“Onions have layers.”
3.3:
Properties
of
Rational
Functions
Rational Functions
A rational function is a function of the form
p(x)
R(x) 
q(x)
where p and q are polynomial functions. The domain consists of
all real numbers except those for which the denominator is zero.

Asymptotes
An asymptote is a line that a function approaches as x or y
goes to ∞ or -∞.
Horizontal asymptotes
The graph of a
function may
cross a HA
Vertical asymptotes
The graph of a
function will
never cross a VA
Oblique asymptotes
The graph of a
function may
cross an OA
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.
x
R(x)  2
x 4
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.
x2  9
G(x)  2
x  4 x  21
Vertical Asymptotes
Find the vertical asymptotes and holes, if any, of the graphs of the
following functions. State the domain of each function
x4 1
F(x)  3
x x

x2
G(x)  2
x  x 12

x 2  2x  24
H(x) 
x 2 16
Horizontal Asymptotes
Given a rational function R(x)  p(x) , a horizontal asymptote
q(x)
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
Bottom heavy: Degree q > p
 q = p
 HA occurs at the quotient of
the leading coefficients
3x 2  3x
F ( x)  2
2 x  5x  3
6
 Always have a HA at y = 0
G(x) 
5
4
3
2x  1
2x 3  4 x 2
2
6
5
4
3
2
2
1
1
-6 -5 -4 -3 -2 -1
-1
-2
1 2 3 4 5 6

-6 -5 -4 -3 -2 -1
-1
-2
-3
-3
-4
-4
-5
-5
-6
-6
1 2 3 4 5 6
Oblique Asymptotes
Given a rational function R(x)  p(x) , an oblique asymptote
q(x)
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.
3
x
H(x)  2
2x  8
8
6
4
2
-8 -7 -6 -5 -4 -3 -2 -1
-2
-4
-6
-8
1 2 3 4 5 6 7 8
Horizontal or Oblique Asymptote?
State whether the following functions contain a horizontal or
oblique asymptote. Then, find it!
x4 1
F(x)  3
x x

x2
G(x)  2
x  x 12

x 2  2x  24
H(x) 
x 2 16
Recap
Let’s recap.. how ‘bout it?!
Type
Vertical
Asymptotes
x=r
Horizontal
Asymptotes
y=b
Oblique
Asymptotes
y = ax + b
Can
function
cross it?
When does a
rational function
have one?
How do you find
it?
3.3:
Properties
of
Rational
Functions
HW:p.196 - 197 #11 – 29 odd, 44,
41-51 Odd,