1. Solving for the Discontinuities of Rational Equations

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

3. Degree  The greatest exponent of an expression  Examples:  f(x) = x 6 – x 2 + 3  f(x) = x 4 – x 9 + x 11 – x 2 + 5  f(x) = 8x + 4  f(x) = 7

4. Important Vocabulary, cont.: 

5. Multiplicity

6. Multiplicity, cont.

7. 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!!!

8. 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

9. Examples

10. Solving for Vertical Asymptotes Examples

11. Your Turn: Solve for the VA

12. 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.

13. 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:

14. Example #1

15. Example #2

16. Example #3

17. Example #4

18. 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.

19. 9.10. 11.12. 13.14.