Q. 3 - 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.

$y=4 \sin (2x-\mathrm{\pi })$

Step-by-Step Solution

Answer

For the function$y=4 \sin (2x-\mathrm{\pi })$

Amplitude $A=4$

Period $T=\mathrm{\pi }$

Phase shift $\frac{\varphi }{\omega }=\frac{\mathrm{\pi }}{2}$

and graph of the function is

Step 1. Given data

The given function is

$y=4 \sin (2x-\mathrm{\pi })$

Step 2. Constants of the function

Compare the function$y=4 \sin (2x-\mathrm{\pi })$ with general sine function form  

$$\begin{gathered} y=A \sin (\omega x-\mathrm{\varphi })+b \\ y=4 \sin (2x-\mathrm{\pi })+0 \end{gathered}$$

so Amplitude

 $$\begin{gathered} A=\left|4\right| \\ A=4 \end{gathered}$$

Period

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

Phase shift

$\frac{\varphi }{\omega }=\frac{\mathrm{\pi }}{2}$

Step 3. Graph of function

Plot the graph of the function$y=4 \sin (2x-\mathrm{\pi })$