MA112
Fall 2010
Dr. Byrne
Worksheet 4.5a: Graphing Simple Rational Functions with Vertical Asymptotes
Identify the vertical asymptote of f ( x) 
x
1
1
at x=0 by graphing the function f ( x)  .
x
x
y
Vertical asymptote:
Approaching the vertical asymptote at x  0
…from the left: x  0 
x
y
x
-0.5
-0.25
-0.1
-0.01
f ( x) 
Approaching the vertical asymptote x  0
…from the right: x  0 
y
0.5
0.25
0.1
0.01
1
x
f ( x) 
Sign of f(x) as x  0  :
1
( )

1
x
Sign of f(x) as x  0  :
1
( )
Behavior of f(x) as x  0  :
Behavior of f(x) as x  0  :
f (x) 
f (x) 

Worksheet 4.5: Identifying Vertical Asymptotes
Vertical asymptotes: For a rational function P ( x) , the zeros of Q (x) may yield vertical asymptotes.
Q ( x)
In particular, the line x=a will be a vertical asymptote if Q (a ) is a zero and P ( a )  0 .
Sketch the following simple rational functions.
f ( x) 
x
1
x
y
f ( x)  
x
Vertical Asymptote:
f ( x) 
x
1
x
Vertical Asymptote:
y
1
x2
Vertical Asymptote:
1
x2
Vertical Asymptote:
y
f ( x) 
x
y
For each graph, specify (a) the vertical asymptotes and (b) the domain of the function.
x
2
f ( x)  2
f ( x)  2
x 2
x x
(a) vertical asymptotes:
(a) vertical asymptotes:
(b) domain:
(b) domain:
(a) vertical asymptotes:
(a) vertical asymptotes:
(b) domain:
(b) domain:
Worksheet 4.5: Graphing Simple Rational Functions P(x)/Q(x)
in the case when has Q(x) has multiple zeros and/or P(x) has zeros
f ( x) 
x
1
( x  3)( x  3)
f ( x) 
x
y
y
Vertical asymptotes:
Vertical asymptotes:
f ( x) 
x
x
x
y
f ( x) 
x
Vertical asymptotes:
x
( x  3)( x  3)
x 1
( x  1)( x  2)
y
Vertical asymptotes: