Q. 5 - 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.

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

Step-by-Step Solution

Answer

Amplitude, $A=2$

Time period, $T=\frac{2\mathrm{\pi }}{3}$

Phase shift= $-\frac{\mathrm{\pi }}{6}$

The graph of the given function is:

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

The general cosine function form is given as:

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

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:

Amplitude, $A=2$

Time period, $T\mathit{=}\frac{\mathit{2}\mathit{\pi }}{\mathit{3}}$

Phase shift:

$$\begin{gathered} \frac{\varphi }{\omega }=\frac{-{\displaystyle \frac{\mathrm{\pi }}{2}}}{3} \\ \frac{\varphi }{\omega }=\frac{-\mathrm{\pi }}{6} \end{gathered}$$

Step 2. Graph the function

The graph of the given function is: