Rational Functions

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Rational Functions

• Asymptotes Rise Their Lovely Heads

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Definition of a Rational Function

• Definition
• Domain CANNOT DIVIDE BY ZERO
• Function in lowest terms cancelling functions
  that are similar in the numerator and denominator

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Graphing y 1/x2
Why does it look like this? What is the
transformation?
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Transformations Review

• H(x) 1/(x 2) 2 1
• (x 2) moves the graph to the right 2
• 1 moves the graph up 1
• Now graph the y 1/x2 graph

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Asymptotes

• Definition
• Three types
• A. Vertical Asymptotes
• B. Horizontal Asymptotes
• C. Oblique Asymptotes

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Vertical Asymptotes

• To find vertical asymptotes
• write the function in lowest terms
• Set the denominator equal to zero
• The asymptote will be located at x this number

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Horizontal Asymptotes

• Rules for finding horizontal asymptotes
• 1.If the numerator and denominator have the same
  degree.
• 2. If the numerator is a smaller degree than the
  denominator
• 3. If the numerator is a larger degree than the
  denominator

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Horizontal Asymptote

1. The equation of the asymptote is y the leading
   coefficients
2. The equation of the asymptote is y 0
3. There is no horizontal asymptote

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Oblique Asymptotes

• If the degree of the numerator is larger than the
  degree of the denominator then the graph has an
  oblique asymptote.
• Divide the numerator into the denominator

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Summary

• See page 223 for all of the rules.

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Graphs

• Step One Find the domain of the rational
  function
• Step Two Write the function in lowest terms
• Step Three Locate the intercepts of the graph
• Step Four Locate the vertical asymptotes
• Step Five Locate the horizontal or oblique
  asymptotes

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Graphs

• Step 6 Identify the behavior of the graph around
  the vertical asymptotes

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Graphs

• Example of function with oblique asymptote
• More Examples

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Constructing a Rational Function Given Asymptotes
and Intercepts

• Example

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Constructing a Rational Function from Its Graph

• Graph

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Applications

• P. 235 problem 46 48