Q. 36 - 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.

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

Step-by-Step Solution

Answer

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

Step 1: Given information

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

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

Comparing, we get

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

Step 3: Sketching the graph

The graph is