Graphing Rational Functions                           Mathematics 4                            July 26, 2011Mathematics 4 ...
Review of Previous Concepts on Rational FunctionsRational Functions    Mathematics 4 ()   Graphing Rational Functions   Ju...
Review of Previous Concepts on Rational FunctionsRational Functions                                                       ...
Review of Previous Concepts on Rational FunctionsRational Functions                                                       ...
Review of Previous Concepts on Rational FunctionsRational Functions                                                       ...
Review of Previous Concepts on Rational FunctionsRational Functions                                                       ...
The Horizontal AsymptoteHorizontal Asymptotes    Mathematics 4 ()    Graphing Rational Functions   July 26, 2011   3 / 61
The Horizontal AsymptoteHorizontal Asymptotes    The horizontal asymptote is an imaginary line that a rational    function...
The Horizontal AsymptoteHorizontal Asymptotes    The horizontal asymptote is an imaginary line that a rational    function...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                             3x + 5                   ...
The Horizontal AsymptoteConsider the function below:                                               3x + 5                 ...
The Horizontal Asymptote                                3x3 + 4x2 + 7x + 2Consider the function f (x) =                   ...
The Horizontal Asymptote                                3x3 + 4x2 + 7x + 2Consider the function f (x) =                   ...
The Horizontal Asymptote                                3x3 + 4x2 + 7x + 2Consider the function f (x) =                   ...
The Horizontal Asymptote                                   3x3 + 4x2 + 7x + 2Consider the function f (x) =                ...
The Horizontal Asymptotedegree of p(x) = degree of q(x)If the degree of the numerator p(x) is equal to the degree of thede...
The Horizontal Asymptote                                       5x + 2Consider the function f (x) =                        ...
The Horizontal Asymptote                                       5x + 2Consider the function f (x) =                        ...
The Horizontal Asymptote                                       5x + 2Consider the function f (x) =                        ...
The Horizontal Asymptote                                         5x + 2Consider the function f (x) =                      ...
The Horizontal Asymptotedegree of p(x) < degree of q(x)If the degree of the numerator p(x) is less than the degree of thed...
The Horizontal AsymptoteSummary                        an xn + ... + a1 x + a0The graph of f (x) =                        ...
The Horizontal AsymptoteSummary                        an xn + ... + a1 x + a0The graph of f (x) =                        ...
The Horizontal AsymptoteSummary                        an xn + ... + a1 x + a0The graph of f (x) =                        ...
The Horizontal AsymptoteSummary                         an xn + ... + a1 x + a0The graph of f (x) =                       ...
Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x):     Mathematics...
Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x):  1   Domain of ...
Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x):  1   Domain of ...
Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x):  1   Domain of ...
Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x):  1   Domain of ...
Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x):  1   Domain of ...
Graphing Rational FunctionsGraphingThe following information is needed to graph a rational function f (x):  1   Domain of ...
Graphing Rational Functions                                                                 x+1                           ...
Graphing Rational Functions                                                                 x+1                           ...
Graphing Rational Functions                                                                 x+1                           ...
Graphing Rational Functions                                                                 x+1                           ...
Graphing Rational Functions                                                                  x+1                          ...
Graphing Rational Functions                                                                 x+1                           ...
Graphing Rational Functions                                                                 x+1                           ...
Graphing Rational Functions                                                                 x+1                           ...
Graphing Rational Functions                                                                 x+1                           ...
Graphing Rational Functions                                                                 x+1                           ...
Graphing Rational Functions                                                                 x+1                           ...
Graphing Rational Functions                                                                                  x+1          ...
Graphing Rational Functions                                                                                  x+1          ...
Graphing Rational Functions                                                                                    x+1        ...
Graphing Rational Functions                                                                                    x+1        ...
Graphing Rational Functions                                                                                    x+1        ...
Graphing Rational Functions                                                                                    x+1        ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                     (x + 3)(x − 1)            ...
Graphing Rational Functions                                                                         (x + 3)(x − 1)        ...
Graphing Rational Functions                                                                         (x + 3)(x − 1)        ...
Graphing Rational Functions                                                                         (x + 3)(x − 1)        ...
Graphing Rational Functions                                                                         (x + 3)(x − 1)        ...
Graphing Rational Functions                                                                         (x + 3)(x − 1)        ...
Graphing Rational Functions                                                                         (x + 3)(x − 1)        ...
Graphing Rational Functions                                                                         (x + 3)(x − 1)        ...
Graphing Rational Functions                                                                         (x + 3)(x − 1)        ...
Assignment: Homework 12 due tomorrow.Mathematics 4 ()              Graphing Rational Functions   July 26, 2011   39 / 61
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                nA rational function f...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                   nA rational functio...
Graphing Rational Functions with Oblique AsymptotesOblique AsymptoteThe oblique asymptote is a line of the form y = mx + b...
Graphing Rational Functions with Oblique AsymptotesOblique AsymptoteThe oblique asymptote is a line of the form y = mx + b...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                            x2 −2x+1Fi...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                            x2 −2x+1Fi...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                            x2 −2x+1Fi...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                            x2 +6x+8Fi...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                            x2 +6x+8Fi...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                            x2 +6x+8Fi...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                            x3 +7x2 −2...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                            x3 +7x2 −2...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                                 x3 +7...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                            3x3 −2x2 +...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                            3x3 −2x2 +...
Graphing Rational Functions with Oblique AsymptotesOblique Asymptote                                                    3x...
Graphing Rational Functions with Oblique Asymptotes                                                          x2 +6x+8     ...
Graphing Rational Functions with Oblique Asymptotes                                                          x2 +6x+8     ...
Graphing Rational Functions with Oblique Asymptotes                                                          x2 +6x+8     ...
Graphing Rational Functions with Oblique Asymptotes                                                          x2 +6x+8     ...
Graphing Rational Functions with Oblique Asymptotes                                                           x2 +6x+8    ...
Graphing Rational Functions with Oblique Asymptotes                                                          x2 +6x+8     ...
Graphing Rational Functions with Oblique Asymptotes                                                          x2 +6x+8     ...
Graphing Rational Functions with Oblique Asymptotes                                                          x2 +6x+8     ...
Graphing Rational Functions with Oblique Asymptotes                                                           x2 +6x+8    ...
Graphing Rational Functions with Oblique Asymptotes                                                           x2 +6x+8    ...
Graphing Rational Functions with Oblique Asymptotes                                                            x2 +6x+8   ...
Graphing Rational Functions with Oblique Asymptotes                                                                       ...
Graphing Rational Functions with Oblique Asymptotes                                                                       ...
Graphing Rational Functions with Oblique Asymptotes                                                                       ...
Math 4 lecture on Graphing Rational Functions
Math 4 lecture on Graphing Rational Functions
Math 4 lecture on Graphing Rational Functions
Math 4 lecture on Graphing Rational Functions
Math 4 lecture on Graphing Rational Functions
Math 4 lecture on Graphing Rational Functions
Math 4 lecture on Graphing Rational Functions
Math 4 lecture on Graphing Rational Functions
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Math 4 lecture on Graphing Rational Functions

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Math 4 lecture on Graphing Rational Functions

  1. 1. Graphing Rational Functions Mathematics 4 July 26, 2011Mathematics 4 () Graphing Rational Functions July 26, 2011 1 / 61
  2. 2. Review of Previous Concepts on Rational FunctionsRational Functions Mathematics 4 () Graphing Rational Functions July 26, 2011 2 / 61
  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 Functions July 26, 2011 2 / 61
  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 Functions July 26, 2011 2 / 61
  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 Functions July 26, 2011 2 / 61
  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 Functions July 26, 2011 2 / 61
  7. 7. The Horizontal AsymptoteHorizontal Asymptotes Mathematics 4 () Graphing Rational Functions July 26, 2011 3 / 61
  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 Functions July 26, 2011 3 / 61
  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 Functions July 26, 2011 3 / 61
  10. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 82. Assignment: Homework 12 due tomorrow.Mathematics 4 () Graphing Rational Functions July 26, 2011 39 / 61
  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 Functions July 26, 2011 40 / 61
  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 Functions July 26, 2011 40 / 61
  85. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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