Question

In Problems 31–34, match the given function to one of the graphs (A)– (D).

Short answer

The graph of the given function is graph (C)

Step-by-step solution

Step 1. Given information   

The trigonometric function is:

$y=-3\sin \left(2x\right)$

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

By comparing given function $-3\sin \left(2x\right)$with standard sinusoidal function $A\sin \left(\omega x\right)$

We get amplitude$A=-3$

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

So the graph of the given function will have time period $\mathrm{\pi }$and amplitude 3

Step 3. To find passing points of the graph 

At y-axis

$x=0$

then $$\begin{gathered} y=-3\sin \left(2.0\right) \\ y=-3\sin o \\ y=-3.0 \\ y=0 \end{gathered}$$

Graph will pass through (0,0)

when $x=\frac{\mathrm{\pi }}{4}$

then $$\begin{gathered} y=-3\sin \left(2.\frac{\mathrm{\pi }}{4}\right) \\ y=-3\sin \left(\frac{\mathrm{\pi }}{2}\right) \\ y=-3.1 \\ y=-3 \end{gathered}$$

So the graph will pass through (0,0) and$\left(\frac{\mathrm{\pi }}{4},-3\right)$

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

graph (C) has amplitude -3 , time period $\mathrm{\pi }$and passing point (0,0) and $\left(\frac{\mathrm{\pi }}{4},-3\right)$

So the graph for given function is graph (C)