4.1 ANALYZING RATIONAL FUNCTIONS ALGEBRAICALLY
1) Determine the asymptotes and/or holes in the graph (if they exist).
p  x
Rational functions have the following form: f  x  
q  x
Vertical asymptotes and holes occur at values of x that make the denominator q  x 
equal to zero (i.e. discontinuities of the function).
To find Vertical Asymptotes (VA)
a) Factor numerator p  x  and denominator q  x  completely and reduce.
b) Set reduced denominator = 0 and solve for x.
c) The real solutions of the equation in part (b) are the vertical asymptotes.
note 1: When finding VA, you must state the equation of the vertical line. Write as
x = #. If there is no VA, you must write “no VA” to show you are
addressing this.
note 2: The graph will never cross its VA.
To find holes
Holes in the function occur if there is a common factor in the numerator p  x  and
denominator q  x  . Pay attention to any factors you divide out when looking for VA.
a) Set common factors = 0 and solve for x. This is the x-value where the hole is at.
b) Find the y-value by plugging the x-value found in part (a) into the reduced
function.
note 3: If there are no common factors in the numerator p  x  and denominator
q  x  , the function does not have any hole. You must write “no holes” to
show you are addressing this.
To find Horizontal Asymptotes (HA)
Compare the degree of the numerator p  x  with the degree of the denominator q  x  :

If degree p  x   degree q  x  , then the HA is y  0 .

If degree p  x   degree q  x  , then the HA is y 

If degree p  x   degree q  x  , then there is no HA. You must write “no HA” to
leading coefficient of p  x 
leading coefficient of q  x 
.
show you are addressing this. (It may have a Slant Asymptote—see next page.)
note 4: The graph may or may not cross its HA. To check if graph crosses its
HA, set reduced f  x  = the value of the HA and solve for x.

The solution to the corresponding equation is the x-value where the graph
crosses its HA. Write “graph crosses its HA at x = ___” to address this.
Math 114
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
Carreon 4.1 (1)
Analyzing Rational Functions Algebraically (cont.)

If the corresponding equation has no solution (i.e. you get a false statement),
then the graph does not cross its HA. Write “graph does not cross its HA” to
address this.
Slant (or Oblique) Asymptotes
Sometimes the graph may have an asymptote that is of the form y = mx + b. These
type of asymptotes are called Slant (or Oblique) Asymptotes.
Slant asymptotes happen when degree of numerator p  x  is exactly 1 degree higher
than degree of denominator q  x  .
To find Slant Asymptotes (SA), divide numerator p  x  by denominator q  x  , and
ignore the remainder. The equation of the SA is y = quotient.
2)
Determine the intercepts of the graph (if they exist).
To find the x-intercepts, set reduced f  x  = 0 and solve for x.
The ordered pairs have the form  x, 0  .
To find the y-intercept, plug in x = 0 to reduced f  x  and simplify. (i.e. compute
f  0 )
The ordered pairs have the form:
 0, y  .
3)
Use dashed lines to sketch all asymptotes found above.
4)
Plot all intercepts found above.
5)
Use your TI to help you accurately sketch the function. Include any holes in
your sketch.
If you are unable to use your TI, you will need to pick numerous x-values that fall
between your vertical asymptotes to plug into your reduced f  x  . From there, you can
plot the corresponding ordered pairs to help you accurately sketch the function.
Math 114
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
Carreon 4.1 (2)
Analyzing Rational Functions Algebraically (cont.)
Problems
Find the slant asymptote of the rational function.
1.
x 2  9 x  20
f ( x) 
x 3
4.
x2
f ( x) 
x 1
2.
5.
2 x6
f ( x)  5
x 1
3.
x2  9
f ( x) 
x2
2 x 2  14 x  7
f ( x) 
x 5
Find the asymptotes, x and y intercepts, holes (if any) of the rational function,
Then make a sketch of the rational function.
6.
4
f ( x) 
x2
9.
3x( x  1)
f ( x) 
x( x 2  4)
12.
7.
3x3  5 x 2  2 x
f ( x) 
x2  4
10.
2 x2  4 x  6
f ( x) 
x2  2 x
8.
x2  1
f ( x)  2
x 1
11.
( x  1) 2
f ( x)  2
( x  1)
( x  1)( x 2  9)
f ( x) 
( x  3)( x 2  1)
Math 114
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
Carreon 4.1 (3)
Analyzing Rational Functions Algebraically (cont.)
Extra Practice
Find the asymptotes, x and y intercepts, holes (if any) of the rational function,
Then make a sketch of the rational function.
12 x 4
1. f ( x) 
(3x  1) 4
25 x 2  1
2. f ( x) 
(16 x 2  1)2
7 x3  1
3. f ( x) 
x5
1  x2
4. f ( x)  2
x 1
4
5. f ( x)  x 
x
1  9 x2
6. f ( x) 
(1  4 x 2 )3
( x  1)( x 2  4)
7. f ( x) 
( x  2)( x 2  1)
6 x 2  3x  1
8. f ( x)  2
3x  5 x  2
9. f ( x)  3x 
Math 114
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
4
x
Carreon 4.1 (4)
Analyzing Rational Functions Algebraically (cont.)
Answers
1. SA : y  x  12
4. SA : y  x  1
6. VA : x  2
2. SA : y  2 x
3. SA : y  x  1
5. SA : y  2 x  24
7. VA : none
HA : y  0
HA : none
SA : none
SA : y  3x  5
holes : none
holes : none
x  int : none
1
x  int : (0, 0), ( 2, 0), ( , 0)
3
y  int : (0, 0)
y  int : (0, 2)
Math 114
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
Carreon 4.1 (5)
Analyzing Rational Functions Algebraically (cont.)
8.
VA : x  1, x  1
9. VA : x  2, x  2.
HA : y  1
HA : y  0
SA : none
SA : none
holes : none
3
holes : (0, )
4
x  int : (1, 0)
x  int : none
y  int : (0, 1)
10. VA : x
 0, x  2
y  int : none
11.
VA : x  1
HA : y  2
HA : y  1
SA : none
SA : none
holes : none
holes : (1, 0)
x  int : (3, 0), (1, 0)
x  int : (1, 0)
y  int : none
y  int : (0, 1)
Math 114
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
Carreon 4.1 (6)
Analyzing Rational Functions Algebraically (cont.)
12.
VA : none
HA : y  1
SA : none
6
holes : (3, )
5
x  int : (3, 0), (1, 0)
y  int : (0, 3)
Math 114
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
Carreon 4.1 (7)
Analyzing Rational Functions Algebraically (cont.)
Extra Practice Answers
1.
1
3
4
HA : y 
27
SA : none
VA : x  
holes : none
x  int : (0, 0)
y  int : (0, 0)
2.
1
1
VA : x  , x  
4
4
HA : y  0
SA : none
holes : none
1
1
x  int : ( , 0), (  , 0)
5
5
y  int : (0, 1)
Math 114
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
Carreon 4.1 (8)
Analyzing Rational Functions Algebraically (cont.)
3.
VA : x  5
HA : none
HA : y  1
SA : none
SA : none
holes : none
holes : none
1
, 0)
7
1
y  int : (0, )
5
x  int : (1, 0), (1, 0)
x  int : ( 3
5.
4. VA : none
VA : x  0
HA : none
y  int : (0,1)
1
1
,x  
2
2
HA : y  0
6. VA : x 
SA : y  x
SA : none
holes : none
holes : none
x  int : (2, 0), (2, 0)
1
1
x  int : ( , 0), (  , 0)
3
3
y  int : (0,1)
y  int : none
Math 114
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
Carreon 4.1 (9)
Analyzing Rational Functions Algebraically (cont.)
7. VA : none
HA : y  1
SA : none
4
holes : (2, )
5
x  int : (1, 0), (2, 0), ( 2, 0)
y  int : (0, 2)
9. VA : x
8. VA : x  2, x  
1
3
HA : y  2
SA : none
holes : none
x  int : none
1
y  int : (0,  )
2
0
HA : none
SA : y  3 x
holes : none
x  int : none
y  int : none
Math 114
©Cerritos College MLC
No part of this work may be reproduced without the prior written consent of the Cerritos College Math Learning Center.
Carreon 4.1 (10)