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. Leading Coefficient  The coefficient of the term with the largest degree  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

5. 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!!!

6. Investigation Conclusion Questions: 4. What observations can you make about a rational equation’s horizontal asymptote when the degree of the numerator and the denominator are the same? 8. What observations can you make about a rational equation’s horizontal asymptote when the degree of the denominator is greater than the degree of the numerator? 12. What observations can you make about a rational equation’s horizontal asymptote when the degree of the numerator is greater than the degree of the denominator?

7. *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: y = ratio of leading coefficients  Degree of Numerator > Degree of Denominator  HA: doesn’t exist

8. Examples

9. Your Turn:  For problems 1 – 4 on the Introduction to Solving Rational Equations Practice, solve for the horizontal asymptote. 1.2. 3.4.

10. 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.

11. 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:

12. Example #1

13. Example #2

14. Example #3

15. Example #4

16. Your Turn:  Complete problems 5 – 12 on the Introduction to Solving for the Discontinuities of Rational Equations Practice handout. BE PREPARED TO SHARE YOUR ANSWERS!!!  Complete problems 13 – 18.

17. 13.14. 15.16. 17.18.

18. Exit Ticket