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Graphing Rational Functions

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The document discusses graphing rational functions of the form f(x) = a(x)/b(x), where a(x) and b(x) are polynomial functions. It provides examples of how the graphs appear depending on whether the degree of the numerator is even or odd. A 7-step process is outlined for sketching the graphs, which involves finding intercepts, roots, vertical asymptotes, horizontal asymptotes, and the sign of the function to sketch the graph. Two examples applying the 7-step method are shown.

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January 19, 2010 Graphing Rational Functions Functions of the form where a(x) and b(x) are polynomial functions. Examples

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January 19, 2010 Graphing Rational Functions Appearance Where n is even, the Where n is odd, the graph looks like this: graph looks like this:

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January 19, 2010 Graphing Rational Functions Sketching (7 steps) Step 1: Find the y-intercept (let x = 0) Step 2: Factor everything. (Use rational roots theorem if necessary.) Step 3: Find the roots of the function by finding the roots of the numerator a(x). Step 4: Find the vertical asymptotes by finding the roots of the denominator b(x).

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January 19, 2010 Graphing Rational Functions Step 5: Find the horizontal asymptotes by dividing each term in the function by the highest power of x, and take the limit as x goes to infinity. (Use the UNfactored form.) You will find that, in general, there are three possible results: i When [degree of numerator < degree of denominator] the horizontal asymptote is y = 0. ii When [degree of numerator = degree of denominator] the H.A. is the ratio leading coefficient of a(x) leading coefficient of b(x) iii When [degree of numerator > degree of denominator] there is no horizontal asymptote; however there may be a slant asymptote or a hole in the graph.

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January 19, 2010 Graphing Rational Functions Sketching (7 steps) Step 6: Determine the sign of the function over the intervals defined by the roots and vertical asymptotes. (Use the factored form.) Step 7: Sketch the graph.

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January 19, 2010 Graphing Rational Functions Sketching: Example 1 of 4 Step 1: Step 5: Step 6: Step 2: Step 3: Step 7: Step 4:

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January 19, 2010 Step 5:

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January 19, 2010 Sketch the graph of

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January 19, 2010

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January 19, 2010

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January 19, 2010 Graphing Rational Functions Sketching: Example 2 of 4 Step 5: Step 1: Step 6: Step 2: Step 7: Step 3: Step 4:

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January 19, 2010 Step 5:

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Graphing Rational Functions

  • 1. January 19, 2010 Graphing Rational Functions Functions of the form where a(x) and b(x) are polynomial functions. Examples
  • 2. January 19, 2010 Graphing Rational Functions Appearance Where n is even, the Where n is odd, the graph looks like this: graph looks like this:
  • 3. January 19, 2010 Graphing Rational Functions Sketching (7 steps) Step 1: Find the y-intercept (let x = 0) Step 2: Factor everything. (Use rational roots theorem if necessary.) Step 3: Find the roots of the function by finding the roots of the numerator a(x). Step 4: Find the vertical asymptotes by finding the roots of the denominator b(x).
  • 4. January 19, 2010 Graphing Rational Functions Step 5: Find the horizontal asymptotes by dividing each term in the function by the highest power of x, and take the limit as x goes to infinity. (Use the UNfactored form.) You will find that, in general, there are three possible results: i When [degree of numerator < degree of denominator] the horizontal asymptote is y = 0. ii When [degree of numerator = degree of denominator] the H.A. is the ratio leading coefficient of a(x) leading coefficient of b(x) iii When [degree of numerator > degree of denominator] there is no horizontal asymptote; however there may be a slant asymptote or a hole in the graph.
  • 5. January 19, 2010 Graphing Rational Functions Sketching (7 steps) Step 6: Determine the sign of the function over the intervals defined by the roots and vertical asymptotes. (Use the factored form.) Step 7: Sketch the graph.
  • 6. January 19, 2010 Graphing Rational Functions Sketching: Example 1 of 4 Step 1: Step 5: Step 6: Step 2: Step 3: Step 7: Step 4:
  • 8. January 19, 2010 Sketch the graph of
  • 11. January 19, 2010 Graphing Rational Functions Sketching: Example 2 of 4 Step 5: Step 1: Step 6: Step 2: Step 7: Step 3: Step 4: