Q. 30 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

The graph of the given function is graph (D) 

Step 1. Given information

The trigonometric function is:

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

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

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

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 $x=0$

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

Graph will pass through (0,-2)

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

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

So the graph for given function is graph (D)