## Precalculus (6th Edition) Blitzer

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}