1. Lesson 21 Finding holes and asymptotes Lesson 21 February 21, 2013

2. Warm ups Lesson 21 Simplify all find the domain on 1 & 2

3. Holes and Asymptotes

4. Holes or points of discontinuity Holes or points of discontinuity occur when in a graph when there are common factors in the numerator and the Denominator In this case the hole would be x = 4 In this case the hole would be x = 5

5. Find the holes in each of these expressions x = -3 x = -4 x = -5

6. Vertical Asymptotes When denominator = 0 Vertical Asymptotes occur if numerator and denominator have no common factors, but the denominator = 0. The graph will have one vertical asymptote at each real zero for the denominator. In this case the vertical Asymptote would be x = -1 In this case the asymptote would be x = 5 and x = -5

7. Find the vertical asymptotes in each of these expressions x = 5 x = 0 x = 0; x = 2

8. Horizontal Asymptotes, the graph of a rational function has at most one horizontal Asymptote We will look at 3 way it will have a horizontal Asymptote a. When higher order is in the denominator b. When the order of the polynomial is the same c. when the higher order of a polynomial is in the numerator

9. When the higher degree is in the denominator When the higher degree is in the denominator (bottom), the horizontal asymptote is y = 0.

10. When the degree of the polynomials are equal in the denominator and numerator

11. When the higher degree is in the numerator When the higher degree is in the numerator (top), the horizontal asymptote is none

12. Find the horizontal asymptotes in each of these expressions y = 0

13. Find all points of discontinuity (holes) and asymptotes

15. Answers Lesson 20

16. Answers Lesson 20 CONT