Alg 1 11.8 pg 694

GOAL 2GRAPHING RATIONAL FUNCTIONSThe inverse variation models you graphed in Lesson 11.3 are a type of rationalfunction. A rational function is a function of the formƒ(x) = p olynomial.polynomialIn this lesson you will learn to graph rational functions whose numerators anddenominators are first-degree polynomials. Using long division, such a functioncan be written in the formay = + k. x º hRATIONAL FUNCTIONS WHOSE GRAPHS ARE HYPERBOLASaThe graph of the rational function y = + k xº his a hyperbola whose center is (h, k).The vertical and horizontal lines through the centerare the asymptotes of the hyperbola. An asymptoteis a line that the graph approaches. While the distancebetween the graph and the line approaches zero, theasymptote is not part of the graph.Branchesof thehyperbolayCenter(h, k)AsymptotesxEXAMPLE 5Graphing a Rational FunctionSketch the graph of y = 1 x +2.1SOLUTION Think of the function as y = + 2. You can see that the centerx º 0is (0, 2). The asymptotes can be drawn as dashed lines through the center. Make atable of values. Then plot the points and connect them with two smooth branches.xyº4 1.75º2 1.5º1 1º0.5 00 undefined0.5 41 32 2.54 2.255y4(0, 2)31y x 2135 xIf you have drawn your graph correctly,it should be symmetric about the center(h, k). For example, the points (0.5, 4) and(º0.5, 0) are the same distance from thecenter (0, 2), but in opposite directions.692 Chapter 11 Rational Equations and Functions

EXAMPLE 6Graphing a Rational Function When h is Negativeº 1Sketch the graph of y = + 1. Describe the domain.x + 2SOLUTIONFor this equation, x º h = x + 2, so h = º2. The graph is a hyperbola withcenter at (º2, 1). Plot this point and draw a horizontal and a vertical asymptotethrough it. Then make a table of values, plot the points, and connect them withtwo smooth branches.xyº6 1.25º4 1.5º3 2º2.5 3º2 undefinedº1.5 º1º1 00 0.52 0.751y x 2 15(2, 1)5213The domain is all real numbersexcept x = º2.y13xEXAMPLE 7Rewriting before GraphingSketch the graph of y = 2 x + 1.x + 2Begin by using long division to rewrite the rational function.2 Quotient Remainderx + 22x+ 1º 3 º 32x + 4y = 2 + x, or y = + 2 x+ 2+ 2º3SOLUTIONThe graph is a hyperbola withcenter at (º2, 2). Plot thispoint and draw a horizontaland a vertical asymptotethrough it. Make a table ofvalues, plot the points, andconnect them with twosmooth branches.. . . . . . . . . .5(2, 2)3533y13y 2x 1x 2xWhen making a table of values, include values of x that are less than h and valuesof x that are greater than h.11.8 Rational Equations and Functions 693

GUIDED PRACTICEVocabulary Check ✓Concept Check ✓Skill Check ✓1. Describe the shape of a hyperbola. What is an asymptote of a hyperbola?In Exercises 2–4, find the least common denominator.2. 1 x , x 23 1 11 1, 3 3. , , 4. , 3 x4 x 6x2 8x2x 2 +6x +9 x + 35. Which of the equations in Exercises 6–11 can you solve using cross multiplying?Explain your answer.Solve the equation. Remember to check for extraneous solutions.6. 3 x = xº 4x 2x 155 7. = 8. + = 12x + 1 x º 26 x 64 449. = 10. = 3 3x º6 x(x +1) x 11. 3 + 4 x + 4 x = º5 x 2 º2xx 2 +4xFind the center of the hyperbola. Draw the asymptotes and sketch the graph.4212. y = º 3 13. y = + 3x + 5x º 2PRACTICE AND APPLICATIONSSTUDENT HELPExtra Practiceto help you masterskills is on p. 807.CROSS MULTIPLYING Solve the equation by cross multiplying.x 714. = 5 3 15. x = 1 4 16. 4 105x = 125(x + 2)5 56 x 7 517. = 18. = 19. x = x + 4 3(x +1)x + 2 4 + 1 x º 3MULTIPLYING BY THE LCD Solve the equation by multiplying each side bythe least common denominator.20. 5 6 9º xx 9 = 21. = + 2x 2x + 9 x + 97 122. º = 2 3x º 12 x º 4 3 23. 1 1 22 + = x º 4 x + 4 x 2 º16178 x + 524. + 1 = º 25. 2 + = x º 4 x 2 + x º20x º 5 x 2 º25STUDENT HELPHOMEWORK HELPExample 1: Exs. 14–19,26–38Example 2: Exs. 20–38Example 3: Exs. 20–38Example 4: Exs. 51, 52Example 5: Exs. 39–50Example 6: Exs. 39–50Example 7: Exs. 39–50CHOOSING A METHOD Solve the equation.26. 1 4 + 4 x = 1 x 27. º 3x º 2 = 28. 1 x + 1 x º 15 º 2 1 = 5xxx 829. º 9 x = 1 9 30. x + 42 = x 31. 2 xx º x 3 = 4 8º 3 2232. = 33. + 1 x + 7 x + 2x + 3 x = 410 34. º 3 3xx + 3 5 = 1 0x+ 13x+ 935. x + 3 56 º 3x85x = 36. + 1 = x º 5 x 2 º 13x +40x + 4 x 2 º 2x º 24x222xx x37. º 1 = 38. º =2 º1 x º 11 x 2 º 5x º 66x + 3 x + 7 x 2 + 10x + 21694 Chapter 11 Rational Equations and Functions

MATCHING GRAPHS Match the function with its graph.A. yB. y C.6621022 x2 2 6 x48482y2x1º 239. y = º º 1 40. y = + 7x + 7x + 341. y = x4 STUDENT HELPHOMEWORK HELPVisit our Web sitewww.mcdougallittell.comfor help with Exs. 52–58.INTERNETORIGAMI CRANESThe art of paperfolding has been practicedin Japan since the seventhcentury A.D. There is anancient Japanese belief thatfolding 1000 cranes willmake your wish come true.REALFOCUS ONAPPLICATIONSLIFEGRAPHING In Exercises 42–50, graph the function. Describe the domain.42. y = 1 x + 4 43. y = 12 º 8 44. y = + 6x º 3x º 43645. y = º + 8 46. y = º 7 47. y = 3 x + 11x + 1x + 9x + 348. y = º2 x + 11x º 549. y = 9 x º 6x º 150. y = º5 x + 19x º 351. BATTING AVERAGE After 50 times at bat, a major league baseballplayer has a batting average of 0.160. How many consecutive hits must theplayer get to raise his batting average to 0.250?52. TEST AVERAGES You have taken 3 tests and have an average of72 points. If you score 100 points on each of the rest of your tests, howmany more tests do you need to take to raise your average to 88 points?FUNDRAISING In Exercises 53 and 54, a library has received a singlelarge contribution of $5000. A walkathon is also being held in which eachsponsor will contribute $10. Let x represent the number of sponsors.53. Write a function that represents the average contribution per person,including the single large contributor. Sketch the graph of the function.54. Explain how such averages could be used to misrepresent actualcontributions.ORIGAMI CRANES In Exercises 55–58, you investigatehow long it would take you and a friend to fold 1000 origamicranes. You take 2 minutes to fold 1 crane. Let x representthe number of minutes it might take a friend to fold 1 crane.55. Working alone, you would take 2000 minutes to fold 1000 cranes, so your1work rate is of the job per minute. Find your work rate per hour.20 006056. The expression is your friend’s work rate per hour. Explain why.10 00x57. Write an expression for the combined work rate per hour of you and yourfriend (the part of the job completed in 1 hour if you both work together).58. Suppose that together you could complete 1000 cranes in 20 hours. Your1combined work rate per hour would be . To find the time your friend takes2 01to fold 1 crane, set the expression in Exercise 57 equal to and solve for x.2 011.8 Rational Equations and Functions 695

INVERSE VARIATION AND RATIONAL FUNCTIONS In Exercises 59 and 60,you will compare the types of graphs in 11.3 with those in this lesson.59. Graph ƒ(x) = 6 x and ƒ(x) = 6 + 1 in the same coordinate plane.xº 260. Writing How are the two graphs in Exercise 59 related? Do you thinkathis relationship will hold for any set of functions ƒ(x) = + k andx º hƒ(x) = a ? Explain.xTestPreparation61. CHANGING CONSTANTS Use a graphing calculator to graphƒ(x) = a x for at least ten values of a. Include positive, negative, andnon-integer values. How does changing the value of a seem to affect theappearance of the graph?9 762. MULTIPLE CHOICE What is the solution of the equation = ?x + 5 x º 5¡ A 5 ¡ B 8 ¡ C 20 ¡ D 40 ¡ E 80463. MULTIPLE CHOICE Choose the graph of the function y = +1. x º 2¡ A y ¡ B y ¡ C6y422x26x42x46★ Challenge¡ D None of the aboveRATIONAL FUNCTIONS In Exercises 64–67, you will look for a pattern.64. Copy and complete the table. Round your answers to the nearest hundredth.x 0 20 40 60 80 100 x 2+ 6x + 2? ? ? ? ? ?x º 2 ? ? ? ? ? ?10x + 2? ? ? ? ? ?x65. What happens to the values of 2 + 610, (x º 2), and as x increases?x +2x + 2x66. Which is the graph of y = x º 2? of y= 2 + 6?x +2y67. CRITICAL THINKING Explain the2 A Brelationship between the two graphsEXTRA CHALLENGE4in Exercise 66. Use the table inwww.mcdougallittell.com Exercise 64 to help you.Exs. 66 and 67x696 Chapter 11 Rational Equations and Functions

MIXED REVIEWFUNCTION VALUES Evaluate the function for x = 0, 1, 2, 3, and 4.(Review 4.8 for 12.1)68. ƒ(x) = 4x 69. ƒ(x) = ºx + 9 70. ƒ(x) = 3x + 171. ƒ(x) = ºx 2 72. ƒ(x) = x 2 º1 73. ƒ(x) =EVALUATING EXPRESSIONS Evaluate the expression. (Review 8.1, 8.2)74. 2 4 • 2 3 75. 6 3 • 6 º1 76. (3 3 ) 2 77. (º4 º2 ) º1RADICAL EXPRESSIONS Simplify the radical expression. (Review 9.2 for 12.1)78. 50 79. 72 80. 1 4 112 81. 1 2 5282. 1 4 64 83. 256 84. 1 625 85. 396586. ACCOUNT BALANCE A principal of $500 is deposited in an accountthat pays 4% interest compounded yearly. Find the balance after 6 years.(Review 8.5)x 2 2QUIZ 3 Self-Test for Lessons 11.7 and 11.8Divide. (Lesson 11.7)1. (x 2 º 8) ÷ (6x) 2. (6a 3 + 5a 2 ) ÷ (10a 2 )3. (x 2 + 16) ÷ (x + 4) 4. (2y 2 + 8y º 5) ÷ (2y º 5)5. (8 + 4z + z 2 ) ÷ (3z º 6) 6. (12x 2 + 17x º 5) ÷ (3x + 2)Solve the equation. (Lesson 11.8)7. 1 2 + 2 t = 1 t 8. 3 x = 92(x +2)1 1 x + 39. + = 10. 7 x º 5 x + 58 º 163 = x 2 º 25s º 2 4In Exercises 11–13, sketch a graph of the function. (Lesson 11.8)11. ƒ(x) = 3 x 12. ƒ(x) = 63 13. ƒ(x) = º 2x º 4x + 114. TELEVISION STATIONS The models are based on data collected by theTelevision Bureau of Advertising from 1988 to 1998 in the United States.Let t represent the number of years since 1988.Number of UHF television stations: U = 14t + 505Total number of commercial television stations: T = 16t + 1048Use long division to find a model for the ratio of the number of UHFtelevision stations to the total number of commercial television stations.(Lesson 11.7)11.8 Rational Equations and Functions 697