Q. 27 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 21–30, match the given function to one of the graphs (A)– (J).  

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

Step-by-Step Solution

Answer

The graph of the given function is graph (C) 

Step 1. Given information

The trigonometric function is: $y=-2\cos \left(\frac{1}{2}x\right)$

Step 2. To determine the amplitude and period of the sinusoidal function.

By comparing given function $-2\cos \left(\frac{1}{2}x\right)$ with standard cosine function: $-A\cos \left(\omega x\right)$

We get amplitude $A=2$

Time period is $$\begin{gathered} T=\frac{2\mathrm{\pi }}{\omega } \\ T=\frac{2\mathrm{\pi }}{{\displaystyle \frac{1}{2}}} \\ T=4\mathrm{\pi } \end{gathered}$$

So the graph of the given function will have time period $4\mathrm{\pi }$ and amplitude 2

Step 3. To find value of function at y-axis.

At y-axis 

$x=0$

When

then $$\begin{gathered} y=-2\cos \left(\frac{1}{2}.0\right) \\ y=-2\cos 0 \\ y=-2.1 \\ y=-2 \end{gathered}$$

graph will pass through (0,2)

Now look at the given graphs from A-J we can see that 

graph (E) has amplitude 2 , time period $4\mathrm{\pi }$ and passing point (0,-2)

So the graph for given function is graph (C)