Solving for the Discontinuities of Rational Equations.

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Presentation transcript:

Solving for the Discontinuities of Rational Equations

Review: 3 Types of Discontinuities  Vertical Asymptotes (VAs)  Horizontal Asymptotes (HAs)  Holes

Degree  The greatest exponent of an expression  Examples:  f(x) = x 6 – x  f(x) = x 4 – x 9 + x 11 – x  f(x) = 8x + 4  f(x) = 7

Important Vocabulary, cont.: 

Multiplicity

Multiplicity, cont.

Horizontal Asymptotes Investigation  Remember:  The horizontal asymptote describes how the graph behaves AT ITS ENDS  Look for the graph to taper to the same y-value on both ends of the graph  Look for dashed, horizontal lines  We DON’T DRAW dashed lines on the X-AXIS or the Y-AXIS!!!

Horizontal Asymptotes  Depend on the degree of the numerator and the denominator  Degree of Numerator < Degree of Denominator  HA: y = 0  Degree of Numerator = Degree of Denominator  HA: ratio of leading coefficients  Degree of Numerator > Degree of Denominator  HA: doesn’t exist

Examples

Solving for Vertical Asymptotes Examples

Your Turn: Solve for the VA

Solving for Vertical Asymptotes and Holes 1. Always factor the numerator and the denominator 1 st ! 2. Identify linear factors in the denominator 3. Figure out where the linear factors in the denominator occur the most to decide if you have a vertical asymptote or a hole 4. Set the linear factors from step 2 equal to zero and solve for x.

Solving for Vertical Asymptotes and Holes, cont. Occur in Denominator Only? Occur Most in Denominator? Occur = # of Times in Numerator and Denominator? Occur Most in Numerator? VA Set factor equal to 0 and solve for x. VA Set factor equal to 0 and solve for x. Hole Set factor equal to 0 and solve for x. Hole Set factor equal to 0 and solve for x. Does the linear factor:

Example #1

Example #2

Example #3

Example #4

Your Turn:  Complete problems 1 – 8 on the Solving for Vertical Asymptotes and Holes Practice handout. BE PREPARED TO SHARE YOUR ANSWERS!!!  Complete problems 9 – 14.