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- 1. Graphing Rational Functions Mathematics 4 July 26, 2011Mathematics 4 () Graphing Rational Functions July 26, 2011 1 / 61
- 2. Review of Previous Concepts on Rational FunctionsRational Functions Mathematics 4 () Graphing Rational Functions July 26, 2011 2 / 61
- 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 Functions July 26, 2011 2 / 61
- 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 Functions July 26, 2011 2 / 61
- 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 Functions July 26, 2011 2 / 61
- 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 Functions July 26, 2011 2 / 61
- 7. The Horizontal AsymptoteHorizontal Asymptotes Mathematics 4 () Graphing Rational Functions July 26, 2011 3 / 61
- 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 Functions July 26, 2011 3 / 61
- 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 Functions July 26, 2011 3 / 61
- 10. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = Mathematics 4 () Graphing Rational Functions July 26, 2011 4 / 61
- 11. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = Mathematics 4 () Graphing Rational Functions July 26, 2011 4 / 61
- 12. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (−1000) = Mathematics 4 () Graphing Rational Functions July 26, 2011 4 / 61
- 13. The Horizontal AsymptoteConsider the function below: 3x + 5 f (x) = x−5 f (−100) = 2.809 f (−1000) = Mathematics 4 () Graphing Rational Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 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 Functions July 26, 2011 4 / 61
- 25. The Horizontal Asymptote 3x3 + 4x2 + 7x + 2Consider the function f (x) = 4x3 − 5x2 + 3x + 1 Mathematics 4 () Graphing Rational Functions July 26, 2011 5 / 61
- 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 Functions July 26, 2011 5 / 61
- 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 Functions July 26, 2011 5 / 61
- 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 Functions July 26, 2011 5 / 61
- 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 Functions July 26, 2011 6 / 61
- 30. The Horizontal Asymptote 5x + 2Consider the function f (x) = 3x3 − x2 − 8x − 2 Mathematics 4 () Graphing Rational Functions July 26, 2011 7 / 61
- 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 Functions July 26, 2011 7 / 61
- 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 Functions July 26, 2011 7 / 61
- 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 Functions July 26, 2011 7 / 61
- 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 Functions July 26, 2011 8 / 61
- 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 Functions July 26, 2011 9 / 61
- 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 Functions July 26, 2011 9 / 61
- 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 Functions July 26, 2011 9 / 61
- 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 Functions July 26, 2011 9 / 61
- 39. Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x): Mathematics 4 () Graphing Rational Functions July 26, 2011 10 / 61
- 40. Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x): 1 Domain of f (x) Mathematics 4 () Graphing Rational Functions July 26, 2011 10 / 61
- 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 Functions July 26, 2011 10 / 61
- 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 Functions July 26, 2011 10 / 61
- 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 Functions July 26, 2011 10 / 61
- 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 Functions July 26, 2011 10 / 61
- 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 Functions July 26, 2011 10 / 61
- 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 Functions July 26, 2011 11 / 61
- 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 Functions July 26, 2011 11 / 61
- 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 Functions July 26, 2011 11 / 61
- 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 Functions July 26, 2011 12 / 61
- 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 Functions July 26, 2011 13 / 61
- 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 Functions July 26, 2011 14 / 61
- 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 Functions July 26, 2011 15 / 61
- 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 Functions July 26, 2011 16 / 61
- 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 Functions July 26, 2011 17 / 61
- 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 Functions July 26, 2011 18 / 61
- 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 Functions July 26, 2011 19 / 61
- 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 Functions July 26, 2011 20 / 61
- 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 Functions July 26, 2011 20 / 61
- 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 Functions July 26, 2011 20 / 61
- 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 Functions July 26, 2011 21 / 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 Functions July 26, 2011 22 / 61
- 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 Functions July 26, 2011 23 / 61
- 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 Functions July 26, 2011 24 / 61
- 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 Functions July 26, 2011 24 / 61
- 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 Functions July 26, 2011 24 / 61
- 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 Functions July 26, 2011 25 / 61
- 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 Functions July 26, 2011 26 / 61
- 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 Functions July 26, 2011 27 / 61
- 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 Functions July 26, 2011 28 / 61
- 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 Functions July 26, 2011 29 / 61
- 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 Functions July 26, 2011 30 / 61
- 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 Functions July 26, 2011 31 / 61
- 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 Functions July 26, 2011 32 / 61
- 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 Functions July 26, 2011 33 / 61
- 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 Functions July 26, 2011 33 / 61
- 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 Functions July 26, 2011 33 / 61
- 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 Functions July 26, 2011 34 / 61
- 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 Functions July 26, 2011 35 / 61
- 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 Functions July 26, 2011 36 / 61
- 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 Functions July 26, 2011 37 / 61
- 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 Functions July 26, 2011 38 / 61
- 82. Assignment: Homework 12 due tomorrow.Mathematics 4 () Graphing Rational Functions July 26, 2011 39 / 61
- 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 Functions July 26, 2011 40 / 61
- 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 Functions July 26, 2011 40 / 61
- 85. Graphing Rational Functions with Oblique AsymptotesOblique AsymptoteThe oblique asymptote is a line of the form y = mx + b. Mathematics 4 () Graphing Rational Functions July 26, 2011 41 / 61
- 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 Functions July 26, 2011 41 / 61
- 87. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x2 −2x+1Find the oblique asymptote of f (x) = x : Mathematics 4 () Graphing Rational Functions July 26, 2011 42 / 61
- 88. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x2 −2x+1Find the oblique asymptote of f (x) = x : Mathematics 4 () Graphing Rational Functions July 26, 2011 42 / 61
- 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 Functions July 26, 2011 42 / 61
- 90. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x2 +6x+8Find the oblique asymptote of f (x) = x+3 : Mathematics 4 () Graphing Rational Functions July 26, 2011 43 / 61
- 91. Graphing Rational Functions with Oblique AsymptotesOblique Asymptote x2 +6x+8Find the oblique asymptote of f (x) = x+3 : Mathematics 4 () Graphing Rational Functions July 26, 2011 43 / 61
- 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 Functions July 26, 2011 43 / 61
- 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 Functions July 26, 2011 44 / 61
- 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 Functions July 26, 2011 44 / 61
- 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 Functions July 26, 2011 44 / 61
- 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 Functions July 26, 2011 45 / 61
- 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 Functions July 26, 2011 45 / 61
- 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 Functions July 26, 2011 45 / 61
- 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 Functions July 26, 2011 46 / 61
- 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 Functions July 26, 2011 46 / 61
- 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 Functions July 26, 2011 46 / 61
- 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 Functions July 26, 2011 47 / 61
- 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 Functions July 26, 2011 48 / 61
- 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 Functions July 26, 2011 49 / 61
- 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 Functions July 26, 2011 50 / 61
- 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 Functions July 26, 2011 51 / 61
- 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 Functions July 26, 2011 52 / 61
- 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 Functions July 26, 2011 53 / 61
- 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 Functions July 26, 2011 54 / 61
- 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 Functions July 26, 2011 55 / 61
- 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 Functions July 26, 2011 55 / 61
- 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 Functions July 26, 2011 55 / 61

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