Answer
See below:
Work Step by Step
Equation (1) is of the form $ y=A\sin \left( Bx-C \right)$.
We can undertake the following steps to determine the graph of the equation:
Step 1. Here, $ A=3,B=2$, and $ C=\pi $, such that
Amplitude $=3$
$\begin{align}
& \text{Period}=\frac{2\pi }{B} \\
& =\frac{2\pi }{2} \\
& =\pi
\end{align}$
$\begin{align}
& \text{Phase shift}=\frac{C}{B} \\
& =\frac{\pi }{2}
\end{align}$
$\text{Quarter period}=\frac{\pi }{4}$
Step 2. Now we will find the $ x $ -values by adding the quarter periods.
$\begin{align}
& x=\frac{\pi }{2} \\
& x=\frac{\pi }{2}+\frac{\pi }{4}=\frac{3\pi }{4} \\
& x=\frac{3\pi }{4}+\frac{\pi }{4}=\pi \\
& x=\pi +\frac{\pi }{4}=\frac{5\pi }{4}
\end{align}$
Step 3. We will evaluate the function at each value of $ x $.
At $ x=\frac{\pi }{2}$,
$\begin{align}
& y=3\sin \left( 2\cdot \frac{\pi }{2}-\pi \right) \\
& =3\sin \left( 0{}^\circ \right) \\
& =0
\end{align}$
Therefore, the coordinates are $\left( \frac{\pi }{2},0 \right)$.
At $ x=\frac{3\pi }{4}$,
$\begin{align}
& y=3\sin \left( 2\cdot \frac{3\pi }{4}-\pi \right) \\
& =3\sin \left( \frac{\pi }{2} \right) \\
& =3
\end{align}$
Therefore, the coordinates are $\left( \frac{3\pi }{4},3 \right)$.
At $ x=\pi $,
$\begin{align}
& y=3\sin \left( 2\cdot \pi -\pi \right) \\
& =3\sin \left( \pi \right) \\
& =0
\end{align}$
Therefore, the coordinates are $\left( \pi,0 \right)$.
At $ x=\frac{5\pi }{4}$,
$\begin{align}
& y=3\sin \left( 2\cdot \frac{3\pi }{2}-\pi \right) \\
& =3\sin \left( 2\pi \right) \\
& =0
\end{align}$
Therefore, the coordinates are $\left( \frac{3\pi }{2},0 \right)$.
Step 4. On connecting these points, the graph can be obtained.
Step 5. Using all the steps above, we produce the final graph.
