Q. 4 - Precalculus Enhanced with Graphing Utilities

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.

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

Step-by-Step Solution

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