MA112
Section 106
Dr. Byrne
Fall 2009
Worksheet:
Homework Problems Section 3.4
Identify the vertical asymptote of f ( x) 
x
1
1
at x=0 by graphing the function f ( x)  .
x
x
y
Approaching the vertical asymptote
from the left: x  0 
x
y
Approaching the vertical asymptote
from the right: x  0 
x
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
-0.01
y
0.6
0.5
0.4
0.3
0.2
0.1
0.01
Sign of f(x) as x  0  :
1

( )
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) 
For each graph, specify (a) the domain, (b) the range and (c) the vertical and horizontal asymptotes
of the function.
(a) domain:
(a) domain:
(a) domain:
(a) domain:
(b) range:
(b) range:
(b) range:
(b) range:
(c) vertical asymptotes:
(c) vertical asymptotes:
(c) vertical asymptotes:
(c) vertical asymptotes:
horizontal asymptotes:
horizontal asymptotes:
horizontal asymptotes:
horizontal asymptotes:
7. Complete the table to determine the vertical and horizontal asymptotes of the function.
Step 1
x 1
f ( x) 
Vertical asymptotes
( x  2)( x  2)
At x =
Step 2
Behavior of x
x  2

x  2

Sign
of f(x)

x  2
x2
Plug in
for x



x 
x  



 

 

 

 

 

 

Behavior of
f(x)

f (x ) 

f (x ) 

f (x ) 

f (x ) 

f (x ) 

f (x ) 
Step 3
8. Complete the table to determine the vertical and horizontal asymptotes of the function.
Step 1
x
f ( x) 
Vertical asymptotes
( x  1)( x  3)
At x =
Step 2
Behavior of x
Plug in
Sign
Behavior of
Step 3
for x
of f(x)
f(x)



x  1
f (x ) 
  
x  1
x3


x  3
x 
x  




 

 

 

 

 


f (x ) 

f (x ) 

f (x ) 

f (x ) 

f (x ) 
9. Complete the table to determine the vertical and horizontal asymptotes of the function.
Step 1
x2 1
f ( x)  2
Vertical asymptotes
x  2x  1
At x =
Step 2
Step 3
Behavior of x
Plug in
Sign
Behavior of
for x
of f(x)
f(x)



x 1
f (x ) 
  
x  1
x 
x  
 
  
 
  
 
  
f (x ) 
f (x ) 
f (x ) 
10. Complete the table to determine the vertical and horizontal asymptotes of the function.
Step 1
x2  4x  4
f ( x) 
Vertical asymptotes
x2  4
At x =
Step 2
Behavior of x
Step 3
Plug in
for x
x  2
x2
Sign
of f(x)


x 
x  



 

 

 

 

Behavior of
f(x)

f (x ) 

f (x ) 

f (x ) 

f (x ) 
Find (a) the domain, (b) the axis intercepts, and (c) the vertical and horizontal asymptotes of the
rational function.
( x  1)( x  1)( x  2)
(3x  1)( x  4)
x 3
13.
15. f ( x) 
f ( x) 
f ( x) 
11.
x  1x  1
x2
x 1
(a) domain:
(a) domain:
(a) domain:
(b) axis intercepts
(b) axis intercepts
(b) axis intercepts
y-intercept:
y-intercept:
y-intercept:
x-intercepts:
x-intercepts:
x-intercepts:
(c) asymptotes
(c) asymptotes
(c) asymptotes
vertical asymptotes:
vertical asymptotes:
vertical asymptotes:
horizontal asymptotes:
horizontal asymptotes:
horizontal asymptotes:
Sketch the graph, labeling any horizontal and vertical asymptotes and axis intercepts.
2
3
18.
f ( x) 
f ( x) 
17.
x 1
x2
Domain:
Domain:
y-intercept:
x-intercepts:
y-intercept:
x-intercepts:
Vertical asymptotes:
Horizontal asymptotes:
Vertical asymptotes:
Horizontal asymptotes:
x
y
x
y
f ( x) 
19.
x
x3
f ( x) 
20.
2x
3x  1
Domain:
Domain:
y-intercept:
x-intercepts:
y-intercept:
x-intercepts:
Vertical asymptotes:
Horizontal asymptotes:
Vertical asymptotes:
Horizontal asymptotes:
x
y
x
y
Sketch the graph, labeling any horizontal and vertical asymptotes and axis intercepts.
2 x  1x  2
x2 9
21.
f ( x) 
f ( x)  2
23.
3x  1x  3
x  16
Domain:
Domain:
y-intercept:
x-intercepts:
y-intercept:
x-intercepts:
Vertical asymptotes:
Horizontal asymptotes:
Vertical asymptotes:
Horizontal asymptotes:
x
y
x
y
48. A drug in the bloodstream has a concentration of c(t ) 
at
at time t  0 hours. The constants
t b
2
a and b depend on the particular drug. For this problem, consider a=3 and b=1.
(a) Sketch a graph of the concentration c (t ).
3t
c(t )  2
t 1
Domain:
t
c(t)
y-intercept:
x-intercepts:
Vertical asymptotes: (none)
Horizontal asymptotes:
(b) Approximate the highest concentration of the drug reached in the bloodstream.
(c) What happens to the drug concentration as time t becomes large?