Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.3 - Polynomial Functions and Their Graphs - Exercise Set - Page 349: 63

Answer

a. $x\to-\infty, y\to\infty$ and $x\to\infty, y\to\infty$ b. $x=-4,1,2$, crosses the x-axis at $x=-4,1$, touches and turns around at $x=2$ c. $y=-16$. d. neither e. See graph

Work Step by Step

Given the function $f(x)=(x-2)^2(x+4)(x-1)$, we have: a. The leading term is $x^4$ with a coefficient of $+1$ and an even power; thus $x\to-\infty, y\to\infty$ and $x\to\infty, y\to\infty$, and the end behaviors are that the curve will rise as $x$ increases (right end) and it will also rise as $x$ decreases (left end). b. The equation is factored; thus the x-intercepts are $x=-4,1,2$ and the graph crosses the x-axis at $x=-4,1$ (odd multiplicity), but touches and turns around at $x=2$ (even multiplicity). c. We can find the y-intercept by letting $x=0$, which gives $y=-16$. d. Test $f(-x)=(-x-2)^2(-x+4)(-x-1)=(x+2)^2(x-4)(x+1)$. As $f(-x)\ne f(x)$ and $f(-x)\ne -f(x)$, the graph is neither symmetric with respect to the y-axis nor with the origin. e. See graph; as $n=4$, the maximum number of turning points will be $3$, which agrees with the graph.
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