Chapter 6.6, Problem 5AYU

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.

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

To determine

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

Answer to Problem 5AYU

Solution:

The amplitude $=2$, period $=\frac{2\pi }{3}$, phase shift $=-\frac{\pi }{6}$.

Explanation of Solution

Given:

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

Calculation:

Compare $y=2\cos \left(3x+\frac{\pi }{2}\right)$ to $y=A\cos \left(\omega x-\varphi \right)$, note that $A=2,\omega =3$ and $\varphi =-\frac{\pi }{2}$. The graph is a cosine curve with amplitude $\left|A\right|=2$, period $T=\frac{2\pi }{\omega }=\frac{2\pi }{3}$ and phase shift $=\frac{\varphi }{\omega }=\frac{-\frac{\pi }{2}}{3}=-\frac{\pi }{6}$.

The graph of $y=2\cos \left(3x+\frac{\pi }{2}\right)$ will lie between $-2$ and 2 on the $y\text{-axis}$.

One cycle will begin at $x=\frac{\varphi }{\omega }=-\frac{\pi }{6}$ and end at $x=\frac{\varphi }{\omega }+\frac{2\pi }{\omega }=-\frac{\pi }{6}+\frac{2\pi }{3}=\frac{\pi }{2}$.

To find five key points, divide the interval $\left[-\frac{\pi }{6},\frac{\pi }{2}\right]$ in to four sub intervals, each of length $\frac{\frac{\pi }{2}+\frac{\pi }{6}}{4}=\frac{\frac{4\pi }{6}}{4}=\frac{\pi }{6}$.

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

$\left(-\frac{\pi }{6}, 2\right), \left(0, 0\right), \left(\frac{\pi }{6}, -2\right), \left(\frac{\pi }{3}, 0\right), \left(\frac{\pi }{2}, 2\right)$

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