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 properties

The general form of a cosine function is $y = A\cos(Bx - C)+D$. For the function $f(t)=2\cos(\frac{\pi}{4}t)$, $A = 2$, $B=\frac{\pi}{4}$, $C = 0$, $D = 0$. The range of the cosine function $y=\cos(x)$ is $[- 1,1]$. So the range of $y = 2\cos(\frac{\pi}{4}t)$ is $[-2,2]$. The minimum value of $y = 2\cos(\frac{\pi}{4}t)$ is $-2$.

Step2: Find when the cosine - function reaches its minimum

We know that $\cos(x)=-1$ when $x=(2n + 1)\pi$, where $n$ is an integer. Set $\frac{\pi}{4}t=(2n + 1)\pi$. Solve for $t$: [ \begin{align*} \frac{\pi}{4}t&=(2n + 1)\pi\ t&=4(2n + 1)\ t&=8n+4 \end{align*} ] When $n = 0$, $t = 4$; when $n = 1$, $t=12$; when $n=-1$, $t=-4$ (but since $t$ represents time, we usually consider non - negative values in this context).

Answer:

The points where the height is at a minimum are $(4,-2)$ and $(12,-2)$ (and in general, $(8n + 4,-2)$ for non - negative integers $n$). On the given graph, you would plot the points $(4,-2)$ and $(12,-2)$.