Essential Question What are some of the characteristics of the graph of a rational function?

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1 8. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..G A..H A..K Grphing Rtionl Functions Essentil Question Wht re some of the chrcteristics of the grph of rtionl function? The prent function for rtionl functions with liner numertor nd liner denomintor is f() = 1. Prent function The grph of this function, shown t the right, is hperbol. Identifing Grphs of Rtionl Functions Work with prtner. Ech function is trnsformtion of the grph of the prent function f() = 1. Mtch the function with its grph. Eplin our resoning. Then describe the trnsformtion.. g() = 1 b. g() = 3 c. g() = d. g() = 1 e. g() = 1 f. g() = 1 + A. B. C. D. E. F. ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in mth, ou need to look closel to discern pttern or structure. Communicte Your Answer. Wht re some of the chrcteristics of the grph of rtionl function? 3. Determine the smptotes, domin, nd rnge of the rtionl function g() = + k. h Section 8. Grphing Rtionl Functions 17

2 8. Lesson Wht You Will Lern Core Vocbulr rtionl function, p. 18 Previous domin rnge smptote long division Grph simple rtionl functions. Trnslte simple rtionl functions. Grph other rtionl functions. Grphing Simple Rtionl Functions A rtionl function hs the form f() = p(), where p() nd q() re polnomils q() nd q() 0. The inverse vrition function f() = is rtionl function. The grph of this function when = 1 is shown below. Core Concept Prent Function for Simple Rtionl Functions STUDY TIP Notice tht 1 0 s nd s. This eplins wh = 0 is horizontl smptote of the grph of f() = 1. You cn lso nlze -vlues s pproches 0 to see wh = 0 is verticl smptote. ANALYZING MATHEMATICAL RELATIONSHIPS Becuse the function is of the form g() = f(), where =, the grph of g is verticl stretch b fctor of of the grph of f. The grph of the prent function f() = 1 is hperbol, which consists of two smmetricl prts clled brnches. The domin is { 0} nd the rnge is { 0}. An function of the form g() = ( 0) hs the sme smptotes, domin, nd rnge s the function f() = 1. Grphing Rtionl Function Grph g() =. Compre the grph with the grph of f() = 1. Step 1 The function is of the form g() =, so the smptotes re = 0 nd = 0. Drw the smptotes. Step Mke tble of vlues nd plot the points. Include both positive nd negtive vlues of Step 3 Drw the two brnches of the hperbol so tht the pss through the plotted points nd pproch the smptotes. verticl smptote = 0 f() = 1 horizontl smptote = 0 The grph of g lies frther from the es thn the grph of f. Both grphs lie in the first nd third qudrnts nd hve the sme smptotes, domin, nd rnge. f g Monitoring Progress Help in English nd Spnish t BigIdesMth.com 1. Grph g() =. Compre the grph with the grph of f() = Chpter 8 Rtionl Functions

3 Trnslting Simple Rtionl Functions Core Concept Grphing Trnsltions of Simple Rtionl Functions To grph rtionl function of the form = + k, follow these steps: h Step 1 Drw the smptotes = h nd = k. = + k h Step Plot points to the left nd to the right of the verticl smptote. = k Step 3 Drw the two brnches of the hperbol so tht the pss through the plotted points nd pproch the smptotes. = h Grphing Trnsltion of Rtionl Function Grph g() = 1. Stte the domin nd rnge. + ANALYZING MATHEMATICAL RELATIONSHIPS Let f() =. Notice tht g is of the form g() = f( h) + k, where h = nd k = 1. So, the grph of g is trnsltion units left nd 1 unit down of the grph of f. Step 1 Drw the smptotes = nd = 1. Step Plot points to the left of the verticl smptote, such s ( 3, 3), (, 1), nd (, 0). Plot points to the right of the verticl smptote, such s ( 1, 5), (0, 3), nd (, ). Step 3 Drw the two brnches of the hperbol so tht the pss through the plotted points nd pproch the smptotes. The domin is { } nd the rnge is { 1}. ( 3, 3) (, 1) (, 0) ( 1, 5) (, ) (0, 3) Monitoring Progress Grph the function. Stte the domin nd rnge. Help in English nd Spnish t BigIdesMth.com. = 3 3. = = Grphing Other Rtionl Functions All rtionl functions of the form = + b lso hve grphs tht re hperbols. c + d The verticl smptote of the grph is the line = d becuse the function is c undefined when the denomintor c + d is zero. The horizontl smptote is the line = c. Section 8. Grphing Rtionl Functions 19

4 Grphing Rtionl Function of the Form = + b c + d Grph f() = + 1. Stte the domin nd rnge. 3 (, 3 5 ) 8 ( ) 0, 1 3 (, 9) (, 13 3 ) 8, 17 5 ( ) 8 1 (, 5) Step 1 Drw the smptotes. Solve 3 = 0 for to find the verticl smptote = 3. The horizontl smptote is the line = c = 1 =. Step Plot points to the left of the verticl smptote, such s (, 5), ( 0, 1 3 ), nd (, 3 5 ). Plot points to the right of the verticl smptote, such s (, 9), (, 13 3 ), nd ( 8, 17 5 ). Step 3 Drw the two brnches of the hperbol so tht the pss through the plotted points nd pproch the smptotes. The domin is (, 3) nd (3, ) nd the rnge is (, ) nd (, ). Rewriting rtionl function m revel properties of the function nd its grph. For emple, rewriting rtionl function in the form = + k revels tht it is h trnsltion of = with verticl smptote = h nd horizontl smptote = k. Rewriting nd Grphing Rtionl Function ANOTHER WAY You will use different method to rewrite g in Emple 5 of Lesson 8.. Rewrite g() = in the form g() = + k. Grph the function. Describe + 1 h the grph of g s trnsformtion of the grph of f() =. Rewrite the function 3 b using long division: + 1 ) The rewritten function is g() = The grph of g is trnsltion 1 unit left nd 3 units up of the grph of f() =. g Monitoring Progress Grph the function. Stte the domin nd rnge. 5. f() = Help in English nd Spnish t BigIdesMth.com. f() = f() = Rewrite g() = + 3 in the form g() = + k. Grph the function. + 1 h Describe the grph of g s trnsformtion of the grph of f() =. 0 Chpter 8 Rtionl Functions

5 Modeling with Mthemtics A 3-D printer builds up lers of mterils to mke three-dimensionl models. Ech deposited ler bonds to the ler below it. A compn decides to mke smll displ models of engine components using 3-D printer. The printer costs $1000. The mteril for ech model costs $50. Estimte how mn models must be printed for the verge cost per model to fll to $90. Wht hppens to the verge cost s more models re printed? USING TECHNOLOGY Becuse the number of models nd verge cost cnnot be negtive, choose viewing window in the first qudrnt. 1. Understnd the Problem You re given the cost of printer nd the cost to crete model using the printer. You re sked to find the number of models for which the verge cost flls to $90.. Mke Pln Write n eqution tht represents the verge cost. Use grphing clcultor to estimte the number of models for which the verge cost is bout $90. Then nlze the horizontl smptote of the grph to determine wht hppens to the verge cost s more models re printed. 3. Solve the Problem Let c be the verge cost (in dollrs) nd m be the number of models printed. (Unit cost)(number printed) + (Cost of printer) 50m c = = Number printed m Use grphing clcultor to grph the function. Using the trce feture, the verge cost flls to $90 per model fter bout 5 models re printed. Becuse the horizontl smptote is c = 50, the verge cost pproches $50 s more models re printed.. Look Bck Use grphing clcultor to crete tbles of vlues for lrge vlues of m. The tbles show tht the verge cost pproches $50 s more models re printed. 00 c = 50m m 0 X= Y= X X=0 Y1 ERROR X X=0 Y1 ERROR Monitoring Progress Help in English nd Spnish t BigIdesMth.com 9. WHAT IF? How do the nswers in Emple 5 chnge when the cost of the 3-D printer is $800? Section 8. Grphing Rtionl Functions 1

6 8. Eercises Dnmic Solutions vilble t BigIdesMth.com Vocbulr nd Core Concept Check 7 1. COMPLETE THE SENTENCE The function = + 3 hs (n) of ll rel numbers + ecept 3 nd (n) of ll rel numbers ecept.. WRITING Is f() = rtionl function? Eplin our resoning. + 1 Monitoring Progress nd Modeling with Mthemtics In Eercises 3 10, grph the function. Compre the grph with the grph of f() = 1. (See Emple 1.) 3. g() = 3. g() = = 1 5. g() = 5. g() = 9 7. g() = g() = g() = g() = 3 In Eercises 11 18, grph the function. Stte the domin nd rnge. (See Emple.) ANALYZING RELATIONSHIPS In Eercises 1, mtch the function with its grph. Eplin our resoning. 11. g() = = h() = h() = = + 1. f() = g() = = g() = h() = f() = 3 1. = A. B. ERROR ANALYSIS In Eercises 19 nd 0, describe nd correct the error in grphing the rtionl function = 8 8 C. D. Chpter 8 Rtionl Functions

7 In Eercises 5 3, grph the function. Stte the domin nd rnge. (See Emple 3.) 5. f() = + 3. = REASONING Wht re the -intercept(s) of the grph 5 of the function = 1? A 1, 1 B 5 7. = f() = h() = h() = g() = = C 1 D 5 5. USING TOOLS The time t (in seconds) it tkes for sound to trvel 1 kilometer cn be modeled b 1000 t = 0.T where T is the ir temperture (in degrees Celsius). In Eercises 33 0, rewrite the function in the form g() = + k. Grph the function. Describe the h grph of g s trnsformtion of the grph of f() =. (See Emple.) 33. g() = g() = g() = g() = g() = g() = 38. g() = g() = You re 1 kilometer from lightning strike. You her the thunder.9 seconds lter. Use grph to find the pproimte ir temperture. b. Find the verge rte of chnge in the time it tkes sound to trvel 1 kilometer s the ir temperture increses from 0 C to 10 C. 1. PROBLEM SOLVING Your school purchses mth softwre progrm. The progrm hs n initil cost of $500 plus $0 for ech student tht uses the progrm. (See Emple 5.). Estimte how mn students must use the progrm for the verge cost per student to fll to $30. b. Wht hppens to the verge cost s more students use the progrm?. PROBLEM SOLVING To join rock climbing gm, ou must p n initil fee of $100 nd monthl fee of $59.. Estimte how mn months ou must purchse membership for the verge cost per month to fll to $9. b. Wht hppens to the verge cost s the number of months tht ou re member increses? 3. USING STRUCTURE Wht is the verticl smptote of the grph of the function = + + 7? A = 7 B = C = D = 7. MODELING WITH MATHEMATICS A business is studing the cost to remove pollutnt from the 15 ground t its site. The function = 1.1 models the estimted cost (in thousnds of dollrs) to remove percent (epressed s deciml) of the pollutnt.. Grph the function. Describe resonble domin nd rnge. b. How much does it cost to remove 0% of the pollutnt? 0% of the pollutnt? 80% of the pollutnt? Does doubling the percentge of the pollutnt removed double the cost? Eplin. USING TOOLS In Eercises 7 50, use grphing clcultor to grph the function. Then determine whether the function is even, odd, or neither. 7. h() = f() = 9 9. = f() = 3 Section 8. Grphing Rtionl Functions 3

8 51. MAKING AN ARGUMENT Your friend clims it is possible for rtionl function to hve two verticl smptotes. Is our friend correct? Justif our nswer. 5. HOW DO YOU SEE IT? Use the grph of f to determine the equtions of the smptotes. Eplin. f 5. ABSTRACT REASONING Describe the intervls where the grph of = is incresing or decresing when () > 0 nd (b) < 0. Eplin our resoning. 57. PROBLEM SOLVING An Internet service provider chrges $50 instlltion fee nd monthl fee of $3. The tble shows the verge monthl costs of competing provider for months of service. Under wht conditions would person choose one provider over the other? Eplin our resoning. 8 Months, Averge monthl cost (dollrs), $ $.9 18 $ DRAWING CONCLUSIONS In wht line(s) is the grph of = 1 smmetric? Wht does this smmetr tell ou bout the inverse of the function f() = 1? 5. THOUGHT PROVOKING There re four bsic tpes of conic sections: prbol, circle, ellipse, nd hperbol. Ech of these cn be represented b the intersection of double-npped cone nd plne. The intersections for prbol, circle, nd ellipse re shown below. Sketch the intersection for hperbol. $ MODELING WITH MATHEMATICS The Doppler effect occurs when the source of sound is moving reltive to listener, so tht the frequenc f (in hertz) herd b the listener is different from the frequenc f s (in hertz) t the source. In both equtions below, r is the speed (in miles per hour) of the sound source. Moving w: 70f s f = 70 + r Approching: 70f s f = 70 r Prbol Circle Ellipse 55. REASONING The grph of the rtionl function f is hperbol. The smptotes of the grph of f intersect t (3, ). The point (, 1) is on the grph. Find nother point on the grph. Eplin our resoning. Mintining Mthemticl Proficienc Fctor the polnomil. (Skills Review Hndbook). An mbulnce siren hs frequenc of 000 hertz. Write two equtions modeling the frequencies herd when the mbulnce is pproching nd when the mbulnce is moving w. b. Grph the equtions in prt () using the domin 0 r 0. c. For n speed r, how does the frequenc herd for n pproching sound source compre with the frequenc herd when the source moves w? Reviewing wht ou lerned in previous grdes nd lessons Simplif the epression. (Section.) / 3/5 5. 5/ 1/ Chpter 8 Rtionl Functions