1. Rational Functions A rational function has the form
where P and Q are polynomial
functions and
Ex 1: Examine
Note:
also

2. The line x = a is a vertical asymptote of the function if
The line y = b is a horizontal asymptote of the function if

3. Ex 2: Graph

4. Ex 3: Graph

5. Let r be the rational function
The vertical asymptotes of r are the lines x = a, where a is a zero of the denominator.
2. (a) If n < m, then r has a horizontal asymptote
y = 0.
(b) If n = m, then r has a horizontal asymptote
(c) If n > m, then r has no horizontal asymptote.

6. Ex 4: Find the asymptotes of

7. Graphing Rational Functions
Factor the numerator and denominator.
2. Find the x-intercepts and y-intercept.
3. Find the vertical asymptotes (if any).
4. Find the horizontal asymptote (if any).
5. Make a small T-chart.
6. Sketch the graph.

8. Ex 5: Graph

9. Ex 6: Graph

10. Ex 7: Graph

11. A slant asymptote is a diagonal line that the graph
A slant asymptote is a diagonal line that the graph of a function approaches as
A slant asymptote occurs when the degree of the numerator is one more than the degree of the denominator.
We calculate a slant asymptote by dividing the numerator by the denominator and ignoring the remainder.

12. Ex 8: Graph

13. Ex 9: Graph

14. Assignment S 4.5: pg
#7-10,16,21-24,40,42,43,54,55