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.

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

Short answer

Amplitude,$A=3$

Time period, $T=-1$

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 sine function form is given as:

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

Where,

$A=$Amplitude

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

localid="1646385012543" $\frac{\varphi }{\omega }=$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:

$$\begin{gathered} T=\frac{2\mathrm{\pi }}{\omega } \\ T=\frac{2\mathrm{\pi }}{-2} \\ T=-1 \end{gathered}$$

Phase shift:

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

Step 2. Graph the function 

The graph of the function is given as: