which of the following is the graph of y = - 3 · cos(3x)?

Explanation:

Step1: Recall cosine - function properties

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

Step2: Determine the amplitude

The amplitude of the function $y = A\cos(Bx)$ is given by $|A|$. Here, $|A|=| - 3|=3$, so the maximum value of the function is $3$ and the minimum value is $-3$.

Step3: Determine the period

The period of the cosine function $y=\cos(Bx)$ is $T=\frac{2\pi}{|B|}$. For $y=-3\cos(3x)$, $B = 3$, so the period $T=\frac{2\pi}{3}$.

The cosine function $y = \cos(x)$ has a maximum value at $x = 0$. The function $y=-3\cos(3x)$ has a minimum value at $x = 0$ (because of the negative sign in front of the cosine). Among the graphs, we look for a graph with an amplitude of 3 and a period of $\frac{2\pi}{3}$ that has a minimum - value at $x = 0$.

Answer:

The graph that has an amplitude of 3 (the distance from the maximum or minimum value to the mid - line $y = 0$ is 3), a period of $\frac{2\pi}{3}$ (the function repeats itself every $\frac{2\pi}{3}$ units along the x - axis), and a minimum value at $x = 0$. Without seeing the specific labels of the options, you should choose the graph that meets these criteria. If we assume the options are labeled as A, B, C, D, you need to visually inspect and pick the one with the correct amplitude, period, and starting point (minimum at $x = 0$).