The Reciprocal Function Family. Objectives To graph reciprocal functions To graph translations of reciprocal functions

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1 - The Reciprocl Function Fmil Objectives To grph reciprocl functions To grph trnsltions of reciprocl functions Content Stndrds F.BF.3 Identif the effect on the grph of replcing f () b f() k, kf(), f(k), nd f( k) for specific vlues of k... A.CED. Crete equtions in two or more vribles to represent reltionships between quntities... Also A.APR. T C A DYNAMIC I V I E S I T Dnmic Activit Grphing Trnsltions of Inverse Vritions For clss prt, the students will shre the cost for the hll rentl. Ech student will lso hve to p $ for food. The cost of the hll rentl is lred grphed. Wht effect does the food cost hve on the grph? Eplin our resoning. Cost per student ($) Cost of Clss Prt (0, 6) (60, ) (0, 3) (00,.) O Number of students Lesson Vocbulr reciprocl function brnch Functions tht model inverse vrition hve the form f () 5, where 0. The belong to fmil whose prent is the reciprocl function f () 5, where 0. Essentil Understnding Trnsformtions of the prent reciprocl function include stretches, compressions (or shrinks), reflections, nd horizontl nd verticl trnsltions. Ke Concept Generl Form of the Reciprocl Function Fmil The generl form of member of the reciprocl function fmil is 5 h k, where h. The inverse vrition functions, 5, re stretches, shrinks, nd reflections of the prent reciprocl function, depending on the vlue of. The grph of the prent reciprocl function 5 is shown t the right. O Lesson - The Reciprocl Function Fmil 507

2 Wht vlues should ou choose for? Choose vlues of tht divide nicel into. Mke tble of points tht re es to grph. Problem Grphing n Inverse Vrition Function Wht is the grph of 5, u 0? Identif the - nd -intercepts nd the smptotes of the grph. Also, stte the domin nd rnge of the function. Step Mke tble of vlues tht Step Grph the points. includes positive nd negtive vlues of O Notice how the -vlues get closer to zero s the bsolute vlues of get lrger. 6 The bsolute vlues of get ver lrge s pproches zero. 6 6 Step 3 Connect the points with smooth curve. cnnot be zero, so there is no -intercept. The numertor is never zero, so is never 0. There is no -intercept. The -is is horizontl smptote. The -is is verticl smptote. Knowing the smptotes provides ou with the bsic shpe of the grph. The domin is the set of ll rel numbers ecept 5 0. The rnge is the set of ll rel numbers ecept O 6 Got It?.. Wht is the grph of 5? Identif the - nd -intercepts nd the smptotes of the grph. Also, stte the domin nd rnge of the function. b. Resoning Would the function 5 6 hve the sme domin nd rnge s 5 or 5? Eplin. Ech prt of the grph of reciprocl function is brnch. The brnches of the prent function 5 re in Qudrnts I nd III. Stretches nd compressions of the prent function remin in the sme qudrnts. Reflections re in Qudrnts II nd IV. 50 Chpter Rtionl Functions

3 Problem Identifing Reciprocl Function Trnsformtions For ech given vlue of, how do the grphs of 5 nd 5 compre? Wht is the effect of on the grph? A 56 The grph (in red) of 5 6 is stretch of the grph of 5 (in blck) b the fctor 6. O B How does the negtive sign ffect the grph? The -vlues hve signs tht re opposite those in prt A. The grph in A reflects cross the -is The grph (in blue) of is shrink of the grph of 5 (in blck) b the fctor. C 5 6 The grph of 5 6 is the stretch b the fctor 6 in prt A followed b reflection cross the -is. 6 Got It?. For ech given vlue of, how do the grphs of 5 nd 5 compre? Wht is the effect of on the grph?. 5 b. 5 c. 5 You cn trnslte n reciprocl function horizontll or verticll just s ou cn other functions. Ke Concept The Reciprocl Function Fmil 5, 0 Prent function Stretch ( u u. ) Shrink (0, u u, ) 5, 0 t Reflection (, 0) cross -is Trnsltion (horizontl b h; verticl b k) with verticl smptote 5 h horizontl smptote 5 k 5 h k; h Combined 5 h k; h Lesson - The Reciprocl Function Fmil 0507_hsmse_cc_00.indd // 5:7:37 PM

4 When ou grph trnslted reciprocl function, good first step is to drw the smptotes. How do ou find the smptotes? The smptotes of 5 (the es) trnslte unit to the left nd units down. Problem 3 Grphing Trnsltion Wht is the grph of 5? Identif the domin nd rnge. Step Drw the smptotes (red). For, 5, h 5 nd k 5. The verticl smptote is 5. The horizontl smptote is 5. Step Trnslte the grph of 5. The grph of 5 contins the points (, ) nd (, ). Trnslte these points unit to the left nd units down to (0, ) nd (, 3), respectivel. Drw the brnches through these points (blue). The domin is the set of ll rel numbers ecept 5. The rnge is the set of ll rel numbers ecept 5. Got It? 3. Wht is the grph of 5 6? Identif the domin nd rnge. If ou know the smptotes of the grph of reciprocl function nd the vlue of, ou cn write the eqution of the function. How cn ou get strted? Identif the smptotes of the grph. Problem Got It? Writing the Eqution of Trnsformtion Multiple Choice Th is grph of function is trnsltion of the grph of 5. Wht is n eqution for the function? The smptotes re 53 nd 5. Thus h 53 nd k h k Use the generl form. Substitute for, h, nd k. (3) 5 3 The correct choice is A. Simplif.. This grph of function is trnsltion of the grph of 5. Wht is n eqution for the function? O O 50 Chpter Rtionl Functions

5 Problem 5 Using Reciprocl Function Clubs The rowing club is renting 57-pssenger bus for d trip. The cost of the bus is $750. Five pssengers will be chperones. If the students who ttend shre the bus cost equll, wht function models the cost per student C with respect to the number of students n who ttend? Wht is the domin of the function? How mn students must ride the bus to mke the cost per student no more thn $0? The bus holds 57 pssengers. The bus costs $750. Five riders re chperones who p nothing for the bus. A function for the cost per student The number of students needed so tht the cost does not eceed $0 per student Write reciprocl function for the sitution. Grph the function nd solve n inequlit using the $0 limit. Is the domin K 5? No; the domin is the possible numbers of students, so onl positive integers mke sense. To shre the cost equll, divide 750 b the number of students, n, who ttend. The function tht models the cost per student is C n. The bus hs cpcit of 57 pssengers nd there will be 5 chperones. The mimum number of students is The domin is the integers from to 5. Use grphing clcultor to solve the inequlit 750 n # 0. Let Y nd Y 5 0. Chnge the window dimensions to get closer look t the grph. Use the intersect feture. For ll vlues greter thn or equl to 3, the cost is less thn $0. The number of people must be whole number. Intersection X = 37.5 Y = 0 If 37, the cost will be more thn $0. At lest 3 students must ride the bus. Got It? 5. The junior clss is renting lser tg fcilit with cpcit of 35 people. The cost for the fcilit is $00. The prt must hve 3 dult chperones.. If ever student who ttends shres the fcilit cost equll, wht function models the cost per student C with respect to the number of students n who ttend? Wht is the domin of the function? How mn students must ttend to mke the cost per student no more thn $7.50? b. The clss wnts to promote the event b giving w 30 spots to students in drwing. How does the model chnge? Now how mn ping students must ttend so the cost for ech is no more thn $7.50? Lesson - The Reciprocl Function Fmil 5

6 Lesson Check Do ou know HOW?. Grph the eqution 5 3. Describe the trnsformtion from the grph of 5 to the grph of the given function Wht re the smptotes of the grph of 5 5 7? Do ou UNDERSTAND? MATHEMATICAL PRACTICES 5. Vocbulr Wht trnsformtion chnges the grph of 5 into the grph of 5? 6. Open Ended Write n eqution of stretch nd reflection of the grph 5 cross the -is. 7. Writing Eplin how ou cn tell if function 5 is stretch or compression of the prent function 5. Prctice nd Problem-Solving Eercises MATHEMATICAL PRACTICES A Prctice Grph ech function. Identif the - nd -intercepts nd the smptotes of the See Problem. grph. Also, stte the domin nd the rnge of the function Grphing Clcultor Grph the equtions 5 nd 5 using the given vlue of. Then identif the effect of on the grph. See Problem. B Appl Sketch the smptotes nd the grph of ech function. Identif the domin nd rnge Write n eqution for the trnsltion of 5 tht hs the given smptotes. See Problem nd nd nd 5 STEM 9. Construction Th e weight P in pounds tht bem cn sfel crr is inversel See Problem 5. proportionl to the distnce D in feet between the supports of the bem. For certin tpe of wooden bem, P D. Wht distnce between supports is needed to crr 00 lb? 30. Think About Pln A high school decided to spend $750 on student cdemic chievement wrds. At lest 5 wrds will be given, the should be equl in vlue, nd ech wrd should not be less thn $50. Write nd sketch function tht models the reltionship between the number of wrds nd the cost c of ech wrd. Wht re the domin nd rnge of the function? Which eqution describes the reltionship between nd c? Wht informtion cn ou use to determine the domin nd rnge? See Problem 3. 5 Chpter Rtionl Functions

7 3. Open-Ended Write n eqution for horizontl trnsltion of 5. Then write n eqution for verticl trnsltion of 5. Identif the horizontl nd verticl smptotes of the grph of ech function. Sketch the grph of ech function Writing Eplin how knowing the smptotes of trnsltion of 5 cn help ou grph the function. Include n emple. e. 3. Multiple Choice The formul p models the reltionship between tmospheric pressure p in inches of mercur nd ltitude in miles. Shr Desert verge lt. 500 ft Klhri Desert verge lt ft Use the dt shown with the photo. At which loction does the model predict the pressure ure to be bout 3.93 in. of mercur? (Hint: mi 5 50 ft.) Mt. Kilimnjro lt. 9,30 ft Shr Desert Klhri Desert Vinson Mssif lt. 6,60 ft Mt. Kilimnjro Vinson Mssif Grphing Clcultor Grph ech pir of functions. Find the pproimte point(s) of intersection , , , 5. Resoning How will the domin nd the rnge of the prent function 5 chnge fter the trnsltion of its grph b 3 units up nd b 5 units to the left? 3.. Gsoline Milege Suppose ou drive n verge of 0,000 miles ech er. Your gsoline milege (mi/gl) vries inversel with the number of gllons of gsoline ou use ech er. Write nd grph model for our verge milege m in terms of the gllons g of gsoline used. b. After ou begin driving on the highw more often, ou use 50 gl less per er. Write nd grph new model to include this informtion. c. Clculte our old nd new milege ssuming tht ou originll used 00 gl of gsoline per er. C Chllenge Resoning Compre ech pir of grphs nd find n points of intersection.. 5 nd 5 P P 5. 5 nd P P nd 5 7. Find two reciprocl functions such tht the minimum distnce from the origin to the grph of ech function is!. k. Write ech eqution in the form 5 b c, nd sketch the grph b. 5 c. 5 d. 5 Lesson - The Reciprocl Function Fmil 0507_hsmse_cc_00.indd // 5::53 PM

8 SAT/ACT Stndrdized Test Prep 9. Wht is n eqution for the trnsltion of 5 tht hs smptotes t 5 3 nd 55? The grph t the right shows which inequlit?,.5 5.5, 5.5 $ 5 5 # 5 O Short Response 5. If p nd q vr inversel, nd p 5 0 when q 5, wht is q when p 5? Which eqution represents the inverse of the grph t the right? 5 log 3 5 log 3 5 log 3 5 log Wht is b if the grph of 5 7b includes the point (, )? O Mied Review Suppose tht nd vr inversel. Write function tht models ech inverse vrition nd find when 55. See Lesson when when when 5 Without grphing, determine whether the function represents eponentil growth or eponentil dec. Then find the -intercept. See Lesson () () (0.) Q R Multipl. See Lesson (5!3 ) 6. (!3)(6 3!3) 63. (!3!5)(!3!5) Get Red! To prepre for Lesson -3, do Eercises Fctor ech epression See Lesson Chpter Rtionl Functions