Question

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

$y=3\sin (3x-\mathrm{\pi })$

Short answer

For the function$y=3\sin (3x-\mathrm{\pi })$

Amplitude$A=3$

Period$T=\frac{2\mathrm{\pi }}{3}$

Phase shift$\frac{\varphi }{\omega }=\frac{\mathrm{\pi }}{3}$

and graph of the function is

Step-by-step solution

Step 1. Given data 

The given function is

$y=3\sin (3x-\mathrm{\pi })$

Step 2. Constants of the function 

Compare the function $y=3\sin (3x-\mathrm{\pi })$with general sine function form

$$\begin{gathered} y=A\sin (\omega x-\varphi )+B \\ y=3\sin (3x-\mathrm{\pi }) \end{gathered}$$

so Amplitude

$$\begin{gathered} A=\left|3\right| \\ A=3 \end{gathered}$$

Period

$$\begin{gathered} T=\frac{2\mathrm{\pi }}{\omega } \\ T=\frac{2\mathrm{\pi }}{3} \end{gathered}$$

Phase shift

$\frac{\varphi }{\omega }=\frac{\mathrm{\pi }}{3}$

Step 3. Graph of function 

Plot the graph of the function $y=3\sin (3x-\mathrm{\pi })$