1. Asymptotes Rise Their Lovely Heads
Rational Functions
Asymptotes Rise Their Lovely Heads

2. Definition of a Rational Function
Domain – CANNOT DIVIDE BY ZERO
Function in lowest terms – cancelling functions that are similar in the numerator and denominator

3. Graphing y = 1/x^2
Why does it look like this? What is the transformation?

4. Transformations Review
H(x) = 1/(x – 2) ^2 + 1
(x – 2) moves the graph to the right 2
+ 1 moves the graph up 1
Now graph the y = 1/x^2 graph

5. Asymptotes Definition Three types: A. Vertical Asymptotes
B. Horizontal Asymptotes
C. Oblique Asymptotes

6. Vertical Asymptotes To find vertical asymptotes
write the function in lowest terms
Set the denominator equal to zero
The asymptote will be located at x = this number

7. Horizontal Asymptotes
Rules for finding horizontal asymptotes:
1.If the numerator and denominator have the same degree.
2. If the numerator is a smaller degree than the denominator
3. If the numerator is a larger degree than the denominator

8. Horizontal Asymptote
The equation of the asymptote is y = the leading coefficients
The equation of the asymptote is y = 0
There is no horizontal asymptote

9. Oblique Asymptotes
If the degree of the numerator is larger than the degree of the denominator then the graph has an oblique asymptote. Divide the numerator into the denominator

10. Summary
See page 223 for all of the rules.

11. Graphs
Step One: Find the domain of the rational function Step Two: Write the function in lowest terms Step Three: Locate the intercepts of the graph Step Four: Locate the vertical asymptotes Step Five: Locate the horizontal or oblique asymptotes

12. Graphs
Step 6: Identify the behavior of the graph around the vertical asymptotes

13. Graphs
Example of function with oblique asymptote
More Examples

14. Constructing a Rational Function Given Asymptotes and Intercepts
Example

15. Constructing a Rational Function from Its Graph

16. Applications
P. 235 problem 46 & 48