Answer
Refer to the graph below.
Work Step by Step
RECALL:
The graph of the function $y=a\cdot \sin{[b(x-d)]}$ is a sinusoidal curve that has:
period = $\frac{2\pi}{b}$
amplitude = $|a|$
phase shift = $|d|$ (to the left when $d\lt0$, to the right when $d\gt0$)
one period is in the interval $[0, 2\pi]$
Write the given equation in the form $y= a \cdot \sin{[b(x-d)]}$ by factoring out $3$ within the sine function:
$y=\sin{[3(x+\frac{\pi}{6})]}$
This function has $a=1, b=3,$ and $d=-\frac{\pi}{6}$
Thus, the graph of this function has:
period = $\frac{2\pi}{3}$
amplitude = $|1| = 1$
phase shift = $|-\frac{\pi}{6}|=\frac{\pi}{6}$, to the left
One period of this function will be in the interval $[0-\frac{\pi}{6}, \frac{2\pi}{3}-\frac{\pi}{6}]=[-\frac{\pi}{6}, \frac{\pi}{2}]$.
Divide this interval into four equal parts to get the key x-values $-\frac{\pi}{6}, 0,
\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}$.
To graph the given function, perform the following steps:
(1) Create a table of values using the key x-values listed above. (Refer to the table below.)
(2) Plot the points from the table of values and connect them using a sinusoidal curve. (Refer to the graph in the answer part above.)
