Answer
See the full explanation below.
Work Step by Step
When the graph of $y=2\cos \frac{1}{3}x$ is shifted two unit downward, then the graph of $y=2\cos \frac{1}{3}x-2$ is obtained.
The period is given below:
$\begin{align}
& \text{Period = }\frac{2\pi }{B} \\
& =\frac{2\pi }{\left( \frac{1}{3} \right)} \\
& =6\pi
\end{align}$
The quarter-period is as follows:
$\begin{align}
& \text{Quarter-period}=\frac{6\pi }{4} \\
& =\frac{3\pi }{2}
\end{align}$
Now, add quarter periods starting from $x=0$ to generate x-values for the key points. The x-value for the first key point is as follows:
$x=0$
And the x-value for the second key point is:
$\begin{align}
& x=0+\frac{3\pi }{2} \\
& =\frac{3\pi }{2}
\end{align}$
And the x-value for the third key point is:
$\begin{align}
& x=\frac{3\pi }{2}+\frac{3\pi }{2} \\
& =3\pi
\end{align}$
And the x-value for the fourth key point is:
$\begin{align}
& x=3\pi +\frac{3\pi }{2} \\
& =\frac{9\pi }{2}
\end{align}$
And the x-value for the fifth key point is:
$\begin{align}
& x=\frac{9\pi }{2}+\frac{3\pi }{2} \\
& =6\pi
\end{align}$
