Graph Simple Rational Functions. is a rational function. The graph of this function when a 5 1 is shown below.

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1 TEKS 8.2 2A.0.A, 2A.0.B, 2A.0.C, 2A.0.F Graph Simple Rational Functions Before You graphed polnomial functions. Now You will graph rational functions. Wh? So ou can find average monthl costs, as in E. 38. Ke Vocabular rational function domain, p. 72 range, p. 72 asmptote, p. 478 A rational function has the form f() 5 p() } q() where p() and q() are polnomials and q() Þ 0. The inverse variation function f() 5 a } is a rational function. The graph of this function when a 5 is shown below. KEY CONCEPT For Your Notebook Parent Function for Simple Rational Functions The graph of the parent function f() 5 } is a hperbola, which consists of two smmetrical parts called branches. The domain and range are all nonzero real numbers. An function of the form g() 5 } a (a Þ 0) has the same asmptotes, domain, and range as the function f() 5 }. vertical asmptote 5 0 f () horizontal asmptote E XAMPLE Graph a rational function of the form 5 a } Graph the function 5 6 }. Compare the graph with the graph of 5 }. INTERPRET TRANSFORMATIONS The graph of 5 6 } is a vertical stretch of the graph of 5 } b a factor of 6. STEP Draw the asmptotes 5 0 and 5 0. Plot points to the left and to the right of the vertical asmptote, such as (23, 22), (22, 23), (2, 3), and (3, 2). Draw the branches of the hperbola so that the pass through the plotted points and approach the asmptotes The graph of 5 } 6 lies farther from the aes than the graph of 5 }. Both graphs lie in the first and third quadrants and have the same asmptotes, domain, and range. 558 Chapter 8 Rational Functions

2 KEY CONCEPT For Your Notebook Graphing Translations of Simple Rational Functions To graph a rational function of the form 5 a } 2 h k, follow these steps: STEP Draw the asmptotes 5 h and 5 k. Plot points to the left and to the right of the vertical asmptote. Draw the two branches of the hperbola so that the pass through the plotted points and approach the asmptotes. 5 k a 5 k 2 h 5 h E XAMPLE 2 Graph a rational function of the form 5 a } 2 h k Graph 5 24 } 2 2. State the domain and range. INTERPRET TRANSFORMATIONS The graph of 5 24 } 2 2 is the graph of 5 24 } translated left 2 units and down unit. STEP Draw the asmptotes 522 and 52. Plot points to the left of the vertical asmptote, such as (23, 3) and (24, ), and points to the right, such as (2, 25) and (0, 23). Draw the two branches of the hperbola so that the pass through the plotted points and approach the asmptotes. (23, 3) (24, ) (2, 25) (0, 23) The domain is all real numbers ecept 22, and the range is all real numbers ecept 2. at classzone.com GUIDED PRACTICE for Eamples and 2 Graph the function. State the domain and range.. f() 5 24 } } } OTHER RATIONAL FUNCTIONS All rational functions of the form 5} a b also c d have graphs that are hperbolas. The vertical asmptote of the graph is the line 52} d, because the function is c undefined when the denominator c d is zero. The horizontal asmptote is the line 5 a } c. 8.2 Graph Simple Rational Functions 559

3 E XAMPLE 3 Graph a rational function of the form 5 a b } c d Graph 5 2 } 2 3. State the domain and range. STEP Draw the asmptotes. Solve for to find the vertical asmptote 5 3. The horizontal asmptote is the line 5 } a 5 } c 4 (4, 9) 3 s6, 3 d Plot points to the left of the vertical asmptote, such as (2, 25) and 0, 2 } 3 2, and points to the right, such as (4, 9) s0, 2 3d 6 (2, 25) and 6, 3 } 3 2. Draw the two branches of the hperbola so that the pass through the plotted points and approach the asmptotes. c The domain is all real numbers ecept 3. The range is all real numbers ecept 2. E XAMPLE 4 TAKS REASONING: Multi-Step Problem 3-D MODELING A 3-D printer builds up laers of material to make three-dimensional models. Each deposited laer bonds to the laer below it. A car compan decides to make small displa models of its vehicles using a 3-D printer. The printer costs $24,000. The material for each model costs $300. Write an equation that gives the average cost per model as a function of the number of models printed. Graph the function. Use the graph to estimate how man models must be printed for the average cost per model to fall to $700. What happens to the average cost as more models are printed? DRAW GRAPHS Because the number of models and average cost cannot be negative, graph onl the branch of the hperbola that lies in the first quadrant. STEP Write a function. Let c be the average cost and m be the number of models printed. c 5 Unit cost p Number printed Cost of printer 300m 24,000 }} 5} Number printed m Graph the function. The asmptotes are the lines m 5 0 and c The average cost falls to $700 per model after 60 models are printed. Interpret the graph. As more models are printed, the average cost per model approaches $300. Cost per model c 200 (60, 700) m Number printed 560 Chapter 8 Rational Functions

4 GUIDED PRACTICE for Eamples 3 and 4 Graph the function. State the domain and range } } f() } WHAT IF? In Eample 4, how do the function and graph change if the cost of the 3-D printer is $2,000? 8.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 2, and 39 5 TAKS PRACTICE AND REASONING Es. 23, 35, 40, 4, 43, and 44 5 MULTIPLE REPRESENTATIONS E. 39. VOCABULARY Cop and complete: The function 5} 7 3 has a(n)? of 4 all real numbers ecept 3 and a(n)? of all real numbers ecept WRITING Is f() 5} a rational function? Eplain our answer. EXAMPLE on p. 558 for Es. 3 0 GRAPHING FUNCTIONS Graph the function. Compare the graph with the graph of 5 } } } } } } 8. f() 5 5 } 9. g() 5 26 } 0. h() 5 23 } EXAMPLE 2 on p. 559 for Es. 23 GRAPHING FUNCTIONS Graph the function. State the domain and range } } } 2 4. f() 5 } } } } 2 8. g() 5 22 } } } } h() 5 } TAKS REASONING What are the asmptotes of the graph of 5} 3 2 3? 8 A 5 8, 5 3 B 5 8, 523 C 528, 5 3 D 528, GRAPHING CALCULATOR Consider the function 5} a k where a 5, 2 h h 5 3, and k 522. Predict the effect on the functions graph of each change in a, h, or k described in parts (a) (c). Use a graphing calculator to check our prediction b graphing the original and revised functions in the same coordinate plane. a. a changes to 23 b. h changes to 2 c. k changes to Graph Simple Rational Functions 56

5 ERROR ANALYSIS Describe and correct the error in the graph } } EXAMPLE 3 on p. 560 for Es GRAPHING FUNCTIONS Graph the function. State the domain and range } } } } } f() } g() 5 5 } h() } TAKS REASONING Write a rational function such that the domain is all real numbers ecept 28 and the range is all real numbers ecept CHALLENGE Show that the equation f() 5 a } 2 h k represents a rational function b writing the right side as a quotient of polnomials. PROBLEM SOLVING EXAMPLE 4 on p. 560 for Es INTERNET SERVICE An Internet service provider charges a $50 installation fee and a monthl fee of $43. Write and graph an equation that gives the average cost per month as a function of the number of months of service. After how man months will the average cost be $53? 38. ROCK CLIMBING GYM To join a rock climbing gm, ou must pa an initial fee of $00 and a monthl fee of $59. Write and graph an equation that gives the average cost per month as a function of the number of months of membership. After how man months will the average cost be $69? 39. MULTIPLE REPRESENTATIONS The time t (in seconds) it takes for sound to travel kilometer can be modeled b t 5} 000 where T is 0.6T 33 the air temperature (in degrees Celsius). a. Evaluating a Function How long does it take for sound to travel 5 kilometers when the air temperature is 258C? Eplain. b. Drawing a Graph Suppose ou are kilometer from a lightning strike, and it takes 3 seconds to hear the thunder. Graph the given function, and use the graph to estimate the air temperature WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS

6 40. TAKS REASONING A business is studing the cost to remove a pollutant from the ground at its site. The function 5} 5 models the estimated. 2 cost (in thousands of dollars) to remove percent (epressed as a decimal) of the pollutant. a. Graph the function. Describe a reasonable domain and range. b. How much does it cost to remove 20% of the pollutant? 40% of the pollutant? 80% of the pollutant? Does doubling the percent of the pollutant removed double the cost? Eplain. 4. TAKS REASONING The Doppler effect occurs when the source of a sound is moving relative to a listener, so that the frequenc f l (in hertz) heard b the listener is different from the frequenc f s (in hertz) at the source. The frequenc heard depends on whether the sound source is approaching or moving awa from the listener. In both equations below, r is the speed (in miles per hour) of the sound source. Moving awa: f / 5 740f s } Approaching: f 740 r / 5 740f s } r a. An ambulance siren has a frequenc of 2000 hertz. Write two equations modeling the frequencies ou hear when the ambulance is approaching and when the ambulance is moving awa. b. Graph the equations from part (a) using the domain 0 r 60. c. For an speed r, how does the frequenc heard for an approaching sound source compare with the frequenc heard when the source moves awa? 42. CHALLENGE A sailboat travels at a speed of 0 knots for 3 hours. It then uses a motor for power, which increases its speed to 5 knots. Write and graph an equation giving the boat s average speed s (in knots) for the entire trip as a function of the time t (in hours) that it uses the motor for power. MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson.5; TAKS Workbook REVIEW Lesson 2.; TAKS Workbook 43. TAKS PRACTICE On Monda, Anna reads one quarter of a novel. On Tuesda, she reads one third of the remaining pages. On Wednesda, she reads one quarter of the remaining pages. On Thursda, she reads the remaining 05 pages. How man pages does the novel have? TAKS Obj. 0 A 29 B 280 C 340 D TAKS PRACTICE Which equation best describes the relationship between and shown in the table? TAKS Obj F 5 4 G 5 4 H J EXTRA PRACTICE for Lesson 8.2, p. 07 ONLINE QUIZ at classzone.com 563

7 Graphing Calculator ACTIVITY Use after Lesson Graph Rational Functions TEKS TEXAS a.5, 2A.0.A, 2A.0.B, 2A.0.F classzone.com Kestrokes QUESTION How can ou use a graphing calculator to graph rational functions? Most graphing calculators have two graphing modes: connected mode and dot mode. Connected mode displas the graph of a rational function as a smooth curve, while dot mode displas the graph as a series of dots. E XAMPLE Graph a rational function Graph 5 3 } 2 3. STEP Enter function Enter the rational function, using parentheses. Use connected mode Graph the function in connected mode. Use dot mode Graph the function in dot mode. Y=(X+3)/(X-3) Y2= Y3= Y4= Y5= Y6= Y7= The graph in Step 2 includes a vertical line at approimatel 5 3. This line is not part of the graph. It is simpl the graphing calculator s attempt at connecting the two branches of the graph. P RACTICE Use a graphing calculator to graph the rational function. Choose a viewing window that displas the important characteristics of the graph } } } } } } f() } 2 8. g() } SKATEBOARDING You are tring to decide whether it is worth joining a skate park. It costs $00 to join and then $4 for each visit. Write a function that gives the average cost per visit after visits. Graph the function. What happens to the average cost as the number of visits increases? What are a reasonable domain and range for the function? 564 Chapter 8 Rational Functions