a buoy marks a channel for boat navigation and bobs up and down with the motion of the waves. in the given…

Explanation:

Step1: Recall cosine - function property

The minimum value of the cosine function $y = \cos(x)$ is - 1.

Step2: Set the argument of cosine for minimum

For $y = 2\cos(\frac{\pi}{4}t)$, we want $\cos(\frac{\pi}{4}t)=-1$. We know that $\cos(x)=-1$ when $x=(2n + 1)\pi$, where $n$ is an integer. So, $\frac{\pi}{4}t=(2n + 1)\pi$.

Step3: Solve for $t$

Divide both sides of $\frac{\pi}{4}t=(2n + 1)\pi$ by $\pi$: $\frac{1}{4}t=2n + 1$. Then multiply both sides by 4 to get $t = 4(2n+1)=8n + 4$. When $n = 0$, $t = 4$ and $f(4)=2\cos(\frac{\pi}{4}\times4)=2\cos(\pi)=-2$. When $n = 1$, $t=12$ and $f(12)=2\cos(\frac{\pi}{4}\times12)=2\cos(3\pi)=-2$.

Answer:

The points $(4,-2)$ and $(12,-2)$ (and in general, the points $(8n + 4,-2)$ for $n\in\mathbb{Z}$) should be plotted.