Chapter 6.6, Problem 8AYU

To determine

To find: The amplitude, period, phase shift of the function, graph the function.

Answer to Problem 8AYU

Solution:

The amplitude $=2$, period $=\pi$, phase shift $=-\frac{\pi }{4}$.

Explanation of Solution

Given:

$y=-2\cos \left(2x-\frac{\pi }{2}\right)$

Calculation:

Compare $y=-2\cos \left(2x-\frac{\pi }{2}\right)$ to $y=A\cos \left(\omega x-\varphi \right)$, note that $A=-2, \omega =2$ and $\varphi =\frac{\pi }{2}$. The graph is a cosine curve with amplitude $\left|A\right|=2$, period $T=\frac{2\pi }{\omega }=\frac{2\pi }{2}=\pi$ and phase shift $=\frac{\varphi }{\omega }=\frac{\frac{\pi }{2}}{2}=\frac{\pi }{4}$.

The graph of $y=-2\cos \left(2x-\frac{\pi }{2}\right)$ will lie between $-2$ and 2 on the $y\text{-axis}$.

One cycle will begin at $x=\frac{\varphi }{\omega }=\frac{\pi }{4}$ and end at $x=\frac{\varphi }{\omega }+\frac{2\pi }{\omega }=\frac{\pi }{4}+\pi =\frac{5\pi }{4}$.

To find five key points, divide the interval $\left[\frac{\pi }{4},\frac{5\pi }{4}\right]$ in to four sub intervals, each of length $\frac{\frac{5\pi }{4}-\frac{\pi }{4}}{4}=\frac{\pi }{4}$.

Use the values of $x$ to determine the five key points on the graph:

$\left(\frac{\pi }{4}, -2\right), \left(\frac{\pi }{2}, 0\right), \left(\frac{3\pi }{4}, 2\right), \left(\pi , 2\right), \left(\frac{5\pi }{4}, -2\right)$

Plot these five points and fill in the graph of the sine function.