Chapter 6.6, Problem 4AYU

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.

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

To determine

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

Answer to Problem 4AYU

Solution:

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

Explanation of Solution

Given:

$y=3\sin \left(3x-\pi \right)$

Calculation:

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

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

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

To find five key points, divide the interval $\left[\frac{\pi }{3}, \pi \right]$ in to four sub intervals, each of length $\frac{\pi -\frac{\pi }{3}}{4}=\frac{\frac{2\pi }{3}}{4}=\frac{\pi }{6}$.

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

$\left(\frac{\pi }{3}, 0\right), \left(\frac{\pi }{2}, 3\right), \left(\frac{2\pi }{3}, 0\right), \left(\frac{5\pi }{6}, -3\right), \left(\pi , 0\right)$

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