College Algebra 9th txtbk

308 CHAPTER 4 POLYNOMIAL AND RATIONAL FUNCTIONS
Although the graph of f does not have a horizontal asymptote, we can still gain some useful
information about the behavior of the graph as x S and as x S if we first perform a
long division:
This shows that
x 1
x 2x 2 3x 4
x 2 2x
x 4
x 2
6
f (x) x2 3x 4
x 2
Quotient
Remainder
x 1 6
x 2
As x S or x S , 6(x 2) S 0 and the graph of f approaches the line y x 1.
This line is called an oblique asymptote for the graph of f. The asymptotes and intercepts
are shown in Figure 9, and the graph of f is sketched in Figure 10.
10
y
10
y
y x 1
10
Oblique
asymptote
y x 1
10
10
Intercepts and asymptotes
x
10
Z Figure 9 Z Figure 10
10
10
x
f(x) x2 3x 4
x 2
Generalizing the results of Example 6, we have Theorem 4.
Z THEOREM 4 Oblique Asymptotes and Rational Functions
If f (x) p(x) q(x), where p(x) and q(x) are polynomials and the degree of p(x) is
1 more than the degree of q(x), then f(x) can be expressed in the form
where the degree of r(x) is less than the degree of q(x). The line
is an oblique asymptote for the graph of f. That is,
[ f (x) (mx b)] S 0
f (x) mx b r(x)
q(x)
y mx b
as x S or x S
MATCHED PROBLEM 6
Graph, including any oblique asymptotes,
f (x) x2 5
x 1 .