Q. 14 - 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. 

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

Step-by-Step Solution

Answer

Amplitude, $A=3$

Time period, $T=-1$

Phase shift$=\frac{\mathrm{\pi }}{4}$

The graph is given as: 

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

The general cosine function form is given as:   

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

Where, 

$A=$Amplitude

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

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

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

Amplitude:

$$\begin{gathered} A=\left|-3\right| \\ A=3 \end{gathered}$$

Time period:

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

Phase shift:

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

The graph is given as:

Step 2. Graph the function

The graph is given as: