Unit 4 Day 4 – Asymptotes
We will learn how to determine if
a rational function has horizontal
or oblique asymptotes
and
I will further analyze rational
functions.
November 7, 2016
Horizontal Asymptotes
A rational function will have a horizontal
asymptote if it’s y values approach a certain
number (limit) as it’s x values approach infinity
or negative infinity.
Horizontal Asymptotes
A horizontal line is an asymptote only to the
far left and the far right of the graph. "Far"
left or "far" right is defined as anything
past the vertical asymptotes or xintercepts.
It is okay to cross a horizontal asymptote in
the middle.
Horizontal Asymptotes
The location of the horizontal asymptote is
determined by looking at the degree of the
numerator (n) and denominator (m).
• If
 x 1
n<m,  x2 
• R is called a Proper Function
• the x-axis, y=0 is the horizontal
asymptote.
• Big On Bottom y = 0
Horizontal Asymptotes
 2x  1 
• If n=m,  x 
• R is called an Improper Function
• the Horizontal Asymptote is the ratio of
the leading coefficients. (y = 2 in this case)
• If
 x2  1 

n>m, 

x


• there is no horizontal asymptote.
• Big On Top, None
•
•
•
•
Horizontal Asymptotes (flipbook)
n = degree of the numerator, m = degree of
the denominator
If n<m,
• the x-axis, y=0 is the horizontal
asymptote. (BOBO)
If n=m,
• then the horizontal asymptote is the ratio
of the leading coefficients.
If n>m,
• there is no horizontal asymptote. (BOTN)
Find the horizontal asymptotes
4x
1. R(x) 
x -3
Horizontal Asymptote: y = 4
3x3 - 3x2  x
2. R(x) 
2
2x  5x  3
x
3. R(x)  3
x 8
Horizontal Asymptote:
none (BOTN)
Horizontal Asymptote: y = 0
(BOBO)
Oblique Asymptotes (flipbook)
• If n = m + 1,
• the line y = ax + b is an oblique (slant)
asymptote. To get this line, we have to
do long division.
• If n > m + 1,
• it has neither a horizontal nor an oblique
asymptote.
Example: find horizontal, and oblique asymptotes, if
any, of each rational function.
x3  8
R(x)  2
x  5x  6
n = 3, m = 2, (BOTN), so no Horizontal Asymptote
n = m + 1, so there is an Oblique Asymptote
x 8
R(x)  2
x  5x  6
3
x+5
x2  5x  6 x3  0x2  0x  8
x3  5x2  6x
 5x2  6x -8
5x2  25x  30
Remainder
19x  38
R(x)  x  5  2
x  5x  6
O. Asymptote
 19x  38