Algebraic, Exponential, and Logarithmic Functions

Transcription

1 Alebraic, Eponential, and Loarithmic Functions Chapter : Functions and Mathematical Models Chapter : Properties o Elementar Functions Chapter : Fittin Functions to Data Chapter : Polnomial and Rational Functions

2 Unit Overview This unit ocuses on a stud o unctions, establishin the alebraic basis or precalculus. Linear, eponential, power, loarithmic, loistic, polnomial and rational unctions are investiated and epanded usin raphical, numerical, and alebraic approaches. In Chapter, students learn overarchin concepts such as transormations, composition, and inverses. An introduction to parametric equations strenthens the concept o inverse unctions. In Chapter, students approach unctions b eaminin shapes o raphs and numerical patterns o evenl spaced data. The eplore characteristics and siniicant eatures o raphs and make eneralizations about dierent unctions. Chapter ocuses on ittin unctions to data usin re-epression and reression. Students make decisions about which unction best its the data b eaminin the shape o the scatter and residual plots, the end behavior, and the correlation coeicient. Chapter completes the unit, ocusin on polnomial and rational unctions. Usin this Unit Each chapter in this unit eamines unctions rom dierent viewpoints. Students need a stron understandin o unctions to be successul in precalculus, calculus, and courses beond calculus. You ma decide to teach Unit and then branch into other parts o the tet. I ou would like to stud trionometr and periodic unctions earl in the course, start with Chapter and then o directl to Unit : Trionometric and Periodic Functions. Dependin on the level o preparation our students have, ou ma be able to omit a portion o Chapter with a cautionar note that the topics in this chapter orm a critical oundation or the rest o the course. Consider omittin Chapter i our students have taken or will be takin statistics. It provides an introduction to ittin unctions to data, but time considerations ma make it necessar to stud this concept in a dierent course. Finall, i our students studied polnomial and rational unctions in second-ear alebra, ou ma preer to dela Chapter until later in the course, as it provides a solid oundation or the discussion o limits in Chapter 6. You ma also want to omit the sections on partial ractions, which will be covered in calculus when students stud interation techniques.

3 Functions and Mathematical Models I ou shoot an arrow into the air, its heiht above the round depends on the number o seconds since ou released it. In this chapter ou will learn was to epress quantitativel the relationship between two variables such as heiht and time. You will deepen what ou have learned in previous courses about unctions and the particular relationships that the describe or eample, how heiht depends on time. CHAPTER OBJECTIVES Work with unctions that are deined raphicall, alebraicall, numericall, or verball. Make connections amon the alebraic equation or a unction, its name, and its raph. Transorm a iven pre-imae unction so that the result is a raph o the imae unction that has been dilated b iven actors and translated b iven amounts. Given two unctions, raph and evaluate the composition o one unction with the other. Given a unction, ind its inverse relation, and tell whether the inverse relation is a unction. Graph parametric equations both b hand and on a rapher, and use parametric equations to raph the inverse o a unction. Given a unction, transorm it b relection and b applin absolute value to the unction or its arument. Start writin a journal to record thins learned about precalculus mathematics and questions concernin concepts which are not quite clear.

4 Chapter Functions and Mathematical Models Overview In this chapter students reresh their memories about unctions the studied previousl in alebra. The uniin concept is that amiliar unctions are built up b transormations o a ew basic parent unctions. For instance, the point-slope orm o the linear unction, m( ), is a dilation o the parent unction,, b a actor o m in the -direction and translations b and in the - and -directions, respectivel. Students stud relections as dilations b a actor o. The stud the ormal deinition o inverses o unctions alon with the composition o unctions. Piecewise unctions and their inverses are introduced usin a raphin calculator to restrict the domain. The Quick Review problems bein in Section -, ivin students a timeeicient review o other concepts and techniques. Students are also introduced to the concept that mathematics can be learned our was: raphicall, alebraicall, numericall, and verball. Usin This Chapter Chapter sets the stae or a stud o precalculus and provides a solid oundation or this course. Section -7, Precalculus Journal, provides a oundation or helpin students learn how to epress themselves mathematicall in oral and written lanuae. Ater Chapter, continue with ittin unctions to real-world data in Chapters, or branch to the stud o periodic unctions in Chapters 9. Teachin Resources Eplorations Eploration -: Paper Cup Analsis Eploration -a: Names o Functions Eploration -b: Restricted Domains and Boolean Variables Eploration -: Transormations rom Graphs Eploration -a: Translations and Dilations, Numericall Eploration -b: Translations and Dilations, Alebraicall Eploration -c: Transormation Review Eploration -: Composition o Functions Eploration -: Parametric Equations Graph Eploration -a: Inverses o Functions Eploration -b: Introduction to Parametric Equations Eploration -6a: Translation, Dilation, and Relection Blackline Masters Sections - to -6, and -8 Supplementar Problems Sections - and - Assessment Resources Test, Sections - to -, Forms A and B Test, Sections - and -, Forms A and B Test, Chapter, Forms A and B Technolo Resources Dnamic Precalculus Eplorations Translation Dilation Sketchpad Presentation Sketches Translation Present.sp Dilation Present.sp Composition Present.sp Inverse Present.sp Relection Present.sp Absolute Value Present.sp Activities Sketchpad: Translation o Functions Sketchpad: Dilation o Functions Fathom: Eplorin Translations and Dilations Fathom: Function Transormations Fathom: Readin the News CAS Activit -a: Findin Polnomial Equations CAS Activit -b: Functions Deined b Two Points CAS Activit -a: Transormed Quadratic Functions A Chapter Interlea: Functions and Mathematical Models

5 Standard Schedule Pacin Guide Da Section Suested Assinment - Functions: Graphicall, Alebraicall, Numericall, and Verball - Tpes o Functions 9 odd, 0 Dilation and Translation o Function RA, Q Q, 6 - Graphs 7 - Composition o Functions RA, Q Q,,,, 7, 9,, 6 Inverse Functions and Parametric RA, Q Q, odd, 6-7 Equations 7, 9,,,,, 7, Relections, Absolute Values, and Other Transormations RA, Q Q,, 7, Precalculus Journal R R6, T T8-8 Chapter Review and Test C, C, Problem Set - Block Schedule Pacin Guide Da Section Suested Assinment Functions: Graphicall, Alebraicall, - Numericall, and Verball - Tpes o Functions 9 odd, 0, - Dilation and Translation o Function Graphs RA, Q Q, odd - Composition o Functions RA, Q Q, 9 odd, Inverse Functions and Parametric - Equations RA, Q Q,,,, 9, - Inverse Functions and Parametric Equations 7, 9,, 9-6 Relections, Absolute Values, and Other Transormations RA, Q Q, 9 odd,, -7 Precalculus Journal -8 Chapter Review and Test R R6, T T8 6-8 Chapter Review and Test - Shapes o Function Graphs Chapter Interlea B

6 Section - Class Time da PLANNING Homework Assinment Problems Teachin Resources Eploration -: Paper Cup Analsis Technolo Resources Reer to paes v and i or a description o Technolo Resources and a ke to the technolo icons. Eploration -: Paper Cup Analsis Activit: Readin the News TEACHING Important Terms and Concepts Function Epressin mathematical ideas raphicall, alebraicall, numericall, and verball Mathematical model Dependent variable Independent variable Domain Rane Asmptote Etrapolation Interpolation Section Notes This section reviews unctions raphicall, numericall, and alebraicall. It beins with an eploration that shows that the heiht o a stack o cups is a unction o the number o cups in the stack. This is a straihtorward introduction to the concept o a unction. It then presents a raph and equation or the relationship between coee temperature and time. Mathematical Overview GRAPHICALLY ALGEBRAICALLY NUMERICALLY VERBALLY Chapter : In previous courses ou have studied linear unctions, quadratic unctions, eponential unctions, power unctions, and others. In precalculus mathematics ou will learn eneral properties that appl to all tpes o unctions. In particular ou will learn how to transorm a unction so that its raph its real-world data. You will ain this knowlede in our was. The raph at riht is the raph o a quadratic unction. The -variable could represent the heiht o an arrow at various times,, ater its release into the air. For larer time values, the quadratic unction shows that is neative. These values ma or ma not be reasonable in the real world. The equation o the unction is.9 0 This table shows correspondin - and -values that satis the equation o the unction. (s) (m) Discuss domain, rane, and asmptote, and point them out on the raph, as shown in Fiure -b on pae. Emphasize domain and rane throuhout the ear so that students become amiliar with how to determine both. Real-world problems alwas have restrictions, and man o the calculus problems students solve net ear will require them to consider domain and rane restrictions. When the variables in a unction stand or thins in the real world, the unction is bein used as a mathematical model. The coeicients in the equation o the unction.9 0 have a realworld meanin. For eample, the coeicient.9 is a constant that is a result o the ravitational acceleration, 0 is the initial velocit, and relects the initial heiht o the arrow (m) 6 The description o an asmptote on pae suests that a raph never crosses an asmptote. This description is helpul or introducin students to asmptotes, but, when eamined more closel, an asmptote describes end behavior. As a unction approaches ininit in the positive or neative direction, it also approaches its horizontal asmptote. The raph ma cross the asmptote, as in the unction (). (s) Chapter : Functions and Mathematical Models

7 - Objective EXPLORATION -: Paper Cup Analsis Functions: Graphicall, Alebraicall, Numericall, and Verball I ou stack paper cups, the heiht increases as the number o cups increases. There is one and onl one heiht or an iven number o cups, so heiht is called a unction o the number o cups. In this course ou ll reresh our memor about some kinds o unctions ou have studied in previous courses. You ll also learn some new kinds o unctions, and ou ll learn properties o unctions so that ou will be comortable with them in later calculus courses. In this section ou ll see that ou can stud unctions in our was. Work with unctions that are deined raphicall, alebraicall, numericall, or verball. In this eploration, ou ll ind an equation or calculatin the heiht o a stack o paper cups.. Obtain several eral paper cups o the same kind.. Let be the number o cups in a stack, and Measure the heiht o stacks containin, let be the heiht o the stack, measured,,, and just cup. Record the heihts in centimeters. Write an equation or as a to the nearest 0. cm in a cop o this table. unction o. State what kind o cup ou used. 6. What is the name o the kind o unction whose equation ou wrote in Problem? Number Heiht (cm) 7. Show that our equation in Problem ives a heiht close to the measured heiht or a stack o cups. 8. Use our equation to predict the heiht o a stack o cups. Round the answer to decimal place. 9. What are the names o the processes o calculatin a value within the rane o the data, as in Problem 7, and outside the rane o data, as in Problem 8?. Plot the points in the table on raph paper. Show the scale ou are usin on the vertical ais.. On averae, b how much did the stack heiht increase or each cup ou added? Show how ou ot our answer.. How tall would ou epect a -cup stack to be? Show how ou et our answer. Would this be twice as tall as a -cup stack?. A cup manuacturer wants to packae this kind o cup in boes that are cm lon and hold one stack o cups. What is the maimum number o cups the bo could hold? Show how ou ot our answer.. What did ou learn as a result o doin this eploration that ou did not know beore? Eploration Notes Eploration - ma be used as a time-eicient wa to reresh students memories about unctions and their use as mathematical models.. Answers will var.. Answers will var.. Answers will var. The stack heiht should increase b the same amount or each additional cup.. Answers will var. No, the -cup stack would not be twice as tall as a -cup stack.. Answers will var. 6. Linear unction 7. Answers will var. Plu in or in our equation. 8. Answers will var. 9. Within: interpolation; Outside: etrapolation. Answers will var.. Answers will var. Section -: 0. () Section -: Functions: Graphicall, Alebraicall, Numericall, and Verball

8 Dierentiatin Instruction Pass out the list o Chapter vocabular, available at or ELL students to look up and translate in their bilinual dictionaries. Consider askin pairs or roups o students to show a connection between two representations o the same concept or eample, alebraic and raphic or raphic and verbal. Have students do all eplorations in pairs or roups. Enlare Fiure -b and pass it out so students can draw on it while ou discuss it. Eample ma conuse ELL students. Consider speciin a speed and a time beore eneralizin the eample. Challenin Vocabular: asmptote, arbitraril, and instantaneousl. For asmptote, draw eamples o vertical, horizontal, and oblique asmptotes. Use the term arbitraril as ou eplain our drawins. Instantaneousl means in an instant. Make eplicit or students the connection between independent variable and domain, and between dependent variable and rane, as well as the dierence between inteer domains and real number domains. A mnemonic such as that domain and rane and and are both in alphabetical order ma help students remember the relationship between them. (min) ( C) Chapter : I ou pour a cup o coee, it cools more rapidl at irst, then less rapidl, inall approachin room temperature. You can show the relationship between coee temperature and time raphicall. Fiure -a shows the temperature,, as a unction o time,. At 0, the coee has just been poured. The raph shows that as time oes on, the temperature levels o, until it is so close to room temperature, 0 C, that ou can t tell the dierence ( C) Room temperature 0 Fiure -a (min) 0 This raph miht have come rom numerical data, ound b eperiment. It actuall came rom an alebraic equation, 0 70 (0.8). From the equation, ou can ind numerical inormation. I ou enter the equation into our rapher and then use the table eature, ou can ind these temperatures, rounded to 0. C. Functions that are used to make predictions and interpretations about somethin in the real world are called mathematical models. Temperature is the dependent variable because the temperature o the coee depends on the time it has been coolin. Time is the independent variable. You cannot chane time simpl b chanin coee temperature! Alwas plot the independent variable on the horizontal ais and the dependent variable on the vertical ais. The set o values the independent ( C) variable o a unction can have is called the domain. In the coee cup eample, 0 80 the domain is the set o nonneative Rane, 0 90 numbers, or 0. The set o values o the dependent variable correspondin to the domain is called the rane o the 60 0 Asmptote 0 unction. I ou don t drink the coee Room temperature (min) (which would end the domain), the 0 0 rane is the set o temperatures between Domain, 0 0 C and 90 C, includin 90 C but not 0 C, or The horizontal Fiure -b line at 0 C is called an asmptote. The word comes rom the Greek asmptotos, meanin not due to coincide. The raph ets arbitraril close to the asmptote but never touches it. Fiure -b shows the domain, rane, and asmptote. Chapter : Functions and Mathematical Models

9 EXAMPLE SOLUTION Eample shows ou how to describe a unction verball. The time it takes ou to et home rom a ootball ame is related to how ast ou drive. Sketch a reasonable raph showin how this time and speed are related. Give the domain and rane o the unction. It seems reasonable to assume that the time it takes depends on the speed ou drive. So ou must plot time on the vertical ais and speed on the horizontal ais. To see what the raph should look like, consider what happens to the time as the speed varies. Pick a speed and plot a point or the correspondin time (Fiure -c). Then pick a aster speed. Because the time will be shorter, plot a point closer to the horizontal ais (Fiure -d). Time Speed Correspondin time A particular speed Time A aster speed Shorter time Speed Fiure -c Fiure -d For a slower speed, the time will be loner. Plot a point arther rom the horizontal ais (Fiure -e). Finall, connect the points with a smooth curve, because it is possible to drive at an speed within the speed limit. The raph never touches either ais, as Fiure - shows. I the speed were zero, ou would never et home. The lenth o time would be ininite. Also, no matter how ast ou drive, it will alwas take ou some time to et home. You cannot arrive home instantaneousl. Technolo Notes Eploration -: Paper Cup Analsis has students collect data on the heihts o stacks o paper cups and then it lines to their data. Students can do this eploration with Fathom. The enter the data into a case table, make a scatter plot, test conjectured unctions b plottin them, and use tracin to make a prediction. Activit: Readin the News in Teachin Mathematics with Fathom has students collect data on how lon it takes to read a newspaper, make a scatter plot, and it a line to the data in order to make predictions. This activit also provides a straihtorward introduction to Fathom. Students can review linear equations b eperimentin with more activities in Chapter o Teachin Mathematics with Fathom. Allow 0 minutes. Time Loner time A slower speed Speed Section -: Time Asmptotes Never touches either ais Speed Fiure -e Fiure - Domain: 0 speed speed limit Rane: time minimum time at speed limit The problem set will help ou see the relationship between variables in the real world and unctions in the mathematical world. CAS Suestions Consider presentin some ke CAS unctionalit. In particular, demonstrate how to deine and evaluate unctions and solve equations. A ke concept when usin a CAS is to reconize that once deined, a unction can be named, manipulated, evaluated, and solved b reerence to its name alone. When usin a CAS, students need to enter the details o the unction epression onl once, reerrin to its name or the rest o the problem. This is a critical technolo-related aspect to thinkin alebraicall about problem solvin. Section -: Functions: Graphicall, Alebraicall, Numericall, and Verball

10 PROBLEM NOTES Problem requires students to analze and interpret a raph, as well as provide verbal descriptions and eplanations o terminolo. a. 0 m; 7. m; it is below the top o the cli. b. 0. s;.8 s;. s c. m d. There is onl one altitude or an iven time; some altitudes correspond to more than one time. e. Domain: 0.; rane: 0. Problem lists a table o values that students must hand-plot on raph paper. Althouh a best-it line is not ormall deined, students are asked to draw one in Problem a. Students requentl think a best-it line needs to o throuh the irst and last points. This is not necessaril true. This problem introduces the terms etrapolation and interpolation. Help students understand that etrapolation involves usin the pattern in the data to estimate values beond the iven values, whereas interpolation involves estimatin values between iven data values. Caution students that etrapolated results ma not be valid because the pattern ma not eist beond the iven data. a. This raph also shows the answer or part b. V (liters) 0 T ( C) 00 0 Eplorator Problem Set -. Archer Problem: An archer climbs a tree near Jacques Charles invented the hdroen balloon. He the ede o a cli, then shoots an arrow hih participated in the irst into the air. The arrow oes up, then comes manned balloon liht back down, oin over the cli and landin in in 78. the valle, 0 m below the top o the cli. The arrow s heiht,, in meters above the top o the cli depends on the time,, in seconds since the archer released it. Fiure - shows the heiht as a unction o time. (m) Chapter : (s) 6 Fiure - a. What was the approimate heiht o the arrow at s? At s? How do ou eplain the act that the heiht is neative at s? b. At what two times was the arrow m above the round? At what time does the arrow land in the valle below the cli? c. How hih was the archer above the round when she released the arrow? d. Wh can ou sa that heiht is a unction o time? Wh is time not a unction o heiht? e. What is the domain o the unction? What is the correspondin rane?. Gas Temperature and Volume Problem: When ou heat a ied amount o as, it epands, increasin its volume. In the late 700s, French chemist Jacques Charles used numerical measurements o the temperature and volume o a as to ind a quantitative relationship between these two variables. Suppose that these temperatures and volumes had been recorded or a ied amount o oen. b. Answers will var. V (00), V(0), and V(T) 0 when T 7. Absolute zero is about 7 C. c. Etrapolation: V(00) and T such that V(T) 0; interpolation: V(0). d. There is onl one volume or a iven temperature; es, because there is onl one temperature or a iven volume. e. Domain: 7; rane: 0. a. On raph paper, plot V T ( C) V (L) as a unction o T. Choose 0 9. scales that o at least rom T 00 to T 00, and 0. rom V 0 to V. You 0.9 should ind that the points.7 lie almost in a straiht line With a ruler, construct the best-ittin line ou can or 0 8. these points. Etend the line to the let until it crosses the T-ais and to the riht to T 00. b. From our raph, read the approimate volumes at T 00 and T 0. Read the approimate temperature at which V 0. How does this temperature compare with absolute zero, the temperature at which molecular motion stops? c. Findin a value o a variable beond all iven data points is called etrapolation. Etrameans beond, and pol- comes rom pole, or end. Findin a value between two iven data points is called interpolation. Which o the three values in part b did ou ind b etrapolation and which b interpolation? d. Wh can ou sa that volume is a unction o temperature? Is temperature also a unction o volume? Eplain. e. Considerin volume to be a unction o temperature, write the domain and the rane o this unction. Problem provides an equation rom which students create both a table and a raph on a rapher. In this problem, the number o months is the independent variable and the balance is the dependent variable. You miht ask students to eplain wh it would be inappropriate to sa that the number o months depends on the number o dollars. 6 Chapter : Functions and Mathematical Models

11 . Mortae Pament Problem: People who bu houses usuall et a loan to pa or most o the house and make paments on the resultin mortae each month. Suppose ou et a $,000 loan and pa it back at $,07.6 per month with an interest rate o 6% per ear (0.% per month). Your balance, B, in dollars, ater n monthl paments is iven b the alebraic equation B, n (.00 n ) a. Make a table o our balances at the end o each months or the irst ears o the mortae. To save time, use the table eature o our rapher to do this. b. How man months will it take ou to pa o the entire mortae? Show how ou et our answer. c. Plot on our rapher the raph o B as a unction o n rom n 0 until the mortae is paid o. Sketch the raph on our paper. d. True or alse: Ater hal the paments have been made, hal the oriinal balance remains to be paid. Show that our conclusion arees with our raph rom part c. e. Give the domain and rane o this unction. Eplain wh the domain contains onl inteers.. Stoppin Distance Problem: The distance our car takes to stop depends on how ast ou are oin when ou appl the brakes. You ma recall rom driver s education that it takes more than twice the distance to stop our car i ou double our speed. a. Sketch a reasonable raph showin our stoppin distance as a unction o speed. b. What is a reasonable domain or this unction? c. Consult a driver s manual, the Internet, or another reerence source to see what the stoppin distance is or the maimum speed ou stated or the domain in part b. Section -: b. Chanin Tbl to shows that the balance becomes neative at the end o month, so the balance will become 0 durin month. c., d. False 0, d. When police investiate an automobile accident, the estimate the speed the car was oin b measurin the lenth o the skid marks. Which are the considerin to be the independent variable, the speed or the lenth o the skid marks? Indicate how this would be done b drawin arrows on our raph rom part a.. Stove Heatin Element Problem: When ou turn on the heatin element o an electric stove, the temperature increases rapidl at irst, then levels o. Sketch a reasonable raph showin temperature as a unction o time. Show the horizontal asmptote. Indicate on the raph the domain and rane. 6. In mathematics ou learn thins in our was alebraicall, raphicall, numericall, and verball. a. In which o Problems was the unction iven alebraicall? Graphicall? Numericall? Verball? b. In which o Problems did ou o rom verbal to raphical? From alebraic to numerical? From numerical to raphical? From raphical to alebraic? From raphical to numerical? From alebraic to raphical? 7 e. Domain: 0, is an inteer; rane: 0,000. Problem is eective or students who learn to phrase the question correctl usin a CAS. Students can use a CAS to evaluate the values requested in Problem a directl. The ke point or Problem b is to reconize that the balance is zero. A CAS uses this act to solve the balance equation or the pament number at which the balance is zero. Because the unction has alread been deined, it can be raphed b tpin the unction name and chanin the domain variable to. Problem ives a description o a situation rom which students draw a reasonable raph and then practice usin vocabular. Problem requires students to draw a reasonable raph to match a iven description. Additional CAS Problems. Use Boolean operators in Problem to determine whether the balance at the midpoint in time is hiher or lower than hal o the oriinal balance.. In how man months does the account balance in Problem become hal o the oriinal balance? Note: This problem is diicult to solve b hand, but it is a classic situation in which the CAS enables the student to remain ocused on the problem without ettin lost in the alebra required to solve it. See pae 97 or answers to Problems a, 6 and CAS Problems and. Section -: Functions: Graphicall, Alebraicall, Numericall, and Verball 7

12 Section - Class Time da PLANNING Homework Assinment Problems 9 odd, 0 Teachin Resources Eploration -a: Names o Functions Eploration -b: Restricted Domains and Boolean Variables Supplementar Problems Technolo Resources Eploration -a: Names o Functions Eploration -b: Restricted Domains and Boolean Variables CAS Activit -a: Findin Polnomial Equations CAS Activit -b: Functions Deined b Two Points TEACHING Important Terms and Concepts Ordered pair Relation Function -intercept -intercept () terminolo Arument o a unction Name o a unction Constant Variable Polnomial unction Quadratic unction Linear unction Direct variation unction Power unction Eponential unction Inverse variation unction Rational alebraic unction Boolean variable Restricted domain Vertical line test Section Notes 8 Chapter : - Objective Tpes o Functions In the previous section ou learned that ou can describe unctions alebraicall, numericall, raphicall, or verball. A unction deined b an alebraic equation oten has a descriptive name. For instance, the unction is called quadratic, rom the latin word quadratum, meanin square, because the unction is a polnomial whose hihest power o is squared and quadranle is one term or a square. In this section ou will reresh our memor about verbal names or alebraicall deined unctions and see what their raphs look like. Make connections amon the alebraic equation or a unction, its name, and its raph. Deinition o Function I ou plot the unction, ou et a raph that rises and then alls, as shown in Fiure -a. For an -value ou pick, there is onl one -value. This is not the case or all raphs. For eample, in Fiure -b there are places where the raph has more than one -value or the same -value. Althouh the two variables are related, the relation is not a unction. A unction No two -values or the same -value -intercept -intercepts Fiure -a Fiure -b Each point on a raph corresponds to an ordered pair o numbers, (, ). A relation is an set o ordered pairs. A unction is a set o ordered pairs or which each value o the independent variable (oten ) in the domain has onl one correspondin value o the dependent variable (oten ) in the rane. So Fiures -a and -b are both raphs o relations, but onl Fiure -a is the raph o a unction. Th e -intercept o a unction is the value o when 0. It ives the place where the raph crosses the -ais (Fiure -a). An -intercept is a value o or which 0. Functions can have more than one -intercept. () Terminolo Bein the section b reviewin the deinition o unction. Make sure students understand that or a relation to be a unction, it must be true that or each value in the domain there is onl one correspondin value in the rane. In other words, the correspondence rom the domain to the rane must be unique. It is possible or the same rane value to correspond to more than one domain Not a unction More than one -value or the same -value You should recall () notation rom previous courses. It is used or, the dependent variable o a unction. With it, ou show what value ou substitute value. To illustrate this, ou miht discuss the unction. For this unction, ever nonzero value in the rane corresponds to two values in the domain. For eample, the rane value 9 corresponds to the domain values and. Students should be amiliar with unction notation rom previous courses. You ma want to discuss briel this notation, emphasizin that the parentheses in () mean substitution, not multiplication. 8 Chapter : Functions and Mathematical Models

13 () Fiure -c () Fiure -d or, the independent variable. For instance, to substitute or in the quadratic unction (), ou would write () () 7 The smbol () is pronounced o or sometimes at. You must reconize that the parentheses mean substitution and not multiplication. This notation is also useul i ou are workin with more than one unction o the same independent variable. For instance, the heiht and velocit o a allin object both depend on time, t, so ou could write the equations o the two unctions this wa: h(t).9t t 70 (or the heiht) v(t) 9.8t (or the velocit) In (), the variable or an value substituted or is called the arument o the unction. It is important to distinuish between and (). The smbol is the name o the unction. The smbol () is the -value o the unction. For instance, i is the square root unction, then () and (9) 9. Note that the releive aiom,, requires that ou substitute the same number or everwhere it appears in an epression or equation. It would be improper ormat to write () 9 i ou have substituted 9 or. Names o Functions Functions are named or the operation perormed on the independent variable. Here are some tpes o unctions ou ma recall rom previous courses, alon with their tpical raphs. In these eamples, the letters a, b, c, m, and n stand or constants. The smbols and () stand or variables, or the independent variable and () or the dependent variable. Polnomial unction, Fiure -c General equation: () a n n a n n... a a 0, where n is a nonneative inteer Verball: () is a polnomial unction o. (I n, is a cubic unction. I n, is a quartic unction.) Features: The raph crosses the -ais up to n times and has up to n vertices (points where the unction chanes direction). The domain is all real numbers. Quadratic unction, Fiure -d (a special case o a polnomial unction) General equation: () a b c, a 0 Verball: () varies quadraticall with, or () is a quadratic unction o. Features: The raph chanes direction at its one verte. The domain is all real numbers. The Names o Functions subsection presents the raphs and equations o eiht tpes o unctions. To help students review these unctions, ou miht assin Eploration -a. Some students have diicult distinuishin between power unctions and eponential unctions. The location o the variable is the ke to the dierence. Power unctions have a variable base, whereas eponential unctions have a variable eponent. The conusion arises Section -: 9 because some students do not distinuish between eponent and power. Actuall, in the epression, or instance, the eponent is onl the. The power is the entire epression,. Thus, has a variable as its eponent, while is a power o. A direct variation unction is a special case o a linear unction, power unction, and polnomial unction. Speciicall, it is a linear unction with -intercept 0, a power unction with an eponent o, and a polnomial unction with onl a linear term. The raph o a direct variation unction is a straiht line throuh the oriin. An inverse variation unction is a power unction with a neative eponent. The raph o an inverse variation unction has asmptotes at both aes. Students ma be amiliar with direct and inverse variation unctions rom their science classes. To help students transer learnin and increase their understandin o both precalculus and science, ou miht ask them or eamples o equations used in science that are direct or inverse variations. For eample, Newton s second law o motion, F ma, is a direct variation i the mass, m, or the acceleration, a, is constant. I the orce, F, is constant, the equation is an inverse variation. Eample on pae illustrates how to plot a unction with a restricted domain usin a rapher. This kind o problem prepares students or the piecewise unctions (unctions composed o two or more unctions) the will encounter in uture courses. Students ma need help understandin the idea o a Boolean variable. A Boolean variable is represented b a condition, not a letter like the variables students are amiliar with. A Boolean variable has onl two possible values: It is equal to i the condition is true, and it is equal to 0 i the condition is alse. In the iven eamples, onl part o the unction is divided b the Boolean variable because it is easier to enter into a rapher. That is, the unctions are written in the orm () 6/ ( and ) rather than () ( 6)/( and ). You miht want to ask students wh the two orms ield the same raph. The are both undeined when the Boolean variable is zero. Section -: Tpes o Functions 9

14 Section Notes (continued) Note: I our students use TI-Nspire raphers, the can plot piecewise unctions b enterin them directl into their raphers usin a template. See the instruction manual or the TI-Nspire or more inormation. Eample also asks students to ive the rane o the unction. Students sometimes mistakenl tr to determine the rane o a unction b substitutin the endpoints o the domain. Point out that, in this eample, the endpoints o the rane do not correspond to the endpoints o the domain. Dierentiatin Instruction For Romance lanuae speakers, a mnemonic or quadratic is the word or square (cuadrado, quadrat, etc.). Intercept and intersect ma be conusin to students because the sound alike and have related meanins. Also, some lanuaes do not make the distinction between eponent and power that is iven in the tet. Make sure students understand how to sa () and what it means. Tie this in to the term arument with eamples such as ( ), ( ) and (). Note that raphers are not allowed in man countries or students this ae. In Problems 8, help ELL students new to the U.S. (or to our school) learn to use their rapher, and consider assinin them a partner who is a stron rapher user. Problems 9 and 0 introduce important concepts that are lanuaeheav. Man ELL students will need help with the vocabular. Problem ma be diicult or ELL students. Consider lettin them answer in their primar lanuae. () () Fiure -e Fiure - a () Fiure -h Chapter : Eploration Notes Linear unction, Fiure -e (another special case o a polnomial unction) General equation: () a b (or () m b) Verball: () varies linearl with, or () is a linear unction o. Features: The straiht-line raph, (), chanes at a constant rate as chanes. The domain is all real numbers. Direct variation unction, Fiure - (a special case o a linear, power, or polnomial unction) General equation: () a or () m 0, or () a Verball: () varies directl with, or () is directl proportional to. Features: The straiht-line raph oes throuh the oriin. The domain is all real numbers. However, or most real-world applications, ou will use the domain 0 (as shown). Power unction, Fiure - (a polnomial unction i b is a nonneative inteer) General equation: () a b (a variable with a constant eponent), a 0, b 0 Verball: () varies directl with the bth power o, or () is directl proportional to the bth power o. Features: The domain depends on the value o b. For positive inteer values o b, the domain is all real numbers; or neative inteer values o b, the domain is 0. In most real-world applications, the domain is 0 i b 0 and 0 i b 0. () I b is positive Eploration -a provides a short summar review o seven o the eiht unctions covered in the tet (the direct variation unction is not included). Students workin in roups with the aid o raphers can complete the seven questions in 0 minutes. It s probabl best to summarize the eploration and answer questions on it beore presentin the eamples rom the tetbook. Fiure - () Eponential unction, Fiure -h General equation: () a b (a constant with a variable eponent), a 0, b 0, b I b is neative Verball: () varies eponentiall with, or () is an eponential unction o. Features: The raph crosses the -ais at (0) a and has the -ais as an asmptote. How about i 0 < b <? For eample, = _ 0. Eploration -b lets students see a restricted domain in a real-world contet. The ravitational attraction or an object above Earth s surace is inversel proportional to the square o the object s distance rom the center o Earth. Below the surace, the attraction is directl proportional to the distance. Students who have raphers usin Boolean variables will ind the eploration helpul. This ollow-up eploration takes about minutes. Chapter : Functions and Mathematical Models

15 () I n is odd Fiure -i () Removable discontinuit Fiure -j Asmptotes Inverse variation unction, Fiure -i (a special case o a power unction) General equation: () a or () a or () a n or () a n, a 0, n 0 Verball: () varies inversel with (or with the nth power o ). Alternativel, () is inversel proportional to (or to the nth power o ). Features: Both o the aes are asmptotes. The domain depends on the value o n. For positive inteer values o n, the domain is 0. For most real-world applications, the domain is 0. Rational alebraic unction, Fiure -j General equation: () n (), where n and d are polnomial unctions d () Verball: () is a rational unction o. Features: A rational unction has a discontinuit (asmptote or missin point) where the denominator is zero; it ma have horizontal or other asmptotes. Restricted Domains and Boolean Variables Suppose that ou want to plot a raph usin onl part o our rapher s window. For instance, let the heiht o a rowin child between aes and be iven b 6, where is ae in ears and is heiht in inches. The domain here is. Some raphers allow ou to enter a restricted domain directl. Other raphers require ou to use Boolean variables. A Boolean variable, named or Geore Boole, an Irish loician and mathematician (8 86), equals i a iven condition is true and 0 i that condition is alse. For instance, the compound statement ( and ) equals i 7 (which is between and ) and equals 0 i or (neither o which is between and ). To plot a raph in a restricted domain usin Boolean variables, divide an term o the equation b the appropriate Boolean variable. For the equation above, enter ( ) 6 / ( and ) What i n is even? For eample, = I is between and, inclusive, the 6 in 6 is divided b, which leaves it unchaned. I is not between and, inclusive, the 6 in 6 is divided b 0 and the rapher plots nothin. CAS Activit -b: Functions Deined b Two Points in the Instructor s Resource Book has students use Solve and Factor commands on a CAS to ind linear, power, and eponential unctions determined b two points. Allow 0 minutes. CAS Suestions Introduce students to the CAS unctionalit that solves sstems o equations at this time. Students can ind intercepts b settin all other variables equal to zero. Usin this idea, -intercepts can be computed with commands like Solve(() = 0, ) or Solve( = (), ) = 0. The smbol on the TI-Nspire CAS substitutes an equation that ollows the smbol into the precedin command. The latter command shows that the student is solvin the oriinal unction or when 0. The command can be used to restrict the domain o a unction in its equation or on its raph. The iure shows the same raph as Eample with the domain restriction entered alonside the unction deinition. Technolo Notes Eploration -a: Names o Functions in the Instructor s Resource Book can be done with Sketchpad. Eploration -b: Restricted Domains and Boolean Variables in the Instructor s Resource Book can be accomplished usin Fathom, but onl i students emplo nested i-then statements rather than Boolean variables. Section -: CAS Activit -a: Findin Polnomial Equations in the Instructor s Resource Book, has students use a CAS to ind equations or polnomials usin sstems o equations. Students use a Solve command to eplore how man coordinates are needed to ind an equation or an nth-deree polnomial. Allow 0 minutes. The command can also be used to deine unctions alebraicall. Once deined, a CAS evaluates values within the domain values, but returns an undeined statement when the input value is outside the domain, even i the input value is deined when the domain is unrestricted. Section -: Tpes o Functions

16 CAS Suestions (continued) A CAS is particularl powerul when manipulatin unctions. The student can ocus on the mathematics, reconizin the orms o unctions and their parameters, while the CAS perorms the calculations reardless o how complicated a solution miht be. For eample, it takes three parameters to deine the eneral equation o a quadratic unction, so three points and a Solve command will suice or computin the equation o an quadratic. The iures show two dierent approaches to computin a quadratic equation containin the points (, ), (, 7), and (, ). The irst approach shows the mathematics more clearl, while the second approach is more sophisticated, allowin the CAS to do the work with ewer obvious inputs. EXAMPLE SOLUTION () Rane Domain Fiure -k EXAMPLE Plot the raph o () in the domain 0. What kind o unction is this? Give the rane. Find a pair o real-world variables that could have a relationship described b a raph o this shape. Enter the equation with restricted domain into our rapher directl. Or, to use Boolean variables, enter () / ( 0 and ) The raph in Fiure -k shows the restricted domain. The unction is quadratic because () equals a second-deree polnomial in. The rane is () 9.. You can ind this interval b tracin to the let endpoint o the raph where (0) and to the hih point where (.) 9.. (At the riht endpoint, () 7, which is between and 9..) The unction could represent the relationship between somethin that rises or a while and then alls, such as a punted ootball s heiht as a unction o time or (i () is multiplied b ) the rade ou could et on a test as a unction o the number o hours ou stud or it. (The rade could be lower or loner times i ou sta up too late and thus are sleep durin the test.) DEFINITION: Boolean Variables A Boolean variable is a variable that has a iven condition attached to it. I the condition is true, the variable equals. I the condition is alse, the variable equals 0. As children row older, their heiht and weiht are related. Sketch a reasonable raph to show this relation and then describe it. Identi what kind o unction has a raph like the one ou drew. SOLUTION Weiht depends on heiht, so weiht is on the vertical ais, as shown on the raph in Fiure -l. The raph curves upward because doublin the heiht more than doubles the weiht. Etendin the raph sends it throuh the oriin, but the domain starts beond the oriin at a value reater than zero, because a person never has zero heiht or weiht. The raph stops at the person s adult heiht and weiht. A power unction has a raph like this. Weiht Birth heiht Heiht Fiure -l Adult heiht PROBLEM NOTES Supplementar Problems or this section are available at keonline. a. Chapter : 0 a. Problems 8 are similar to Eample. 8 b. () b. 0 ().8 c. Linear c. Power d. Answers will var. d. Answers will var. Chapter : Functions and Mathematical Models

17 Problem Set - For Problems, a. Plot the raph on our rapher usin the iven domain. Sketch the result on our paper. b. Give the rane o the unction.. Eponential unction (). with the domain. Eponential unction () with the domain c. Name the kind o unction.. Inverse-square variation unction () with d. Describe a pair o real-world variables that the domain 0 could be related b a raph o this shape. 6. Direct variation unction () with the. () domain: 0 domain 0. () 0. domain: 0 7. Rational unction. () ( )( ) domain: 0 with the domain 6,,. h () 0. 6 domain: For Problems 8, a. Plot the raph usin a window set to show the entire raph, when possible. Sketch the result. b. Give the -intercept and an -intercepts and the locations o an asmptotes. c. Give the rane.. Quadratic (polnomial) unction () with the domain 0 6. Quadratic (polnomial) unction () 6 0 with the domain Cubic (polnomial) unction () 7 with the domain 7 8. Quartic (polnomial) unction () 8 6 with the domain 9. Power unction () / with the domain 0 8. Power unction () 0.. with the domain 0 9. Linear unction () 0.7 with the domain. Linear unction () 6 with the domain a Rational unction with the domain 6, For Problems 9 8, name the tpe o unction that has the raph shown a. 8. Section -: 0 0 The intercepts or Problems 8 can be computed quickl usin a Solve command on a CAS. a. b. -intercept at ; no -intercepts; no asmptotes c a. 6b. -intercept at 0; no -intercepts; no asmptotes 6c. 6 7a. 6 0 (, 6) 0 (, ) 0 0 7b. -intercept at ; -intercepts at, and 6; no asmptotes 7c Problems 9 8 require students to name tpes o unctions rom their raphs. 9. Eponential 0. Linear. Linear. Eponential. Quadratic. Cubic. Power 6. Inverse variation 7. Rational 8. Direct variation (.6, 0.7) 8 b. (). b h() 6. c. Inverse variation d. Answers will var. c. Eponential d. Answers will var. See paes or answers to Problems 8 8. Section -: Tpes o Functions

18 Problem Notes (continued) Problems 9 are similar to Eample. Problems 8 require students to identi which relations are unctions.. Function; no -value has more than one correspondin -value.. Not a unction; some -values on the let have two correspondin -values.. Not a unction; there is at least one -value with more than one correspondin -value. 6. Function; no -value has more than one correspondin -value. 7. Not a unction; there is at least one -value with more than one correspondin -value. 8. Not a unction; the -value in the middle has ininitel man correspondin -values. Problem 9 presents the vertical line test, which all students should know and use. Be sure to discuss this problem when ou review the homework. To reinorce the concept, question students periodicall in uture assinments about whether raphs pass the vertical line test. 9a. A vertical line throuh a iven -value crosses the raph at the -values that correspond to that -value. So, i a vertical line crosses the raph more than once, it means that that -value has more than one -value. 9b. In Problem, an vertical line crosses the raph at most once, but in Problem, an vertical line between the two endpoints crosses the raph twice. Problems 0 and require students to demonstrate understandin o some o the terminolo associated with unctions. Problem is a research problem that requires students to write. See pae 97 or answers to Problems 9, 0 and CAS Problems. For Problems 9, a. Sketch a reasonable raph showin how the variables are related. b. Identi the tpe o unction it could be (quadratic, power, eponential, and so on). 9. The weiht and lenth o a do. 0. The temperature o a cup o coee and the time since the coee was poured.. The purchase price o a house in a particular neihborhood as a unction o the number o square eet o loor space in the house, includin a ied amount or the lot on which the house was built.. The heiht o a punted ootball as a unction o the number o seconds since it was kicked. For Problems 8, tell whether the relation raphed is a unction. Eplain how ou made our decision Vertical Line Test Problem: There is a raphical wa to tell whether a relation is a unction. It is called the vertical line test. Chapter : PROPERTY: The Vertical Line Test Additional CAS Problems I an vertical line cuts the raph o a relation in more than one place, then the relation is not a unction.. Determine an equation or a quadratic unction whose raph contains the points (, ), (0.,.6), and (, 77).. What is the -intercept o the th-deree polnomial that contains the points (, ), (, 0), (, 6), (, 6), and (, 90)? Comment on the results. Fiure -m illustrates the test. Not a unction A unction A vertical line cuts more than once. No vertical line cuts more than once. Fiure -m a. Based on the deinition o unction, eplain how the vertical line test distinuishes between relations that are unctions and relations that are not unctions. b. Sketch the raphs in Problems and. On our sketch, show how the vertical line test tells ou that the relation in Problem is a unction but the relation in Problem is not. 0. Eplain wh a unction can have more than one -intercept but onl one -intercept.. What is the arument o the unction ( )?. Research Problem: Look up Geore Boole on the Internet or in another reerence source. Describe several o Boole s accomplishments that ou discover. Include our source. Note: These points deine a cubic polnomial, which the CAS reveals b ivin a leadin coeicient o zero. The -intercept is automaticall seen as the constant term in the CAS results. The students don t need to know that this is not a proper quartic polnomial.. How man -intercepts does the raph o the polnomial in CAS Problem have? Chapter : Functions and Mathematical Models

19 - Objective Dilation and Translation o Function Graphs Each o these two raphs shows the unit semicircle and a transormation o it. The let raph shows the semicircle dilated (maniied) b a actor o in the -direction and b a actor o in the -direction. The riht raph shows the unit semicircle translated b units in the -direction and b units in the -direction. Pre-imae unction Imae unction Fiure -a Pre-imae unction Section -: Imae unction h The transormed unctions, and h, in Fiure -a are called imaes o the unction. The oriinal unction,, is called the pre-imae. In this section ou will learn how to transorm the equation o a unction so that its raph will be dilated and translated b iven amounts in the - and -directions. Transorm a iven pre-imae unction so that the result is a raph o the imae unction that has been dilated b iven actors and translated b iven amounts. Dilations To et the vertical dilation in the let raph o Fiure -a, multipl each -coordinate b. Fiure -b shows the imae, (). Fiure -b () () Section - Class Time das PLANNING Homework Assinment Da : Readin Analsis (alwas assin these questions), Q Q (alwas assin these), Problems 6 Da : Problems 7 Teachin Resources Eploration -: Transormations rom Graphs Eploration -a: Translations and Dilations, Numericall Eploration -b: Translations and Dilations, Alebraicall Eploration -c: Transormation Review Blackline Masters Problems 0 Test, Sections - to -, Forms A and B Technolo Resources Dilation Translation Presentation Sketch: Translation Present.sp Presentation Sketch: Dilation Present.sp Activit: Translation o Functions Activit: Dilation o Functions Activit: Eplorin Translations and Dilations Activit: Function Transormations CAS Activit -a: Transormed Quadratic Functions Section -: Dilation and Translation o Function Graphs

20 TEACHING Important Terms and Concepts Dilation Translation Transormation Inside transormation Outside transormation Imae Pre-imae Section Notes It is recommended that ou spend two das on this section. Bein the irst da o instruction b assinin Eploration -a as a small-roup activit. Then cover the material throuh Eample. On the second da, start b assinin Eploration -. Then cover the remainin material in the section. Dependin on the speciic rapher our students use, ou ma want to discuss the idea o usin a riendl rapher window. A riendl window is one that ives nice coordinate values when a unction is traced. When ou trace a unction, the cursor moves one piel at a time. Settin the window so that Xma Xmin is a multiple o the number o piels in the horizontal direction ensures that the -coordinates displaed when the unction is traced will be nice numbers. For eample, the TI-8/8 Plus calculators displa 9 piels in the horizontal direction. I Xmin 0 and Xma 9 or i Xmin 7 and Xma 7, the -coordinate will chane b with each trace step. I Xmin 9. and Xma 9., the -coordinate will chane b 0. with each trace step. I the coeicients in a unction are rational, ou can also set Ymin and Yma so that nice -coordinates are displaed. The TI-8/8 Plus displas 6 piels in the vertical direction. So, or eample, usin the settins Ymin 0 and Yma 6 or Ymin 6. and Yma 6. will ive nice -coordinates when the 6 Chapter : Functions and Mathematical Models EXAMPLE () Fiure -d SOLUTION () 6 Chapter : The horizontal dilation is trickier. Each value o the arument must be times what it was in the pre-imae to enerate the same -values. Substitute v or the arument o. (v) Let v represent the oriinal -values. v Th e -values o the dilated imae must be times the -values o the pre-imae. v Solve or v. R e p l a c e v with _ or the arument to obtain the equation o the dilated imae. Fiure -c shows the raph o the imae, _ () Fiure -c ( ) Puttin the two transormations toether ives the equation or () shown in Fiure -a. () unction is traced. (Note: The -coordinate is calculated b evaluatin the unction or the -coordinate. So, i the coeicients in a unction are irrational, the -coordinates will not be nice, even in a riendl window.) The documentation or each rapher will eplain more about riendl windows. It is important or students to be amiliar with both orms o the dilation rule, _ a () _ b and () a _ b. The orm () a _ b is the more common, The equation o the pre-imae unction in Fiure -a is (). Conirm on our rapher that () _ is the transormed imae unction a. B direct substitution into the equation b. B usin the rapher s built-in variables eature () a. (/) Substitute / as the arument o, and multipl the entire epression b. Enter: () () (/) The raph in Fiure -d shows a dilation b in the -direction and b in the -direction. Use the rid-on eature to make the rid points appear. Use equal scales on the two aes so the raphs have the correct proportions. b. Enter: () (/) is the unction name in this ormat, not the unction value. This raph is the same as the raph o () in Fiure -d. and it is the orm used when a unction is entered into a rapher. However, the orm _ a () _ b allows students to reconize that the same thin happens to as to. That is, both and are divided b their respective scale actors. (To make this clearer, ou ma want to write the rule as _ a _ b rather than _ a () _ b ). Similarl, the translation rule can be written as h() c ( d) or h() c ( d). The second orm

21 Note that the transormation in _ is applied to the arument o unction, inside the parentheses. The transormation in () is applied outside the parentheses, to the value o the unction. For this reason the transormations are iven the names inside transormation and outside transormation, respectivel. An inside transormation aects the raph in the horizontal direction, and an outside transormation aects the raph in the vertical direction. You ma ask, Wh do ou multipl b the -dilation and divide b the -dilation? You can see the reason b substitutin or () and dividin both sides o the equation b : Divide both sides b or multipl b _. You actuall divide b both dilation actors, b the -dilation and b the -dilation. Translations The translations in Fiure -a that transorm () to h() are shown aain in Fiure -e. To iure out what translation has been done, ask oursel, To where did the point at the oriin move? As ou can see, the center o the semicircle, initiall at the oriin, has moved to the point (, ). So there is a horizontal translation o units and a vertical translation o units. Pre-imae, unction Fiure -e Imae, unction h To et a vertical translation o units, add to each -value: () To et a horizontal translation o units, note that what was happenin at 0 in unction has to be happenin at in unction h. Aain, substitutin v or the arument o ives h() (v) v v h() ( ) Substitute as the arument o. Section -: illustrates that the vertical translation is subtracted rom just as the horizontal translation is subtracted rom. You want to avoid students comin to the alse conclusion that the two variables are treated in dierent was. Point out that i a unction is onl dilated, the -dilation is the number ou can substitute or to make the arument equal to. I the unction is onl translated, the -translation is the number ou can 7 substitute or to make the arument equal to zero. Stress that multiplin or dividin variable can lead to dilations and that addin to or subtractin rom variable leads to translations. Encourain students to describe transormations in words (such as Dilate the unction b in the -direction ) helps them move rom the raph to an equation and vice versa. Should ou venture into doin more complicated transormations involvin combinations o relections, dilations, and translations, caution students to be mindul o the order o operations: The must appl relections and dilations irst and then translations. Students will have siniicant additional eposure to transormations o sinusoidal unctions in Section 6-. I our students use TI-Nspire raphers, consider deinin a unction in a raphical window and usin sliders to demonstrate eneral dilations and translations. Dierentiatin Instruction Help ELL students understand dilation b usin epand and shrink. Have students discuss paes 8 and 9 in pairs. Have them raph () and (/) i () on white boards, and show ou their raphs. Have them describe the vertical and horizontal dilations or each unction, and then raph a composition o the two unctions. Make sure ELL students understand the rapher s abilit to store and use variables. You can do this b circulatin, usin the overhead, or b havin them et help rom their partner. For Problems 9 8, instruct students to answer in complete sentences. When doin Eploration -, oral reportin will help students veri their understandin. Section -: Dilation and Translation o Function Graphs 7

22 Eploration Notes For notes on Eploration - see pae 0. Eploration -a introduces vertical and horizontal translations and dilations b havin students numericall determine imae raphs or a iven pre-imae unction. This activit works well as a small-roup discover section at the beinnin o the irst da o instruction. Allow students 0 minutes to complete the activit. Eploration -b and Eploration -c can be used either as reviews or as quizzes ater students complete Section -. Eploration -b takes about 0 minutes and Eploration -c takes about minutes. Technolo Notes Problem asks students to eplore two Dnamic Precalculus Eplorations at The Translation eploration allows students to manipulate sliders to observe how a unction plot is translated both alebraicall and eometricall; the Dilation eploration allows them to observe how it is dilated. Presentation Sketch: Translation Present.sp at uses sliders to translate a unction plot both eometricall and alebraicall. This sketch is related to the activit Translation o Functions. CAS Activit -a: Transormed Quadratic Functions in the Instructor s Resource Book has students use a CAS to ind equations or transormed quadratic unctions in standard, actored, and verte orms. Allow 0 minutes. EXAMPLE SOLUTION () Fiure - () The equation o the pre-imae unction in Fiure -e is (). Conirm on our rapher that () ( ) is the transormed imae unction b a. Direct substitution into the equation b. Usin the rapher s built-in variables eature () a. ( ) Substitute or the arument. Add to the epression. Enter: () () ( ) The raph in Fiure - shows an -translation o units and a -translation o units. b. Enter: () ( ) The raph is the same as that or () in Fiure -. Aain ou ma ask, Wh do ou subtract an -translation and add a -translation? The answer aain lies in associatin the -translation with the -variable. You actuall subtract both translations: ( ) ( ) Subtract rom both sides. The reason or writin the transormed equation with b itsel is to make it easier to calculate the dependent variable, either b pencil and paper or on our rapher. This bo summarizes the dilations and translations o a unction and its raph. PROPERTY: Dilations and Translations The unction iven b () b or, equivalentl, () a a b represents a dilation b a actor o a in the -direction and b a actor o b in the -direction. The unction h iven b 8 Chapter : Functions and Mathematical Models h() c ( d) or, equivalentl, h() c ( d) represents a translation b c units in the -direction and b d units in the -direction. Note: I the unction is onl dilated, the -dilation is the number ou can substitute or to make the arument equal. I the unction is onl translated, the -translation is the number to substitute or to make the arument equal zero. 8 Chapter : Functions and Mathematical Models

23 EXAMPLE SOLUTION The three raphs in Fiure - show three dierent transormations o the pre-imae raph to imae raphs (). Eplain verball what transormations were done. Write an equation or () in terms o the unction. Fiure - Let raph: Vertical dilation b a actor o Equation: () () Note: Each point on the raph o is times as ar rom the -ais as the correspondin point on the raph o. Note that the vertical dilation moved points above the -ais arther up and moved points below the -ais arther down. Middle raph: Vertical translation b 6 units Equation: () 6 () Note: The vertical dilation moved all points on the raph o up b the same amount, 6 units. Also note that the act that () is three times () is purel coincidental and is not true at other values o. Riht raph: Horizontal dilation b a actor o and vertical translation b 7 units Equation: () 7 Note: Each point on the raph o is twice as ar rom the -ais as the correspondin point on the raph o. The horizontal dilation moved points to the riht o the -ais arther to the riht and moved points to the let o the -ais arther to the let. In this eploration, iven a pre-imae and an imae raph, ou ll identi the transormation. Section -: 9 Presentation Sketch: Dilation Present.sp at uses sliders to translate a unction plot both eometricall and alebraicall. This sketch is related to the activit Dilation o Functions. Activit: Translation o Functions in the Instructor s Resource Book ives students an opportunit to translate points and unction plots both eometricall and alebraicall. It takes minutes, dependin on students amiliarit with Sketchpad. Activit: Dilation o Functions in the Instructor s Resource Book asks students to eplore stretchin and compressin unctions both horizontall and verticall, both alebraicall and eometricall. Allow 0 0 minutes. Activit: Eplorin Translations and Dilations in the Instructor s Resource Book provides hands-on eperience with translations and dilations o the parent quadratic unction. Allow 0 0 minutes. Activit: Function Transormations in Teachin Mathematics with Fathom leads students who are airl new to Fathom throuh a stud o translations, dilations, and relections o the parent quadratic unction. It will take 0 0 minutes, dependin on student eperience. Optionall, ou can use a prepared document. Section -: Dilation and Translation o Function Graphs 9

24 CAS Suestions Consider the equation. The ive, the subtraction, and the eponent can all be seen as operations on the variable. Once this equation is deined as a unction (e.., () ), operations can be perormed on the unction, (). This is particularl helpul when eplorin transormations o unctions. The raph o ( ) is a transormation o the unction oriinall deined b (). EXPLORATION -: Transormations rom Graphs For Problems 6, identi the transormation. o (dotted) to et (solid). Describe the transormation verball, and ive an equation or ()... The unction manipulations shown in Eample are easil accomplished with a CAS, but sometimes the alebraic results are in unepected orms. A Boolean operator can be used to veri the alebraic equivalence between the epected result and CAS result. Alebraic equivalence is a ke motivation behind mathematical smbol manipulation, and it is central to CAS mathematics Graphs coincide. Graphs coincide. 7. What did ou learn as a result o doin this eploration that ou did not know beore? Additional Eploration Notes Eploration - presents si raphs that illustrate various transormations and combinations o transormations. Problem is an absolute value transormation, () (), and Problem 6 is a horizontal dilation b a actor o. These two problems preview Section -6. You miht want to use this eploration to bein the second da o instruction. Allow small roups o students 0 minutes to complete the activit. 0 Chapter : Functions and Mathematical Models. Vertical translation b 6; () () 6. Horizontal translation b ; () ( ). Vertical dilation b ; () (). Horizontal dilation b ; (). Relection across the -ais o that part o the raph that is below the -ais; () () 6. Relection across the -ais; () () 7. Answers will var. 0 Chapter : Functions and Mathematical Models

25 Problem Set - Readin Analsis From now on most problem sets will bein with an assinment that requires ou to spend ten minutes or so readin the section and answerin some questions to see how well ou understand what ou have read. This will help develop our abilit to read a tetbook, a ver important skill to have in collee. It should also make workin the problems in the problem set easier. From what ou have read in this section, what do ou consider to be the main idea? What is the major dierence on the imae raph between a translation and a dilation, and what operation causes each transormation? How can ou tell whether a translation or a dilation will be in the -direction or the -direction? min Quick Review From now on there will be ten short problems at the beinnin o most problem sets. Some o the problems are intended or review o skills rom previous sections or chapters. Others are intended to test our eneral knowlede. Speed is the ke here, not detailed work. Tr to do all ten problems in less than ive minutes. Q. 7 is a particular eample o a? unction. Q. Write the eneral equation o a power unction. Q. Write the eneral equation o an eponential unction. Q. Calculate the product: ( 7)( 8) Q. Epand: ( ) Q6. Sketch the raph o a relation that is not a unction. Q7. Sketch the raph o _. Q8. Sketch an isosceles trianle. Q9. Find 0% o 000. Q. Quadratic Q. a b, a 0, b 0 Q. a b, a 0, b 0, b Q. 6 Q. 9 0 Q6. Q. Which one o these is not the equation o a unction? A. B. () C. ( ) D. E. / For Problems 6, let () 9. a. Write the equation or ( ) in terms o. b. Plot the raphs o and on the same screen. Use a window with inteers rom about to as rid points. Use the same scale on both aes. Sketch the result. c. Describe how () was transormed to et ( ), includin whether the transormation was an inside or an outside transormation.. ( ) (). () (). ( ) ( ). ( ). ( ) 6. ( ) ( ) For Problems 7, Section -: a. Describe how the pre-imae unction (dashed) was transormed to et the raph o the imae unction (solid). b. Write an equation or ( ) in terms o unction. Q7. Q8. 6 PROBLEM NOTES The Readin Analsis questions bein in Sec tion -. These questions are desined to help students learn to read a mathematics tetbook. Students need to develop the skills to read dense material slowl and careull. Encourae them to read with a pencil in hand and a calculator nearb. Problems Q Q are short problems that do not require detailed work. Some students like the opportunit to solve problems mentall instead o showin all the steps on paper. You ma ind that students are more willin to show detailed work on other problems i ou allow them to show minimal work on the dail Quick Review problems. Problems 6 allow students to see the eects o si dierent transormations on the same unction. In Problems 6, the new raphs are obtained b deinin () as iven. In a calculator screen, alebraic versions o each o the iven unctions can also be obtained. Encourae students to interpret and predict the outcomes o the transormations beore allowin technolo to create the imae raphs. Problems 7 require students to identi transormations based on raphs o the pre-imae and imae unctions. Because students are not iven eplicit ormulas or the pre-imae, the must write the equation or the imae in terms o. 7a. -translation b 7 7b. () 7 () 8a. -translation b 8b. () ( ) Q Q. D See pae 97 or answers to Problems 6. Section -: Dilation and Translation o Function Graphs

26 Problem Notes (continued) 9a. -dilation b 9b. () a. -dilation b b. () () a. -translation b 6, -dilation b b. () ( 6) a. -dilation b, -translation b b. (). No. The domain o () is, but the domain o the raph is.. No. The domain o () is, but the domain o the raph is. Problems 0 do not ive students an eplicit ormula or the pre-imae unction and so require students to perorm transormations without plottin points b calculatin their coordinates. When an eplicit ormula or the pre-imae unction is iven, students are more likel to approach the problem b usin the ormula to plot points and miss the undamental idea o transormin unctions. A blackline master or these problems is available in the Instructor s Resource Book. a. b. -translation b 6 Problem requires students to use Dnamic Precalculus Eplorations at to investiate dilations and translations. These are Java-based sketches that do not require additional sotware.. Answers will var. See pae 976 or answers to Problems 6 0 and CAS Problems The equation o in Problem is ()..( ). Enter this equation and the equation or () into our rapher and plot the raphs. Does the result aree with the iure in Problem?. The equation o in Problem is ()..( ). Enter this equation and the equation or () into our rapher and plot the raphs. Does the result aree with the iure in Problem? Chapter : Functions and Mathematical Models Additional CAS Problems. For some unctions, two dierent transormations on a parent unction can be used to obtain the same inal imae. Let (). Determine a horizontal translation that produces the same raph as (). Compare the manitude and direction o each translation numericall and raphicall. Fiure -h shows the raph o the pre-imae unction. For Problems 0, a. Sketch the raph o the imae unction on a cop o Fiure -h. b. Identi the transormation(s) that are done. Fiure -h. ( ) ( 6) 6. () 7. ( ) () 8. () () 9. ( ) ( 6) 0. (). Dnamic Transormations Problem: Go to and ind the Dnamic Precalculus Eplorations or Chapter. Complete the Translation eploration and the Dilation eploration, and eplain in writin what ou learned.. Dei ne () = and Solve () + k = ( + h) or h. Interpret the result.. What is the eect o a on the raph o a ()? What is the eect o b in the raph o (b )? Dei ne () =. Solve a () = (b ) or b and eplain what the result means.. Will there be a solution or b or the equation a () (b ) or all unctions ()? Eplain. Chapter : Functions and Mathematical Models

27 - Composition o Functions Radius increases. Section - Class Time da PLANNING Objective Fiure -a I ou drop a pebble into a pond, a circular ripple etends out rom the drop point (Fiure -a). The radius o the circle is a unction o time. The area enclosed b the circular ripple is a unction o the radius. Thus area is a unction o time throuh this chain o unctions: In this case the area is a composite unction o time. In this section ou will learn some o the mathematics o composite unctions. Given two unctions, raph and evaluate the composition o one unction with the other. In this eploration, ou ll ind the composition o one unction with another. EXPLORATION -: Composition o Functions. The iure shows two linear unctions, and.. Read values o () rom the raph and write Write the domain and rane o each unction. them in a cop o this table. I the value o is out o the domain, write none. 7 ( ) 6 0 () () continued Eploration Notes Eploration - can be used to introduce the idea o composition o unctions. Blackline masters or Problems and are available in the Instructor s Resource Book. Allow students minutes to complete the eploration.. : Domain: ; Rane: 6 : Domain: 8; Rane: 0 Section -: Homework Assinment Readin Analsis (RA), Q Q, Problems,,, 7, 9,, Teachin Resources Eploration -: Composition o Functions Blackline Masters Eploration Problems and Eamples and Problems and 6 Supplementar Problems Technolo Resources Presentation Sketch: Composition Present.sp TEACHING Important Terms and Concepts Composite unction Input Output Inside unction Outside unction Notation or a composite unction: ( ()), + (), ( + )() Domain and rane o a composite unction See pae 976 or answers to Eploration -, Problem. Section -: Composition o Functions

28 Section Notes This section eplores compositions o unctions throuh application problems and the use o a rapher. It is important that students understand the underlin concept o composition. The irst two eamples do not ive eplicit unctions, so students cannot depend on their alebraic skills to solve the problems. Make sure it is clear that the inside unction is applied irst and then the outside unction is applied to the result. A blackline master or Eample is available in the Instructor s Resource Book. Eample on pae 7 shows how to evaluate a composite unction at various values and then ind an eplicit ormula or ( ()). Eample on pae 8 shows how to ind the domain and rane o a composite unction. Make sure students understand that or a value to be in the domain o a composite unction +, must be in the domain o the inside unction, and () must be in the domain o the outside unction. A blackline master or Eample is available in the Instructor s Resource Book. In Eample d, the rane o + is ound b substitutin the endpoints o the domain, # # 7, into the equation o the composite unction. Emphasize that this technique works because + is linear (a linear unction does not chane directions, so the least and reatest -values occur at the endpoints o the domain). Usin the endpoints ives the correct rane or an unction that is monotone increasin or decreasin on the entire domain. For other tpes o unctions, students should eamine the raph to determine the rane. See pae 976 or answers to Eploration, Problem. EXPLORATION, continued. The smbol ( ()) is read o o. It means ind the value o () irst, and then ind o the result. For instance, ().. So ( ()) (.).. Put another column into the table or values o ( ()). Write none where appropriate.. Show in the table an instance where () is deined but ( ()) is not deined.. Plot the values o ( ()) on a cop o the iure in Problem. I the points do not lie in a straiht line, o back and check our work. 6. The unction in Problem is called the composition o with, which can be written. What are the domain and rane o? 7. Write equations or unctions and. Chapter : Smbols or Composite Functions 8. Enter into our rapher the equations o and as () and (), respectivel. Use Boolean variables or enter the domain directl, dependin on our rapher, to make the unctions have the proper domains. Then plot the raphs. Does the result aree with the iven iure? 9. Enter into () b enterin (). Plot this raph. Does it aree with the raph ou drew in Problem?. B suitable alebraic operations on the equations in Problem 7, ind an equation or ( ()). Simpli the equation as much as possible.. What did ou learn as a result o doin this eploration that ou did not know beore? Suppose that the radius o the ripple is increasin at the constant rate o 8 in./s. Then r 8t where r is the radius in inches and t is the number o seconds. I t, then r 8 0 in. The area o the circular reion is iven b a r where a is the area in square inches and r is the radius in inches. At time t, when the radius is 0, the area is iven b a in or about t. The s is the input or the radius unction, and the 0 in. is the output. Fiure -b shows that the output o the radius unction becomes the input or the area unction. The output o the area unction is Input or radius n. Eploration Notes, continued. The lines in which and. 6. Domain: # # 8 Rane: # # 6 8 Output rom radius n. s 0 in. 0 in. 07 in r 8t a r Radius unction Input or area n. Fiure -b Area unction Output rom area n. 7. () 7, # # (), # # 8 8. Yes () () 8 Chapter : Functions and Mathematical Models

29 Mathematicians oten use () terminolo or composite unctions. For these radius and area unctions, ou can write r () 8 is the input or unction r. a() is the input or unction a. Th e r and a become the names o the unctions, and the r() and a() are the values, or outputs, o the unctions. The simpl stands or the input o the unction. You must keep in mind that the input or unction r is the time and the input or unction a is the radius. Combinin the smbols leads to this wa o writin a composite unction: area a(r ()) Th e is the input or the radius unction, and the r() is the input or the area unction. The notation a(r()) is pronounced a o r o. Function r is called the inside unction because it appears inside a pair o parentheses. Function a is called the outside unction. Fiure -c shows this smbol and its meanins. The names parallel the terms inside transormation and outside transormation that ou learned in the previous section. Dierentiatin Instruction Because the notations ( ()) and + () are not standard in other countries, check that ELL students understand that the two notations are essentiall interchaneable. In Eample, make sure ELL students understand that () and () are dierent unctions than the () and () used beore this eample. Also, make sure the understand the note about domain. Outside unction area a(r()) Inside unction Input or unction r Input or unction a Output o unction r Fiure -c The two unction names are sometimes combined this wa: a r() or (a r)() The smbol a r is pronounced a composition r. The parentheses in the epression (a r) indicate that a r is the name o the unction. EXAMPLE Composite Functions rom Graphs Eample shows ou how to ind a value o the composite unction ( ()) rom raphs o the two unctions and. Functions and are raphed in Fiure -d. Find ( (0)), showin on copies o the raphs how ou ound this value. 6 () Fiure -d () 6 9. Section -: () () Yes. (()) () Answers will var. () 8 Section -: Composition o Functions

30 Technolo Notes Presentation Sketch: Composition Present.sp at uses dnaraphs to show composition o unctions. This sketch also includes a pae that illustrates domains and ranes or unction composition. SOLUTION First ind the value o the inside unction, (0). As shown on the let in Fiure -e, (0).8 Use this output o unction as the input or unction, as shown on the riht in Fiure -e. Note that the in () is simpl the input or unction and is not the same number as the in (). (.8) 80 ( (0)) 80 () 6 (0) Fiure -e () (.8) 80 6 EXAMPLE 6 Chapter : SOLUTION Composite Functions rom Tables Eample shows ou how to ind values o a composite unction when the two unctions are deined numericall. Functions and are deined onl or the inteer values o in the table. () () a. Find ( ()) or the si values o in the table. b. Find ( ()), and show that it does not equal ( ()). a. To ind ( ()), irst ind the value o the inside unction, (), b indin in the -column and () in the () column (third column). () Then use as the input or the outside unction b indin in the -column and () in the () column (second column). () 0 ( ()) 0 6 Chapter : Functions and Mathematical Models

31 Find the other values the same wa. Here is a compact wa to arrane our work. ( ()) () 0 ( )) () 6 ( ()) () ( ()) () ( ()) (7), which does not eist ( (6)) () b. ( ()) (), which is not the same as ( ()) 6 Note that in order to ind a value o a composite unction such as ( ()), the value o () must be in the domain o the outside unction,. Because () 7 in Eample and there is no value or (7), the value o ( ()) is undeined. CAS Suestions You can use the CAS to calculate compositions o unctions once the unctions have been deined alebraicall. The iure below shows the area computation rom pae. Notice the use o composition, the inclusion o units or the area computation, and the presentation o the correct units in the inal problem. EXAMPLE SOLUTION Composite Functions rom Equations Eample shows ou how to ind values o a composite unction i ou know the equations o the two unctions. Let be the linear unction (), and let be the eponential unction (). a. Find ( ()), ( (0)), and ( ( )). b. Find ( ( )) and show that it is not the same as ( ( )). c. Find an equation or h() ( ()) eplicitl in terms o. Show that h() arees with the value ou ound or ( ()). a. () 6, and (6) 6, so ( ()) Writin the same steps more compactl or the other two values o ives ( (0)) 0 () 8 ( ( )) (0.) (0.) 6. b. ( ( )) ( ) (), which does not equal 6. rom part a c. h() ( ()) The equation is h(). So h() 6, which arees with part a. Eample shows ou that ou can compose a unction with itsel or compose more than two unctions. Section -: 7 The iure below shows Eample c. The domain restriction o the composite unction is clear, even thouh it is not directl stated in the alebraic orm. Students should be epected to produce the alebraic results without usin a CAS. A CAS can provide instant alebraic eedback. Students can do Eample on pae 8 b hand and et veriication o their results rom their CAS, ainin eicient and valuable eedback. Section -: Composition o Functions 7

32 EXAMPLE SOLUTION EXAMPLE Let be the linear unction (), and let be the eponential unction (), as in Eample. Find these values. a. ( ()) b. (( ( ))) a. ( ()) ( ) () 8 b. ( ( ( ))) ( ( )) ( ( )) (0.06) Domain and Rane o a Composite Function In Eample, ou saw that the value o the inside unction sometimes is not in the domain o the outside unction. Eample shows ou how to ind the domain o a composite unction and the correspondin rane under this condition. The let raph in Fiure - shows unction with domain 7, and the riht raph shows unction with domain. 6 7 Fiure - a. Show on copies o these raphs what happens when ou tr to ind (6), ( (8)), and ( ()). b. Make a table o values o () and ( ()) or inteer values o rom throuh 8. I there is no value, write none. From the table, what does the domain o unction seem to be? c. The equations o unctions and are (), or 7 () 8, or Plot (), (), and () on our rapher, with the rapher s rid showin. Does the domain o conirm what ou ound numericall in part b? What is the rane o? d. Find the domain o alebraicall and show that it arees with part c. 8 Chapter : 8 Chapter : Functions and Mathematical Models

33 SOLUTION a. The let raph in Fiure - shows that () and (6) but (8) does not eist, because 8 is outside the domain o unction. The riht raph shows the two output values o unction, and, used as inputs or unction. From the raph, () and ( ) does not eist, because is outside the domain o unction. Summarize the results: ( (6)) () ( (8)) does not eist because 8 is outside the domain o. ( ()) ( ), which does not eist because is outside the domain o. (6) () No (8) 6 No ( ) () Fiure - b. () (()) none none none 0 none none none 6 () (()) 7 Fiure -h The domain o seems to be 7. c. Enter: ( ) /( and 7) or () Use Boolean variables or enter the domain directl, dependin on our rapher, to restrict the domain. Enter: ( ) 8/( and ) or () Enter: ( ) ( ) or (()) and become unction names in this ormat. Graph (Fiure -h), showin (()) solid stle. The domain o is 7, in areement with part b. From the raph, the rane o is 0 6. Section -: 9 Section -: Composition o Functions 9

34 PROBLEM NOTES Supplementar Problems or this section are available at keonline. Q. -dilation b Q. -translation b Q. ( ) Q. () Q. is the base, not the eponent. Q6. () a b c, a 0 Q7. Q8. () Q9. 0 Q. C Problem is similar to the eample o droppin a pebble into a pond. Students should assume that the radius o the spot o liht increases at a constant rate o 7 cm/s. In calculus, students will encounter related-rate problems in which rate chanes but not at a constant rate. a. cm; cm b..9 cm ; cm c. The area depends on the radius, which in turn depends on the time. d. r(t) 7t; a r(t) r(t) ; a r(t) ( 7t ) ; a().9 c m ; a(7) c m d. To calculate the domain alebraicall, irst observe that () must be within the domain o. () Write () in the domain o. Substitute or (). 8 Add to all three members o the inequalit. Net observe that must also be in the domain o, speciicall, 7. The domain o is the intersection o these two intervals. Number-line raphs (Fiure -i) will help ou visualize the intersection. the domain o is Fiure -i Intersection DEFINITION AND PROPERTIES: Composite Function The composite unction (pronounced composition or o ) is the unction ( )() ( ( )) Function, the inside unction, is evaluated irst, usin as its input. Function, the outside unction, is evaluated net, usin ( ) as its input (the output o unction ). The domain o is the set o all values o in the domain o or which ( ) is in the domain o. The iure shows this relationship. Note: Horizontal dilations and translations are eamples o composite unctions because the are inside transormations applied to. For instance, the horizontal translation () ( ) is actuall a composite unction with the inside unction (). 0 Chapter : 0 Chapter : Functions and Mathematical Models

35 Problem Set - Readin Analsis From what ou have read in this section, what do ou consider to be the main idea? Think o a realworld eample, other than the one in the tet, in which the value o one variable depends on the value o a second variable and the value o the second variable depends on the value o a third variable. In our eample, which is the inside unction and which is the outside unction? For there to be a value o the composite unction, what must be true o the value o the inside unction? min Quick Review Q. What transormation o is represented b () ()? Q. What transormation o is represented b h() ()? Q. I is a horizontal translation o b units, then ()?. Q. I p is a horizontal dilation o b a actor o 0., then p ()?. Q. Wh is not an eponential unction, even thouh it has an eponent? Q6. Write the eneral equation o a quadratic unction. Q7. For what value o will the raph o have a discontinuit? Q8. Sketch the raph o (). Q9. Find 0% o 00. Q. Which o these is a horizontal dilation b a actor o? A. ( ) () B. ( ) 0. () C. ( ) (0.) D. ( ) (). Flashliht Problem: You shine a lashliht, makin a circular spot o liht on the wall with radius cm. As ou back awa rom the wall, the radius increases at a rate o 7 cm/s. a. Find the radius at times s and 7 s ater ou start backin awa. b. Use the radius at times s and 7 s to ind the area o the spot o liht at these times. c. Wh can it be said that the area is a composite unction o time? d. Let t be the number o seconds since ou started backin awa. Let r(t) be the radius o the spot o liht, in centimeters. Let a(r(t)) be the area o the spot, in square centimeters. Write an equation or r(t) as a unction o t. Write another equation or a(r(t)) as a unction o r(t). Write a third equation or a r(t)) eplicitl in terms o t. Show that the last equation ives the correct area or times t s and t 7 s.. Bacteria Culture Problem: When ou row a culture o bacteria in a petri dish, the area o the culture is a measure o the number o bacteria present. Suppose that the area o the culture, A(t), in square millimeters, is iven b this unction o time t, in hours: A(t) 9(. t ) a. Find A(0), A(), and A(), the area at times t 0 h, h, and h, respectivel. b. Assume that the bacteria culture is circular. Find the radius o the culture at the three times in part a. c. Wh can it be said that the radius is a composite unction o time? d. Let R be the radius unction, with input A(t), the area o the culture. Write an equation or R(A(t)), the radius as a unction o area. Then write an equation or R(A(t)) eplicitl in terms o t. Show that this equation ives the correct answer or the radius at time t h.. Shoe Size Problem: The size shoe a person wears, S(), is a unction o the lenth o the person s oot. The lenth o the oot, L(), is a unction o the person s ae. a. Sketch reasonable raphs o unctions S and L. Label the aes o each raph with the name o the variable represented. Section -: Problem asks students to determine whether the area and radius o a bacterial culture are rowin at an increasin rate or at a decreasin rate. This problem is a ood preview o the related-rate problems students will solve in calculus. Students takin AP Biolo ma do eperiments similar to the one described in Problem. You ma want to ask the AP Biolo teacher to suest problems similar to Problem that the students will actuall perorm in AP Biolo. a. A(0) 9 m m ; A().9 m m ; A().6 m m b. R(0).69 mm; R().79 mm; R().78 mm c. The radius depends on the area (essentiall the number o bacteria), which in turn depends on the time. d. R A(t) A(t). A(t) 9(. ) t, so R A(t) 9(. ) t. R A() 9(. ).79 mm Problem involves a step unction. a. Answers will var. Note that shoe size is a discrete raph, because shoe sizes come onl in hal units. Sample answer: S() (size) (in.) L() (in.) (r) Section -: Composition o Functions

36 Problem Notes (continued) b. In S(), represents oot lenth (in inches, or the precedin raph). In L(), represents ae (in ears). The composite unction S L() ives shoe size as a unction o ae ( represents ae). L S() would be meaninless with the iven unctions L and S. Because is substituted into S, must represent oot lenth. S then ives shoe size. But this is substituted into L, which epects to have an ae, not a shoe size, substituted into it. c. Answers will var. Sample Answer: S(L()) (r) Problems and 6 ask students to answer questions about a composite unction rom raphs without usin eplicit ormulas or the unctions. A blackline master or these problems is available in the Instructor s Resource Book. a. h() 7 6 h() 6 7 b. p h() p(). 7 6 p() 6 7 c. p h().; p h() 6a. () () 6 b. What does represent in unction S? What does represent in unction L? What does the composite unction S(L()) represent? What, i an, real-world meanin does L(S()) have? c. Let () (S L)(). Sketch a reasonable raph o unction. Label the aes o the raph with the name o the variable represented.. Traic Problem: The lenth o time, T(), it takes ou to travel a mile on the reewa depends on the speed at which ou travel. The speed, S(), depends on the number o other cars on that mile o reewa. a. Sketch reasonable raphs o unctions T and S. Label the aes o each raph with the name o the variable represented. b. What does represent in unction T? What does represent in unction S? What does the composite unction T(S()) represent? What, i an, real-world meanin does S(T()) have? c. Let () (T S)(). Sketch a reasonable raph o unction. Label the aes o the raph with the name o the variable represented.. Composite Function Graphicall, Problem : Functions h and p are deined b the raphs in Fiure -j, in the domains shown. h() Fiure -j p() 6 7 a. Find h(). On a cop o the raphs, draw arrows to show how ou ound this value. b. Use the output o h() to ind p(h()). Draw arrows to show how ou ound this value. c. Find p(h()) and p(h()) b irst indin h() and h() and then usin these values as inputs or unction p. Chapter : 6b. () () c. () 7; () d. Find h(p()) b irst indin p() and then usin the result as the input or unction h. Draw arrows to show how ou ound this value. Show that h(p()) p(h()). e. Eplain wh there is no value o h(p(0)), even thouh there is a value o p(0). 6. Composite Function Graphicall, Problem : Functions and are deined b the raphs in Fiure -k, in the domains shown () 6 Fiure -k () a. Find the approimate value o (). On a cop o the raphs, show how ou ound this value. b. Use the output o () to ind the approimate value o ( ()). Draw arrows to show how ou ound this value. c. Find approimate values o ( ()) and ( ()) b irst indin () and () and then usin these values as inputs or unction. d. Eplain wh there is no value o ( (6)). e. Tr to ind ( ()) b irst indin () and then usin the result as the input or unction. Draw arrows to illustrate wh there is no value o ( ()). 7. Composite Function Numericall, Problem : Functions and consist o the discrete points in the table, and onl these points. Find the values o the composite unctions, or eplain wh no such value eists. () () 7 6d. 6 is not in the domain o. 6e. () is not in the domain o () () Chapter : Functions and Mathematical Models

37 a. Find () and ( ()). b. Find () and ( ()). c. Find () and ( ()). d. Find () and ( ()). e. Find ( ()).. Find ( ()).. Find ( ()). h. Find ( ( ())). 8. Composite Function Numericall, Problem : Functions u and v consist o the discrete points in the table, and onl these points. Find the values o the composite unctions, or eplain wh no such value eists. a. Find v() and u(v()). b. Find v(6) and u(v(6)). c. Find v() and u(v()). d. Find u() and v(u()). e. Find v(u()).. Find v(v()) 6 8 u() 8 6 v() 6 8. Find u(u(6)). h. Find v(v(v(8))) 9. Composite Function Alebraicall, Problem : Let and be deined b () 9 8 () a. Make a table showin values o () or each inteer value o in the domain o. In another column, show the correspondin values o ( ()). I there is no such value, write none. b. From our table in part a, what does the domain o the composite unction seem to be? Conirm (or reute) our conclusion b indin the domain alebraicall. c. Eplain wh ( (6)) is undeined. Eplain wh ( ()) is undeined, even thouh () is deined. d. Repeat parts a and b or the composite unction. e. Fiure -l shows unctions and. Enter the two unctions as () and (). Then enter ( ()) as () (), and ( ()) as (). Plot the raphs usin the window Problems 7 and 8 use numeric inormation to ind values o composite unctions. 7a. () ; () 7b. () ; () 7c. () 7; () is undeined. 7d. () ; () 7e. () 7. () 7. () is undeined. 7h. () shown, with the rapher s rid showin and thick stle or the two composite unction raphs. Sketch the result. Do the domains o the composite unctions rom the raph aree with our results in parts b and d? Section -: Fiure -l. Find ( ()). Eplain wh ( ()) is undeined.. Composite Function Alebraicall, Problem : Let and be deined b () 8 6 () 0 7 a. Make a table showin values o () or each inteer value o in the domain o. In another column, show the correspondin values o ( ()). I there is no such value, write none. b. From our table in part a, what does the domain o the composite unction seem to be? Conirm (or reute) our conclusion b indin the domain alebraicall. c. Show wh ( ()) is deined but ( ()) is undeined. d. Fiure -m shows the raphs o and. Enter these equations as () and (). Then enter ( ()) as () () (). Plot the three raphs with the rapher s rid showin. Sketch the result. Does the domain o the composite unction aree with our calculation in part b? Fiure -m 8a. v() 6; u v() 8b. v(6) ; u v(6) 8 8c. v() ; u v() is undeined because v() is not in the domain o u. 8d. u() 8; v u() 8e. v u() 8. v v() 8. u u(6) 8h. v v v(8) v v() v(6) Problems 9 and are similar to Eample. Consider supplementin Problems 9 with a CAS veriication. 9b. 9c. 6 is not in the domain o, so () is undeined. (), but is not in the domain o. 9e. 8 The domains o the composite unctions match the calculations in parts b and d. 9. () () ; () 7, and 7 is not in the domain o. a. () (()) none 6 none 7 none b. 0 c. () 8; () (), but is not in the domain o, so () is undeined. d. See paes or answers to Problems, d, e, 9a, and 9d. Section -: Composition o Functions

38 Problem Notes (continued) e. (), with the domain 0 ound in part b. The raph coincides with the raph in part d. Problem requires students to work with square root unctions. Because the square root o a neative number doesn t eist amon real numbers, the square root unctions in these problems have restricted domains. a. () ; (7) 7; () ; (8) 8; Conjecture: For all values o, () (). b. (9) is undeined. (9) 9 9. No. c. is deined onl or nonneative, so + is deined onl or nonneative. d. e. () ( ) i 0 i, 0 e. Find an equation or ( ( )) eplicitl in terms o. Enter this equation as ( ) and plot it on the same screen as the other three unctions. What similarities and what dierences do ou see or ( ) and ( )?. Square and Square Root Functions: Let and be deined b (), where is an real number ( ), where the values o make ( ) a real number a. Find ( ()), ( (7)), ( ()), and ( (8)). What do ou notice in each case? Make a conjecture: For all values o, ( ( ))? and ( ())?. b. Test our conjecture b indin ( ( 9)) and ( ( 9)). Does our conjecture hold or neative values o? c. Plot (), ( ), and ( ( )) on the same screen. Use approimatel equal scales on both aes, as in Fiure -n. Eplain wh ( ( )), but onl or nonneative values o. Fiure -n d. Deactivate ( ( )), and plot (), ( ), and ( ()) on the same screen. Sketch the result. e. Eplain wh ( ()) or nonneative values o, but ( ()) (the opposite o ) or neative values o. What other amiliar unction has this propert?. Horizontal Translation and Dilation Problem: Let,, and h be deined b () ( ) or all real values o h() _ or all real values o a. ( ( )) ( ). What transormation is applied to unction b composin it with? b. (h()) _. What transormation is applied to unction b composin it with h? c. Plot the raphs o,, and h. Sketch the results. Do the raphs conirm our conclusions in parts a and b? For Problems and, ind what transormation will turn the dashed raph ( ) into the solid raph ( )... Both raphs coincide.. Linear Function and Its Inverse Problem: Let and be deined b () _ (). a. Find ( (6)), ( ( )), ( ()), and ( ( 8)). What do ou notice in each case? b. Plot the raphs o,,, and on the same screen. How are the raphs o and related? How are the raphs o and related to their parent raphs, and? c. Show that ( ( )) and ( ()) both equal. d. Functions and in this problem are said to be inverses o each other. Whatever does to, undoes. Let h() 7. Find an equation or the inverse unction o h. Problem connects Sections - and -. a. Translation units to the riht b. Horizontal dilation b a actor o Problems prepare students or later sections, so it is important to assin them.. I the dotted raph is (),, then the solid raph is () (),. In terms o See pae 977 or answers to Problems c, and CAS Problems. Chapter : composition o unctions, the solid raph is () h(), where h(). Problem asks students to compose a unction and its inverse and to observe that the two unctions undo one another. Additional CAS Problems. Ever transormation is mathematicall equivalent to a composition o unctions. Let (). I () is an other unction, eplain the meanin o ( ()), ( ()), and ( ()).. When a linear unction is composed with another linear unction, what tpe o unction is the result? Prove our claim.. What is the slope o the composition o two linear unctions? Under what conditions is the composition o two linear unctions a decreasin unction? Chapter : Functions and Mathematical Models

39 - Objective t (h) d (mi) Inverse Functions and Parametric Equations The photoraph shows a hihwa crew paintin a center stripe. From records o previous work the crew has done, it is possible to predict how much o the stripe the crew will have painted at an time durin a normal eiht-hour shit. It ma also be possible to tell how lon the crew has been workin b how much stripe has been painted. The input or the distance unction is time, and the input or the time unction is distance. I a new relation is ormed b interchanin the input and output variables in a iven relation, the two relations are called inverses o each other. I both relations turn out to be unctions, the are called inverse unctions. I not, the relation and its inverse can still be plotted easil usin parametric equations in which both and are unctions o some third variable, such as time. Given a unction, ind its inverse relation, and tell whether the inverse relation is a unction. Graph parametric equations both b hand and on a rapher, and use parametric equations to raph the inverse o a unction. Inverse o a Function Numericall Suppose that the distance d, in miles, a particular hihwa crew paints in an eiht-hour shit is iven numericall b this unction o t, in hours, it has been on the job. Let d (t). You can see that () 0., () 0.6,..., (8).0. The input or unction is the number o hours, and the output is the number o miles. As lon as the crew does not stop paintin durin the eiht-hour shit, the number o hours it has been paintin is a unction o the distance. Let t (d). You can see that (0.), (0. 6),..., () 8. The input or unction is the number o miles, and the output is the number o hours. The input and output or unctions and have been interchaned, and thus the two unctions are inverses o each other. Section -: Section - Class Time das PLANNING Homework Assinment Da : RA, Q Q, Problems odd, 6 Da : Problems 7, 9,,,,, 7, 8 Teachin Resources Eploration -: Parametric Equations Graph Eploration -a: Inverses o Functions Eploration -b: Introduction to Parametric Equations Blackline Masters Problem Problem Problem 6 Problem 7 Problem 8 Test, Sections - to -, Forms A and B Technolo Resources Presentation Sketch: Inverse Present.sp TEACHING Important Terms and Concepts Increasin Inverse Inverse unction Inverse unction notation: () and relect across the line Invertible unction One-to-one unction Strictl increasin unction Strictl decreasin unction Parametric equations Thereore smbol, Q.E.D. Section -: Inverse Functions and Parametric Equations

40 Section Notes This section discusses the inverse o unctions and parametric equations. You should emphasize that when a unction s variables are inter chaned, ou et the inverse relation, which ma or ma not be a unction. Discussin Eample helps lead to the deinition o an invertible unction. In Eample, note that when t, there is not a point on the (, ) raph. Eample shows how to raph a unction and its inverse usin parametric equations. Students oten think parametric raphin is trul bizarre, but ou should reinorce that because t, can be replaced b t in the irst set o parametric equations. In the second set o parametric equations, simpl interchane the equations or and in the irst set. Eample provides an eample or an invertible unction and shows an important eature o these unctions: ( ()). Discussin the concept o a one-to-one unction is ver important in relation to inverse unctions. It will help students understandin to ive additional eamples or invertible unctions: (), or other power unctions with odd eponents, and () e and its transormed orms. Also ive additional eamples or noninvertible unctions, such as (), or other power unctions with even eponents, and (). You miht want to start our section with an eplanation o wh the concept o inverse is useul b reerrin to Problem 7, the Brakin Distance Problem. You can use the eample o police investiators to show that the would need to ind the inverse o a unction to ind the speed rom the skid mark evidence. 6 Chapter : Smbols or the Inverse o a Function I unction is the inverse o unction, the smbol is oten used or the name o unction. In the hihwa stripe eample, ou can write (0.), (0.6),..., () 8. Note that () does not mean the reciprocal o (). () 8 and (), not 8 The used with the name o a unction means the unction inverse, whereas the used with a number, as in, means the multiplicative inverse o that number. Inverse o a Function Graphicall Fiure -a shows a raph o the data or the hihwa stripe eample. Note that the points seem to lie in a straiht line. Connectin the points is reasonable i ou assume that the crew paints continuousl. The line meets the t-ais at about t 0. h, indicatin that it takes the crew about hal an hour at the beinnin o the shit to redirect traic and set up the equipment beore it can start paintin d (t) t Because man students use the word inverse to reer to the reciprocal, or multiplicative inverse, o a number, the ma mistakenl think that the inverse o a unction () is (). Eplain that () is a unctional inverse, whereas is a multiplicative () inverse. Also emphasize that () is read inverse o t (d) d Fiure -a Fiure -b Fiure -c Fiure -b shows the inverse unction, t (d). Note that ever vertical eature on the raph o is a horizontal eature on the raph o, and vice versa. For instance, the raph o meets the vertical ais at 0.. Fiure -c shows both raphs on the same set o aes. In this iure, is used or the input variable and or the output variable. Keep in mind that or unction represents hours and or unction represents miles. The raphs are relections o each other across the line whose equation is. Inverse o a Function Alebraicall In the hihwa stripe eample, the linear unction that its the raph o unction in Fiure -c is 0.( 0.) or, equivalentl, slope 0., -intercept 0. Later in the course, students oten mistakenl believe trionometric inverses are equal to recip rocal trionometric unctions (or eample, sin csc ). I ou reserve the term inverse to mean unctional inverse and reer to the multiplicative inverse as the reciprocal, ou can avoid errors later. 6 Chapter : Functions and Mathematical Models

41 The linear unction that its the raph o is. 0. slope., -intercept 0. I ou know the equation o a unction, ou can transorm it alebraicall to ind the equation o the inverse relation b irst interchanin the variables. Function: Inverse: The equation o the inverse relation can be solved or in terms o Solve or in terms o. To distinuish between the unction and its inverse, ou can write () and (). 0. Bear in mind that the used as the input or unction is not the same as the used as the input or unction. One is time, and the other is distance. An interestin thin happens i ou take the composition o a unction and its inverse. In the hihwa stripe eample, (). and (.) ( ()) You et the oriinal input,, back aain. This result should not be surprisin to ou. The composite unction ( ()) means How man hours does it take the crew to paint the distance it can paint in our hours? There is a similar meanin or ( ()). For instance, (.) (). In this case. is the oriinal input o the inside unction. Dierentiatin Instruction Students ma be conused that both the word inverse and the notation as an eponent have more than one mathematical meanin. Point out the two distinct meanins o each, and have students write these in their notes. Clari the lanuae in Eample or ELL students. Because an equivalent to sketch does not eist in all lanuaes, show ELL students how to sketch a curve, and eplain the dierence between sketch and plot. As ELL students do Eploration -, monitor their work or understandin. Because o the dierence in topic order rom countr to countr, ou ma need to show ELL students how to invert a unction alebraicall and raphicall. For Problems and, clari the meanin o show that. Invertibilit and the Domain o an Inverse Relation In the hihwa stripe eample, the lenth o stripe painted durin the irst hal hour was zero because it took some time at the beinnin o the shit or the crew to divert traic and prepare the equipment. The raph o unction in Fiure -d includes times at the beinnin o the shit, alon with its inverse relation. Fiure -d Section -: A memorable wa to illustrate that the raph o the inverse o a unction is the mirror imae o the raph o the unction is to have students hold a small mirror so that one ede is on the line. The raph o the inverse will appear in the mirror. Eploration -a can be used to introduce students to inverses o unctions. 7 Section -: Inverse Functions and Parametric Equations 7

42 Eploration Notes Eploration -a introduces students to inversion o unctions b havin them ind the inverses o a linear, a quadratic, and an eponential unction. Students do this b a combination o numerical, raphical, and alebraic techniques. Alon the wa, the demon strate that whether or not the inverse relation is a unction, it is a mirror imae o the parent unction in the line. Allow about 0 minutes. Eploration -b introduces students to parametric equations b havin them raph parametric equations both on paper and on their rapher. Allow 0 minutes. Technolo Notes Presentation Sketch: Inverse Present.sp at demonstrates properties o inverses on a dnaraph. It also includes a pae that demonstrates the relationship between the raph o a unction and that o its inverse. The inal pae allows eploration o linear unctions that are their own inverses. EXAMPLE SOLUTION Note that the inverse relation has multiple values o when equals zero. Thus the inverse relation is not a unction. You cannot answer the question How lon has the crew been workin when the distance painted is zero? I the domain o unction is restricted to times no less than a hal hour, the inverse relation is a unction. In this case, unction is said to be invertible. I is invertible, ou are allowed to use the smbol or the inverse unction. I the domain o is 0. 8, then there is eactl one distance or each time and one time or each distance. Function is said to be a one-to-one unction. An one-to-one unction is invertible. A unction that is strictl increasin, such as the hihwa stripe unction, or strictl decreasin is a one-to-one unction and thus is invertible. The hihwa stripe problem ives eamples o operations with unctions rom the real world. Eample shows ou how to operate with a unction and its inverse in a strictl mathematical contet. Given () 0. a. Make a table o values or ( ), ( ), (0), (), and (). From the numbers in the table, eplain wh ou cannot ind a unique value o i ().. How does this result tell ou that unction is not invertible? b. Plot the ive points in part a on raph paper. Connect the points with a smooth curve. On the same aes, plot the ive points or the inverse relation and connect them with another smooth curve. How does the raph o the inverse relation conirm that unction is not invertible? c. Find an equation or the inverse relation. Plot unction and its inverse on the same screen on our rapher. Show that the two raphs are relections o each other across the line. a. (). 0. I ()., there are two dierent values o, and. You cannot uniquel determine the value o. Function is not invertible because there will be two values o or the same value o i the variables are interchaned. 8 Chapter : 8 Chapter : Functions and Mathematical Models

43 Fiure -e () () () Fiure - b. Fiure -e shows the raphs o unction and its inverse relation. The inverse relation is not a unction because there are two values o or each value o. The inverse relation ails the vertical line test. c. Function: 0. Use or (). Inverse: 0. Interchane and. ( ) 0. ( ) ( ) ( ) ( ) ( ) Take the square root o both sides. Enter () as ( ). Enter as ( ). Fiure - shows the raphs o unction and its inverse relation. The raphs are relections o each other across the line. Parametric Equations Enter the two branches o the inverse relation as ( ) and ( ). There is a simple wa to plot the raph o the inverse o a unction with the help o parametric equations. Here, and are both epressed in terms o some third variable, usuall t (because time is oten the independent variable in real-world applications). In this eploration, ou will see how to raph a relation speciied b parametric equations, both b hand and on our rapher. EXPLORATION -: Parametric Equations Graph Let and be unctions o a third variable, t, as. For the relation ou plotted in Problem, is speciied b these equations: a unction o? Eplain. t t Inverse o. Make a table o values o t,, and or each inteer value o t rom to.. On raph paper, plot the points ou ound in Problem. Connect the points with a smooth curve in the order o increasin values o t.. Set our rapher to parametric mode. Enter the two equations. Use a window with t and a t-step o 0.. Set and 6 6. Then plot the raph. Does our rapher s raph aree with our pencil-and-paper raph? I not, make chanes until the two raphs aree.. What did ou learn as a result o doin this eploration that ou did not know beore? Additional Eploration Notes Eploration - introduces students to raphin parametric equations. Students eamine two unctions, (t, (t)) where is a unction o t and (t, (t)) where is a unction o t but is not a unction o. I our students have a solid understandin o the deinition o unctions, ou ma want to point out that (, ) is, however, a unction o t since a value o t corresponds to eactl one point (, ). It ma clari thins i ou write t, t, and so on, net to the discrete points on our raph and draw an arrow to indicate the path as t oes rom to. Parametric curves have a dnamic qualit to them. The discrete points indicate time passin and a direction o motion. When ou do Problem in the eploration, spend time eplainin how to ind reasonable windows or t,, and as this is not alwas obvious to students. Emphasize that a rapher in parametric mode can raph curves that are not unctions while a rapher in unction mode can onl raph unctions.. The relation is not a unction because there is more than one value o or some values o.. Grapher raph arees with raph on paper.. Answers will var. Section -:. t Section -: Inverse Functions and Parametric Equations 9

44 CAS Suestions To solve or the inverse o an equation, deine the oriinal unction as (). Then echane the variables and solve or the new -variable. Note: This procedure or indin the inverse can be used either to solve an application problem or as a tool once students understand what to do. Students need to understand the alebraic steps involved in transormin the oriinal unction to its inverse. It is important to use available tools to enhance students understandin o the mathematics. Consider havin students use a CAS to complete the alebraic steps to ind the inverse beore teachin them the shortcut described above. The alebraic parallel to the noninvertibilit o () 0. in Eample is that there is no sinle alebraic orm or the inverse. Notice two thins in the iure. First, two dierent equations are iven or the inverse. This shows that a sinle inverse equation does not eist. Second, each inverse is iven a domain (which is the rane o the oriinal unction). Students are oten careless in notin domain restrictions when indin inverses, but a CAS notes them meticulousl, encourain students to be careul in their computations. EXAMPLE SOLUTION EXAMPLE Eample shows ou in eneral what parametric equations are. On raph paper, plot the raph o these parametric equations b irst calculatin values o and or inteer values o t rom throuh 7. (t) t (t) t The table shows values o t,, and. Fiure - shows the raph o the parametric equations. Note that is not a unction o because there are two values o or some values o. t t 7 t t Fiure - The independent variable t in parametric equations is called the parameter. The word comes rom the Greek para- meanin alonside, as in parallel, and meter, meanin measure. The values o t do not show up on the raph in Fiure - unless ou write them in. Your rapher is prorammed to plot parametric equations. For the equations in Eample, use parametric mode and enter (t) abs(t ) (t) (t ) t None Use a window with t 7, 0, and 0 and a convenient t-step such as 0.. The raph will be similar to the raph in Fiure -. Eample shows ou how to use parametric equations to plot inverse relations on our rapher. Plot the raph o 0. or in the domain and its inverse usin parametric equations. What do ou observe about the domain and rane o the unction and its inverse? 0 Chapter : 0 Chapter : Functions and Mathematical Models

45 SOLUTION Function Inverse relation Fiure -h EXAMPLE SOLUTION Put our rapher in parametric mode. Then enter (t) t (t) 0. t Because t, this is equivalent to 0.. (t) 0. t (t) t For the inverse, interchane the equations or and. Use a window with t. Use a convenient t-step, such as 0.. The result is shown in Fiure -h. The rane o the inverse relation is the same as the domain o the unction, and vice versa. The rane and the domain are interchaned. Eample shows ou how to demonstrate alebraicall that ( ()) and ( ). Let (). a. Find an equation or the inverse o, and eplain how that equation conirms that is an invertible unction. b. Demonstrate that ( ()) and (). a. Function: Inverse: Because the equation or the inverse relation has the orm m b, the inverse is a linear unction. Because the inverse relation is a unction, is invertible, so the equation can be written ( ) b. ( ()) ( ) Substitute or (). ( ) Substitute as the input or unction. Show that ( ()) equals. Also, ( ) Show that () equals. ( ()) and ( ), q.e.d. Note: The three-dot mark stands or thereore. The letters q.e.d. stand or the Latin words quod erat demonstrandum, meanin which was to be demonstrated. The bo on the net pae summarizes the inormation o this section reardin inverses o unctions. Consider deinin the inverse unction usin a name like inv instead o to avoid conusion with the reciprocal notation. A CAS allows the student to compute + and + and show the results, or test the inverse unction usin Boolean operators to show that the composition actuall provides the epected results. Remember that it is eas to switch between unction and parametric modes without leavin a raphin window. A siniicant advantae o the TI-Nspire over previous models is that dierent raph tpes can be raphed on the same screen. The parts o parametric unctions can also be deined on a calculator screen and then raphed b name in parametric mode. While it is unnecessar here, usin this method allows students to manipulate the unction b name. Section -: Section -: Inverse Functions and Parametric Equations

46 PROBLEM NOTES Q. Inside Q. Outside Q. (m + d)() Q. 8 Q. Q6. Q7. Q8. Q9. Q. () Problems and are applications o inverses to real-world situations. The emphasize inverse unction notation and the relection o a unction and its inverse across the line. a. () psi; () 6 psi; ().7 psi b. The air leaks out o the tire as time passes, so the pressure is constantl ettin lower. Thus, is a decreasin unction and hence is invertible. () min, which answers the question At what time was the pressure psi? (6) min, which answers the question At what time was the pressure 6 psi? c. Somewhere between and 0 min, all the air oes out o the tire, and the pressure remains zero. So it is not possible to ive a unique time correspondin to a pressure o 0 psi; (0) cannot be deined. d. The raph o the inverse relation is dotted. The two raphs are relections o each other over the line. (The coincidentall happen to be ver close over most o their lenth.) () Problem Set - From what ou have read in this section, what do ou consider to be the main idea? Wh is it possible to ind the inverse o a unction even i the unction is not invertible? Under what conditions are ou permitted to use the smbol or the inverse o unction? How does the meanin o in the unction name dier rom the meanin o in a numerical epression such as 7? min Quick Review Q. In the composite unction m(d()), unction d is called the? unction. Q. In the composite unction m(d()), unction m is called the? unction. Q. Give another smbol or m(d()). Q. I () and (), ind ( ()). Chapter : DEFINITIONS AND PROPERTIES: Function Inverses inverse o a relation in two variables is ormed b interchanin the two variables. relections o each other across the line. is also a unction, then is invertible. is invertible and (), then ou can write the inverse o as ( ). Interchane the variables, solve or, and plot the resultin equation(s), or use parametric mode, as in Eample. is invertible, then the compositions o and a r e ( ()), provided is in the domain o and () is in the domain o ( ), provided is in the domain o and ( ) is in the domain o unctions are one-to-one unctions. Readin Analsis Q. Find ( ()) or the unctions in Problem Q. Q6. Find ( ()) or the unctions in Problem Q. Q7.? Q8. Identi the unction raphed. Q9. I (), ind (0). Q. I (), ind an equation or (), a horizontal translation o () b units.. Punctured Tire Problem: Suppose that our car runs over a nail. The table shows the pressure, in pounds per square inch (psi), o the air inside the tire as a unction o, the number o minutes that have elapsed since the nail punctured the tire e. As an input or, represents time in minutes. As an input or, it represents pressure in psi. Chapter : Functions and Mathematical Models

47 (min) (psi) a. Let (). Find (), (), and (). b. Wh is it reasonable to assume that is an invertible unction i is in the domain 0? Find () and (6), and ive their real-world meanins. c. Wh is unction not invertible on the whole interval 0? What do ou suppose happens between min and 0 min that causes to not be invertible? d. Plot the eiht iven points or unction and the eiht correspondin points or the inverse relation. Connect each set o points with a smooth curve. Draw and eplain how the two raphs are related to this line. e. Suppose is restricted to the domain 0. What is the dierence in the meanin o as an input or unction and as an input or unction?. Cricket Chirpin Problem: The rate at which crickets chirp is a unction o the temperature o the air around them. Suppose that the ollowin data have been measured or chirps, c, per minute,, at temperatures in derees Fahrenheit,. ( F) (c/min) a. Let c(). Find c(0), c(0), and c(60). b. For temperatures o 0 F and above, the chirpin rate seems to be a one-to-one unction o time. How does this act impl that unction c is invertible or 0? Find the values o c (0) and c (80). How do these values dier in meanin rom c(0) and c(80)? c. Wh is unction c not invertible or in the interval 0 80? What is true in this real-world situation that makes c not invertible? d. On raph paper, plot the seven iven points or unction c and the correspondin points or the inverse relation. Connect each set o points with a line or a smooth curve. Draw the line and eplain how the two raphs are related to this line. e. Suppose c is restricted to the domain What is the dierence in the meanin o as an input or unction c and as an input or unction c?. Punted Football Problem: Fiure -i shows the heiht o a punted ootball, in meters, as a unction o time, in tenths o a second since it was punted. On a cop o the iure, sketch the raph o the inverse relation and show that the two raphs are relections across the line. How does the raph o the inverse relation reveal that the heiht unction is not invertible? 0 0 Fiure -i a. c(0) c/min; c(0) 0 c/min; c(60) c/min b. An one-to-one unction is invertible. c (0) 0 F; c (80) 70 F; these ive the temperature correspondin to 0 c/min and 80 c/min. B contrast, c(0) and c(80) ive the number o chirps/min correspondin to 0 F and 80 F. c. The cricket does not bein chirpin until the temperature is at least 0 F. For 0 0, the number o chirps/min remains zero, so c (0) cannot be deined. d The raphs are relections o each other across the line. e. As the input to c, represents temperature in F. As the input to c, it represents the number o chirps/min. Problems and 8 provide practice in sketchin the inverse relation or a iven raph. A blackline master or these problems is available in the Instructor s Resource Book.. 0 c() 0 Throuhout most o its domain, the inverse relation has two -values or ever -value. Section -: Inverse Functions and Parametric Equations

48 Problem Notes (continued) Problem requires students to appl the idea o invertibilit to a new situation. The unction presented in this problem is made up o a discrete set o points (a set o points with aps between them). Althouh the unction is not strictl increasin or strictl decreasin, it is one-to-one and is thereore invertible.. No -value comes rom more than one -value. Also, no horizontal line passes throuh more than one point o the unction.. Function. Discrete Function Problem: Fiure -j shows a unction that consists o a discrete set o points. Show that the unction is one-to-one and thus is invertible, even thouh the unction is increasin in some parts o the domain and decreasin in other parts. For Problems 8, sketch the line and the inverse relation on a cop o the iven iure. Be sure that the inverse relation is a relection o the unction raph across the line. Tell whether the inverse relation is a unction Fiure -j. t 7 t t. 7 t t t. t t (t ). t t t t. t t t t t t. Two Paths Problem: Two particles (small objects) move alon the paths shown in Fiure -k. The paths are iven b these parametric equations, where and are distance in meters and t is time in seconds. Particle : t Particle :.t 7 t.t 6 Particle 6. Not a unction 7. Not a unction 8. Not a unction For Problems 9, Chapter : a. Plot the parametric equations on raph paper usin the iven domain or t. Connect the points with lines or smooth curves. b. Tell whether is a unction o. c. Conirm our results b rapher, usin the iven domain or t. 9. t t t equations, raphin b hand ma seem tedious and unnecessar, but students develop a stroner understandin o parametric equations when the ind the (t, (t)), (t, (t)), and (t, (t), (t)) points. Problems 9 parts c can be solved Problems 9 ask the students to raph usin a CAS b checkin whether the the parametric equations b hand, -equation or each parametric equation decide i is a unction o, and then use can be solved or t in a sinle equation. I a rapher to check. Once students learn so, is a unction o. how to use the rapher or parametric Chapter : Functions and Mathematical Models Particle Fiure -k a. The paths intersect at two points. For each point, determine whether the particles reach that point at the same time or at dierent times. Give numbers to support our conclusion. b. Conirm our answer to part a raphicall b plottin the two sets o parametric equations dnamicall on the same screen, settin our rapher to simultaneous mode. Write a sentence or two eplainin how our raph conirms our answer to part a. Problems and 6 are application problems involvin parametric equations. a. Paths intersect simultaneousl at point (, ) when t s. Paths intersect at point (, 6) but not simultaneousl. b. Grapher raph conirms that the paths intersect simultaneousl onl at point (, ) when t s. 6a. The -values are equal at t 6 h. Freihter: 0 mi; Cutter: mi. The -values are not equal at t 6 h.

49 6. Two Ships Problem: At time t 0 h, a reihter is at the point (90, ) to the east-northeast o a lihthouse located at the oriin o a Cartesian coordinate sstem, where and are distance in miles. At time t h, a Coast Guard cutter starts rom the lihthouse to intercept the reihter. Fiure -l shows the raph o these parametric equations representin the ships paths: Freihter: 90 t Cutter: 8(t ) t (t ) 0 Cutter 0 Lihthouse Fiure -l Freihter 0 a. Find the value o t at which the -values o the two paths are equal. At this value o t, are the two -values equal? b. Do the two ships arrive at the intersection point at the same time? I so, how can ou tell? I not, which ship arrives at the intersection point irst? For Problems 7 8, plot the unction in the iven domain usin parametric mode. On the same screen, plot the inverse relation. Tell whether the inverse relation is a unction. Sketch the raphs. 7. () 6 8. () () 0 0. (). () is an real number.. () 0. is an real number.. () 6. () 8. () 8 6. () 6 7. () 8. () 0.06 For Problems 9, write an equation or the inverse relation b interchanin the variables and solvin or in terms o. Then plot the unction and its inverse on the same screen, usin unction mode. Sketch the result, showin that the unction and its inverse are relections across the line. Tell whether the inverse relation is a unction Show that () _ is its own inverse unction.. Show that () is its own inverse unction.. Cost o Ownin a Car Problem: Suppose that ou have ied costs (car paments, insurance, and so on) o $00 per month and operatin costs o $0.0 per mile ou drive. The monthl cost o ownin the car is iven b the linear unction c() where is the number o miles ou drive the car in a iven month and c() is the number o dollars per month ou spend. a. Find c(00). Eplain the real-world meanin o the answer. b. Find an equation or c (), where now stands or the number o dollars ou spend instead o the number o miles ou drive. Eplain wh ou can use the smbol c or the inverse relation. Use the equation o c () to ind c (78), and eplain its realworld meanin. Section -: 6b. The ships do not arrive at the intersection point at the same time because the two -values are not equal when the two -values are equal. Freihter arrives at.8 mi when t.87 h. Cutter arrives at.8 mi when t.98 h. Freihter arrives at the intersection point 0.07 h, or about minutes, beore cutter. c. Plot () c() and () c () on the same screen, usin unction mode. Use a window with 0 00 and use equal scales on the two aes. Sketch the two raphs, showin how the are related to the line. Problems 7 8 require students to write equations or, and then raph inverses o, iven unctions. Problems 7 8 can be solved b alebraicall reversin the variables and solvin or. (See CAS Suestions or additional inormation.) Problem 7b could be solved without indin the equation o the inverse unction b reconizin that 00 is an input o the inverse unction and thereore an output value o the oriinal unction. Problems 9 require students to write equations or, and then raph the inverses o, iven unctions and decide whether the inverses are unctions. Problems and present two unctions that are their own inverses. Problems and can be solved on a CAS usin compositions o unctions.. ( ()) () (/), 0. ( ()) () () or all. a. c (00) 900. I ou drive 00 mi in a month, our monthl cost is $900. b. c (). 0. c () is a unction because no input produces more than one output. c (78) 6. You would have a monthl cost o $78 i ou drove 6 mi in a month. c c() c () Problems 7 are application problems with multiple questions and require students to use several skills. Problems structured like this appear in the reeresponse section o the AP Calculus test. I most o our students will take AP Calculus net ear, tr creatin similar problems on our chapter tests in addition to assinin these kinds o problems or homework. See paes or answers to Problems 9, and 7. Section -: Inverse Functions and Parametric Equations

50 Problem Notes (continued) 6a. A(0).88 ; A(0) ; A().9 ; Deer that weih 0, 0, and lb have hides o areas approimatel., 8.6, and.9 t, respectivel. 6b. False. A(0) A(0) 6c. A () (.). 6d The two curves are relections o each other across the line. 7a. d (). Because the 0.07 domain o d is 0, the rane o d is d () 0. A () A() 0 00 = 7b. d (00) 9. This means that a 00-t skid mark is caused b a car movin at a speed o about 9 mi/h. 7d. Because the domain o d now contains neative numbers, the rane o the inverse relation contains neative numbers. Now, because the rane o the inverse relation contains neative numbers,, which is not a 0.07 unction. Problem 8 introduces the horizontal line test to determine invertibilit. It is important to amiliarize students with this visual clue to determinin the invertibilit o unctions. 6. Deer Problem: The surace area o a deer s bod is approimatel proportional to the _ power o the deer s weiht. (This is true because the area is proportional to the square o the lenth and the weiht is proportional to the cube o the lenth.) Suppose that the particular equation or area as a unction o weiht is iven b the power unction A() 0. / where is the weiht in pounds and A() is the surace area measured in square eet. a. Find A(0), A(0), and A(). Eplain the real-world meanin o the answers. b. True or alse: A deer twice the weiht o another deer has a surace area twice that o the other deer. Give numerical evidence to support our answer. c. Find an equation or A ( ), where now stands or area instead o weiht. d. Plot A and A on the same screen usin unction mode. Use a window with 0 0. How are the two raphs related to the line? 7. Brakin Distance Problem: The lenth o skid marks, d() eet, let b a car brakin to a stop is a direct square power unction o, the speed in miles per hour when the brakes were applied. Based on inormation in the Teas Drivers Handbook (00), d() is iven approimatel b d() 0.07 or 0 The raph o this unction is shown in Fiure -m. 0 d() (t) (mi/h) 0 0 Fiure -m a. When police oicers investiate automobile accidents, the use the lenth o the skid marks to calculate the speed o the car at the time it started to brake. Write an equation or the inverse unction, d ( ), where is now the lenth o the skid marks. Eplain wh ou need to take onl the positive square root. b. Find d (00). What does this number represent in the contet o this problem? c. Suppose that the domain o unction d started at 0 instead o zero. With our rapher in parametric mode, plot the raphs o unction d and its inverse relation. Use the window shown in Fiure -m with 0 t 70. Sketch the result. d. Eplain wh the inverse o unction d in part c is not a unction. What relationship do ou notice between the domain and rane o d and its inverse? 8. Horizontal Line Test Problem: The vertical line test o Section -, Problem 9, helps ou see raphicall that a relation is a unction i no vertical line crosses the raph more than once. Th e horizontal line test allows ou to tell whether a unction is invertible. Sketch two raphs, one or an invertible unction and one or a noninvertible unction, that illustrate this test. PROPERTY: The Horizontal Line Test I a horizontal line cuts the raph o a unction in more than one place, then the unction is not invertible because it is not one-to-one. Additional CAS Problems. Find the equations o the inverse unctions (i the eist) or unctions o the orm a, where a can be an nonzero inteer. See pae 979 or answers to Problems 7c and 8, and CAS Problems and. 6 Chapter : a. For what values o a do a unction in this orm and its inverse have the same equation? b. For what values o a are unctions in this orm invertible? From our knowlede o eponents, eplain wh our answer is reasonable.. A student once claimed that () and () were inverse unctions. a. Find ( ()) and ( ()). b. Based on the results to part a, eplain wh the student s claim is not completel correct. c. Under what domain restrictions are and inverses? d. Graph and under the conditions ou determined in part c. How does the raph conirm our answer to part c? 6 Chapter : Functions and Mathematical Models

51 Horizontal relection ( ) Fiure -6a -6 () Vertical relection () Objective EXAMPLE SOLUTION Relections, Absolute Values, and Other Transormations In Section -, ou learned that i (), then multiplin b a nonzero constant causes a horizontal dilation. Suppose that the constant is. Each -value will be /( ) or times what it was in the pre-imae. Fiure -6a shows that the resultin imae is a horizontal relection o the raph across the -ais. The new raph is the same size and shape, simpl a mirror imae o the oriinal. Similarl, a vertical dilation b a actor o relects the raph verticall across the -ais. In this section ou will learn special transormations o unctions that relect their raphs in various was. You will also learn what happens when ou take the absolute value o a unction or o the independent variable. Finall, ou will learn about odd and even unctions. Given a unction, transorm it b relection and b applin absolute value to the unction or its arument. Relections Across the -ais and -ais Eample shows ou how to plot the raphs in Fiure -6a. The pre-imae unction () in Fiure -6a is () 8 7, where. a. Write an equation or the relection o this unction across the -ais. b. Write an equation or the relection o this unction across the -ais. c. Plot the pre-imae and the two relections on the same screen. a. A relection across the -ais is a horizontal dilation b a actor o. So ( ) ( ) 8( ) 7 Substitute or. 8 7 Domain: or Multipl all three sides o the inequalit b. The inequalities reverse. b. () For a relection across the -ais, ind the opposite o (). 8 7 The domain remains. Section -6: 7 Section -6 Class Time da PLANNING Homework Assinment RA, Q Q, Problems, 7, 9 Teachin Resources Eploration -6a: Translation, Dilation, and Relection Blackline Masters Problem Problem Problem Problem Problem 7 Technolo Resources Dilation Presentation Sketch: Relection Present.sp Presentation Sketch: Absolute Value Present.sp Activit: Eplorin Translations and Dilations TEACHING Important Terms and Concepts Relection Relection across the -ais Relection across the -ais Displacement Piecewise unction Absolute value transormations Even unction Odd unction Step discontinuit Greatest inteer unction, Section -6: Relections, Absolute Values, and Other Transormations 7

52 Section Notes This section discusses relections across the - and -aes, absolute value transormations, and even and odd unctions. You can teach this section in one da. Start b discussin Eample, and consider challenin students to sketch the raphs o unctions with which the are amiliar and then to raph () and (). Because () is a horizontal dilation b a actor o, students ma mistakenl think it is a relection across the horizontal ais. Make sure students understand that () represents a relection across the vertical ais, or -ais. Similarl, make sure the understand that althouh () is a vertical dilation, it is a relection across the horizontal ais, or -ais. Beore discussin absolute value transormations, ou ma need to review the deinition o absolute value. The act that or, 0 is troublin to man students because the mistakenl think that alwas represents a neative number. Usin the words the opposite o instead o neative can help clari the deinition or students. Tr sain, I the number inside the absolute value sin is neative, then its absolute value is the opposite o the neative number. You miht illustrate with an eample: I, then () (). The value o () is the opposite o, which is. Spend some time in class eplainin the dierence between the () and transormations. Use the raphs o amiliar unctions to demonstrate how the () transormation aects the rane and the transormation aects the domain o. Finall, ou can introduce the discussion o even and odd unctions b raisin (t) Basket level (t) Floor level Fiure -6b Graphs coincide. Basket level Floor level Fiure -6c 8 Chapter : (s) () (s) c. () 8 7 / ( and ) () 8 7 / ( and ) () 8 7 / ( and ) The raphs are shown in Fiure -6a. You can check the alebraic solutions b plottin () ( ) and () () usin thick stle. The raphs should overla () and (). PROPERTY: Relections Across the Coordinate Aes ( ) () is a vertical relection o unction across the -ais. ( ) ( ) is a horizontal relection o unction across the -ais. Absolute Value Transormations Suppose ou shoot a basketball. While in the air, it is above the basket level sometimes and below it at other times. Fiure -6b shows (), the displacement rom the level o the basket as a unction o time. I the ball is above the basket, its displacement is positive; i the ball is below the basket, its displacement is neative. Distance, however, is the manitude (or size) o the displacement, which is never neative. Distance equals the absolute value o the displacement. The solid raph in Fiure -6c is the raph o () (). Takin the absolute value o () retains the non-neative values o and relects the neative values verticall across the -ais. Fiure -6d shows what happens or (), or which ou take the absolute value o the arument (this is a dierent unction than in the last eample). For positive values o,, so () () and the raphs coincide. For neative values o,, so () ( ), across the -ais o the part o unction where 0. Notice that the raph o or the neative values o is not a part o the raph o. The equation or () can be written this wa: () () ( ) these questions: For which unctions will the () transormation result in the oriinal unction, and or which does it ive the opposite o the oriinal unction? In other words, or which unctions is () (), and or which unctions is () ()? Ask students to ive eamples (such as k,,, ), and perhaps have them discuss the question in small roups. Once students have reconized the i 0 i 0 Divide b a Boolean variable or enter the domain directl, dependin on our rapher, to restrict the domain. 6 and coincide. Fiure -6d properties o unctions that satis these equations (unctions smmetrical to the -ais or to the oriin), ou can introduce the terms even and odd. I ou run out o time, ou could assin this question as homework. Trionometric unctions are either odd or even unctions. For eample, sin() sin and cos() cos. Studin odd and even polnomial unctions now will prepare students or 8 Chapter : Functions and Mathematical Models

53 Odd unction - or -relection Fiure -6 Because there are two dierent rules or () in dierent pieces o the domain, is called a piecewise unction o. PROPERTY: Absolute Value Transormations The transormation () () across the -ais i () is neative unchaned i () is nonneative The transormation () unchaned or nonneative values o to the correspondin neative values o or neative values o Even Functions and Odd Functions Fiure -6e shows the raph o (), a polnomial unction with onl even eponents. (The number equals 0, which has an even eponent.) Fiure -6 shows the raph o () 6, a polnomial unction with onl odd eponents. What smmetries do ou observe? Even unction a 8 a ( a) (a () + () + 6 Fiure -6e Fiure -6 Section -6: Odd unction 8 ( a) (a) (a) a a (a) Relectin the raph o the even unction () horizontall across the -ais leaves the raph unchaned. You can see this alebraicall iven the propert o powers with even eponents. ( ) ( ) ( ) Substitute or. ( ) ( ) () Neative number raised to an even power. Fiure -6 shows that relectin the raph o the odd unction () 6 horizontall across the -ais has the same eect as relectin it verticall across the -ais. Alebraicall, ( ) ( ) 6( ) Substitute or. ( ) 6 ( ) () studin trionometric unctions and their properties. Eploration -6a ma be used as a ollow-up assinment or quiz to see i students have learned about the transormations o this section. Neative number raised to an odd power. 9 Dierentiatin Instruction Discussin displacement and distance usin an enlarement o Fiures -6c and -6d will help clari the dierence between these two concepts. I ou assin Problem, help ELL students understand the vocabular. Have ELL students write a summar o the section rather than doin the Readin Analsis questions. I the write in their primar lanuae, have them translate a couple o sentences or ou. Eploration Notes Eploration -6a presents students with eiht transormations o the same unction. (The unction is cubic with a restricted domain, but this act is not mentioned in the eploration.) In each case, students are presented with an equation or in terms o unction, and the are asked to name the transormation and sketch the transormed raph. Allow about 0 minutes. Technolo Notes Problem asks students to use a Dnamic Precalculus Eploration at The Dilation eploration allows students to eplore relections across the - and -aes b chanin one or both o the dilation sliders to. Presentation Sketch: Relection Present.sp at demonstrates the eects o relectin the raph o a square root unction across coordinate aes. The presentation demonstrates the eects on the coordinates as well as on the alebraic description o the unction. Presentation Sketch: Absolute Value Present.sp at demonstrates the raphical eect o composin a quadratic unction with absolute value. Section -6: Relections, Absolute Values, and Other Transormations 9

54 Technolo Notes (continued) Activit: Eplorin Translations and Dilations in the Instructor s Resource Book ocuses on translations and dilations, but ou could etend it to include investiations o relections. CAS Suestions Functions can be deined and transormed directl. For eample, to raph the relection imae o () 7 across the -ais, deine () on a calculator screen and raph (). To veri whether unctions are even or odd usin a CAS, deine the unction and use Boolean operators to determine i () () or () (). A Boolean true result will be displaed onl i the equation is universall true or the iven unction. I the equation is not universall true, then the unction is not even or odd, and the CAS thinks an equation is bein deined and displas the result. Note: Perormin the alebraic manipulations required to determine i a unction is even or odd is aruabl a dierent skill than understandin that () () or all even unctions and () () or all odd unctions. A CAS allows ou to assess students understandin independent o their abilit to perorm the alebraic manipulations. Problem Set -6 An unction havin the propert ( ) () is called an even unction. An unction havin the propert ( ) () is called an odd unction. These names appl even i the equation or the unction does not have eponents. DEFINITION: Even Function and Odd Function The unction is an even unction i and onl i ( ) () or all in the domain. The unction is an odd unction i and onl i ( ) () or all in the domain. Note: For odd unctions, relection across the -ais ives the same imae as relection across the -ais. For even unctions, relection across the -ais is the same as the pre-imae. So odd unctions are smmetric about the oriin, and even unctions are smmetric across the -ais. Most unctions do not possess the propert o oddness or evenness. Readin Analsis Q8. Write the deinition o a one-to-one unction. From what ou have read in this section, what do ou consider to be the main idea? Reread the pararaph on pae 8 that discusses Fiure -6d. Use numerical values rom the raph to uide oursel throuh this pararaph. Eplain in our own words what the sentence about neative values o means. Wh is part o the raph o lost in the raph o? Write down speciic questions about what ou ma not understand, and ind someone who can answer them. min Quick Review Q. I (), then ()?. Q. I (), then ()?. Q. I (), then ()?. Q. I (), write the equation or the inverse relation. Q. Eplain wh the inverse relation in Problem Q is not a unction. Q6. I (), then (8)?. Q7. I the inverse relation or unction is also a unction, then is called?. Q9. Give a number or which. Q. Give a number or which. For Problems, sketch the raphs o a. ( ) () b. h() ( ) c. a() () d. v().. () () () () Q. Q. Q. Q. PROBLEM NOTES Q. There are two -values or ever positive -value. Q6. 0 Chapter : Q7. Invertible Q8. A unction or which each -value in the rane corresponds to onl one -value Q9. Sample Answer: Q. Sample Answer: Problems provide practice in applin the transormations introduced in this section to the raphs o unctions. Blackline masters or these problems are available in the Instructor s Resource Book. a. 0 Chapter : Functions and Mathematical Models

55 . The equation or the unction in Problem is () 6 or 6. Plot the unction as () on our rapher. Plot () usin thick stle. Does the result conirm our answer to Problem, part d? 6. The equation or the unction in Problem is () ( ) or 7. Plot the unction as () on our rapher. Plot () usin thick stle. Does the result conirm our answer to Problem, part d? 7. Absolute Value Transormations Problem: Fiure -6h shows the raph o () 0.( ). in the domain 6. Fiure -6h a. Plot the raph o () (). On the same screen, plot () () usin thick stle. Sketch the result and describe how this transormation chanes the raph o. b. Deactivate (). On the same screen as (), plot the raph o () usin thick stle. Sketch the result and describe how this transormation chanes the raph o. c. Use the equation or unction to ind the value o () and the value o. Show that both results aree with our raphs in parts a and b. Eplain wh is in the domain o even thouh it is not in the domain o itsel. d. Fiure -6i shows the raph o a unction, but ou don t know the equation or the unction. On a cop o this iure, sketch the raph o (), usin the conclusion ou reached in part a. On another cop o this Section -6: b. c. iure, sketch the raph o, usin the conclusion ou reached in part b. Fiure -6i 8. Displacement vs. Distance Absolute Value Problem: Calvin s car runs out o as as he is oin uphill. He continues to coast uphill or a while, stops, then starts rollin backward without applin the brakes. His displacement,, in meters, rom a as station on the hill as a unction o time,, in seconds, is iven b 0. 0 a. Plot the raph o this unction. Sketch the result. b. Find Calvin s displacement at s and at 0 s. What is the real-world meanin o his neative displacement at s? c. What is Calvin s distance rom the as station at times s and 0 s? Eplain wh both values are positive. d. Deine Calvin s distance rom the as station. Sketch the raph o distance versus time. e. I Calvin keeps movin as indicated in this problem, when will he pass the as station as he rolls back down the hill? 9. Even Function and Odd Function Problem: Fiure -6j shows the raph o the even unction (). Fiure -6k on the net pae shows the raph o the odd unction () Fiure -6j d. a. b. c. Problems and 6 ask students to use their raphers to plot the unctions in Problems and and to veri the results or parts d.. The raphs match. 6. The raphs match. Problems 6 and 7 can be handled usin an rapher, but a CAS allows the unctions to be deined and transormed directl. A blackline master or Problem 7d is available in the Instructor s Resource Book. Problems 7 provide practice or and promote understandin o the concepts o absolute value transormations and even and odd unctions. I ou have time ater discussin the concepts with them, have students work in small roups to solve these problems in class. See paes or answers to Problems d, 7 and 8. Section -6: Relections, Absolute Values, and Other Transormations

56 Problem Notes (continued) 9a. Problems 9c, 9d, and can be solved usin direct deinitions. 8 8 The polnomial unction () is the sum o even powers o. A neative number raised to an even power is equal to the absolute value o that number raised to the same power. So, or, the same correspondin -value occurs, and thereore () (). 9b. 8 A neative number raised to an odd power is equal to the opposite o the absolute value o that number raised to the same power. Because each term in () is a monomial in raised to an odd power, () has the same eect on () as (). 9c. Function h is odd; unction j is even. 9d Fiure -6k a. On the same screen, plot () () and () ( ). Use thick stle or (). Based on the properties o neative numbers raised to even powers, eplain wh the two raphs are identical. b. Deactivate () and (). On the same screen, plot () (), () ( ), and () (). Use thick stle or (). Based on the properties o neative numbers raised to odd powers, eplain wh the raphs o () and () are identical. c. Even unctions have the propert ( ) (). Odd unctions have the propert ( ) (). Fiure -6l shows two unctions, h and j, but ou don t know the equation o either unction. Tell which unction is an even unction and which is an odd unction. h j Fiure -6l d. Let e(). Sketch the raph. Based on the raph, is unction e an odd unction, an even unction, or neither? Conirm our answer alebraicall b indin e( ).. Absolute Value Function Odd or Even? Plot the raph o (). Sketch the result. Based on the raph, is unction an odd unction, an even unction, or neither? Conirm our answer alebraicall b indin ( ).. Step Discontinuit Problem: Fiure -6m shows the raph o () The raph has a step discontinuit at 0, where () jumps instantaneousl rom to. a. Plot the raph o () (). Use a window that includes 0 as a rid point. Does our raph aree with the iure? b. Fiure -6n is a vertical dilation o unction with vertical and horizontal translations. Enter an equation or this unction as (), usin operations on the variable (). Use a window that includes as a rid point. When ou have duplicated the raph in Fiure -6n, write an equation or the transormed unction in terms o unction. 8 Fiure -6m Fiure -6n Fiure -6o c. Fiure -6o shows the raph o the quadratic unction ( ) to which somethin has been added or subtracted to ive it a step discontinuit o units at. Find an equation o the unction. Veri that our equation is correct b plottin it on our rapher. The unction e() is neither odd nor even. e() e(). Chapter : Problems are important to assin and discuss with students because o the siniicance o the concepts o discontinuit, piecewise unctions, the reatest inteer unction, and some interestin applications. The unction is an even unction. () () Chapter : Functions and Mathematical Models

57 . Step Functions The Postae Stamp Problem: Fiure -6p shows the raph o the reatest inteer unction, (). In this unction, is the reatest inteer less than or equal to. For instance,.9,, and.. Fiure -6p a. Plot the reatest inteer unction usin dot stle so that points will not be connected. Most raphers use the smbol int() or. Trace to.9,, and.. What do ou ind or the three -values? b. In the ear 00, the postae or a irst-class letter was 7 cents or weihts up to oz and cents more or each additional ounce or raction o an ounce. Sketch the raph o this unction c. Usin a transormation o the reatest inteer unction, write an equation or the 00 postae as a unction o the weiht. Plot it on our rapher. Does the raph aree with the one ou sketched in part b? d. In 00, irst-class postae rates applied onl until the letter reached the weiht at which the postae would eceed $.. What is the domain o the unction in part c? e. Check the Internet or another source to ind the irst-class postae rates this ear. What dierences do ou ind rom the 00 rates? Cite the source ou used.. Piecewise Functions Weiht Above and Below Earth s Surace Problem: When ou are above the surace o Earth, our weiht is inversel proportional to the square o our distance rom the center o Earth. This is because the arther a. The raphs match. Section -6: b. () ; () ( ) c. () ( ) The raphs match. awa ou are, the weaker the ravitational orce between Earth and ou. When ou are below the surace o Earth, our weiht is directl proportional to our distance rom the center. At the center ou would be weihtless because Earth s ravit would pull ou equall in all directions. Fiure -6q shows the raph o the weiht unction or a -lb person. The radius o Earth is about 000 mi. The weiht is called a piecewise unction o the distance because it is iven b dierent equations in dierent pieces o the domain. Each piece is called a branch o the unction. The equation o the unction can be written a i b i (lb) 000 Fiure -6q 8000 (mi) a. Find the values o a and b that make when 000 or each branch. b. Plot the raph o. Use piecewise unctions or Boolean variables to restrict the domain o the raph. c. Find i 000 and i 000. d. Find the two distances rom the center at which the weiht would be 0 lb.. Dnamic Relection Problem: Go to and open the Dilation eploration. Set slider c equal to and slider d equal to and describe what ou observe. Then set slider c equal to and slider d equal to and describe what ou observe. Finall, set both sliders equal to and describe what ou observe. Eplain how relections are related to dilations. a. (.9), (), (.) b Price (cents) Weiht (oz) c. Dilated b a actor o ; translated up 7 cents; 0, 0 7, 0 The raphs match. d. 0 e. Answers will var. a. a 0.07; b,00,000,000 b c. (000). lb; (000) 96 lb d. () 0.. _ mi; () mi Problem reers students to and ives them opportunit to work with relections dnamicall.. Answers will var. Additional CAS Problems. For an unction (), is () ( ) even, odd, or neither? Eplain our response both alebraicall and with respect to the transormation involved.. For an unction (), is h() () even, odd, or neither? Eplain our response both alebraicall and with respect to the transormation involved. See pae 980 or answers to CAS Problems and. Section -6: Relections, Absolute Values, and Other Transormations

58 Section -7 PLANNING Class Time da TEACHING -7 Precalculus Journal In this chapter ou have been learnin mathematics raphicall, numericall, and alebraicall. An important abilit to develop or an subject ou stud is to verbalize what ou have learned, both orall and in writin. To ain verbal practice, ou should start a journal. In it ou will record topics ou have studied and topics about which ou are still unsure. The word journal comes rom the same word as the French jour, meanin da. Journe has the same root and means a da s travel. Your journal will ive ou a written record o our travel throuh mathematics. Section Notes This section introduces writin in a journal. In recent ears, collee and hih school mathematics curricula and standardized tests have placed increased emphasis on havin students verbalize mathematical concepts and ideas. Students will improve their writin skills i ou require them to write periodicall. In addition to assinin journal entr problems or homework, ou ma want to include writin questions on chapter tests. Establish uidelines or how journal entries will be raded. Here is a sample. Answer the writin prompt. Write at least our complete sentences. Be speciic. Use at least three new vocabular words. Eplain their meanins, and ive eamples. Give real-world eamples or the applications o the new concepts and procedures. Eemplar journal entries receive etra credit. Vaue, ramblin statements receive no credit. Have students read this section, includin the sample journal entr. Discuss with students whether the sample meets all o our uidelines. Assin Problem. Allow 0 minutes o class time or students to write. Then call on volunteers to read their entries. Point out somethin in each volunteer s entr that meets the uidelines, is unusuall well written, or sheds new insiht. Avoid neative comments. Chapter : Functions and Mathematical Models Objective Problem Set -7 Start writin a journal in which ou can record thins ou have learned about precalculus mathematics and questions ou have concernin concepts about which ou are not quite clear. You should use a bound notebook or a spiral notebook with lare inde cards or paes so that our journal will hold up well under dail use. Researchers use such notebooks to record their indins in the laborator or in the ield. Each entr should start with the date and a title or the topic. A tpical entr miht look like this sample. Topic: Inverse o a Function 9/ I ve learned that ou invert a unction b interchanin the variables. Sometimes an inverse is a relation that is not a unction. I it is a unction, the inverse o = () is = (). At irst, I thouht this meant () but ater losin points on a quiz, I realized that wasn t correct. The raphs o and are relections o each other across the line =, like this:. Start a journal or recordin our thouhts about precalculus mathematics. The irst entr should invert the raph o a unction include thins such as these: Sketches o raphs rom real-world as its potential useulness to ou inormation Familiar kinds o unctions rom overcame previous courses An topics about which ou are still unsure Chapter : You ma want to assin the Section -8 notes, especiall when the have inall Chapter Review problems the same da ou mastered diicult topics. eplain Section -7 so students who inish their journal entries earl can start workin on those problems. Alternativel, ou ma want to assin the journal as homework ollowin our test on Chapter. Be sure our students realize that the journal is not the place to write their class notes. Rather, it is the place to record briel some thins the can distill rom their Consider allowin students to use their journals on tests. For instance, the test on trionometric identities in Chapter 7 could have a part where students use the properties and techniques or provin identities that the have recorded in their journals. Periodicall ask students to read rom their journals. The last ive minutes o class are a

59 -8 Chapter Review and Test In this chapter ou saw how ou can use unctions as mathematical models alebraicall, raphicall, numericall, and verball. Functions describe a relationship between two variable quantities, such as distance and time or a movin object. Functions deined alebraicall are named accordin to the wa the independent variable appears in the equation. I is an eponent, the unction is an eponential unction, and so orth. You can transorm the raphs o unctions b dilatin and translatin them in the - and -directions. Some o these transormations relect the raph across the - or -ais or the line. A ood understandin o unctions will prepare ou or later courses in calculus, in which ou will learn how to ind the rate o chane o as varies. You ma continue our stud o precalculus mathematics either with periodic unctions in Chapters throuh 9, which will probabl be quite new to ou, or with the ittin o other unctions to real-world data in Chapters throuh, which ma be more amiliar to ou rom previous courses. The Review Problems are numbered accordin to the sections o this chapter. Answers are provided at the back o the book. The Concept Problems allow ou to appl our knowlede to new situations. Answers are not provided, and, in some chapters, ou ma be required to do research to ind answers to open-ended problems. The Chapter Test is more like a tpical classroom test our instructor miht ive ou. It has a calculator part and a noncalculator part, and the answers are not provided. Section -8 PLANNING Class Time das (includin da or testin) Homework Assinment Da : R R6, T T8 Da (ater Chapter Test): Problems C C and Problem Set - Teachin Resources Blackline Masters Problems R, R, R6 Problem C Problems T T6 Test, Chapter, Forms A and B TEACHING Review Problems R. Punctured Tire Problem: For parts a d, suppose b. The alebraic equation or the unction in that our car runs over a nail. The tire s air Fiure -8a is pressure,, in pounds per square inch (psi), 0. 7 decreases with time,, in minutes, as the air leaks out. A raph o pressure versus time is Make a table o numerical values o pressure shown in Fiure -8a. or times o 0,,,, and min. c. Suppose the equation in part b ives (psi) reasonable answers until the pressure drops to psi. At that pressure, the tire comes loose rom the rim and the pressure drops to zero. What is the domain o the unction described b this equation? What is the (min) correspondin rane? 6 8 d. The raph in Fiure -8a ets closer and Fiure -8a closer to the -ais but never quite touches a. Find raphicall the pressure ater min. it. What special name is iven to the -ais Approimatel how man minutes can ou in this case? drive beore the pressure reaches psi? Section -7 (continued) ood time or this activit. It s ood to keep track o which students have participated so that all students have a turn. Dierentiatin Instruction This section will be challenin or some students, but it will also help them tremendousl to relect on and eplain Section -8: what the ve learned and what the do and do not understand ull. Because students ma never have written journals in an course, the ma need help ettin started. Let ELL students write in their primar lanuae. You can check their work b askin them to translate pieces.. Answer will var. Important Terms and Concepts Sine unction Periodic unction Period o a periodic unction Section Notes The last section o each chapter includes a set o review problems, numbered accordin to the sections in the chapter, and a sample chapter test, which can also be used or review. Most chapters also have a set o concept problems that etend the concepts in the chapter or introduce concepts or the net chapter. Concept problems make ecellent roup activities or projects. Answers or the review problems are provided in the back o the book so that students can monitor their own proress. Answers are not provided or the chapter test or concept problems. You ma want to assin Section -7 and Section -8 on the same da. This would also be a ood time to assin an problems ou will have due the da ater a chapter test. See pae 980 or answers to Problem Ra d. Section -8: Chapter Review and Test

60 Dierentiatin Instruction Go over the review problems in class, perhaps b havin students present their solutions. You miht assin students to write up their solutions beore class starts. Because man cultures norms hihl value helpin peers, ELL students oten help each other on tests. You can limit this tendenc b makin multiple versions o the test. Consider ivin a roup test the da beore the individual test, so that students can learn rom each other as the review, and the can identi what the don t know prior to the individual test. Give a cop o the test to each roup member, have them work toether, then randoml choose one paper rom the roup to rade. Grade the test on the spot, so students know what the need to review urther. Make this test worth _ the value o the individual test, or less. ELL students ma not be used to the tpe o eam iven in this course. Doin the chapter test in the book will help them et used to the ormat and tpe o questions the will be epected to answer. ELL students ma need more time to take the test. ELL students will beneit rom havin access to their bilinual dictionaries while takin the test. PROBLEM NOTES e. Earthquake Problem: Earthquakes happen when rock plates slide past each other. The stress between plates that builds up over a number o ears is relieved b the quake in a ew seconds. Then the stress starts buildin up aain. Sketch a reasonable raph showin stress as a unction o time. In 989, a manitude 7. earthquake struck Northern Caliornia, destroin houses in San Francisco's Marina district. R. For parts a e, name the kind o unction or each equation iven. a. () 7 b. () 7 c. (). d. (). e. (). Name a pair o real-world variables that could be related b the unction in part a.. I the domain o the unction in part a is, what is the rane? h. In a lu epidemic, the number o people currentl inected depends on time. Sketch a reasonable raph o the number o people inected as a unction o time. What kind o unction has a raph that most closel resembles the one ou drew? i. For Fiures -8b throuh -8d, what kind o unction has the raph shown? Fiure -8b Fiure -8c Fiure -8d j. Eplain how ou know that the relation raphed in Fiure -8e is a unction but the relation raphed in Fiure -8 is not a unction. Fiure -8e Fiure -8 R. a. For unctions and in Fiure -8, identi how the pre-imae unction (dashed) was transormed to et the imae unction (solid). Write an equation or () in terms o iven that the equation o is () Conirm the result b plottin the imae and the pre-imae on the same screen on our rapher. Fiure -8 Fiure -8h b. I () ( ), eplain how unction was transormed to et unction. Usin the pre-imae in Fiure -8h, sketch the raph o on a cop o this iure. R. Heiht and Weiht Problem: For parts a e, the weiht o a rowin child depends on his or her heiht, and the heiht depends on ae. Assume that the child is 0 in. when born and rows in. per ear. a. Write an equation or h(t) (in inches) as a unction o t (in ears). b. Assume that the weiht unction W is iven b the power unction W(h(t)) 0.00h(t ).. Find h(), and use the result to calculate the predicted weiht o the child at ae. c. Plot the raph o W(h(t)). Sketch the result. Ra. Rb. Rc. Rd. Re. Linear Polnomial (cubic) Eponential Power Rational R. Answers will var; e.., number o items manuactured and total manuacturin cost. R. () 7 Rh. 6 Chapter : A quadratic unction (with a neative -coeicient) its this pattern. Ri. -8b: eponential; -8c: polnomial (probabl quadratic); -8d: power Rj. Fiure -8e passes the vertical line test, so no -value corresponds to more than one -value. Fiure -8 ails the vertical line test, so more than one -value corresponds to the same -value. Problems R and R both require students to answer questions about unctions based on their raphs. A blackline master or these problems is available in the Instructor s Resource Book. 6 Chapter : Functions and Mathematical Models

61 d. Assumin that the heiht increases at the constant rate o in. per ear, does the weiht also increase at a constant rate? Eplain how ou arrived at our answer. e. What is a reasonable domain or t or the composite unction W h?. Composite Functions Numericall Problem: Functions and are deined onl or the values o in the table. () () Find these values, or eplain wh the are undeined: ( ()), ( ()), ( ()), ( (6)), ( (6)), ( ()), and ( ()). Two Linear Functions Problem: For parts j, let unctions and be deined b () 8 () 6. Plot the raphs o,, and ( ()) on the same screen. Sketch the results. h. Find ( ()). i. Show that ( ()) is undeined, even thouh () is deined. j. Calculate the domain o the composite unction and show that it arees with the raph ou plotted in part. R. Fiure -8i shows the raph o () in the domain. () Fiure -8i a. On a cop o the iure, sketch the raph o the inverse relation. Eplain wh the inverse is not a unction. b. Plot the raphs o and its inverse relation on the same screen usin parametric equations. Also plot the line. How are the raphs o and its inverse relation related to the line? How are the domain and rane o the inverse relation related to the domain and rane o unction? c. Write an equation or the inverse o the unction b interchanin the variables. Solve the new equation or in terms o. How does this solution reveal that there are two dierent -values or some -values? d. On a cop o Fiure -8j, sketch the raph o the inverse relation. What propert does the unction raph have that allows ou to conclude that the unction is invertible? What are the vertical lines at and at called? Fiure -8j Rc. Rd. No; the raph is curved. Re. Sample answer: 0 t R. () 6; () ; () (8), which is undeined; (6) ; (6) 8; () () which is undeined; () R. Rh. () Ri. () is undeined because () is not in the domain o. Rj. The domain is 7_, which arees with the raph. Ra. The inverse does not pass the vertical line test. Rb. 0 8 t Ra. Horizontal dilation b a actor o, vertical translation b ; () Rb. Horizontal translation b, vertical dilation b a actor o Section -8: 7 Problem Rj can be deined alebraicall with domain restrictions and the inal domain read rom the raph o the composition. This is not a proo, but it does oer ver stron evidence or the inal result. Ra. h(t) t 0 Rb. h() in.; W h() 9 lb The raphs are each other s relections across the line. The domain o corresponds to the rane o the inverse relation. The rane o corresponds to the domain o the inverse relation. Problem Rc can be solved directl on a CAS. See pae 98 or answers to Problems Re, Rc and Rd. Section -8: Chapter Review and Test 7

62 Problem Notes (continued) Re. Grapher raph arees with raph on paper; Not a unction because ever in the domain has multiple values o. R. The curve is invertible because it is increasin. As the input to v, represents radius in meters. As the input to v, it represents volume in cubic meters. I 0 is a particular input to v, then 0, v( 0 ) is a point on the raph o v(). Pluin the output, v( 0 ), into v ives the point v( 0 ), v v( 0 ) on the raph o v (). But the raph o v () is just the raph o v() with all the - and -values echaned, so this point is actuall v( 0 ), 0. Thus, v v( 0 ) 0. R. Since no corresponds to more than one in the oriinal unction, no corresponds to more than one in the inverse relation, so the inverse relation is a unction. Sample raph: A blackline master or Problem R6a is available in the Instructor s Resource Book. See pae 98 or answers to Problem R6a. e. Plot these parametric equations on raph paper, usin each inteer value o t rom to. Conirm the results b plottin on our rapher. Is a unction o? Eplain. t t. Spherical Balloon Problem: The table shows the volume o a spherical balloon, v(), in cubic meters, as a unction o its radius,, in meters. 8 Chapter : (m) v () ( m ) Plot unction v on raph paper b plottin v() or these points and connectin the points with a smooth curve. What evidence do ou have that unction v is invertible? Plot the raph o v () on the same aes. What is the dierence in the meanin o as the input or unction v and as the input or unction v? Eplain wh v (v()) equals. Echo, the irst communication satellite developed b NASA, was a iant metal balloon that loated in orbit. It was used to bounce sound sinals rom one place on Earth to another.. Sketch the raph o a one-to-one unction. Eplain wh it is invertible. Problem R6b can be deined in piecewise mode on a TI-Nspire or TI-Nspire CAS. Usin separate () and () deinitions is unnecessar. R6b. The raph arees with Fiure -8k; each o the raphs arees with those in part a. R6c. Because power unctions with odd powers satis the propert () () and power unctions with even powers satis the propert () () R6d. R6. a. On our copies o () in Fiure -8k, sketch the raphs o these our unctions: (), ( ), (), and. Fiure -8k () b. Function in part a is deined piecewise b () 6 7 Plot the two branches o this unction as () and () on our rapher. Does the raph aree with Fiure -8k? Plot b plottin () () and () (). Does the raph aree with our result in the correspondin portion o part a? c. Eplain wh unctions with the propert ( ) () are called odd unctions and unctions with the propert ( ) () are called even unctions. d. Plot the raph o () 0. Use a window that includes as a rid point. Sketch the result. Name the eature that appears at. R7. In Section -7 ou started a precalculus journal. In what was do ou think keepin this journal will help ou? How could ou use the completed journal at the end o the course? What is our responsibilit throuhout the ear to ensure that writin the journal has been a worthwhile project? Discontinuit R7. Answers will var. 6 8 Chapter : Functions and Mathematical Models

63 Concept Problems C. Four Transormations Problem: Fiure -8l shows a pre-imae unction (dashed) and a transormed imae unction (solid). a. The sine unction is an eample o a periodic unction. Wh do ou think this name is iven to the sine unction? Dilations and translations were perormed in b. Th e period o a periodic unction is the both directions to et the raph or. Fiure dierence in -values rom a point on the out what the transormations were. Write an equation or ( ) in terms o. Let () raph to the point where the raph irst starts repeatin itsel. Approimatel what does the with domain. Plot the raph o period o the sine unction seem to be? on our rapher. Does our rapher aree with the iure? c. Is the sine unction an odd unction, an even unction, or neither? How can ou tell? d. On a cop o Fiure -8m, sketch a vertical dilation o the sine unction raph b a actor o. What is the equation o this transormed unction? Check our answer b plottin the sine raph and the transormed imae raph on the same screen. e. Fiure -8n shows a two-step transormation o the sine raph in Fiure -8m. Name the two transormations. Write an equation or the transormed unction, and check our answer b plottin both unctions on our rapher. Fiure -8l C. Sine Function Problem: I ou enter ( ) sin() into our rapher and plot the raph, the result resembles Fiure -8m. (Your rapher should be in radian mode.) The unction is called the () sine unction (pronounced sin ), which ou () will stud startin in Chapter. C. Horizontal dilation b, vertical dilation b, horizontal translation b, vertical translation b ; () Problem C introduces the sine unction, which students will stud in Chapter. In the problem, students look at the periodic behavior o the unction, determine whether it is odd or even, and eplore transormations o the unction. A blackline master or this problem is available in the Instructor s Resource Book. Ca. Answers will var. The unction repeats itsel periodicall. Cb. About 6., or Cc. Odd. It is its own relection throuh the oriin, so () (). Cd. Fiure -8m () Fiure -8n. Let () sin. What transormation would ( ) sin _ be? Check our answer b plottin both unctions on our rapher. sin() Ce. Horizontal translation, vertical translation ; sin( ) C. Horizontal dilation b Section -8: 9 Section -8: Chapter Review and Test 9

64 Problem Notes (continued) T. Eponential T. Linear T. Polnomial (quadratic) T. Power Chapter Test Part : No calculators allowed (T T) For Problems T T, name the tpe o unction that each raph shows. T. () T. () T. T. All ecept T; Functions that are not one-to-one are not invertible; that is, their inverses are not unctions. T6. Answers will var. Sample Answer: T. () T. () Temperature T. T7. Odd T8. Neither Time T9. Horizontal dilation b ; () T. Horizontal translation b, vertical translation b ; () ( ) T. Horizontal translation b 6, vertical dilation b ; () ( 6) T. Domain: 7; rane: 9 Blackline masters or Problems T T6 are available in the Instructor s Resource Book. Problems T T6 can be solved directl i students take the time to ind a domain-restricted alebraic representation o the iven raph. T. Vertical dilation b T. Which o the unctions in Problems T T are one-to-one unctions? What conclusion can ou make about a unction that is not one-to-one? T6. When ou turn on the hot water aucet, the time the water has been runnin and the temperature o the water are related. Sketch a reasonable raph o this unction. For Problems T7 and T8, tell whether the unction is odd, even, or neither. T7. () T8. () For Problems T9 T, describe how the raph o (dashed) was transormed to et the raph o (solid). Write an equation or () in terms o. T9. 60 Chapter : T. Horizontal dilation b Part : Graphin calculators allowed (T T9) T. Fiure -8o shows the raph o a unction, (). Give the domain and the rane o. Fiure -8o T. Horizontal translation b, vertical translation b 60 Chapter : Functions and Mathematical Models

65 For Problems T T6, sketch the indicated transormations on copies o Fiure -8o. Describe the transormations. T. () T. T. ( ) T6. The inverse relation o () T7. Eplain wh the inverse relation in Problem T6 is not a unction. T8. Let (). Let (). Find ( ()). Find (). Eplain wh () is not a real number, even thouh () is a real number. T9. Use the absolute value unction to write a sinle equation or the discontinuous unction raphed in Fiure -8p. Check our answer b plottin it on our rapher. 8 Fiure -8p T0. Plot these parametric equations on raph paper, usin each inteer value o t rom to. Conirm the results b plottin them on our rapher. Is a unction o? Eplain. t t Wild Oats Problem: Problems T T8 reer to the competition o wild oats, a kind o weed, with the wheat crop. Based on data in A. C. Madett s book Applications o Mathematics: A Nationwide Surve, the percent loss in wheat crop, L(), is approimatel L(). 0. where is the number o wild oat plants per square meter o land. T6. Relection throuh the line T. Describe how L() varies with. What kind o unction is L? T. Find L(). Eplain verball what this number means. T. Suppose the wheat crop is reduced to 60% o what it would be without the wild oats. How man wild oats per square meter are there? T. Let L(). Find an equation or L (). For what kind o calculations would the equation L () be more useul than L()? T. Find L (0). Eplain its real-world meanin. T6. Based on our answer to Problem T, what would be a reasonable domain and rane or L? T7. Plot () L() and () L () on the same screen. Use equal scales or the two aes. Use the domain and rane rom Problem T. Sketch the results alon with the line. T8. How can ou tell that the inverse relation is a unction? T9. What did ou learn as a result o takin this test that ou didn t know beore? Section -8: 6 Problem T7 can be solved directl i students ind the alebraic representation or Problems T T6. T7. The raph ails the vertical line test. (The pre-imae raph ails the horizontal line test it is not one-to-one.) T8. () () ; () ; () (), which is not deined, because is not in the domain o. T9. Horizontal translation b, vertical translation b, and vertical dilation b o ; T0. Grapher raph arees with raph on paper. Function, because it passes the vertical line test. Problem T, T, and T can be ound usin substitution or a Solve command on a CAS. T. L() varies proportionatel to the 0. power o. Power unction. T. L().( ) 0..8 I there are wild oat plants per square meter o land, the percentae loss to the wheat crop will be about %. T. About 9 plants per square meter T I ou know the percentae loss and want to ind the number o wild oat plants per square meter. T. L (0) I the crop loss is 0% (i.e., the total crop is lost), there must have been about 70 wild oat plants (or more) per square meter. T 6. Domain: 0 70; rane: 0 0 T7. 00 L () L() 00 T8. It passes the vertical line test. (The oriinal unction passes the horizontal line test it is one-to-one.) T9. Answers will var. Section -8: Chapter Review and Test 6