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Math 110 Quiz 5 Oct. 19, 2006
Name:
1)
Let
R
(
x
) =
x
2

5
x
2
+4
x
, a rational function.
(a) Determine the
x
and
y
intercepts of
R
(
x
).
(b) Give the equations of any vertical asymptotes.
(c) Give the equations of any horizontal asymptotes.
(d) Sketch a graph of
R
(
x
), plotting additional points if necessary. Be sure that your graph indicates inter
cepts and asymptotes.
Solutions:
(a) To ﬁnd
x
intercepts we solve
R
(
x
) = 0, in other words
x
2

5 = 0. This gives
x
=
±
√
5, so the
x
intercepts
are
(
±
√
5
,
0)
To ﬁnd the
y
intercept we ﬁnd
R
(0).
R
(0) =

5
0
, which is undeﬁned. Hence there are no
y
intercepts. In fact, the
y
axis is a vertical asymptote as we’ll see below.
(b) To ﬁnd vertical asymptotes, we solve
x
2
+ 4
x
=
x
(
x
+ 4) = 0. Hence
x
= 0
and
x
=

4
are ver
tical asymptotes.
(c) To ﬁnd horizontal asymptotes, we compare the degrees of the numerator and denominator. Since they
are both quadratic, we have a horizontal asymptote at the ratio of the leading coeﬃcients:
y
= 1
(d)
1
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The population of the world doubles roughly every 38 years. In 1995, the world population was 5.8
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 Spring '07
 carlson
 Equations, Asymptotes

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