Chapter 6.6, Problem 7AYU

In Problems 3-14, find the amplitude, period, and phase shift of each function. Graph each function. Be sure to label key points. Show at least two periods.

$y\text{ = }-\text{3sin}\left(2x+\frac{\pi }{2}\right)$

To determine

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

Answer to Problem 7AYU

Solution:

The amplitude $=3$, period $=\pi$, phase shift $=-\frac{\pi }{4}$.

Explanation of Solution

Given:

$y=-3\sin \left(2x+\frac{\pi }{2}\right)$

Calculation:

Compare $y=-3\sin \left(2x+\frac{\pi }{2}\right)$ to $y=A\sin \left(\omega x-\varphi \right)$, note that $A=-3, \omega =2$ and $\varphi =-\frac{\pi }{2}$. The graph is a sine curve with amplitude $\left|A\right|=3$, 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 $y=-3\sin \left(2x+\frac{\pi }{2}\right)$ will lie between $-3$ and 3 on the $y\text{-axis}$.

One cycle will begin at $x=\frac{\varphi }{\omega }=-\frac{\pi }{4}$ and end at $x=\frac{\varphi }{\omega }+\frac{2\pi }{\omega }=-\frac{\pi }{4}+\pi =\frac{3\pi }{4}$.

To find five key points, divide the interval $\left[-\frac{\pi }{4},\frac{3\pi }{4}\right]$ in to four sub intervals, each of length $\frac{\frac{3\pi }{4}+\frac{\pi }{4}}{4}=\frac{\pi }{4}$.

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

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

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