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# Math 4 graphing rational functions

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### Math 4 graphing rational functions

1. 1. Graphing Rational Functions1 Mathematics 4 January 3, 2012Mathematics 4 () Graphing Rational Functions1 January 3, 2012 1/1
2. 2. Review of Previous Concepts on Rational FunctionsRational Functions Mathematics 4 () Graphing Rational Functions1 January 3, 2012 2/1
3. 3. Review of Previous Concepts on Rational FunctionsRational Functions p(x) A rational function can be expressed as f (x) = where p(x) q(x) and q(x) are polynomial functions. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 2/1
4. 4. Review of Previous Concepts on Rational FunctionsRational Functions p(x) A rational function can be expressed as f (x) = where p(x) q(x) and q(x) are polynomial functions. The domain of f (x) is R excluding the values where the denominator q(x) is zero. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 2/1
5. 5. Review of Previous Concepts on Rational FunctionsRational Functions p(x) A rational function can be expressed as f (x) = where p(x) q(x) and q(x) are polynomial functions. The domain of f (x) is R excluding the values where the denominator q(x) is zero. The zeros of the numerator p(x) are the zeros of the rational function f (x). Mathematics 4 () Graphing Rational Functions1 January 3, 2012 2/1
6. 6. Review of Previous Concepts on Rational FunctionsRational Functions p(x) A rational function can be expressed as f (x) = where p(x) q(x) and q(x) are polynomial functions. The domain of f (x) is R excluding the values where the denominator q(x) is zero. The zeros of the numerator p(x) are the zeros of the rational function f (x). If f (x) is in lowest terms, the zeros of the denominator q(x) determine the vertical asymptotes of the function. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 2/1
7. 7. The Horizontal AsymptoteHorizontal Asymptotes Mathematics 4 () Graphing Rational Functions1 January 3, 2012 3/1
8. 8. The Horizontal AsymptoteHorizontal Asymptotes The horizontal asymptote is an imaginary line that a rational function approaches as x → −∞ and x → ∞. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 3/1
9. 9. The Horizontal AsymptoteHorizontal Asymptotes The horizontal asymptote is an imaginary line that a rational function approaches as x → −∞ and x → ∞. The graph of a rational function CAN cross its horizontal asymptotes. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 3/1
10. 10. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
11. 11. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
12. 12. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (−1000) = Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
13. 13. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (−1000) = Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
14. 14. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (−1000) = 2.98009 f (−10000) = Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
15. 15. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (−1000) = 2.98009 f (−10000) = Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
16. 16. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (100) = f (−1000) = 2.98009 f (−10000) = 2.998 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
17. 17. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (100) = f (−1000) = 2.98009 f (−10000) = 2.998 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
18. 18. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (100) = 3.2105 f (−1000) = 2.98009 f (1000) = f (−10000) = 2.998 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
19. 19. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (100) = 3.2105 f (−1000) = 2.98009 f (1000) = f (−10000) = 2.998 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
20. 20. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (100) = 3.2105 f (−1000) = 2.98009 f (1000) = 3.0201 f (−10000) = 2.998 f (10000) = Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
21. 21. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (100) = 3.2105 f (−1000) = 2.98009 f (1000) = 3.0201 f (−10000) = 2.998 f (10000) = Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
22. 22. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (100) = 3.2105 f (−1000) = 2.98009 f (1000) = 3.0201 f (−10000) = 2.998 f (10000) = 3.002 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
23. 23. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (100) = 3.2105 f (−1000) = 2.98009 f (1000) = 3.0201 f (−10000) = 2.998 f (10000) = 3.002As x → −∞ and as x → ∞, the value of f (x) becomes closer to 3. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
24. 24. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (100) = 3.2105 f (−1000) = 2.98009 f (1000) = 3.0201 f (−10000) = 2.998 f (10000) = 3.002As x → −∞ and as x → ∞, the value of f (x) becomes closer to 3. The horizontal asymptote is the line y = 3. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 4/1
25. 25. The Horizontal Asymptote 3x3 + 4x2 + 7x + 2Consider the function f (x) = 4x3 − 5x2 + 3x + 1 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 5/1
26. 26. The Horizontal Asymptote 3x3 + 4x2 + 7x + 2Consider the function f (x) = 4x3 − 5x2 + 3x + 1Let’s analyze the function as x → −∞: (please refer to the blackboard) Mathematics 4 () Graphing Rational Functions1 January 3, 2012 5/1
27. 27. The Horizontal Asymptote 3x3 + 4x2 + 7x + 2Consider the function f (x) = 4x3 − 5x2 + 3x + 1Let’s analyze the function as x → −∞: (please refer to the blackboard)Now analyze the function as x → ∞: (please refer to the blackboard) Mathematics 4 () Graphing Rational Functions1 January 3, 2012 5/1
28. 28. The Horizontal Asymptote 3x3 + 4x2 + 7x + 2Consider the function f (x) = 4x3 − 5x2 + 3x + 1Let’s analyze the function as x → −∞: (please refer to the blackboard)Now analyze the function as x → ∞: (please refer to the blackboard) The horizontal asymptote is y = 3 . 4 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 5/1
29. 29. The Horizontal Asymptotedegree of p(x) = degree of q(x)If the degree of the numerator p(x) is equal to the degree of thedenominator q(x), the horizontal asymptote is an y= bnwhere an is the leading coeﬃcient of p(x) and q(x) is the leadingcoeﬃcient of q(x). Mathematics 4 () Graphing Rational Functions1 January 3, 2012 6/1
30. 30. The Horizontal Asymptote 5x + 2Consider the function f (x) = 3x3 − x2 − 8x − 2 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 7/1
31. 31. The Horizontal Asymptote 5x + 2Consider the function f (x) = 3x3 − x2 − 8x − 2Let’s analyze the function as x → −∞: (please refer to the blackboard) Mathematics 4 () Graphing Rational Functions1 January 3, 2012 7/1
32. 32. The Horizontal Asymptote 5x + 2Consider the function f (x) = 3x3 − x2 − 8x − 2Let’s analyze the function as x → −∞: (please refer to the blackboard)Now analyze the function as x → ∞: (please refer to the blackboard) Mathematics 4 () Graphing Rational Functions1 January 3, 2012 7/1
33. 33. The Horizontal Asymptote 5x + 2Consider the function f (x) = 3x3 − x2 − 8x − 2Let’s analyze the function as x → −∞: (please refer to the blackboard)Now analyze the function as x → ∞: (please refer to the blackboard) The horizontal asymptote is y = 0. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 7/1
34. 34. The Horizontal Asymptotedegree of p(x) < degree of q(x)If the degree of the numerator p(x) is less than the degree of thedenominator q(x), the horizontal asymptote is y=0 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 8/1
35. 35. The Horizontal AsymptoteSummary an xn + ... + a1 x + a0The graph of f (x) = has: bm xm + ... + b1 x + b0 its HA at the x-axis n<m Mathematics 4 () Graphing Rational Functions1 January 3, 2012 9/1
36. 36. The Horizontal AsymptoteSummary an xn + ... + a1 x + a0The graph of f (x) = has: bm xm + ... + b1 x + b0 its HA at the x-axis n<m Mathematics 4 () Graphing Rational Functions1 January 3, 2012 9/1
37. 37. The Horizontal AsymptoteSummary an xn + ... + a1 x + a0The graph of f (x) = has: bm xm + ... + b1 x + b0 its HA at the x-axis n<m an its HA at y = n=m bm Mathematics 4 () Graphing Rational Functions1 January 3, 2012 9/1
38. 38. The Horizontal AsymptoteSummary an xn + ... + a1 x + a0The graph of f (x) = has: bm xm + ... + b1 x + b0 its HA at the x-axis n<m an its HA at y = n=m bm no horizontal asymptote n>m Mathematics 4 () Graphing Rational Functions1 January 3, 2012 9/1
39. 39. Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x): Mathematics 4 () Graphing Rational Functions1 January 3, 2012 10 / 1
40. 40. Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x): 1 Domain of f (x) Mathematics 4 () Graphing Rational Functions1 January 3, 2012 10 / 1
41. 41. Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x): 1 Domain of f (x) 2 Zeros of f (x) Mathematics 4 () Graphing Rational Functions1 January 3, 2012 10 / 1
42. 42. Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x): 1 Domain of f (x) 2 Zeros of f (x) 3 The vertical asymptotes (if any) Mathematics 4 () Graphing Rational Functions1 January 3, 2012 10 / 1
43. 43. Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x): 1 Domain of f (x) 2 Zeros of f (x) 3 The vertical asymptotes (if any) 4 The horizontal asymptote (if any) Mathematics 4 () Graphing Rational Functions1 January 3, 2012 10 / 1
44. 44. Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x): 1 Domain of f (x) 2 Zeros of f (x) 3 The vertical asymptotes (if any) 4 The horizontal asymptote (if any) 5 The y-intercept Mathematics 4 () Graphing Rational Functions1 January 3, 2012 10 / 1
45. 45. Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x): 1 Domain of f (x) 2 Zeros of f (x) 3 The vertical asymptotes (if any) 4 The horizontal asymptote (if any) 5 The y-intercept 6 The table of signs Mathematics 4 () Graphing Rational Functions1 January 3, 2012 10 / 1
46. 46. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 Domain: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 11 / 1
47. 47. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 Domain: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 11 / 1
48. 48. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 Domain: x=1 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 11 / 1
49. 49. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 Zeros: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 12 / 1
50. 50. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 Zeros: x = −1 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 13 / 1
51. 51. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 Vertical asymptotes: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 14 / 1
52. 52. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 Vertical asymptotes: x=1 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 15 / 1
53. 53. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 Horizontal asymptote: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 16 / 1
54. 54. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 Horizontal asymptote: y=1 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 17 / 1
55. 55. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 y-intercept: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 18 / 1
56. 56. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 y-intercept: −1 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 19 / 1
57. 57. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 −1 1 x+1 - 0 + + + Table of signs: x−1 - - - 0 + Mathematics 4 () Graphing Rational Functions1 January 3, 2012 20 / 1
58. 58. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 −1 1 x+1 - 0 + + + Table of signs: x−1 - - - 0 + Mathematics 4 () Graphing Rational Functions1 January 3, 2012 20 / 1
59. 59. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 −1 1 x+1 - 0 + + + Table of signs: x−1 - - - 0 + f (x) + 0 − VA + Mathematics 4 () Graphing Rational Functions1 January 3, 2012 20 / 1
60. 60. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 −1 1 x+1 - 0 + + + Table of signs: x−1 - - - 0 + f (x) + 0 − VA + Mathematics 4 () Graphing Rational Functions1 January 3, 2012 21 / 1
61. 61. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 −1 1 x+1 - 0 + + + Table of signs: x−1 - - - 0 + f (x) + 0 − VA + Mathematics 4 () Graphing Rational Functions1 January 3, 2012 22 / 1
62. 62. Graphing Rational Functions x+1 Example: f (x) = x−1 y 5 4 3 2 1 x −5 −4 −3 −2 −1 1 2 3 4 −1 −2 −3 −1 1 x+1 - 0 + + + Table of signs: x−1 - - - 0 + f (x) + 0 − VA + Mathematics 4 () Graphing Rational Functions1 January 3, 2012 23 / 1
63. 63. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 Domain: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 24 / 1
64. 64. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 Domain: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 24 / 1
65. 65. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 Domain: x = −1, 3 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 24 / 1
66. 66. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 Zeros Mathematics 4 () Graphing Rational Functions1 January 3, 2012 25 / 1
67. 67. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 Zeros −3, 1 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 26 / 1
68. 68. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 y-intercept Mathematics 4 () Graphing Rational Functions1 January 3, 2012 27 / 1
69. 69. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 y-intercept: −1 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 28 / 1
70. 70. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 vertical asymptote: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 29 / 1
71. 71. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 vertical asymptote: x = −1, x = 3 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 30 / 1
72. 72. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 horizontal asymptote: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 31 / 1
73. 73. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 horizontal asymptote: y = −1 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 32 / 1
74. 74. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 −3 −1 1 3 x+3 - 0 + + + + + + + x−1 - - - - - 0 + + + Table of signs: x+1 - - - 0 + + + + + 3−x + + + + + + + 0 - Mathematics 4 () Graphing Rational Functions1 January 3, 2012 33 / 1
75. 75. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 −3 −1 1 3 x+3 - 0 + + + + + + + x−1 - - - - - 0 + + + Table of signs: x+1 - - - 0 + + + + + 3−x + + + + + + + 0 - Mathematics 4 () Graphing Rational Functions1 January 3, 2012 33 / 1
76. 76. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 −3 −1 1 3 x+3 - 0 + + + + + + + x−1 - - - - - 0 + + + Table of signs: x+1 - - - 0 + + + + + 3−x + + + + + + + 0 - f (x) − 0 + VA − 0 + VA − Mathematics 4 () Graphing Rational Functions1 January 3, 2012 33 / 1
77. 77. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 −3 −1 1 3 x+3 - 0 + + + + + + + x−1 - - - - - 0 + + + Table of signs: x+1 - - - 0 + + + + + 3−x + + + + + + + 0 - f (x) − 0 + VA − 0 + VA − Mathematics 4 () Graphing Rational Functions1 January 3, 2012 34 / 1
78. 78. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 −3 −1 1 3 x+3 - 0 + + + + + + + x−1 - - - - - 0 + + + Table of signs: x+1 - - - 0 + + + + + 3−x + + + + + + + 0 - f (x) − 0 + VA − 0 + VA − Mathematics 4 () Graphing Rational Functions1 January 3, 2012 35 / 1
79. 79. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 −3 −1 1 3 x+3 - 0 + + + + + + + x−1 - - - - - 0 + + + Table of signs: x+1 - - - 0 + + + + + 3−x + + + + + + + 0 - f (x) − 0 + VA − 0 + VA − Mathematics 4 () Graphing Rational Functions1 January 3, 2012 36 / 1
80. 80. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 −3 −1 1 3 x+3 - 0 + + + + + + + x−1 - - - - - 0 + + + Table of signs: x+1 - - - 0 + + + + + 3−x + + + + + + + 0 - f (x) − 0 + VA − 0 + VA − Mathematics 4 () Graphing Rational Functions1 January 3, 2012 37 / 1
81. 81. Graphing Rational Functions (x + 3)(x − 1) Example: f (x) = (x + 1)(3 − x) 5 y 4 3 2 1 x −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 −1 −2 −3 −4 −5 −3 −1 1 3 x+3 - 0 + + + + + + + x−1 - - - - - 0 + + + Table of signs: x+1 - - - 0 + + + + + 3−x + + + + + + + 0 - f (x) − 0 + VA − 0 + VA − Mathematics 4 () Graphing Rational Functions1 January 3, 2012 38 / 1
82. 82. Assignment: Homework 12 due tomorrow.Mathematics 4 () Graphing Rational Functions1 January 3, 2012 39 / 1
83. 83. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote nA rational function f (x) = bmx m+...+a11x+a0 has an oblique asymptote if it an x +...+b x+b0is in lowest terms and n = m + 1. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 40 / 1
84. 84. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote nA rational function f (x) = bmx m+...+a11x+a0 has an oblique asymptote if it an x +...+b x+b0is in lowest terms and n = m + 1.Functions with oblique asymptotes x2 − 2x + 1f (x) = x x 2 + 6x + 8f (x) = x+3 x3 + 7x2 − 20x − 96f (x) = x2 − 10x + 16 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 40 / 1
85. 85. Graphing Rational Functions with Oblique AsymptotesOblique AsymptoteThe oblique asymptote is a line of the form y = mx + b. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 41 / 1
86. 86. Graphing Rational Functions with Oblique AsymptotesOblique AsymptoteThe oblique asymptote is a line of the form y = mx + b.Finding the oblique asymptoteTake the polynomial part of the long division of the rational function. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 41 / 1
87. 87. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x2 −2x+1Find the oblique asymptote of f (x) = x : Mathematics 4 () Graphing Rational Functions1 January 3, 2012 42 / 1
88. 88. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x2 −2x+1Find the oblique asymptote of f (x) = x : Mathematics 4 () Graphing Rational Functions1 January 3, 2012 42 / 1
89. 89. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x2 −2x+1Find the oblique asymptote of f (x) = x : x−2 x x2 − 2x + 1 − x2 − 2x 2xThe oblique asymptote is y = x − 2. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 42 / 1
90. 90. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x2 +6x+8Find the oblique asymptote of f (x) = x+3 : Mathematics 4 () Graphing Rational Functions1 January 3, 2012 43 / 1
91. 91. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x2 +6x+8Find the oblique asymptote of f (x) = x+3 : Mathematics 4 () Graphing Rational Functions1 January 3, 2012 43 / 1
92. 92. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x2 +6x+8Find the oblique asymptote of f (x) = x+3 : x+3 x+3 x2+ 6x + 8 − x2 − 3x 3x + 8 − 3x − 9 −1The oblique asymptote is y = x + 3. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 43 / 1
93. 93. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x3 +7x2 −20x−96Find the oblique asymptote of f (x) = x2 −10x+16 : Mathematics 4 () Graphing Rational Functions1 January 3, 2012 44 / 1
94. 94. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x3 +7x2 −20x−96Find the oblique asymptote of f (x) = x2 −10x+16 : Mathematics 4 () Graphing Rational Functions1 January 3, 2012 44 / 1
95. 95. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x3 +7x2 −20x−96Find the oblique asymptote of f (x) = x2 −10x+16 : x + 17 x2 − 10x + 16 x3+ 7x2 − 20x − 96 − x3 + 10x2 − 16x 17x2 − 36x − 96 − 17x2 + 170x − 272 134x − 368The oblique asymptote is y = x + 17. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 44 / 1
96. 96. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote 3x3 −2x2 +4x−3Find the oblique asymptote of f (x) = x2 +3x+3 : Mathematics 4 () Graphing Rational Functions1 January 3, 2012 45 / 1
97. 97. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote 3x3 −2x2 +4x−3Find the oblique asymptote of f (x) = x2 +3x+3 : Mathematics 4 () Graphing Rational Functions1 January 3, 2012 45 / 1
98. 98. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote 3x3 −2x2 +4x−3Find the oblique asymptote of f (x) = x2 +3x+3 : 3x − 11 x2 + 3x + 3 3x3 − 2x2 + 4x − 3 − 3x3 − 9x2 − 9x − 11x2 − 5x − 3 11x2 + 33x + 33 28x + 30The oblique asymptote is y = 3x − 11. Mathematics 4 () Graphing Rational Functions1 January 3, 2012 45 / 1
99. 99. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 Domain: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 46 / 1
100. 100. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 Domain: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 46 / 1
101. 101. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 Domain: x = −3 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 46 / 1
102. 102. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 Zeros: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 47 / 1
103. 103. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 Zeros: −4, −2 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 48 / 1
104. 104. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 y-intercept: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 49 / 1
105. 105. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 8 y-intercept: 3 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 50 / 1
106. 106. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 vertical asymptotes: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 51 / 1
107. 107. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 vertical asymptotes: x = −3 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 52 / 1
108. 108. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 oblique asymptote: Mathematics 4 () Graphing Rational Functions1 January 3, 2012 53 / 1
109. 109. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 oblique asymptote: y =x+3 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 54 / 1
110. 110. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 −4 −3 −2 x+2 - - - - - 0 Table of signs: x+4 - 0 x+3 - - - 0 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 55 / 1
111. 111. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 −4 −3 −2 x+2 - - - - - 0 Table of signs: x+4 - 0 x+3 - - - 0 Mathematics 4 () Graphing Rational Functions1 January 3, 2012 55 / 1
112. 112. Graphing Rational Functions with Oblique Asymptotes x2 +6x+8 Example: f (x) = x+3 y 4 3 2 1 x −7 −6 −5 −4 −3 −2 −1 1 2 −1 −2 −3 −4 −4 −3 −2 x+2 - - - - - 0 Table of signs: x+4 - 0 x+3 - - - 0 f (x) − 0 + VA − 0 + Mathematics 4 () Graphing Rational Functions1 January 3, 2012 55 / 1