Q. 33 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

The graph of the given function is graph (B)  

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

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

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)