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.

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

Short answer

Amplitude: $A=3$

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

Phase shift=$-\frac{\mathrm{\pi }}{2}$

The graph of the function is given as:

Step-by-step solution

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

The general cosine function form is given as:

$y=A\cos \left(\omega x-\varphi \right)+B$

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:

$A=3$

Time period,

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

Phase shift:

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

Step 2. Graph the function

The graph is given as: