Q. 7 - 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.  

$y=-3\sin \left(2x+\frac{\mathrm{\pi }}{2}\right)$

Step-by-Step Solution

Answer

Amplitude: $A=-3$

Time period: $\mathrm{T}=\mathrm{\pi }$$T=\mathrm{\pi }$

Phase shift: $-\frac{\mathrm{\pi }}{4}$

The graph is given as:

Step 1. Find the amplitude, period, and phase shift.

The general sine function form is given as: 

$y=A\sin \left(\omega t-\varphi \right)+B$, where

A=Amplitude

Time period, T=$\frac{2\mathrm{\pi }}{\omega }$

Phase shift, $\frac{\varphi }{\omega }$

Comparing the given function with the general function, we get:

$A=-3$

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

Phase shift:

$$\begin{gathered} \frac{\varphi }{\omega }=\frac{{\displaystyle \frac{-\mathrm{\pi }}{2}}}{2} \\ =-\frac{\mathrm{\pi }}{4} \end{gathered}$$

Step 2. Graph the function

The graph is given as: