College Algebra Lecture Notes
Section 3.5
Page 1 of 6
Section 3.5: Graphing Rational Functions
Big Idea: A rational function is a ratio of polynomials, so our knowledge of graphing
polynomials can be leveraged into use for graphing rational functions.
Big Skill: You should be able to graph rational functions by identifying intercepts, vertical
asymptotes, and end behavior.
A. Rational Function and Asymptotes
Rational Functions
A rational function V(x) has the form
p  x
V  x 
d  x
Where p and d are polynomials and d  0.
The domain of V is all reals except for the zeroes of d.
The Reciprocal Function y 
1
x
The Reciprocal Square Function y 
1
x2
Horizontal Asymptotes
Given a constant k, the line y = k is a horizontal asymptote for a function V if as x increases or
decreases without bound, V(x) approaches k:
as x  ,V  x   k or as x  ,V  x   k
College Algebra Lecture Notes
Section 3.5
Page 2 of 6
Vertical Asymptotes
Given a constant h, the line x = h is a vertical asymptote for a function V if as x approaches h,
V(x) increases or decreases without bound:
as x  h ,V  x    or as x  h ,V  x   

The horizontal and vertical asymptotes for the reciprocal and squared reciprocal functions
1
1
k
are moved by shifting linear transformations: f  x  
 k or g  x  
2
 x  h
 x  h
Practice:
1. Graph y  f  x  
1
1
 3.
 1 and y  g  x  
2
 x  2
 x  5
College Algebra Lecture Notes
Section 3.5
Page 3 of 6
B. Vertical Asymptotes and the Domain
Vertical Asymptotes of a Rational Function
p  x
Given V  x  
is a rational function in simplest form, vertical asymptotes will occur at the
d  x
real zeroes of d.
Practice:
2x
x2  4 x  3
2. Find the vertical asymptotes of f  x   2
and g  x   2
.
x 4
x  2x 1
C. Vertical Asymptotes and Multiplicity
 If the multiplicity of a zero in the denominator is odd, then the function will grow to
opposite infinities on either side of the vertical asymptote.
 If the multiplicity of a zero in the denominator is even, then the function will grow to the
same infinity on either side of the vertical asymptote.
Practice:
3. Note the behavior of the function near the vertical asymptotes below.

y


 



x




y = 1/( (x+2)^2(x-1)
)



College Algebra Lecture Notes
Section 3.5
Page 4 of 6
D. Finding Horizontal Asymptotes
Horizontal Asymptotes
p  x  ax n 

Given V  x  
is a rational function in simplest form, with leading terms as
d  x  bx m 
indicated, then:
 If n < m, there is a horizontal asymptote at y = 0.
a
 If n = m, there is a horizontal asymptote at y  .
b
 If n > m, there is no horizontal asymptote. The end behavior of the graph is that it is
asymptotic to the quotient function you obtain when performing long division of the
polynomials (this is true in all three cases, actually).
Practice:
4. Find the horizontal asymptotes of f  x  
h  x 
4 x3  5
.
2x2  2x  1
2x
2 x2  4 x  3
g
x

,
, and


x2  4
x2  2 x  1
College Algebra Lecture Notes
Section 3.5
Page 5 of 6
E. The Graph of a Rational Function
Guidelines for Graphing Rational Functions
p  x
Given V  x  
is a rational function in simplest form:
d  x







Find the y intercept: y = V(0) .
Find any vertical asymptotes where d = 0.
Find x intercepts where p = 0.
Find any horizontal asymptotes y = k.
Determine if the graph crosses the horizontal asymptote by solving V(x) = k.
Compute midinterval points as needed.
Use this information to make the graph.
Practice:
5. Graph f  x  
x2  5x  6
.
2 x 2  x  15
College Algebra Lecture Notes
6. Graph f  x  
x3  1
.
x2  5
E. Applications of Rational Functions
Practice:
7. .
Section 3.5
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