic: 5 5 odd, 9,, 5 7 Average: 5 9 odd, 9, 5 7 Advanced: 0 even, 67 (optional: 68 7) All: Practice Quiz ( 5) Answer 8. I I 6 d d Teacher to Teacher Susan Nelson Spring H.S., Spring, TX "I have m students do a data gathering activit called Rotations where we have the student do a regression for the diameter of a lid versus the number of rotations it takes to move across a fied length of masking tape." Lesson 9- Direct, Joint, and Inverse Variation 97

35 Assess pen-ended Assessment Modeling Have students write equations for various variations in their life (time spent studing, hours of sleep, and so on). Ask them to write eamples and eplain them. Intervention Make sure that students understand the essential differences between direct and inverse variation. For eample, ask them if the number of candles on a birthda cake varies directl or indirectl with the age of the birthda person. Then ask how the length of the candle remaining varies with the time the candle has burned. New Getting Read for Lesson 9-5 PREREQUISITE SKILL In Lesson 9-5, students will identif equations and graphs as different tpes of functions. Use Eercises 68 7 to determine our students familiarit with identifing equations as step, constant, absolute value, or piecewise functions. Assessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons 9- and 9-. Lesson numbers are given to the right of eercises or instruction lines so students can review concepts not et mastered. Standardized Test Practice Maintain Your Skills Mied Review t 6. t (t ) ( t ) Getting Read for the Net Lesson P ractice Quiz 98 Chapter 9 Rational Epressions and Equations 55. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How is variation used to find the total cost given the unit cost? Include the following in our answer: an eplanation of wh the equation for the total cost is a direct variation, and a problem involving unit cost and total cost of an item and its solution. 56. If the ratio of a to b is to 5, what is the ratio of 5a to b? D A B C 9 8 D 57. Suppose b varies inversel as the square of a. If a is multiplied b 9, which of the following is true for the value of b? C A It is multiplied b. B It is multiplied b 9. C It is multiplied b. 8 D It is multiplied b. Determine the equations of an vertical asmptotes and the values of for an holes in the graph of each rational function. (Lesson 9-) 58. ƒ() 59. ƒ() 60. ƒ() asmp.: ; hole: asmp.:, hole: Simplif each epression. (Lesson 9-) m t m t t 6. m (m ) m 5 m m 5 6. ASTRNMY The distance from Earth to the Sun is approimatel 9,000,000 miles. Write this number in scientific notation. (Lesson 5-) State the slope and the -intercept of the graph of each equation. (Lesson -) ; ; ; PREREQUISITE SKILL Identif each function as S for step, C for constant, A for absolute value, or P for piecewise. (To review special functions, see Lesson -6.) 68. h() C 69. g() A 70. ƒ() S 7. ƒ() if 0 P 7. h() A 7. g() C if 0 Graph each rational function. (Lesson 9-). See pp. 59A 59D.. ƒ(). ƒ() 6 9 Answer Find each value. (Lesson 9-). If varies inversel as and when 7, find when. 9. If varies directl as and when 5, find when.. 5. If varies jointl as and z and 80 when 5 and z, find when 0 and z A direct variation can be used to determine the total cost when the cost per unit is known. Answers should include the following. Since the total cost T is the cost per unit u times the number of units n or T un, the relationship is a direct variation. In this equation u is the constant of variation. Sample answer: The school store sells pencils for 0 each. John wants to bu 5 pencils. What is the total cost of the pencils? ($.00) Lessons 9- and 9-98 Chapter 9 Rational Epressions and Equations

36 Classes of Functions Lesson Notes Constant Function The general equation of a constant function is a, where a is an number. Its graph is a horizontal line that crosses the -ais at a. Greatest Integer Function Identif graphs as different tpes of functions. Identif equations as different tpes of functions. The purpose of the 00 Mars dsse Mission is to stud conditions on Mars. The findings will help NASA prepare for a possible mission with human eplorers. The graph at the right compares a person s weight on Earth with his or her weight on Mars. This graph represents a direct variation, which ou studied in the previous lesson. Mars Weight in Pounds Earth IDENTIFY GRAPHS In this book, ou have studied several tpes of graphs representing special functions. The following is a summar of these graphs. can graphs of functions be used to determine a person s weight on a different planet? Direct Variation Function The general equation of a direct variation function is a, where a is a nonzero constant. Its graph is a line that passes through the origin and is neither horizontal nor vertical. Absolute Value Function Special Functions Identit Function The identit function is a special case of the direct variation function in which the constant is. Its graph passes through all points with coordinates (a, a). Quadratic Function Focus 5-Minute Check Transparenc 9-5 Use as a quiz or review of Lesson 9-. Mathematical Background notes are available for this lesson on p. 70D. Building on Prior Knowledge In previous course material, students have learned about different kinds of functions. In this lesson, students will revisit different functions and group them into logical categories based on their characteristics. can graphs of functions be used to determine a person s weight on a different planet? Ask students: According to the graph, what is the approimate weight on Mars of a person who weighs 50 pounds on Earth? about 0 lb According to the graph, what is the approimate weight on Earth of a person who would weigh 0 pounds on Mars? about 75 lb If an equation includes an epression inside the greatest integer smbol, the function is a greatest integer function. Its graph looks like steps. An equation with a direct variation epression inside absolute value smbols is an absolute value function. Its graph is in the shape of a V. The general equation of a quadratic function is a b c, where a 0. Its graph is a parabola. (continued on the net page) Lesson 9-5 Classes of Functions 99 Resource Manager Workbook and Reproducible Masters Chapter 9 Resource Masters Stud Guide and Intervention, pp. 5 5 Skills Practice, p. 5 Practice, p. 5 Reading to Learn Mathematics, p. 55 Enrichment, p. 56 Assessment, p. 568 Transparencies 5-Minute Check Transparenc 9-5 Answer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 99

37 Teach IDENTIFY GRAPHS In-Class Eample Power Point Square Root Function Rational Function Special Functions Inverse Variation Function Identif the tpe of function represented b each graph. a. If an equation includes an epression inside the radical sign, the function is a square root function. Its graph is a curve that starts at a point and continues in onl one direction. The general equation for a rational function is p ( ), where p() and q( ) q() are polnomial functions. Its graph has one or more asmptotes and/or holes. The inverse variation function a is a special case of the rational function where p() is a constant and q(). Its graph has two asmptotes, 0 and 0. b. absolute value function Eample Identif a Function Given the Graph Identif the tpe of function represented b each graph. a. b. In-Class Eample quadratic function IDENTIFY EQUATINS Power Point SHIPPING CHARGES A chart gives the shipping rates for an Internet compan. The charge $.50 to ship less than pound, $.95 for pound and over up to pounds, and $5.0 for pounds and over up to pounds. Which graph depicts these rates? c, the step or greatest integer function a. b. c Rocketr A rocket-powered airplane called the X-5 set an altitude record for airplanes b fling 67 miles above Earth. Source: World Book Encclopedia 500 Chapter 9 Rational Epressions and Equations The graph has a starting point and curves in one direction. The graph represents a square root function. IDENTIFY EQUATINS If ou can identif an equation as a tpe of function, ou can determine the shape of the graph. Eample Match Equation with Graph RCKETRY Emil launched a to rocket from ground level. The height above the ground level h, in feet, after t seconds is given b the formula h(t) 6t 80t. Which graph depicts the height of the rocket during its flight? a. h (t ) b. h(t ) c. h (t ) t t 6 The graph appears to be a direct variation since it is a straight line passing through the origin. However, the hole indicates that it represents a rational function. t 6 The function includes a second-degree polnomial. Therefore, it is a quadratic function, and its graph is a parabola. Graph b is the onl parabola. Therefore, the answer is graph b. Teacher to Teacher Deedee S. Adams 80 0 ford H.S., ford, AL "I have m students pla Simon Sas b having them all stand and graph different tpes of functions with their arms. Students sit down if the don't illustrate the correct graph." 500 Chapter 9 Rational Epressions and Equations

38 7- See pages Eample Chapter 7 Graphing Polnomial Functions Concept Summar The Location Principle: Since zeros of a function are located at -intercepts, there is also a zero between each pair of these zeros. Turning points of a function are called relative maima and relative minima. Graph f() 0 b making a table of values. Make a table of values for several f() f() 6 values of and plot the points. Connect the points with a smooth 8 curve Stud Guide and Review Eercises For Eercises 8, complete each of the following. a. Graph each function b making a table of values. b. Determine consecutive values of between which each real zero is located. c. Estimate the -coordinates at which the relative maima and relative minima occur. See Eample on page See margin.. h() 6 9. f() 7 5. p() 5 6. g() 7. r() 8. f() 8 6 f () 0 In-Class Eample Power Point Identif the tpe of function represented b each equation. Then graph the equation. a. constant function b. 9 square root function Practice/Appl 7- Solving Equations Using Quadratic Techniques See pages Concept Summar Solve polnomial equations b using quadratic techniques. Eample 9. 5,, , 0, 5., i., Solve ( 5) 0 ( 9)( 6) 0 riginal equation Factor out the GCF. Factor the trinomial. 0 or 9 0 or 6 0 Zero Product Propert Eercises Solve each equation. See Eample on page m m 0m. a 6 0. r 9r , 5 Chapter 7 Stud Guide and Review 0 Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter 9. add sketches to illustrate each special function graph. include an other item(s) that the find helpful in mastering the skills in this lesson. Answer. Sample answer: P Differentiated Instruction Interpersonal Have students work with a partner or in small groups to do quick sketches of graphs and identif the tpe of function the graph could represent. Have each group make a list of the identifing characteristics of the graph; then ask groups to echange and compare their lists. d This graph is a rational function. It has an asmptote at. Lesson 9-5 Classes of Functions 50

39 About the Eercises rganization b bjective Identif Graphs: Identif Equations: dd/even Assignments Eercises 0 are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 9 odd,, 7 6 Average: 9 odd,, 7 6 Advanced: 0 even, 55 (optional: 56 6) 9. See margin for graphs. Application indicates increased difficult Practice and Appl Homework Help For See Eercises Eamples 8 9, 6 0 Etra Practice See page 88. Identif the tpe of function represented b each equation. Then graph the equation. 9. identit or 0.. direct variation quadratic absolute value. GEMETRY Write the equation for the area of a circle. Identif the equation as a tpe of function. Describe the graph of the function. A r ; quadratic; the graph is a parabola. Identif the tpe of function represented b each graph absolute value square root rational Answers 9. direct variation quadratic constant 0.. You can use functions to determine the relationship between primar and secondar earthquake waves. Visit com/webquest to continue work on our WebQuest project. 0. See pp. 59A 59D for graphs.. constant. direct variation 5. square root 6. inverse variation or rational 7. rational 8. greatest integer 9. absolute value 0. quadratic 50 Chapter 9 Rational Epressions and Equations Match each graph with an equation at the right b e.. g a Identif the tpe of function represented b each equation. Then graph the equation a. b. c. d. e. 0.5 f. g Chapter 9 Rational Epressions and Equations Cost (cents) unces 6. The graph is similar to the graph of the greatest integer function because both graphs look like a series of steps. In the graph of the postage rates, the solid dots are on the right and the circles are on the left. However, in the greatest integer function, the circles are on the right and the solid dots are on the left.

40 HEALTH For Eercises, use the following information. A woman painting a room will burn an average of.5 Calories per minute.. Write an equation for the number of Calories burned in m minutes. C.5m. Identif the equation in Eercise as a tpe of function. direct variation. Describe the graph of the function. a line slanting to the right and passing through the origin Stud Stud Guide and and Intervention Intervention, p. 5 (shown) and p NAME DATE PERID Classes of Functions Identif Graphs You should be familiar with the graphs of the following functions. Function Description of Graph Constant a horizontal line that crosses the -ais at a Direct Variation a line that passes through the origin and is neither horizontal nor vertical Identit a line that passes through the point (a, a), where a is an real number Greatest Integer a step function Absolute Value V-shaped graph Quadratic a parabola Square Root a curve that starts at a point and curves in onl one direction. ARCHITECTURE The shape of the Gatewa Arch of the Jefferson National Epansion Memorial in St. Louis, Missouri, resembles the graph of the function f() , where is in feet. Describe the shape of the Gatewa Arch. similar to a parabola Rational Inverse Variation Eercises a graph with one or more asmptotes and/or holes a graph with curved branches and asmptotes, 0 and 0 (special case of rational function) Identif the function represented b each graph.... Architecture The Gatewa Arch is 60 feet high and is the tallest monument in the United States. Source: World Book Encclopedia Standardized Test Practice MAIL For Eercises 5 and 6, use the following information. In 00, the cost to mail a first-class letter was for an weight up to and including ounce. Each additional ounce or part of an ounce added to the cost. 5. Make a graph showing the postal rates to mail an letter from 0 to 8 ounces. See pp. 59A 59D. 6. Compare our graph in Eercise 5 to the graph of the greatest integer function. See pp. 59A 59D. 7. CRITICAL THINKING Identif each table of values as a tpe of function. a. f() b. f() c. f() d. f() undefined undefined 0 0 undefined absolute value quadratic greatest integer square root 8. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See pp. 59A 59D. How can graphs of functions be used to determine a person s weight on a different planet? Include the following in our answer: an eplanation of wh the graph comparing weight on Earth and Mars represents a direct variation function, and an equation and a graph comparing a person s weight on Earth and Venus if a person s weight on Venus is 0.9 of his or her weight on Earth. 9. The curve at the right could be part of the graph of which function? C A B C D Lesson 9-5 Classes of Functions 50 quadratic rational direct variation constant absolute value greatest integer identit square root inverse variation 9-5 NAME 5 DATE PERID Practice (Average) Skills Practice, p. 5 and Practice, p. 5 (shown) Classes of Functions Identif the tpe of function represented b each graph.... rational square root absolute value Match each graph with an equation below. A. B. C. D.. D 5. C 6. A Identif the tpe of function represented b each equation. Then graph the equation constant quadratic rational 0. BUSINESS A startup compan uses the function P. 7 to predict its profit or loss during its first 7 ears of operation. Describe the shape of the graph of the function. The graph is U-shaped; it is a parabola.. PARKING A parking lot charges $0 to park for the first da or part of a da. After that, it charges an additional $8 per da or part of a da. Describe the graph and find the cost of parking for 6 das. The graph looks like a series of steps, similar to a greatest integer function, but with open circles on the left and closed circles on the right; $58. Reading to Learn Mathematics, p NAME 5 DATE PERID Reading to Learn Mathematics Classes of Functions Pre-Activit How can graphs of functions be used to determine a person s weight on a different planet? Read the introduction to Lesson 9-5 at the top of page 99 in our tetbook. Based on the graph, estimate the weight on Mars of a child who weighs 0 pounds on Earth. about 5 pounds Although the graph does not etend far enough to the right to read it directl from the graph, use the weight ou found above and our knowledge that this graph represents direct variation to estimate the weight on Mars of a woman who weighs 0 pounds on Earth. about 5 pounds Reading the Lesson. Match each graph below with the tpe of function it represents. Some tpes ma be used more than once and others not at all. I. square root II. quadratic III. absolute value IV. rational V. greatest integer VI. constant VII. identit a. III b. I c. VI ELL Lesson 9-5 NAME DATE PERID 9-5 Enrichment, p. 56 Partial Fractions It is sometimes an advantage to rewrite a rational epression as the sum of two or more fractions. For eample, ou might do this in a calculus course while carring out a procedure called integration. You can resolve a rational epression into partial fractions if two conditions are met: () The degree of the numerator must be less than the degree of the denominator; and () The factors of the denominator must be known. Eample Resolve into partial fractions. The denominator has two factors, a linear factor,, and a quadratic factor,. Start b writing the following equation. Notice that the degree of the numerators of each partial fraction is less than its denominator. A B C d. II e. IV f. V Helping You Remember. How can the smbolic definition of absolute value that ou learned in Lesson - help ou to remember the graph of the function f()? Sample answer: Using the definition of absolute value, f() if 0 and f() if 0. Therefore, the graph is made up of pieces of two lines, one with slope and one with slope, meeting at the origin. This forms a V-shaped graph with verte at the origin. Lesson 9-5 Lesson 9-5 Classes of Functions 50

41 Assess pen-ended Assessment Modeling Have students use string on a coordinate grid to model some of the nine different tpes of functions in this lesson. Ask them to give an eample of an equation that might have that sort of graph. Intervention Help students associate the graphs and their functions b grouping the 9 tpes into groups, those which involve straight lines and those which involve curves. New Getting Read for Lesson 9-6 PREREQUISITE SKILL Students will solve rational equations in Lesson 9-6. These equations often contain fractions that are simplified b finding the LCD. Use Eercises 56 6 to determine our students familiarit with finding LCMs of polnomials. Assessment ptions Quiz (Lessons 9- and 9-5) is available on p. 568 of the Chapter 9 Resource Masters. Answers 5. (8, ); 8, 7 8 ; 8; 8 ; up; unit ( ) ( 8) 6.,,, ; ; ; right; units Maintain Your Skills Mied Review impossible a b c (d ) 57 Getting Read for the Net Lesson 0. If g(), which of the following is the graph of g? D A C. If varies directl as and 5 when, find when. (Lesson 9) 5 Graph each rational function. (Lesson 9-). See pp. 59A 59D. 8. ƒ(). ƒ(). ƒ() 5 ( ) ( ) Identif the coordinates of the verte and focus, the equations of the ais of smmetr and directri, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. (Lesson 8-) 5 7. See margin. 5. ( ) ( 8) Find each product, if possible. (Lesson -) Solve each sstem of equations b using either substitution or elimination. (Lesson -) a b 5. s t 0 9 (7, 5) s t 6 (, ) a b, Determine the value of r so that a line through the points with the given coordinates has the given slope. (Lesson -) 5. (r, ), (, 6); slope 8 5. (r, 6), (8, ); slope 55. Evaluate [(7 ) 5 ] 6. (Lesson -) 67 PREREQUISITE SKILL Find the LCM of each set of polnomials. (To review least common multiples of polnomials, see Lesson 9-.) 56. 5ab c, 6a, bc 57. 9, 5, d 0, d , ( )( ) a a, a a 6 (a )(a )(a ) t 9t 5, t t 0 (t 5)(t 6)(t ) 50 Chapter 9 Rational Epressions and Equations 7. (5, ); 5, ; ; ; right; units B D 8 50 Chapter 9 Rational Epressions and Equations

42 Solving Rational Equations and Inequalities Lesson Notes Vocabular rational equation rational inequalit Solve rational equations. Solve rational inequalities. are rational equations used to solve problems involving unit price? The Coast to Coast Phone Compan advertises 5 a minute for long-distance calls. However, it also charges a monthl fee of $5. If the customer has minutes in long distance calls last month, the bill in cents will be The actual cost per minute is To find how man long-distance minutes a person would need to make the actual cost per minute 6, ou would need to solve the equation The equation is an eample of SLVE RATINAL EQUATINS a rational equation. In general, an equation that contains one or more rational epressions is called a rational equation. Rational equations are easier to solve if the fractions are eliminated. You can eliminate the fractions b multipling each side of the equation b the least common denominator (LCD). Remember that when ou multipl each side b the LCD, each term on each side must be multiplied b the LCD. Eample Solve a Rational Equation 9 Solve. Check our solution. 8 z The LCD for the three denominators is 8(z ). 9 8 z 8(z ) 9 8 z z riginal equation 8(z ) Multipl each side b 8(z + ). 8(z ) 8(z ) 8(z ) Distributive Propert (9z 8) 8 z Simplif. 9z 0 z Wh pa more for long distance? Pa onl 5 a minute for calls to anwhere in the U.S. at an time!* Simplif. * Plus $5 monthl fee 60 z Subtract 9z and from each side. 5 z Divide each side b. CAST to CAST Phone Compan Lesson 9-6 Solving Rational Equations and Inequalities 505 Workbook and Reproducible Masters Chapter 9 Resource Masters Stud Guide and Intervention, pp Skills Practice, p. 59 Practice, p. 550 Reading to Learn Mathematics, p. 55 Enrichment, p. 55 Assessment, p. 568 Focus 5-Minute Check Transparenc 9-6 Use as a quiz or review of Lesson 9-5. Mathematical Background notes are available for this lesson on p. 70D. Building on Prior Knowledge In Chapter, students reviewed techniques for solving linear equations and inequalities. In this lesson, students will appl those same techniques to solving rational equations and inequalities. are rational equations used to solve problems involving unit price? Ask students: Wh does the equation use 500 instead of 5 for the monthl fee? The fee and the per minute cost are both epressed in cents. If a person makes 00 minutes of calls for a given month, how much did the monthl fee add to the per minute cost for these calls? The fee adds 5 cents per minute. Resource Manager Transparencies 5-Minute Check Transparenc 9-6 Answer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 505

43 Teach SLVE RATINAL EQUATINS In-Class Eamples Power Point 5 Solve. Check our solution. 5 Teaching Tip Discuss with students that, when checking a solution, it is not possible to simpl substitute the solution for the variable and then multipl both sides of the equation b the LCD because the truth of the equation is being tested; it cannot be assumed. Solve p p p 7 p p p. Check our solution. p Teaching Tip Remind students that solutions must alwas be checked in the original equation, rather than in an of the steps of the solution. 9 CHECK 8 z The solution is 5. riginal equation z Simplif Simplif. The solution is correct. When solving a rational equation, an possible solution that results in a zero in the denominator must be ecluded from our list of solutions. Eample CHECK Cec Elimination of a Possible Solution Solve r r 5 r r r. Check our solution. r The LCD is (r ). r r 5 r r r riginal equation r (r )r r 5 r (r r Multipl each side b the LCD, ) r r (r ). (r )r (r r ) (r ) 5 (r r ) r Distributive Propert r r (r r) (r 5) (r )(r r ) Simplif. r r r 5 r r Simplif. r r 0 Subtract (r r ) from each side. (r )(r ) 0 Factor. r 0 or r 0 Zero Product Propert r r r r 5 r r r riginal equation r 5 r 8 Simplif. Stud Tip Etraneous Solutions Multipling each side of an equation b the LCD of rational epressions can ield results that are not solutions of the original equation. These solutions are called etraneous solutions. 7 7 r r 5 r r r r ( ) 5 ( ) () ( ) r 0 0 riginal equation Simplif. Since r results in a zero in the denominator, eliminate from the list of solutions. The solution is. 506 Chapter 9 Rational Epressions and Equations 506 Chapter 9 Rational Epressions and Equations

44 Tunnels The Chunnel is a tunnel under the English Channel that connects England with France. It is miles long with miles of the tunnel under water. Source: Some real-world problems can be solved with rational equations. Eample Work Problem TUNNELS When building the Chunnel, the English and French each started drilling on opposite sides of the English Channel. The two sections became one in 990. The French used more advanced drilling machiner than the English. Suppose the English could drill the Chunnel in 6. ears and the French could drill it in 5.8 ears. How long would it have taken the two countries to drill the tunnel? In ear, the English could complete of the tunnel. 6. In ears, the English could complete 6. or of the tunnel. 6. In t ears, the English could complete 6. t or t of the tunnel. 6. Likewise, in t ears, the French could complete 5.8 t or t of the tunnel. 5.8 Together, the completed the whole tunnel. Part completed part completed entire b the English plus b the French equals tunnel. t 6. t 5.8 Solve the equation. t 6. t 5.8 riginal equation t t () Multipl each side b t t Distributive Propert.9t.t 7.98 Simplif. 6t 7.98 Simplif. t.00 Divide each side b 6. It would have taken about ears to build the Chunnel. Eample Rate problems frequentl involve rational equations. Rate Problem NAVIGATIN The speed of the current in the Puget sound is 5 miles per hour. A barge travels 6 miles with the current and returns in 0 hours. What is the speed of the barge in still water? WRDS The formula that relates distance, time, and rate is d rt or d t. r VARIABLES Let r be the speed of the barge in still water. Then the speed of the barge with the current is r 5, and the speed of the barge against the current is r 5. Time going with time going against the current plus the current equals total time. 6 6 EQUATIN 0 r 5 r 5 In-Class Eamples Power Point MWING LAWNS Tim and Ashle mow lawns together. Tim working alone could complete the job in.5 hours, and Ashle could complete it alone in.7 hours. How long does it take to complete the job when the work together? about h Teaching Tip Suggest to students that when the problem involves completing part of a job and working together, students think about what part of the work gets done in one da, or one ear whatever is one unit of the time. SWIMMING Janine swims for 5 hours in a stream that has a current of mile per hour. She leaves her dock and swims upstream for miles and then back to her dock. What is her swimming speed in still water? about.5 mi/h Teaching Tip Make sure that students understand the difference between the solutions to the quadratic equation (of which there are two) and the solution to the problem (of which there is onl one). (continued on the net page) Lesson 9-6 Solving Rational Equations and Inequalities 507 Lesson 9-6 Solving Rational Equations and Inequalities 507

45 SLVE RATINAL INEQUALITIES In-Class Eample 5 Power Point Solve. s 0 or s 9 s s 5 6 Teaching Tip Suggest that students also verif whether the boundar indicated b the solution of the equation is or is not in the solution set of the inequalit. Stud Tip Look Back To review the Quadratic Formula, see Lesson 6-5. Solve the equation r 5 r 5 riginal equation (r 6 6 5) r 5 r (r 5 5)0 Multipl each side b (r 5). (r 5) (r 5) (r 6 5) (r 6 5) (r 5) Distributive Propert r 5 r 5 (78r 90) (78r 90) r 800 Simplif. 56r r 800 Simplif. 0 r Subtract 56r from 56r 800 each side. 0 8r 9r 00 Divide each side b. Use the Quadratic Formula to solve for r. b bac a Quadratic Formula (9) (9) )(00 (8) r (8) 9 79 r 6 r r 8 or.5 r, a 8, b 9, and c 00 Simplif. Simplif. Simplif. Since the speed must be positive, the answer is 8 miles per hour. SLVE RATINAL INEQUALITIES Inequalities that contain one or more rational epressions are called rational inequalities. To solve rational inequalities, complete the following steps. Step State the ecluded values. Step Solve the related equation. Step Use the values determined in Steps and to divide a number line into regions. Test a value in each region to determine which regions satisf the original inequalit. Eample 5 Solve a Rational Inequalit Solve a 5 8a. Step Values that make a denominator equal to 0 are ecluded from the domain. For this inequalit, the ecluded value is 0. Step Solve the related equation. 5 a 8 a Related equation 8a 5 a 8 a 8a Multipl each side b 8a. 5 a Simplif. 7 a Add. a Divide each side b. 508 Chapter 9 Rational Epressions and Equations 508 Chapter 9 Rational Epressions and Equations

46 Step Draw vertical lines at the ecluded value and at the solution to separate the number line into regions. ecluded value solution of related equation Practice/Appl Concept Check. Sample answer: 5 a. ( );. Jeff; when Dustin multiplied b a, he forgot to multipl the b a. Guided Practice GUIDED PRACTICE KEY Eercises Eamples 9,, 5 0, Application Now test a sample value in each region to determine if the values in the region satisf the inequalit. Test a. 5? ( ) 8( ) 5 8? 7 8 a 0 is not a solution. The solution is 0 a. 0 Jeff - = a 6a - 9 = a a = 9 Test a. 5? ( ) 8( ) 5 8? a = a is a solution. Dustin - a = - 9 = a -7 = a -.5 = a Test a. 5? ( ) 8( ) 8 5 6? 7 6 a is not a solution.. PEN ENDED Write a rational equation that can be solved b first multipling each side b 5(a ).. State the number b which ou would multipl each side of in order to solve the equation. What value(s) of cannot be a solution?. FIND THE ERRR Jeff and Dustin are solving a. Who is correct? Eplain our reasoning. Solve each equation or inequalit. Check our solutions.. d 5. t 8 0, 6 t v 6 v 6, 8. c c 9. v v v 0 or v 7 0. WRK A bricklaer can build a wall of a certain size in 5 hours. Another bricklaer can do the same job in hours. If the bricklaers work together, how long would it take to do the job? 9 h Lesson 9-6 Solving Rational Equations and Inequalities 509 Differentiated Instruction Logical Have students think about the difference between pure mathematics, such as solving an equation, and applied mathematics, such as solving a real-world problem. Ask them to list some was in which these two are alike and some was in which the are different. Stud Notebook Have students complete the definitions/eamples for the remaining terms on their Vocabular Builder worksheets for Chapter 9. add the steps for solving rational inequalities given in this lesson to their notebooks, along with an eample from their work. write a list of cautions, or checks, that must be done as part of working with problems such as those in the lesson. include an other item(s) that the find helpful in mastering the skills in this lesson. FIND THE ERRR Ask students what value of a can be ecluded at the beginning of the problem. The value of a cannot be 0. About the Eercises rganization b bjective Solve Rational Equations:, 7, 8, 9 Solve Rational Inequalities: 5, 6, 9 dd/even Assignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. Assignment Guide Basic: 7 odd,,, 0 5 Average: odd, 7 5 Advanced: even, 6, 8 5 Lesson 9-6 Solving Rational Equations and Inequalities 509

47 Stud Stud Guide and and Intervention Intervention, p. 57 (shown) and p NAME DATE PERID Solving Rational Equations and Inequalities Solve Rational Equations A rational equation contains one or more rational epressions. To solve a rational equation, first multipl each side b the least common denominator of all of the denominators. Be sure to eclude an solution that would produce a denominator of zero. Eample Solve riginal equation ) (0) 9( ( ) Multipl Distributive Propert 5 5 Subtract and 9 from each side. Divide each side b ( ) 0( ) Multipl each side b 0( ). Check riginal equation Simplif Simplif Eercises 5 5 Simplif. Solve each equation. t t m m m m 7. NAVIGATIN The current in a river is 6 miles per hour. In her motorboat Marissa can travel miles upstream or 6 miles downstream in the same amount of time. What is the speed of her motorboat in still water? mph 8. WRK Adam, Bethan, and Carlos own a painting compan. To paint a particular house alone, Adam estimates that it would take him das, Bethan estimates 5 das, and Carlos 6 das. If these estimates are accurate, how long should it take the three of them to paint the house if the work together? about das NAME 57 DATE PERID 9-6 Skills Practice Practice, p. 59 and (Average) Practice, Solving Rational p. 550 Equations (shown) and Inequalities Solve each equation or inequalit. Check our solutions.. 6., p 0 8., p p s 5s. s s s all reals ecept t 5 or t t t h h h 5 7 w w a a a or 65. p 0 or p p p 5 6. g b b g g b b 5. g 6. b d 7. 6 d 8. 5 d d 6 9. n n n c 0. c 5,5 c c 5 v 5v k k v v. 7., k 7k v v r r 6 6a r r r 6 a 7 a all reals ecept and 7. BASKETBALL Kiana has made 9 of 9 free throws so far this season. Her goal is to make 60% of her free throws. If Kiana makes her net free throws in a row, the function 9 f() represents Kiana s new ratio of free throws made. How man successful free 9 throws in a row will raise Kiana s percent made to 60%? 6 p q f 8. PTICS The lens equation relates the distance p of an object from a lens, the distance q of the image of the object from the lens, and the focal length f of the lens. What is the distance of an object from a lens if the image of the object is 5 centimeters from the lens and the focal length of the lens is centimeters? 0 cm Reading to Learn Mathematics, p NAME 550 DATE PERID Reading to Learn Mathematics Solving Rational Equations and Inequalities Pre-Activit How are rational equations used to solve problems involving unit price? Read the introduction to Lesson 9-6 at the top of page 505 in our tetbook. If ou increase total number of minutes of long-distance calls from March to April, will our long-distance phone bill increase or decrease? increase Will our actual cost per minute increase or decrease? decrease Reading the Lesson. When solving a rational equation, an possible solution that results in 0 in the denominator must be ecluded from the list of solutions. 6 z z. Suppose that on a quiz ou are asked to solve the rational inequalit 0. Complete the steps of the solution. Step The ecluded values are and Step The related equation is z z. To solve this equation, multipl both sides b the LCD, which is z(z ). Solving this equation will show that the onl solution is. Step Divide a number line into regions using the ecluded values and the solution of the related equation. Draw dashed vertical lines on the number line below to show these regions z z Consider the following values of for various test values of z. 6 6 z z z z If z 5, 0.. If z,. 6 6 z z z z If z, 9. If z, 5. Using this information and our number line, write the solution of the inequalit. z or z 0 Helping You Remember ELL. How are the processes of adding rational epressions with different denominators and of solving rational epressions alike, and how are the different? Sample answer: The are alike because both use the LCD of all the rational epressions in the problem. The are different because in an addition problem, the LCD remains after the fractions are added, while in solving a rational equation, the LCD is eliminated. Lesson 9-6 indicates increased difficult Practice Application and Appl Homework Help For Eercises 9. t 0 or t. p 0 or p 50 Chapter 9 Rational Epressions and Equations 9-6 Enrichment, p. 55 Limits NAME DATE PERID Sequences of numbers with a rational epression for the general term often approach some number as a finite limit. For eample, the reciprocals of the positive integers approach 0 as n gets larger and larger. This is written using the notation shown below. The smbol stands for infinit and n means that n is getting larger and larger, or n goes to infinit.,,,,,, lim 0 n n n Eample See Eamples 0,, 5 9, Etra Practice See page 89. n Find lim n (n ) It is not immediatel apparent whether the sequence approaches a limit or not. But notice what happens if we divide the numerator and denominator of the general term b n. Solve each equation or inequalit. Check our solutions... p p 5. s 5 6 6, s. a 6 a, a a m m 9 7. t t t 8. w w w 9. 5 w w t 6 t 0. 7 b 5 b 0 b p p. b b n n b b b. n 9 n n 5. d d d 6. d 7. b b 7 n 5b 6 b b 7 8. n n 7 n 8 n q q z 6 5 q q z z. NUMBER THERY The ratio of 8 less than a number to 8 more than that number is to 5. What is the number?. NUMBER THERY The sum of a number and 8 times its reciprocal is 6. Find the number(s). or. ACTIVITIES The band has 0 more members than the school chorale. If each group had 0 more members, the ratio of their membership would be :. How man members are in each group? band, 80 members; chorale, 50 members PHYSICS For Eercises and 5, use the following information. The distance a spring stretches is related to the mass attached to the spring. This is represented b d km, where d is the distance, m is the mass, and k is the spring constant. When two springs with spring constants k and k are attached in a series, the resulting spring constant k is found b the equation k. k k. If one spring with constant of centimeters per gram is attached in a series with another spring with constant of 8 centimeters per gram, find the resultant spring constant..8 cm/g 5. If a 5-gram object is hung from the series of springs, how far will the springs stretch? cm 6. CYCLING n a particular da, the wind added kilometers per hour to Alfonso s rate when he was ccling with the wind and subtracted kilometers per hour from his rate on his return trip. Alfonso found that in the same amount of time he could ccle 6 kilometers with the wind, he could go onl kilometers against the wind. What is his normal biccling speed with no wind? 5 km/h Spring k cm/g Spring k 8 cm/g d 5 g Spring Spring 50 Chapter 9 Rational Epressions and Equations

48 Chemist Man chemists work for manufacturers developing products or doing qualit control to ensure the products meet industr and government standards. nline Research For information about a career as a chemist, visit: com/careers Standardized Test Practice 7. CHEMISTRY Kiara adds an 80% acid solution to 5 milliliters of solution that is 0% acid. The function that represents the percent of acid in the resulting solution is ƒ() 5(0.0 ) (0.80), where is the amount of 80% solution added. 5 How much 80% solution should be added to create a solution that is 50% acid? 5 ml STATISTICS For Eercises 8 and 9, use the following information. A number is the harmonic mean of and z if is the average of and z. 8. Find if 8 and z Find if 5 and z CRITICAL THINKING Solve for a if a b c. b bc. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. See margin. How are rational equations used to solve problems involving unit price? Include the following in our answer: an eplanation of how to solve , and the reason wh the actual price per minute could never be 5. st. If T, what is the value of s when t 5 and T 0? s B t A 0 B 0 C 5 D. Amanda wanted to determine the average of her 6 test scores. She added the scores correctl to get T, but divided b 7 instead of 6. Her average was less than the actual average. Which equation could be used to determine the value of T? C 6T 7T T 7 T A B 6 C T 7 T 6 T 6 T D 7 Assess pen-ended Assessment Writing Have students write their own real-world problems similar to some the have seen in this lesson, but using their own data. Then solve them. Intervention Make sure ever student is clear about wh some values must be ecluded, even though the appear as solutions in the course of working a problem. Point out that multipling each side of an equation b the variable ma introduce etraneous roots. New Assessment ptions Quiz (Lesson 9-6) is available on p. 568 of the Chapter 9 Resource Masters. Maintain Your Skills Mied Review Identif the tpe of function represented b each equation. Then graph the equation. (Lesson 9-5) 6. See margin for graphs.. quad. 5. sq. root direct var. Answers. 5. { or } 5. { 0 } 5. b b 7. If varies inversel as and when 9, find when 6. (Lesson 9-) 6 8. If varies directl as and = 9 when =, find when = 5. (Lesson 9-).75 Find the distance between each pair of points with the given coordinates. (Lesson 8-) 9. (5, 7), (9, ) 50. (, 5), (7, ) 5 5. (, ), (5, 8) 7 0 Solve each inequalit. (Lesson 6-7) 5. ( )( ) b b Lesson 9-6 Solving Rational Equations and Inequalities 5. If something has a general fee and cost per unit, rational equations can be used to determine how man units a person must bu in order for the actual unit price to be a given number. Answers should include the following. To solve , multipl each side of the equation b to eliminate the rational epression. Then subtract 5 from each side. Therefore, 500. A person would need to make 500 minutes of long distance minutes to make the actual unit price 6. Since the cost is 5 per minute plus $5.00 per month, the actual cost per minute could never be 5 or less Lesson 9-6 Solving Rational Equations and Inequalities 5

49 Graphing Calculator Investigation A Follow-Up of Lesson 9-6 Getting Started Know Your Calculator To see the vertical asmptote for the graph of, students should check to see that the calculator is in Connected mode. Using Parentheses When students enter functions on the Y= list, the should use parentheses around an numerator or denominator that is not a single number or variable. Teach After the enter the two equations, have students enter the CALC menu and select 5:Intersect. Then have them move the cursor and press ENTER to identif each of the graphs. In response to Guess?, move the cursor to an estimated point of intersection and press ENTER. Assess Ask students to describe some was ou can identif ecluded values. You can use a graphing calculator to solve rational equations. You need to graph both sides of the equation and locate the point(s) of intersection. You can also use a graphing calculator to confirm solutions that ou have found algebraicall. Eample Use a graphing calculator to solve. First, rewrite as two functions, and. Net, graph the two functions on our calculator. KEYSTRKES: ( X,T,,n ) ZM 6 Notice that because the calculator is in connected mode, a vertical line is shown connecting the two branches of the hperbola. This line is not part of the graph. Net, locate the point(s) of intersection. KEYSTRKES: nd CALC 5 Select one graph and press ENTER. Select the other graph, press ENTER, and press ENTER again. The solution is. Check this solution b substitution. A Follow-Up of Lesson 9-6 Solving Rational Equations b Graphing Eercises Use a graphing calculator to solve each equation all real numbers ecept 5. 6., no real solution [0, 0] scl: b [0, 0] scl: Solve each equation algebraicall. Then, confirm our solution(s) using a graphing calculator.. 7 or about.56 and Investigating Slope-Intercept Form 5 Chapter 9 Rational Epressions and Equations 5 Chapter 9 Rational Epressions and Equations

50 Stud Guide and Review Vocabular and Concept Check asmptote (p. 85) comple fraction (p. 75) constant of variation (p. 9) continuit (p. 85) State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence.. The equation has a(n) asmptote at. false; point discontinuit. The equation is an eample of a direct variation equation. true. The equation is a(n) polnomial equation. false; rational. The graph of has a(n) variation at. false; asmptote 5. The equation b is a(n) inverse variation equation. true a 6. n the graph of 5, there is a break in continuit at. false; 9- See pages Eamples. ( )( ) ( ) direct variation (p. 9) inverse variation (p. 9) joint variation (p. 9) point discontinuit (p. 85) Multipling and Dividing Rational Epressions Concept Summar Multipling and dividing rational epressions is similar to multipling and dividing fractions. Eercises Simplif each epression. See Eamples 7 on pages 7 and ab c bc c a 8. a a 6 b b a b 6b 9. 6b(a b) ( )( 6) n. 6n 9. 0 n n n 8 rational equation (p. 505) rational epression (p. 7) rational function (p. 85) rational inequalit (p. 508) Simplif 8 6. Simplif p 7p 9 p. p p 8 6 p 7p 9 p p p p 7p p p 9 p p(7 p) (7 p) p (7 p)(7 p) For more information about Foldables, see Teaching Mathematics with Foldables. TM Chapter 9 Stud Guide and Review 5 Suggest that students think of a concept map as a visual organizer that is related to a linear outline, but better shows interrelated ideas. Remind students that different people will organize, remember, and stud differentl, so the should make a Foldable that works well for them, rather than coping someone else s wa of doing notes. Encourage students to refer to their Foldables while completing the Stud Guide and Review and to use them in preparing for the Chapter Test. Vocabular and Concept Check This alphabetical list of vocabular terms in Chapter 9 includes a page reference where each term was introduced. Assessment A vocabular test/review for Chapter 9 is available on p. 566 of the Chapter 9 Resource Masters. Lesson-b-Lesson Review For each lesson, the main ideas are summarized, additional eamples review concepts, and practice eercises are provided. Vocabular PuzzleMaker ELL The Vocabular PuzzleMaker software improves students mathematics vocabular using four puzzle formats crossword, scramble, word search using a word list, and word search using clues. Students can work on a computer screen or from a printed handout. MindJogger Videoquizzes ELL MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams in a game show format to gain points for correct answers. The questions are presented in three rounds. Round Concepts (5 questions) Round Skills ( questions) Round Problem Solving ( questions) Chapter 9 Stud Guide and Review 5

51 Stud Guide and Review Chapter 9 Stud Guide and Review Answers 9. f () 9- See pages Adding and Subtracting Rational Epressions Concept Summar To add or subtract rational epressions, find a common denominator. To simplif comple fractions, simplif the numerator and the denominator separatel, and then simplif the resulting epression. 0.. f( ) f () f() f () Eample 9 Simplif. 9 9 Factor the denominators. ( )( ) ( ) 9 The LCD is ( )( ). ( )( ) ( )( ) ( ) 9 Subtract the numerators. ( ) ( ) 9 Distributive Propert ( ) ( ) 5 Simplif. ( )( ) 7 Eercises Simplif each epression. See Eamples and on page ( ). 6 7( ) b 5 b b 0b 8. m m 8 m 6m 9 9 m f() 9- See pages Graphing Rational Functions (m m 7) (m )(m ) Concept Summar Functions are undefined at an value where the denominator is zero. An asmptote is a line that the graph of the function approaches, but never crosses.. f() f () Eample 5 Graph ƒ(). ( ) The function is undefined for 0 and. 5 Since is in simplest form, 0 and ( ) are vertical asmptotes. Draw the two asmptotes and sketch the graph. f() 5 ( ) f(). f () 9. See margin. Eercises Graph each rational function. See Eamples on pages ƒ() 0. ƒ(). ƒ(). ƒ() 5. ƒ(). ƒ() ( ) ( ) 5 Chapter 9 Rational Epressions and Equations 5 f() ( )( ). f () f() 5 Chapter 9 Rational Epressions and Equations

52 Chapter 9 Stud Guide and Review Stud Guide and Review 9- See pages Direct, Joint, and Inverse Variation Concept Summar Direct Variation: There is a nonzero constant k such that k. Joint Variation: There is a number k such that kz, where 0 and z 0. Inverse Variation: There is a nonzero constant k such that k or k. Eample If varies inversel as and when 6, find when. 6 (6) ( ) Inverse variation, 6, Cross multipl. 8 Simplif. 7 7 When, the value of is 7 7. Eercises Find each value. See Eamples on pages 9 and If varies directl as and when 7, find when If varies inversel as and 9 when.5, find when If varies inversel as and 8 when 8, find when If varies directl as and 8 when 8, find when If varies jointl as and z and and z when 6, find when 5 and z Classes of Functions See pages Concept Summar The following is a list of special functions. constant function greatest integer function square root function direct variation function absolute value function rational function identit function quadratic function inverse variation function Eamples Identif the tpe of function represented b each graph. The graph has a parabolic shape, therefore it is a quadratic function. The graph has a stair-step pattern, therefore it is a greatest integer function. Chapter 9 Stud Guide and Review 55 Chapter 9 Stud Guide and Review 55

53 Stud Guide and Review Etra Practice, see pages Mied Problem Solving, see page 870. Eercises Identif the tpe of function represented b each graph. See Eample on page Answers Practice Test. f () f() 9-6 See pages Eample square root absolute value direct variation Solving Rational Equations and Inequalities Concept Summar Eliminate fractions in rational equations b multipling each side of the equation b the LCD. Possible solutions to a rational equation must eclude values that result in zero in the denominator. To solve rational inequalities, find the ecluded values, solve the related equation, and use these values to divide a number line into regions. Then test a value in each region to determine which regions satisf the original inequalit. Solve 0. The LCD is ( ). 0 riginal equation ( ) ( )(0) Multipl each side b ( ). ( ) ( ) ( )(0) Distributive Propert () ( ) 0 Simplif. 0 Distributive Propert 0 Simplif. Add to each side. The solution is. Divide each side b. Eercises Solve each equation or inequalit. Check our solutions. See Eamples,, and 5 on pages 505, 506, 508, and r r 0 r b b 6 b 0 56 Chapter 9 Rational Epressions and Equations. f () f() ( )( ) 56 Chapter 9 Rational Epressions and Equations

54 Practice Test Vocabular and Concepts Match each eample with the correct term.. z c a. inverse variation equation. 5 b b. direct variation equation. 7 a c. joint variation equation Skills and Applications Simplif each epression.. a ab a b a 5b 5b 5. ( ) ( ( )( 5) 0 )( 7) ( ) ( ) ( ) Identif the tpe of function represented b each graph. ( )( ) 0. inverse variation. greatest integer Graph each rational function.. See margin.. ƒ(). ƒ() ( ) ( ) Solve each equation or inequalit z t t or t 0 t , m 9 9. m If varies inversel as and 9 when, find when If g varies directl as w and g 0 when w, find w when g. 5. Suppose varies jointl as and z. If 0 when 50 and z 5, find when.5 and z AUT MAINTENANCE When air is pumped into a tire, the pressure required varies inversel as the volume of the air. If the pressure is 0 pounds per square inch when the volume is 0 cubic inches, find the pressure when the volume is 00 cubic inches. lb/in. ELECTRICITY The current I in a circuit varies inversel with the resistance R. a. Use the table at the right to write an equation relating the current and the resistance. I 6 I b. What is the constant of variation? 6 R R STANDARDIZED TEST PRACTICE If m, n 7m, p, q p, and r, find. n D q A r B q C p D r E q Chapter 9 Practice Test 57 Assessment ptions Vocabular Test A vocabular test/review for Chapter 9 can be found on p. 566 of the Chapter 9 Resource Masters. Chapter Tests There are si Chapter 9 Tests and an pen- Ended Assessment task available in the Chapter 9 Resource Masters. Chapter 9 Tests Form Tpe Level Pages MC basic A MC average B MC average C FR average D FR average FR advanced MC = multiple-choice questions FR = free-response questions pen-ended Assessment Performance tasks for Chapter 9 can be found on p. 565 of the Chapter 9 Resource Masters. A sample scoring rubric for these tasks appears on p. A5. TestCheck and Worksheet Builder This networkable software has three modules for assessment. Worksheet Builder to make worksheets and tests. Student Module to take tests on-screen. Management Sstem to keep student records. Portfolio Suggestion Introduction In this lesson, ou have been working with several kinds of variation. Ask Students Write an application problem for each of the three tpes of variations (direct, inverse, and joint), and show our steps for each problem. Then describe how our problems are modeled b each of the variations, and how our answers relate to the solutions for the problems. Chapter 9 Practice Test 57

55 Standardized Test Practice These two pages contain practice questions in the various formats that can be found on the most frequentl given standardized tests. A practice answer sheet for these two pages can be found on p. A of the Chapter 9 Resource Masters. Part Multiple Choice Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. Best Bikes has 5000 bikes in stock on Ma. B the end of Ma, 0 percent of the bikes have been sold. B the end of June, 0 percent of the remaining bikes have been sold. How man bikes remain unsold? C 5. In a hardware store, n nails cost c cents. Which of the following epresses the cost of k nails? B A C nck n k c 6. If 5w w 9, then D A C B D k c n n c n w. B w. w. D w. Standardized Test Practice NAME DATE PERID Standardized Test Practice 9 Student Recording Sheet (Use with pages Sheet, ofp. the Student AEdition.) Part Multiple Choice Select the best answer from the choices given and fill in the corresponding oval. A B C D A B C D 7 A B C D A B C D 5 A B C D 8 A B C D A B C D 6 A B C D 9 A B C D Part Short Response/Grid In Solve the problem and write our answer in the blank. For Questions 0, also enter our answer b writing each number or smbol in a bo. Then fill in the corresponding oval for that number or smbol / / / / / / Answers A C 000 B D 000. In ABC, if AB is equal to 8, then BC is equal to C A 8 5 B A C C. 8 B.. D The graphs show a driver s distance d from a designated point as a function of time t. The driver passed the designated point at 60 mph and continued at that speed for hours. Then she slowed to 50 mph for hour. She stopped for gas and lunch for hour and then drove at 60 mph for hour. Which graph best represents this trip? B A d t B d t / / / / / / / / In the figure, the slope of AC is and mc 0. What is the length of BC? D C d D d Part Quantitative Comparison Select the best answer from the choices given and fill in the corresponding oval. A B C D A B C D 5 A B C D A B C D A B C D B(, 0) t t Additional Practice See pp in the Chapter 9 Resource Masters for additional standardized test practice. A C 0 0 A(, ) B D C Which equation has roots of n, n, and? D A 8n 0 B 8n 0 C n 8n 0 D n 8n 0. Given that k, which of the following could be k? B 9. What point is on the graph of and has a -coordinate of 5? A A 5 B A (, 5) B (7, 5) C D C (5, ) D (, 5) 58 Chapter 9 Standardized Test Practice Log n for Test Practice The Princeton Review offers additional test-taking tips and practice problems at their web site. Visit or TestCheck and Worksheet Builder Special banks of standardized test questions similar to those on the SAT, ACT, TIMSS 8, NAEP 8, and Algebra End-of-Course tests can be found on this CD-RM. 58 Chapter 9 Rational Epressions and Equations

56 Aligned and verified b Part Short Response/Grid In Part Quantitative Comparison Record our answers on the answer sheet provided b our teacher or on a sheet of paper. 0. In the figure, what is the equation of the circle Q that is circumscribed around the square ABCD? ( 5) 8 B C Compare the quantit in Column A and the quantit in Column B. Then determine whether: A the quantit in Column A is greater, B the quantit in Column B is greater, C the two quantities are equal, or D the relationship cannot be determined from the information given. A Q D. A Column A Column B the number of the number of distinct prime distinct prime factors of 05 factors of 89. Find one possible value for k such that k is an integer between 0 and 0 that has a remainder of when it is divided b and that has a remainder of when divided b. 6 or 8. 7 D. The coordinates of the vertices of a triangle are (, ), (0, ), and (a, b). If the area of the triangle is 6 square units, what is a possible value for b? or 5. If ( )( ) 6, what is a possible value of? or. If the average of five consecutive even integers is 76, what is the greatest of these integers? In Ma, Hank s Camping Suppl Store sold 5 tents. In June, it sold 90 tents. What is the percent increase in the number of tents sold? If n 6, what is the value of n? 0 7. If 5 and 0, what is the value of ( )? 0 8. If a 8 a, 8 then what is the value of a? a 9. If 80 5, what is the value of? 0. What is the -intercept of the graph of 6?. D t. 0 A 5. 0 B Test-Taking Tip Questions 5 In quantitative comparison questions that involve variables, make sure ou consider all of the possible values of the variables before ou make a comparison. Consider positive and negative integers, positive and negative fractions, and 0. t Chapter 9 Standardized Test Practice 59 Chapter 9 Standardized Test Practice 59