Chapter 6.6, Problem 6AYU

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.

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

To determine

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

Answer to Problem 6AYU

Solution:

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

Explanation of Solution

Given:

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

Calculation:

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

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

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

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

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

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

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