Unformatted text preview: s, end behavior is determined by the leading term, so in
the denominator, the x2 term wins out over the x term. 254 Rational Functions
y
8
7
6
5 (+) (−) 0 (+) 0 (−) −2 −1 5
2 4 (+) 3 3 1
−9 −8 −7 −6 −5 −4 −3 −1
−1 1 2 4 5 6 7 8 9 x −2
−3
−4 Our next example gives us not only a hole in the graph, but also some slightly diﬀerent end behavior.
Example 4.2.3. Sketch a detailed graph of h(x) = 2x3 + 5 x2 + 4 x + 1
.
x2 + 3 x + 2 Solution.
1. For domain, you know the drill. Solving x2 + 3x + 2 = 0 gives x = −2 and x = −1. Our
answer is (−∞, −2) ∪ (−2, −1) ∪ (−1, ∞).
2. To reduce h(x), we need to factor the numerator and denominator. To factor the numerator,
we use the techniques12 set forth in Section 3.3 and we get
1 !
¡
(2x + 1)(x + 1)
2x3 + 5x2 + 4x + 1
(2x + 1)(x + 1)2
(2x + 1)(x + 1)2
h(x) =
=
=
$
$=
2 + 3x + 2
(x +
x
(x + 2)(x + 1)
(x + 2)$$ 1)
x+2 We will use this reduced formula for h(x) as long as we’re not substituting x = −1....
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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