Functions
AII.7 e
2009
Objectives:
• Find the Vertical Asymptotes
• Find the Horizontal Asymptotes
Rational Functions
A rational function f(x) is a function that can
be written as
p ( x)
f ( x) 
q ( x)
where p(x) and q(x) are polynomial functions
and q(x)  0 .
A rational function can have more than one
vertical asymptote, but it can have at most one
horizontal asymptote.
Vertical Asymptotes
If p(x) and q(x) have no common
factors, then f(x) has vertical
asymptote(s) when q(x) = 0. Thus
the graph has vertical asymptotes
at the zeros of the denominator.
Vertical Asymptotes
V.A. is x = a, where a represents real zeros of q(x).
Example:
Find the vertical asymptote of
f ( x) 
2x
2
x 1
2
q
(
x
)

x
 1  ( x  1)( x  1) the zeros
Since
are 1 and -1. Thus the vertical asymptotes
are x = 1 and x = -1.
.
Horizontal Asymptotes
A rational function f(x) is a function that can
be written as
p ( x)
f ( x) 
q ( x)
where p(x) and q(x) are polynomial functions
and q(x)  0 .
The horizontal asymptote is
determined by looking at the
degrees of p(x) and q(x).
Horizontal Asymptotes
f ( x) 
p ( x)
q ( x)
a. If the degree of p(x) is less than the degree of
q(x), then the horizontal asymptote is y = 0.
b. If the degree of p(x) is equal to the degree of
q(x), then the horizontal asymptote is
leading coefficien t of p( x)
y
.
leading coefficien t of q( x)
c. If the degree of p(x) is greater than the degree
of q(x), then there is no horizontal asymptote.
Horizontal Asymptotes
• deg of p(x) < deg of q(x), then H.A. is y = 0
f ( x) 
deg of p(x) = deg of q(x), then H.A. is
p ( x) •
q ( x)
Example:
•
leading coefficient of p( x)
y
.
leading coefficient of q( x)
deg of p(x) > deg of q(x), then no H.A.
Find the horizontal asymptote: f ( x) 
Degree of numerator = 1
Degree of denominator = 2
3x
x2 1
Since the degree of the numerator is
less than the degree of the denominator,
horizontal asymptote is y = 0.
.
Horizontal Asymptotes
• deg of p(x) < deg of q(x), then H.A. is y = 0
f ( x) 
p ( x)
q ( x)
• deg of p(x) = deg of q(x), then H.A. is
leading coefficient of p( x)
leading coefficient of q( x)
• deg of p(x) > deg of q(x), then no H.A.
y
Example:
3x  1
.
Find the horizontal asymptote: f ( x) 
2x 1
Degree of numerator = 1
Degree of denominator = 1
Since the degree of the numerator is
equal to the degree of the denominator,
horizontal asymptote is y  3 .
2
Horizontal Asymptotes
• deg of p(x) < deg of q(x), then H.A. is y = 0
f ( x) 
p ( x)
q ( x)
• deg of p(x) = deg of q(x), then H.A. is
y
leading coefficient of p( x)
leading coefficient of q( x)
• deg of p(x) > deg of q(x), then no H.A.
Example:
3 x2  1
.
Find the horizontal asymptote: f ( x ) 
2x  1
Degree of numerator = 2
Degree of denominator = 1
Since the degree of the numerator is greater
than the degree of the denominator, there is
no horizontal asymptote.
Vertical & Horizontal Asymptotes
V.A. :
x = a, where a
represents real
zeros of q(x).
Practice:
H.A. :
• deg of p(x) < deg of q(x), then H.A. is y = 0
• deg of p(x) = deg of q(x), then H.A. is
leading coefficient of p( x)
y
leading coefficient of q( x)
• deg of p(x) > deg of q(x), then no H.A.
Find the vertical and horizontal asymptotes:
3x2  1
f ( x) 
2x  1
Answer Now
Vertical & Horizontal Asymptotes
V.A. :
x = a, where a
represents real
zeros of q(x).
H.A. :
• deg of p(x) < deg of q(x), then H.A. is y = 0
• deg of p(x) = deg of q(x), then H.A. is
y
Practice:
leading coefficient of p( x)
leading coefficient of q( x)
• deg of p(x) > deg of q(x), then no H.A.
Find the vertical and horizontal asymptotes:
3 x2  1
f ( x) 
.
2x  1
1
V.A. : x =
2
H.A.: none
Vertical & Horizontal Asymptotes
V.A. :
x = a, where a
represents real
zeros of q(x).
H.A. :
• deg of p(x) < deg of q(x), then H.A. is y = 0
• deg of p(x) = deg of q(x), then H.A. is
y
Practice:
leading coefficient of p( x)
leading coefficient of q( x)
• deg of p(x) > deg of q(x), then no H.A.
Find the vertical and horizontal asymptotes:
f ( x) 
2x
x 1
2
Answer Now
Vertical & Horizontal Asymptotes
V.A. :
x = a, where a
represents real
zeros of q(x).
H.A. :
• deg of p(x) < deg of q(x), then H.A. is y = 0
• deg of p(x) = deg of q(x), then H.A. is
y
leading coefficient of p( x)
leading coefficient of q( x)
• deg of p(x) > deg of q(x), then no H.A.
Practice:
Find the vertical and horizontal asymptotes:
f ( x) 
x 1
2
2x
x2  1
V.A. : none
H.A.: y = 0
is not factorable and thus has no real roots.
Vertical & Horizontal Asymptotes
V.A. :
x = a, where a
represents real
zeros of q(x).
H.A. :
• deg of p(x) < deg of q(x), then H.A. is y = 0
• deg of p(x) = deg of q(x), then H.A. is
y
Practice:
leading coefficient of p( x)
leading coefficient of q( x)
• deg of p(x) > deg of q(x), then no H.A.
Find the vertical and horizontal asymptotes:
2x  1
f ( x) 
x 1
Answer Now
Vertical & Horizontal Asymptotes
V.A. :
x = a, where a
represents real
zeros of q(x).
H.A. :
• deg of p(x) < deg of q(x), then H.A. is y = 0
• deg of p(x) = deg of q(x), then H.A. is
y
Practice:
leading coefficient of p( x)
leading coefficient of q( x)
• deg of p(x) > deg of q(x), then no H.A.
Find the vertical and horizontal asymptotes:
2x  1
f ( x) 
x 1
V.A. : x = -1
H.A.: y = 2