1. Warm-up Determine the:
1) degree 2) x and y intercepts 3) zeros, their multiplicity, and behavior of the graph at each 4) end behavior
Graph it.

2. 3.3 Properties of Rational Functions

3. 1. Rational Functions A rational function is a function of the form
where p and q are polynomial functions.

4. 2. Domain and Intercepts
1. Find the domain: (Denominator cannot be zero)
2. Find the y-intercept
Find x-intercepts. 4

5. Domain and Intercepts 1. Find the domain:) 2. Find the y-intercept
Find x-intercepts. 5

6. The graph of a function will never cross a Vertical Asymptote
3. Vertical Asymptotes
is the
vertical asymptote for f(x) when: as
The graph of a function will never cross a Vertical Asymptote 6

7. 4. Finding Vertical Asymptotes of
If (c is a zero of the denominator)
and
If (c is not a zero of numerator)
Then is a vertical asymptote of

8. A hole in a graph
A hole occurs at x = c when both:
and

9. More examples
Find the vertical asymptotes and holes, if any, of the graphs of the following functions.

10. 5a. Horizontal Asymptotes
A horizontal asymptote describes the end behavior
of the graph! as
The graph of a function may cross a Horizontal Asymptote (usually near the center of the graph).
A rational function’s graph will have only one horizontal asymptote. 10

11. 5b. Oblique Asymptotes oblique asymptote
When the y-values get close to a line that is not horizontal as the x values approach , then we say the graph has a
oblique asymptote
y = mx + b
If an asymptote is neither vertical or horizontal, it is called an
Oblique asymptote
An oblique asymptote describes the
end behavior
of the graph!
The graph of a function may cross an OA

12. Horizontal Asymptotes
n = degree of numerator
m=degree of denominator
Asymptotes
Example
bottom heavy
(denominator is higher degree)
n<m
y=0
equal
(same degree)
n=m

13. Oblique Asymptotes y = quotient in long division Asymptotes Example
top heavy
(degree of numerator is larger)
m > n
y = quotient in long division

14. Horizontal Asymptotes
Case 1: Bottom Heavy
Case 2: Equal degrees

15. Horizontal or Oblique Asymptote?
State whether the following functions contain a horizontal or oblique asymptote. Then, find it!

16. Application Problems Ex 1. A population P after t years is modeled by
What is the horizontal asymptote for this function?
Ex. 2. We wish to enclose an area that is 800 sq. ft. The minimum amount of fencing required to enclose 3 sides can be modeled with
Are there vertical or horizontal asymptotes?

17. Practice

18. Write an equation for the rational function with the following properties:
Vertical Asymptote at x=0, -2, 3.
x-intercepts at x=.5, 1, 4.
Horizontal Asymptote y=1/3
A hole at x = 2.