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3.7 Graphs of Rational Functions

Learn how to graph rational functions and determine their vertical, horizontal, and slant asymptotes. This includes identifying the location of asymptotes and understanding point discontinuities.

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3.7 Graphs of Rational Functions

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  1. 3.7 Graphs of Rational Functions Objectives: Graph rational functions. Determine vertical, horizontal, and slant asymptotes.

  2. Asymptote: A line that a graph approaches but never intersects. Vertical Asymptote: The line x=a is a vertical asymptote for a function f(x) if f(x)→ ∞ or f(x) → -∞ as x → a from either the left or the right . Horizontal Asymptote: The line y=b is a horizontal asymptote for a function f(x) if f(x)→ b as x → ∞ as x → - ∞. See example 1 to see methods of how to find horizontal and vertical asymptotes.

  3. Ex. 1) Determine the asymptotes for the graph of f(x) = x/(x-1) Ex. 2) Use the parent graph f(x)=1/x to graph each function. Describe the transformation(s) that take place. Identify the new location of each asymptote. a.) g(x)=1/(x-1) b.) h(x)=-2/x c.) k(x) = 7/(x + 5) d.) m(x) = 1/(x – 3) + 2

  4. Slant asymptote: The oblique line l is a slant asymptote for a function f(x) if the graph of f(x) approaches l as x→∞ or as x →-∞. (Top heavy rational expression.) -bottom heavy → horizontal at y= 0 See example 3 in book on pg. 183. Ex. 4) Determine the slant asymptote for f(x)=(4x² + 6x – 37) / (x + 4)

  5. Point Discontinuity: →hole in graph -when numerator and denominator contain a common linear factor. Ex. 5) Graph y = (x + 2)(x – 3) x(x – 4)²(x + 2)

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