Precalculus (6th Edition) Blitzer

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

Chapter 4 - Section 4.5 - Graphs of Sine and Cosine Functions - Exercise Set - Page 596: 44

Answer

Amplitude of the given function is $1$ , period is $2\pi $, and phase shift is $-\frac{\pi }{2}$.
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Work Step by Step

For the standard function in the form $y=A\cos \left( Bx+C \right)$ $\begin{align} & \text{Amplitude}=\left| A \right| \\ & \text{Period}=\frac{2\pi }{\left| B \right|} \end{align}$ And, $\text{Phase}\,\text{Shift}=-\frac{C}{B}$ Then, compare the given function $y=\cos \left( x+\frac{\pi }{2} \right)$ with the standard function as follows: $A=1,\text{ }B=1,\text{ and }C=\frac{\pi }{2}$ So, $\begin{align} & \text{Amplitude=}\left| 1 \right| \\ & =1 \\ & \text{Period=}\frac{2\pi }{\left| 1 \right|} \\ & =2\pi \end{align}$ And, $\begin{align} & \text{Phase}\,\text{Shift}=-\frac{\frac{\pi }{2}}{1} \\ & =-\frac{\pi }{2} \end{align}$ And calculate quarter period $\begin{align} & \text{Quarter}\,\text{Period}=\frac{\text{Period}}{4} \\ & =\frac{2\pi }{4} \\ & =\frac{\pi }{2} \end{align}$ Thus, the cycle begins at $x=-\frac{\pi }{2}$. Find the x-value and y-value at multiples of quarter periods. Now, plot the obtained coordinates to get the graph of the function $y=\cos \left( x+\frac{\pi }{2} \right)$
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