Rational Functions
Section 3.5
Apply Notation Describing Infinite Behavior of a Function
Let p(x) and q(x) be polynomials where q(x) ≠ 0. A function f defined by
Rational Function:
f ( x) 
p ( x)
is called a rational function.
q ( x)
Notation
Meaning
x approaches c from the right (but will not equal c).

xc
x  c
x approaches c from the left (but will not equal c).
x 
x  
1.
The graph of f  x  
x approaches infinity (x increases without bound).
x approaches negative infinity (x decreases without bound).
x 1
is given. Complete the statements.
x 3
As x   , f  x   ______
As x   , f  x   ______
As x  3 , f  x   ______

As x  3 , f  x   ______
Identify Vertical Asymptotes
Vertical Asymptote:
As x  c  , f  x   
The line x = c is a vertical asymptote of the graph of a function f if
f (x) approaches infinity or negative infinity as x approaches c from either side.
As x  c  , f  x   
As x  c  , f  x   
As x  c  , f  x   
Let f ( x) 
p ( x)
where p(x) and q(x) have no common factors other than 1. To locate the vertical asymptotes of
q ( x)
f  x  , determine the real numbers x where the denominator is zero, but the numerator is nonzero.
Identify the vertical asymptotes (if any).
2.
f  x 
x 3
x  4x  5
2
3.
f  x 
x 3
x  2x  3
2
4.
f  x  
Identify Horizontal Asymptotes
Horizontal Asymptote: The line y = d is a horizontal asymptote of the graph of a function f
if f (x) approaches d as x approaches infinity or negative infinity.
As x  , f  x   d
As x  , f  x   d
Let f be a rational function defined by f ( x) 
an x n  an1 x n1  an 2 x n2  ...  a1 x  a0
bm x m  bm1 x m1  bm2 x m2  ...  b1 x  b0
where n is the degree of the numerator and m is the degree of the denominator.
1.
If n > m, f has no horizontal asymptote.
2.
If n < m, then the line y = 0 (the x-axis) is the horizontal asymptote of f.
3.
If n = m, then the line y 
an
is the horizontal asymptote of f.
bm
8
x  25
2
Identify the horizontal asymptotes (if any).
5.
3x
f  x  2
x 9
6.
3x 2
f  x  2
x 9
7.
3x3
f  x  2
x 9
The graph of a rational function may not cross a vertical asymptote. The graph may cross a horizontal asymptote.
8.
Determine if the function f  x  
2 x2  4
crosses its horizontal asymptote.
x 2  3x  4
Identify Slant Asymptotes
Slant Asymptote: A rational function will have a slant asymptote if the degree of the numerator
is exactly one greater than the degree of the denominator.
To find an equation of a slant asymptote, divide the numerator of the function by the denominator.
The quotient will be linear and the slant asymptote will be of the form y = quotient.
9.
Determine the slant asymptote of f  x  
3x3
.
x2  9
Graph Rational Functions
f ( x) 
p ( x)
where p(x) and q(x) are polynomials with no common factors.
q ( x)
Intercepts:
Determine the x- and y-intercepts
Asymptotes: Determine if the function has vertical asymptotes and graph them as dashed lines.
Determine if the function has a horizontal asymptote or a slant asymptote,
and graph the asymptote as a dashed line.
Determine if the function crosses the horizontal or slant asymptote.
Symmetry:
f is symmetric to the y-axis if f (–x) = f (x).
f is symmetric to the origin if f (–x) = –f (x).
Plot points:
Plot at least one point on the intervals defined by the x-intercepts,
vertical asymptotes, and points where the function crosses a horizontal
or slant asymptote.
10. Graph f  x  
x2  x  6
.
x2  5x  6
11. Graph f  x  
2x2
.
x2  4
12. Graph f  x  
2 x2  8x  8
.
x 3
13. Use transformations to graph f  x  
1
1 .
x 3
Use Rational Functions in Applications
14. Young's rule is a formula for calculating the approximate dosage for a child, given the adult dosage. For an
adult dosage of d mg, the child's dosage C(x) may be calculated by
C  x 
dx
x  12
x is the child's age in years and 2  x  13 .
a.
What is the horizontal asymptote of the function? What does it mean in the context of this problem?
b.
What is the vertical asymptote of the function?
c.
For an adult dosage of 50 mg, what is the approximate dosage for a 9-year-old child?
Graph of C  x  
50 x
x  12