Q. 29 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

The graph of the given function is graph (J)  

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 sine 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 2

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 (0,2)

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

then

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

graph will pass through $\left(\frac{\mathrm{\pi }}{4},3\right)$

Now look at the given graphs from A-J in which 

amplitude is 3, time period is  $\mathrm{\pi }$and passing point (0,2) and $\left(\frac{\mathrm{\pi }}{4},3\right)$

So the graph for given function is graph (J)