Math1314 Rational Functions
Math1314 Rational Functions
Math1314 Rational Functions
P ( x)
r ( x) =
Q ( x)
where P and Q are polynomials. We assume that P(x) and Q(x) have no factors in
common, and Q(x) is not the zero polynomial. The domain of the rational function r(x) is
the set of all real numbers such that the denominator Q(x) is not zero.
1
Example 1: Sketch a graph of the rational function r ( x ) = .
x
Solution:
Step 1: First we note that the function r is not defined for x = 0. The
number 0 cannot be used as a value of x, but for graphing it is
helpful to find the values of f (x) for some values of x close to 0.
Step 1:
r (x) → – ∞ as x → 0–
“y approaches negative infinity as x approaches 0 from the left”
The second table shows that as x approaches 0 from the right, the
values of r (x) increase without bound. In symbols,
r (x) → ∞ as x → 0+
“y approaches infinity as x approaches 0 from the right”
These tables show that as |x| becomes large, the value of r (x)
gets closer and closer to zero.
Definition of Asymptotes:
y → ∞ or y → – ∞ as x → a+ or x → a–
y→ b as x → ∞ or x → – ∞
Asymptotes are not part of the graph of a function, but are important aids in graphing the
function. Typically, they are drawn as dashed lines to distinguish them from the graph of
the function. The following is the procedure for finding asymptotes.
an x n + an −1 x n −1 + ... + a1 x + a0
r ( x) =
bm x m + bm −1 x m −1 + ... + b1 x + b0
2x 1
(a) r ( x ) = (b) s ( x ) =
x+3 x +9
2
x4 x3 + 2 x
(c) t ( x ) = (d) r ( x ) =
( x − 4 )( x + 1) x2
Solution (a):
Step 1: We start by finding the vertical asymptotes. These are the lines
2x
x = a, where a is a zero of the denominator of r ( x ) = .
x+3
Since x = –3 is the only zero of the denominator of r, we have
one vertical asymptote at x = –3.
Step 2: Next we will find the horizontal, if it exists. The degree of the
2x
numerator of r ( x ) = is n = 1, and the degree of the
x+3
denominator is m = 1. Since n = m, r has a horizontal asymptote
at
an 2
y= = = 2.
bm 1
Solution (b):
Solution (c):
Solution (d):
5. Sketch the Graph. Graph the information provided by the first four
steps. Then plot as many additional points as needed to fill in the rest
of the graph of the function.
Note that while a graph may never cross a vertical asymptote, it can cross a horizontal
asymptote, as long as some part of the graph follows the asymptote
Example 3: Find the intercepts and asymptotes and sketch a graph of the function
3x
s ( x) = 2 .
x − 16
Solution:
3x
s ( x) =
x − 16
2
3x
=
( x + 4 )( x − 4 )
Step 5: Sketch the Graph: Graph the information provided by the first
four steps. Then plot as many additional points as needed to fill
in the rest of the graph of the function.