1. Horizontal & Vertical Asymptotes Today we are going to look further at the behavior of the graphs of different functions. Now we are going to determine if the graphs have horizontal and/or vertical asymptotes.

2. Vertical Asympotes A vertical asymptote comes from the denominator. It exists at the value(s) that will make the denominator zero. For example— f(x)= 9 x – 2 For example— f(x) = x 2 – 9 6 Because there are no variables in the denominator, there are no vertical asymptotes.

3. Horizontal Asymptotes To check for horizontal asymptotes there are 3 rules you must memorize.

4. Rule #1 If the degree of the numerator is < the degree of the denominator, then the HA is y = 0. For example— f(x) = 5 x – 2 HA @ y = 0, VA @ x = 2 f(x) = 5x x 2 + 4  HA @ y = 0 ← degree is 0 ← degree is 1 ← degree is 2

5. Rule #2 For example— F(x) = 6x 5x + 10 If the degree of the numerator = degree of denominator, Then divide the leading coefficients. ← degree is 1  HA @ y = 6/5 VA @ x = -2

6. Rule #3 If the degree of the numerator  degree of denominator then there is none. In this case there may be asymptotes but they are oblique or parabolic. In calc. you will learn another way to find horizontal asymptotes when studying the limits of functions.

7. Example— f(x) = x – 2 x + 2  VA @ x = -2, HA @ y = 1 f(x) = 2(x 2 – 9) x 2 – 4  VA @ x = -2, 2, HA @ y = 2