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Exploring Rational Functions: Vertical Asymptotes, Holes & More

Learn about rational functions with vertical asymptotes, holes, horizontal asymptotes, and slant asymptotes. Understand how to determine and graph these features in rational functions.

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Exploring Rational Functions: Vertical Asymptotes, Holes & More

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  1. Section 5.6 - Rational Functions

  2. Rational Functions A rational function is a function of the form where and are polynomial functions and . The domain of the rational function is the set of all inputs for which .

  3. Graph Features • Vertical asymptotes • Holes • Horizontal Asymptotes • Slant Asymptotes

  4. Vertical Asymptotes and Holes The vertical line is a vertical asymptote for the graph of if any of the following is true: as , or as , or as , or as • To find Vertical Asymptotes and Holes: • Find the domain of the function • Factor the numerator and denominator. • Factors in the denominator that do NOT cancel give vertical asymptotes. • Factors in the denominator that do cancel give holes.

  5. Vertical Asymptotes Determine the vertical asymptotes for the graph of the function. Vertical Asymptote:

  6. Vertical Asymptotes Determine vertical asymptotes , if they exist. Hole at

  7. Vertical Asymptotes Determine the vertical asymptotes for the graph of the function. Vertical Asymptote:

  8. Horizontal Asymptotes The horizontal line is a horizontal asymptote for the graph of if either or both of the following are true: as , or as • Determining a Horizontal Asymptote: • Degree on top the degree on bottom: • Degree on top the degree on bottom: • Degree on top the degree on bottom: No horizontal asymptote

  9. Horizontal Asymptotes Determine the horizontal asymptote for the graph of the function. Since the degree on top is less than the degree on bottom, we have a horizontal asymptote at

  10. Horizontal Asymptotes Determine the horizontal asymptote for the graph of the function. Since the degree on top the degree on bottom, we have a horizontal asymptote at or ***Note that the software does not indicate the hole in the graph at

  11. Horizontal Asymptotes Determine the horizontal asymptote for the graph of the function. Since the degree on top the degree on bottom, we have a horizontal asymptote at or

  12. Slant Asymptotes Sometimes a line that is not horizontal is as asymptote. Such a line is called an oblique asymptote or slant asymptote. • To find a Slant Asymptote (In the reduced function, if the degree on top is exactly one more than the degree on bottom) : • Perform polynomial division. • The quotient, without the remainder will give the slant asymptote. Horizontal asymptotes and slant asymptotes can NOT occur together!

  13. Slant Asymptotes Determine the oblique asymptote for the graph of the function. Since the degree on top is exactly one more than the degree on bottom, we have a slant asymptote. Slant Asymptote:

  14. Graphs of Rational Functions • Vertical Asymptotes: These occur at x-values that are not in the domain of the function and not in the domain of the reduced function. • Holes: These occur at x-values that are not in the domain of the function, but are in the domain of the reduced function. • Horizontal Asymptotes: • Oblique or Slant Asymptote: These occur when the degree of the numerator is equal to the degree of the denominator plus 1 (in the reduced function). The graph will NEVER cross a vertical asymptote. It may cross other types!

  15. Graph Can Cross Horizontal and Slant Asymptotes 8. Domain: Vertical Asymptotes: Holes: None Horizontal Asymptote: Slant Asymptote: None If you would like to find where the graph crosses the horizontal asymptote, you can set the function equal to 1 and solve!

  16. Graph Rational Function To find the x-intercept, let y = 0. Domain: Vert. Asym: Holes: Hor. Asym: Slant Asym: To find the y-intercept, let x = 0.

  17. Graph Rational Function Domain: Vert. Asym: Slant Asym:

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