Chapter 1 | Functions and Graphs
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Example 1.14 Graphing Polynomial Functions
For the following functions a. and b., i. describe the behavior of f ( x ) as x →±∞, ii. find all zeros of f , and iii. sketch a graph of f .
a. f ( x ) =−2 x 2 +4 x −1 b. f ( x ) = x 3 −3 x 2 −4 x
Solution a. The function f ( x ) =−2 x 2 +4 x −1 is a quadratic function. i. Because a =−2<0, as x →±∞, f ( x )→−∞.
ii. To find the zeros of f , use the quadratic formula. The zeros are
2 − 4(−2)(−1) 2(−2)
−4±2 2
2± 2
x = −4± 4
= −4± 8 −4 =
−4 = 2 . iii. To sketch the graph of f , use the information from your previous answers and combine it with the fact that the graph is a parabola opening downward.
b. The function f ( x ) = x 3 −3 x 2 −4 x is a cubic function.
i. Because a =1>0, as x →∞, f ( x )→∞. As x →−∞, f ( x )→−∞. ii. To find the zeros of f , we need to factor the polynomial. First, when we factor x out of all the terms, we find f ( x ) = x ( x 2 −3 x −4). Then, when we factor the quadratic function x 2 −3 x −4, we find f ( x ) = x ( x −4)( x +1).
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