Question

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

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

Short answer

The graph of the given function is graph (B)

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

Graph will pass through (0,0)

when role="math" localid="1646496063332" $x=\frac{\mathrm{\pi }}{4}$

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

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 (B) 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 (B)