Precalculus (6th Edition) Blitzer

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

Chapter 11 - Section 11.2 - Finding Limits Using Properties of Limits - Exercise Set - Page 1155: 89

Answer

The amplitude of $ y=-4\cos \frac{\pi }{2}x $ is 4 and the period of $ y=-4\cos \frac{\pi }{2}x $ is 4.
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Work Step by Step

Consider the given equation, $ y=-4\cos \frac{\pi }{2}x $ The equation is of the form $ y=A\cos Bx $ with $ A=-4$ and $ B=\frac{\pi }{2}$. Amplitude of $ y=-4\cos \frac{\pi }{2}x $ is given by $\left| A \right|=\left| -4 \right|=4$ Period of $ y=-4\cos \frac{\pi }{2}x $ is given by $\frac{2\pi }{B}=\frac{2\pi }{\frac{\pi }{2}}=2\pi \times \frac{2}{\pi }=4$ Thus, the amplitude of $ y=-4\cos \frac{\pi }{2}x $ is 4 and the period of $ y=-4\cos \frac{\pi }{2}x $ is 4. Graph: Step1: Identify the amplitude and the period of $ y=-4\cos \frac{\pi }{2}x $. The amplitude of $ y=-4\cos \frac{\pi }{2}x $ is 4 means that the maximum value of $ y $ is 4 and the minimum is $-4$. The period of $ y=-4\cos \frac{\pi }{2}x $ is 4 means that each cycle is of length of 4. Step 2: Find the values of $ x $ for the first five key points. Begin by dividing the period 4, by 4. $\frac{\text{period}}{4}=\frac{4}{4}=1$ Start with the value of $ x $ where the cycle begins, ${{x}_{1}}=0$. Adding quarter periods 1, the five key points are: $\begin{align} & {{x}_{1}}=0 \\ & {{x}_{2}}=0+1=1 \\ & {{x}_{3}}=1+1=2 \\ & {{x}_{4}}=2+1=3 \\ & {{x}_{5}}=3+1=4 \\ \end{align}$
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