Section 3-6
1
Math 150 Lecture Notes
Rational Functions
A rational function is a function of the form )(
)(
)( xR
xP
xr = where P and Q are polynomials.
The domain of a rational function consists of all real number x except those for which the
denominator is zero.
The line x = a is a vertical asymptote of the function f if f(x) approaches ± ∞ as x approaches a
from the right or left.
The line y = b is a horizontal asymptote of the function f if f(x) approaches b as x approaches ± ∞.
Asymptotes of Rational Functions
Let r be the rational function
01
1
1
01
1
1
)( bxbxbxb
axaxaxa
xr
m
m
m
m
n
n
n
n
+⋅⋅⋅++
++⋅⋅⋅++
=
−
−
−
−
1. The vertical asymptotes of r are the lines x = a where a is a zero of the denominator.
2. a. If n < m, then r has horizontal asymptote y = 0.
b. If n = m, then r has horizontal asymptote y =
m
n
b
a
.
c. If n > m, then r has no horizontal asymptote.
To Sketch a Graph of a Rational Function
1. Factor the numerator and denominator. Note that a factor common to numerator and
denominator indicates where there is a “hole” in the graph.
2. Find x- and y-intercepts and graph them.
3. Find any vertical asymptotes and graph them with dotted lines.
4. Find the horizontal/oblique asymptote and graph it with a dotted line.
5. Make a table of values including test points to determine behavior near asymptotes and plot
additional points as needed to determine the rest of the graph.
