1. Short Run Behavior of Rational Functions
Lesson 9.5

2. Zeros of Rational Functions
We know that
So we look for the zeros of P(x), the numerator
Consider
What are the roots of the numerator?
Graph the function to double check

3. Zeros of Rational Functions
Note the zeros of the function when graphed
r(x) = 0 when x = ± 3

4. Vertical Asymptotes
A vertical asymptote happens when the function R(x) is not defined
This happens when the denominator is zero
Thus we look for the roots of the denominator
Where does this happen for r(x)?

5. Vertical Asymptotes Finding the roots of the denominator
View the graph to verify

6. Summary The zeros of r(x) are where the numerator has zeros
The vertical asymptotes of r(x) are where the denominator has zeros

7. Drawing the Graph of a Rational Function
Check the long run behavior
Based on leading terms
Asymptotic to 0, to a/b, or to y=(a/b)x
Determine zeros of the numerator
These will be the zeros of the function
Determine the zeros of the denominator
This gives the vertical asymptotes
Consider

8. Given the Graph, Find the Function
Consider the graph given with tic marks = 1
What are the zeros of the function?
What vertical asymptotes exist?
What horizontal asymptotes exist?
Now … what is the rational function?

9. Look for the Hole
What happens when both the numerator and denominator are 0 at the same place?
Consider
We end up with which is indeterminate
Thus the function has a point for which it is not defined … a “hole”

10. Look for the Hole
Note that when graphed and traced at x = -2, the calculator shows no value
Note also, that it does not display a gap in the line

11. Assignment
Lesson 9.5
Page 420
Exercises 1 – 41 EOO