1. Asymptotes

2. Rational function
A rational function is one that can be written in the form
where p(x) and q(x) are polynomials and q(x) is not the zero polynomial. In general, the domain of a rational function includes all real numbers except x-values which make the denominator zero.

3. Find domain
Find the domain of each of the following:

4. Answer
Set denominator not equal to 0

5. Vertical Asymptotes
At the value(s) for which the domain is undefined, there will be one or more vertical asymptotes. List the vertical asymptotes for the following.

6. Answer
Set denominator = 0

7. Graphically The figure below shows the graph of The
equation of the vertical asymptote is x=-2.

8. Summary
The graph of a function has vertical asymptote(s) at the discontinuity of the function. (set the denominator = 0)
If the degree of the numerator is = to the degree of the denominator, the function has a horizontal asymptote. ( y = ratio of lead coefficients)
If the degree of the numerator is < the degree of the denominator, the function has a horizontal asymptote. (y = 0, the x-axis).
If the degree to the numerator is > the degree of the denominator, the function does not have a horizontal asymptote.

9. Find asymptotes
Name the vertical and horizontal asymptotes for

10. Answer Domain: Degree of numerator: 1 Degree of denominator: 1
HA: y = 2
VA: x = -1

11. Find asymptotes
Name the vertical and horizontal asymptotes for

12. Answer Domain: Degree of numerator: 2 Degree of denominator: 2
HA: y = 5/2
VA: x = -2,2

13. Find asymptotes
Name the vertical and horizontal asymptotes for

14. Answer Domain: Degree of numerator: 0 Degree of denominator: 2
HA: y = 0
VA: none

15. Find asymptotes
Name the vertical and horizontal asymptotes for

16. Answer Domain: Degree of numerator: 1 Degree of denominator: 2
HA: y = 0
VA: x=3,-3

17. Find asymptotes
Name the vertical and horizontal asymptotes for

18. Answer Domain: Degree of numerator: 2 Degree of denominator: 1
HA: none
VA: x=-2

19. Find asymptotes
Name the vertical and horizontal asymptotes for

20. Answer Domain: Degree of numerator: 3 Degree of denominator: 1
HA: none
VA: x=-1/3