graph y = -3 cos (π/6 x) - 6 in the interactive widget. note that one moveable point always defines an…

Explanation:

Step1: Identify the general form of cosine - function

The general form of a cosine function is $y = A\cos(Bx - C)+D$. For the function $y=-3\cos(\frac{\pi}{6}x)-6$, we have $A = - 3$, $B=\frac{\pi}{6}$, $C = 0$, and $D=-6$.

Step2: Find the amplitude

The amplitude is $|A|$. Since $A=-3$, the amplitude $|A| = 3$.

Step3: Find the period

The period of the cosine - function $y = A\cos(Bx - C)+D$ is given by $T=\frac{2\pi}{|B|}$. Here, $B=\frac{\pi}{6}$, so $T=\frac{2\pi}{\frac{\pi}{6}}=12$.

Step4: Find the mid - line

The mid - line of the function is given by $y = D$. Since $D=-6$, the mid - line is $y=-6$.

Step5: Find the maximum and minimum values

The maximum value of $y = A\cos(Bx - C)+D$ is $y_{max}=|A|+D$. Substituting $|A| = 3$ and $D=-6$, we get $y_{max}=3+( - 6)=-3$. The minimum value is $y_{min}=-|A|+D=-3+( - 6)=-9$.

Step6: Plot key points

We know that for $y = \cos(x)$, the key points in one period are $(0,1),(\frac{\pi}{2},0),(\pi,-1),(\frac{3\pi}{2},0),(2\pi,1)$. For $y=-3\cos(\frac{\pi}{6}x)-6$, when $x = 0$, $y=-3\cos(0)-6=-3 - 6=-9$; when $x = 3$, $y=-3\cos(\frac{\pi}{2})-6=-6$; when $x = 6$, $y=-3\cos(\pi)-6=3 - 6=-3$; when $x = 9$, $y=-3\cos(\frac{3\pi}{2})-6=-6$; when $x = 12$, $y=-3\cos(2\pi)-6=-3 - 6=-9$.

Answer:

To graph the function $y=-3\cos(\frac{\pi}{6}x)-6$, first draw the mid - line $y = - 6$. The amplitude is 3, so the maximum value of the function is $y=-3$ and the minimum value is $y=-9$. The period is 12. Plot key points such as $(0, - 9),(3,-6),(6,-3),(9,-6),(12,-9)$ and then connect them with a smooth cosine - like curve.