2. Packet #10 Rational Functions
Math 160
Packet #10
Rational Functions

3. A rational function is a polynomial over a polynomial (that is 𝑃 𝑥 𝑄 𝑥 ). Here are some examples:

7. Most of the time, rational functions have vertical asymptotes and a horizontal asymptote. An asymptote is basically a line/curve to which the graph of the function gets closer and closer.

8. Most of the time, rational functions have vertical asymptotes and a horizontal asymptote. An asymptote is basically a line/curve to which the graph of the function gets closer and closer.

9. Visually, here are the four types of vertical asymptotes:

10. When 𝑥→ 𝑎 − , 𝑦→∞

11. When 𝑥→ 𝑎 + , 𝑦→∞

12. When 𝑥→ 𝑎 − , 𝑦→−∞

13. When 𝑥→ 𝑎 + , 𝑦→−∞

14. Note: 𝑥→ 𝑎 − means “𝑥 approaches 𝑎 from the left” 𝑥→ 𝑎 + means “𝑥 approaches 𝑎 from the right”

15. For rational functions, vertical asymptotes happen where the denominator is zero.

16. Ex 1. Find the vertical asymptotes of the following functions
Ex 1. Find the vertical asymptotes of the following functions. For each vertical asymptote, determine happens to 𝑦 as 𝑥 approaches from the left and from the right (that is, does 𝑦→∞ or 𝑦→−∞?). 𝑓 𝑥 = 𝑥−1 𝑥+2 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2

17. Here are the two types of horizontal asymptotes:

18. When 𝑥→∞, 𝑦→𝑏

19. When 𝑥→−∞, 𝑦→𝑏

20. What is the horizontal asymptote (if any) of 𝑃 𝑥 𝑄 𝑥
What is the horizontal asymptote (if any) of 𝑃 𝑥 𝑄 𝑥 ? If deg(P) < deg (Q), then horizontal asymptote 𝑦=0. If deg(P) = deg(Q), then horizontal asymptote 𝑦= leading coefficient of 𝑃 leading coefficient of 𝑄 . If deg(P) > deg(Q), then no horizontal asymptotes.

21. What is the horizontal asymptote (if any) of 𝑃 𝑥 𝑄 𝑥
What is the horizontal asymptote (if any) of 𝑃 𝑥 𝑄 𝑥 ? If deg(P) < deg (Q), then horizontal asymptote 𝑦=0. If deg(P) = deg(Q), then horizontal asymptote 𝑦= leading coefficient of 𝑃 leading coefficient of 𝑄 . If deg(P) > deg(Q), then no horizontal asymptotes.

22. What is the horizontal asymptote (if any) of 𝑃 𝑥 𝑄 𝑥
What is the horizontal asymptote (if any) of 𝑃 𝑥 𝑄 𝑥 ? If deg(P) < deg (Q), then horizontal asymptote 𝑦=0. If deg(P) = deg(Q), then horizontal asymptote 𝑦= leading coefficient of 𝑃 leading coefficient of 𝑄 . If deg(P) > deg(Q), then no horizontal asymptotes.

23. What is the horizontal asymptote (if any) of 𝑃 𝑥 𝑄 𝑥
What is the horizontal asymptote (if any) of 𝑃 𝑥 𝑄 𝑥 ? If deg(P) < deg (Q), then horizontal asymptote 𝑦=0. If deg(P) = deg(Q), then horizontal asymptote 𝑦= leading coefficient of 𝑃 leading coefficient of 𝑄 . If deg(P) > deg(Q), then no horizontal asymptotes.

24. Ex 2. Find the horizontal asymptotes (if any) of the following functions. 𝑓 𝑥 = 8𝑥+5 5 𝑥 2 −3𝑥+1 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2 ℎ 𝑥 = 6 𝑥 3 − 𝑥 𝑥 2 +4𝑥−7

25. Ex 2. Find the horizontal asymptotes (if any) of the following functions. 𝑓 𝑥 = 8𝑥+5 5 𝑥 2 −3𝑥+1 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2 ℎ 𝑥 = 6 𝑥 3 − 𝑥 𝑥 2 +4𝑥−7

26. Ex 2. Find the horizontal asymptotes (if any) of the following functions. 𝑓 𝑥 = 8𝑥+5 5 𝑥 2 −3𝑥+1 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2 ℎ 𝑥 = 6 𝑥 3 − 𝑥 𝑥 2 +4𝑥−7

27. Ex 2. Find the horizontal asymptotes (if any) of the following functions. 𝑓 𝑥 = 8𝑥+5 5 𝑥 2 −3𝑥+1 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2 ℎ 𝑥 = 6 𝑥 3 − 𝑥 𝑥 2 +4𝑥−7

28. Ex 2. Find the horizontal asymptotes (if any) of the following functions. 𝑓 𝑥 = 8𝑥+5 5 𝑥 2 −3𝑥+1 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2 ℎ 𝑥 = 6 𝑥 3 − 𝑥 𝑥 2 +4𝑥−7

29. Ex 2. Find the horizontal asymptotes (if any) of the following functions. 𝑓 𝑥 = 8𝑥+5 5 𝑥 2 −3𝑥+1 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2 ℎ 𝑥 = 6 𝑥 3 − 𝑥 𝑥 2 +4𝑥−7

30. Ex 2. Find the horizontal asymptotes (if any) of the following functions. 𝑓 𝑥 = 8𝑥+5 5 𝑥 2 −3𝑥+1 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2 ℎ 𝑥 = 6 𝑥 3 − 𝑥 𝑥 2 +4𝑥−7

31. Ex 2. Find the horizontal asymptotes (if any) of the following functions. 𝑓 𝑥 = 8𝑥+5 5 𝑥 2 −3𝑥+1 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2 ℎ 𝑥 = 6 𝑥 3 − 𝑥 𝑥 2 +4𝑥−7

32. Ex 2. Find the horizontal asymptotes (if any) of the following functions. 𝑓 𝑥 = 8𝑥+5 5 𝑥 2 −3𝑥+1 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2 ℎ 𝑥 = 6 𝑥 3 − 𝑥 𝑥 2 +4𝑥−7

33. Ex 2. Find the horizontal asymptotes (if any) of the following functions. 𝑓 𝑥 = 8𝑥+5 5 𝑥 2 −3𝑥+1 𝑔 𝑥 = 3 𝑥 2 −2𝑥−1 2 𝑥 2 +3𝑥−2 ℎ 𝑥 = 6 𝑥 3 − 𝑥 𝑥 2 +4𝑥−7

34. Graphing Rational Functions 1. Factor numerator and denominator. 2
Graphing Rational Functions 1. Factor numerator and denominator. 2. Find 𝒙-intercepts and 𝒚-intercept. 3. Find vertical asymptotes (where denominator is 0). 4. Find horizontal asymptotes. 5. Graph (plot additional points as needed).

35. Graphing Rational Functions 1. Factor numerator and denominator. 2
Graphing Rational Functions 1. Factor numerator and denominator. 2. Find 𝒙-intercepts and 𝒚-intercept. 3. Find vertical asymptotes (where denominator is 0). 4. Find horizontal asymptotes. 5. Graph (plot additional points as needed).

36. Graphing Rational Functions 1. Factor numerator and denominator. 2
Graphing Rational Functions 1. Factor numerator and denominator. 2. Find 𝒙-intercepts and 𝒚-intercept. 3. Find vertical asymptotes (where denominator is 0). 4. Find horizontal asymptotes. 5. Graph (plot additional points as needed).

37. Graphing Rational Functions 1. Factor numerator and denominator. 2
Graphing Rational Functions 1. Factor numerator and denominator. 2. Find 𝒙-intercepts and 𝒚-intercept. 3. Find vertical asymptotes (where denominator is 0). 4. Find horizontal asymptotes. 5. Graph (plot additional points as needed).

38. Graphing Rational Functions 1. Factor numerator and denominator. 2
Graphing Rational Functions 1. Factor numerator and denominator. 2. Find 𝒙-intercepts and 𝒚-intercept. 3. Find vertical asymptotes (where denominator is 0). 4. Find horizontal or slant asymptotes (if any). 5. Graph (plot additional points as needed).

39. Graphing Rational Functions 1. Factor numerator and denominator. 2
Graphing Rational Functions 1. Factor numerator and denominator. 2. Find 𝒙-intercepts and 𝒚-intercept. 3. Find vertical asymptotes (where denominator is 0). 4. Find horizontal or slant asymptotes (if any). 5. Graph (plot additional points as needed).

40. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

41. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

42. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

43. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

44. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

45. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

46. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

47. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

48. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

49. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

50. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

51. Ex 3. Graph 𝑓 𝑥 = 2 𝑥 2 +7𝑥−4 𝑥 2 +𝑥−2

52. Ex 4. Graph 𝑔 𝑥 = 𝑥−3 𝑥 2 +4𝑥+4

53. Ex 4. Graph 𝑔 𝑥 = 𝑥−3 𝑥 2 +4𝑥+4

54. Ex 4. Graph 𝑔 𝑥 = 𝑥−3 𝑥 2 +4𝑥+4

55. Ex 4. Graph 𝑔 𝑥 = 𝑥−3 𝑥 2 +4𝑥+4

56. Ex 4. Graph 𝑔 𝑥 = 𝑥−3 𝑥 2 +4𝑥+4

57. Ex 4. Graph 𝑔 𝑥 = 𝑥−3 𝑥 2 +4𝑥+4

58. Ex 4. Graph 𝑔 𝑥 = 𝑥−3 𝑥 2 +4𝑥+4

59. Ex 4. Graph 𝑔 𝑥 = 𝑥−3 𝑥 2 +4𝑥+4

60. Ex 4. Graph 𝑔 𝑥 = 𝑥−3 𝑥 2 +4𝑥+4

61. Sometimes rational functions have slant asymptotes (also called oblique asymptotes), which are neither vertical nor horizontal.

62. Ex 5. Graph ℎ 𝑥 = 𝑥 2 −4𝑥−5 𝑥−3

63. Ex 5. Graph ℎ 𝑥 = 𝑥 2 −4𝑥−5 𝑥−3

64. Ex 5. Graph ℎ 𝑥 = 𝑥 2 −4𝑥−5 𝑥−3

65. Ex 5. Graph ℎ 𝑥 = 𝑥 2 −4𝑥−5 𝑥−3

66. Ex 5. Graph ℎ 𝑥 = 𝑥 2 −4𝑥−5 𝑥−3

67. Ex 5. Graph ℎ 𝑥 = 𝑥 2 −4𝑥−5 𝑥−3

68. Ex 5. Graph ℎ 𝑥 = 𝑥 2 −4𝑥−5 𝑥−3

69. Ex 5. Graph ℎ 𝑥 = 𝑥 2 −4𝑥−5 𝑥−3

70. Ex 5. Graph ℎ 𝑥 = 𝑥 2 −4𝑥−5 𝑥−3