1. Jeff Bivin -- LZHS Graphing Rational Functions Jeffrey Bivin Lake Zurich High School Jeff.bivin@lz95.org Last Updated: February 18, 2008

2. Jeff Bivin -- LZHS Graph: xy 11 2 5 10 2 x = 0 y = 0

3. Jeff Bivin -- LZHS Graph: xy -2 -5 -10 -2 -10 x = 0 y = 0

4. Jeff Bivin -- LZHS Graph: x = 0 y = 0 DOMAIN RANGE

5. Jeff Bivin -- LZHS

6. Graph: x = 2 y = 3 x – 2 = 0 x = 2 2 3

7. Jeff Bivin -- LZHS Graph: x = 2 y = 3 y-intercept 0

8. Jeff Bivin -- LZHS Graph: x = 2 y = 3 x-intercept 0

9. Jeff Bivin -- LZHS Graph: x = 2 y = 3 DOMAIN RANGE

10. Jeff Bivin -- LZHS

11. Graph: x = -1 y = -2 x + 1 = 0 x = -1 1 2 Vertical stretch

12. Jeff Bivin -- LZHS Graph: x = -1 y = -2 y-intercept 0

13. Jeff Bivin -- LZHS Graph: x = -1 y = -2 x-intercept 0

14. Jeff Bivin -- LZHS Graph: x = -1 y = -2 DOMAIN RANGE

15. Jeff Bivin -- LZHS

16. Graph: x = 1 y = 2 x - 1 = 0 x = 1 Horizontal asymptote Vertical asymptote 1

17. Jeff Bivin -- LZHS Graph: x = 1 y = 2 y-intercept 0 0

18. Jeff Bivin -- LZHS Graph: x = 1 y = 2 x-intercept 0

19. Jeff Bivin -- LZHS Graph: x = 1 y = 2 DOMAIN RANGE

20. Jeff Bivin -- LZHS

21. Graph: 2x - 3 = 0 Horizontal asymptote Vertical asymptote 2x = 3

22. Jeff Bivin -- LZHS Graph: y-intercept 0 0

23. Jeff Bivin -- LZHS Graph: x-intercept

24. Jeff Bivin -- LZHS Graph: DOMAIN RANGE

25. Jeff Bivin -- LZHS

26. Graph: x - 2 = 0 Horizontal asymptote Vertical asymptote x = 2 y = 0 x + 3 = 0 x = -3 x = 2x = -3

27. Jeff Bivin -- LZHS Graph: y = 0 x = 2x = -3 y-intercept 0 00

28. Jeff Bivin -- LZHS Graph: y = 0 x = 2x = -3 x-intercept

29. Jeff Bivin -- LZHS Graph: y = 0 x = 2x = -3 let x = -5

30. Jeff Bivin -- LZHS Graph: y = 0 x = 2x = -3 let x = 3

31. Jeff Bivin -- LZHS Graph: y = 0 x = 2x = -3 DOMAIN RANGE

32. Jeff Bivin -- LZHS

33. Graph: x - 2 = 0 Horizontal asymptote Vertical asymptote x = 2 y = 0 x + 2 = 0 x = -2 x = 2x = -2

34. Jeff Bivin -- LZHS Graph: y = 0 x = 2 y-intercept 00 x = -2

35. Jeff Bivin -- LZHS Graph: y = 0 x = 2x = -2 x-intercept Ø

36. Jeff Bivin -- LZHS Graph: y = 0 x = 2x = -2 let x = -3

37. Jeff Bivin -- LZHS Graph: y = 0 x = 2x = -2 let x = 3

38. Jeff Bivin -- LZHS Graph: y = 0 x = 2x = -2 DOMAIN RANGE

39. Jeff Bivin -- LZHS

40. Graph: x + 4 = 0 Horizontal asymptote Vertical asymptote x = -4 y = 0 x - 1 = 0 x = 1 x = -4

41. Jeff Bivin -- LZHS Graph: y = 0 x = 1x = -4 y-intercept

42. Jeff Bivin -- LZHS Graph: y = 0 x = 1x = -4 x-intercept

43. Jeff Bivin -- LZHS Graph: y = 0 x = 1x = -4 let x = -5

44. Jeff Bivin -- LZHS Graph: y = 0 x = 1x = -4 let x = 2

45. Jeff Bivin -- LZHS

46. Graph: x - 1 = 0 Horizontal asymptote Vertical asymptote x = 1 x + 4 = 0 x = -4 x - 3 = 0 Holes x = 3 x + 1 = 0 x = -1

47. Jeff Bivin -- LZHS Graph: y = 0 x = 1x = -4 Horizontal asymptote Vertical asymptote x = 1x = -4 Holes

48. Jeff Bivin -- LZHS Graph: y = 0 x = 1x = -4 y-intercept

49. Jeff Bivin -- LZHS Graph: y = 0 x = 1x = -4 x-intercept

50. Jeff Bivin -- LZHS Graph: y = 0 x = 1x = -4 VA HA Holes y-int x-int x = 1 x = -4

51. Jeff Bivin -- LZHS Graph: y = 0 x = 1x = -4 let x = -∞

52. Jeff Bivin -- LZHS

53. Graph: x - 1 = 0 Horizontal asymptote Vertical asymptote x = 1 x + 3 = 0 x = -3

54. Jeff Bivin -- LZHS Graph: x - 2 = 0 Holes x = 2 x + 2 = 0 x = -2 x + 6 = 0 x = -6

55. Jeff Bivin -- LZHS Graph: y-intercept

56. Jeff Bivin -- LZHS Graph: x-intercept(s)

57. Jeff Bivin -- LZHS Graph: y = 1 x = 1x = -3 VA HA Holes y-int x-int x = 1 x = -3

58. Jeff Bivin -- LZHS Graph: Asymptote Crossing(s)

59. Jeff Bivin -- LZHS Graph: y = 1 x = 1x = -3 VA HA Holes y-int x-int x = 1 x = -3 AX

60. Jeff Bivin -- LZHS Graph: y = 1 x = 1x = -3 DOMAIN RANGE

61. Jeff Bivin -- LZHS

62. Graph: x - 3 = 0 Horizontal asymptote Vertical asymptote x = 3 x + 2 = 0 x = -2

63. Jeff Bivin -- LZHS Graph: y-interceptx-intercept Ø

64. Jeff Bivin -- LZHS Graph: y = 0 x = 3x = -2 VA HA Holesnone y-int x-intnone x = 3 x = -2

65. Jeff Bivin -- LZHS Graph: y = 0 x = 3x = -2 let x = -∞ let x = +∞

66. Jeff Bivin -- LZHS

67. Graph: Horizontal asymptote Vertical asymptote

68. Jeff Bivin -- LZHS Graph: y-interceptx-intercept Ø

69. Jeff Bivin -- LZHS Graph: y = 0 VAnone HA Holesnone y-int x-intnone

70. Jeff Bivin -- LZHS

71. Graph: Hole x + 1 = 0 x = -1 y-intercept y = 0 - 2 y = -2 x-intercept 0 = x - 2 2 = x

72. Jeff Bivin -- LZHS Graph: VAnone HAnone Holes y-int x-int

73. Jeff Bivin -- LZHS

74. Graph: Horizontal asymptote: none Slant asymptote Vertical asymptote x + 1 = 0 x = -1

75. Jeff Bivin -- LZHS Graph: y-interceptx-intercept(s)

76. Jeff Bivin -- LZHS Graph: y = x+1 x = -1 VA HAnone SA Holesnone y-int x-int AX x = -1 y=x+1

77. Jeff Bivin -- LZHS Graph: Asymptote crossing Ø

78. Jeff Bivin -- LZHS Graph: y = x+1 x = -1 VA HAnone SA Holesnone y-int x-int AXnone x = -1 y=x+1

79. Jeff Bivin -- LZHS

80. Graph: x + 1 = 0 Horizontal asymptote Vertical asymptote x = -1 Hole No holes x + 1 = 0 x = -1 Since both a vertical asymptote and a hole seem to be at x = - 1, the asymptote wins.

81. Jeff Bivin -- LZHS Graph: y-interceptx-intercept none

82. Jeff Bivin -- LZHS Graph: x = -1 VA HA SAnone Holesnone y-int x-intnone AXnone x = -1 y = 0