Q. 37 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 35-38, find the amplitude, period, and phase shift of each function. Graph each function. Show at least two periods.

$y=\frac{1}{2}\sin \left(\frac{3}{2}x-\pi \right)$

Step-by-Step Solution

Answer

The amplitude, time period , phase shift is $\frac{1}{2},\frac{4\mathrm{\pi }}{3},\frac{2\mathrm{\pi }}{3}$ respectively and the graph is 

Step 1: Given information

Given function $y=\frac{1}{2}\sin \left(\frac{3}{2}x-\pi \right)$

Step 2: Comparing with y = A sin ( b x - ∅ ) + B and calculating

Comparing, we get

$$\begin{gathered} A=\frac{1}{2} \\ T=\frac{2\mathrm{\pi }}{{\displaystyle \frac{3}{2}}} \\ T=\frac{4\mathrm{\pi }}{3} \end{gathered}$$

The phase shift is $$\begin{gathered} \frac{\varphi }{\omega }=\frac{\pi }{\frac{3}{2}} \\ \frac{\varphi }{\omega }=\frac{2\mathrm{\pi }}{3} \end{gathered}$$

Step 3: Sketching the graph

The graph is