1. “This is the most magnificent discarded living room set I've ever seen

2. 3.3: Properties of
Rational Functions

3. Rational Functions A rational function is a function of the form
where p and q are polynomial functions. The domain consists of all real numbers except those for which the denominator is zero.

4. Horizontal asymptotes
An asymptote is a line that a function approaches as x or y goes to ∞ or -∞.
Horizontal asymptotes
Vertical asymptotes
Oblique asymptotes
The graph of a function may cross a HA
The graph of a function will never cross a VA
The graph of a function may cross an OA

5. Vertical Asymptotes
Given a rational function R(x), a vertical asymptote x = r occurs for all values of r for which the denominator is zero but the numerator is not.

6. Vertical Asymptotes
Given a rational function R(x), a vertical asymptote x = r occurs for all values of r for which the denominator is zero but the numerator is not.

7. Vertical Asymptotes
Find the vertical asymptotes and holes, if any, of the graphs of the following functions. State the domain of each function

8. Horizontal Asymptotes
Given a rational function , a horizontal asymptote
y = b occurs if the degrees of p and q are equal OR if the degree of q > p (in other words, if R(x) is bottom heavy)
Equal: Degree q = p
Bottom heavy: Degree q > p
 HA occurs at the quotient of the leading coefficients
 Always have a HA at y = 0

9. Oblique Asymptotes Given a rational function , an oblique asymptote
y = ax + b occurs if the degrees of q < p (in other words, if
R(x) is top heavy). Use long division to find the OA.

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

11. When does a rational function have one?
Recap
Let’s recap.. how ‘bout it?!
Type
Can function cross it?
When does a rational function have one?
How do you find it?
Vertical
Asymptotes
x = r
Horizontal Asymptotes
y = b
Oblique Asymptotes
y = ax + b

12. 3.3: Properties of
Rational Functions
HW #5: p.196
#11-29 odd, 44,
41-51 odd