College Algebra, 2013a

306 Rational Functions
Theorem 4.1. Location of Vertical Asymptotes and Holes: a Suppose r is a rational
function which can be written as r(x) = p(x)
q(x) where p and q have no common zeros.b Let c be a
real number which is not in the domain of r.
( )
ˆ If q(c) ≠ 0, then the graph of y = r(x) has a hole at .
c, p(c)
q(c)
ˆ If q(c) = 0, then the line x = c is a vertical asymptote of the graph of y = r(x).
a Or, ‘How to tell your asymptote from a hole in the graph.’
b In other words, r(x) is in lowest terms.
In English, Theorem 4.1 says that if x = c is not in the domain of r but, when we simplify r(x), it
no longer makes the denominator 0, then we have a hole at x = c. Otherwise, the line x = c is a
vertical asymptote of the graph of y = r(x).
Example 4.1.2. Find the vertical asymptotes of, and/or holes in, the graphs of the following
rational functions. Verify your answers using a graphing calculator, and describe the behavior of
the graph near them using proper notation.
1. f(x) = 2x
x 2 − 3
3. h(x) = x2 − x − 6
x 2 +9
2. g(x) = x2 − x − 6
x 2 − 9
4. r(x) = x2 − x − 6
x 2 +4x +4
Solution.
1. To use Theorem 4.1, we first find all of the real numbers which aren’t in the domain of f. To
do so, we solve x 2 − 3 = 0 and get x = ± √ 3. Since the expression f(x) is in lowest terms,
there is no cancellation possible, and we conclude that the lines x = − √ 3andx = √ 3are
vertical asymptotes to the graph of y = f(x). The calculator verifies this claim, and from the
graph, we see that as x →− √ 3 − , f(x) →−∞,asx →− √ 3 + , f(x) →∞,asx → √ 3 − ,
f(x) →−∞, and finally as x → √ 3 + , f(x) →∞.
= (x−3)(x+2)
(x−3)(x+3) = x+2
2. Solving x 2 − 9=0givesx = ±3. In lowest terms g(x) = x2 −x−6
x 2 −9
x+3 . Since
x = −3 continues to make trouble in the denominator, we know the line x = −3 isavertical
asymptote of the graph of y = g(x). Since x = 3 no longer produces a 0 in the denominator,
we have a hole at x = 3. To find the y-coordinate of the hole, we substitute x =3into x+2
x+3
and find the hole is at ( 3, 5 6)
. When we graph y = g(x) using a calculator, we clearly see the
vertical asymptote at x = −3, but everything seems calm near x = 3. Hence, as x →−3 − ,
g(x) →∞,asx →−3 + , g(x) →−∞,asx → 3 − , g(x) → 5 −
6 ,andasx → 3 + , g(x) → 5 6
+ .