1. 9.3 Rational Functions and Their Graphs

3. If the graph is not continuous at x = a then the function has a point of discontinuity at x = a.

4. Ex 1 Find any points of discontinuity.

5. Ex 2 Find any points of discontinuity.

6. Vertical Asymptotes
There is a point of discontinuity for each real zero of Q(x).
If P(x) and Q(x) have no common real zeros, then the graph has a VA at each real zero of Q(x).
If P(x) and Q(x) have a common real zero, a, then there is a hole in the graph or a VA at x = a.

7. Ex 3 Find the VA or holes.

8. Ex 4 Find the VA or holes.

9. Ex 5 Find the VA or holes.

10. Horizontal Asymptotes (There is at most 1 HA per graph.)
If the degree of the denominator is > the degree of the numerator then there is a HA at y = 0.
If the degree of the numerator is > the degree of the denominator then there is NO HA.
If the degree of the numerator = the degree of the denominator then the HA is y = a/b where a is the leading coefficient of the numerator & b is the LC of the denominator.

11. Ex 6 Find the VA, HA, and holes.

12. Ex 7 Sketch the graph and identify the VA, HA, and holes.

13. The zero of the numerator is the x-intercept!!