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.

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

Short answer

Amplitude: $A=2$

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

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

The graph is given as:

Step-by-step solution

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

The general cosine function form is given as:

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

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:

$$\begin{gathered} A=\left|-2\right| \\ =2 \end{gathered}$$

Time period,

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

Phase Shift:

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

Step 2. Graph the function 

The graph is given as: