College Algebra, 2013a

330 Rational Functions
ˆ The behavior of y = h(x) as x →∞: If x →∞,then 3
x+2
≈ very small (+). This means
h(x) ≈ 2x − 1 + very small (+), or that the graph of y = h(x) is a little bit above the
line y =2x − 1asx →∞.
Graphically we have
4
3
2
1
y
−1
x
−2
−3
−4
6. To make our sign diagram, we place an ‘”’ abovex = −2 andx = −1 and a ‘0’ above x = − 1 2 .
On our four test intervals, we find h(x) is (+) on (−2, −1) and ( − 1 2 , ∞) and h(x) is(−) on
(−∞, −2) and ( −1, − 1 2)
. Putting all of our work together yields the graph below.
(−) ” (+) ” (−) 0 (+)
−2 −1 − 1 2
9
8
7
6
5
4
3
2
1
−4 −3 −1 −1 1 2 3 4
−2
−3
−4
−5
−6
−7
−8
−9
−10
−11
−12
−13
−14
y
x
We could ask whether the graph of y = h(x) crosses its slant asymptote. From the formula
h(x) =2x − 1+ 3
x+2 , x ≠ −1, we see that if h(x) =2x − 1, we would have 3
x+2
= 0. Since this will
never happen, we conclude the graph never crosses its slant asymptote. 14
14 But rest assured, some graphs do!