use the drawing tools to form the correct answer on the graph.\na buoy marks a channel for boat navigation…

Explanation:

Step1: Recall cosine - function property

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

Step2: Set the argument of the cosine function

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

Step3: Solve for $t$

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

Answer:

The points where $f(t)$ is at a minimum are of the form $(8n + 4,-2)$ for $n\in\mathbb{Z}$. On the given non - negative $t$ axis, the points are $(4,-2),(12,-2),(20,-2),\cdots$ (you can plot $(4,-2)$ and $(12,-2)$ within the visible range of the provided graph).