Question

graph the function over a one - period interval. y = 3 cot 3x choose the correct graph.

Explanation:

Step1: Find the period of the cotangent function

The period of the cotangent function $y = A\cot(Bx)$ is given by $\frac{\pi}{|B|}$. For the function $y = 3\cot(3x)$, $B = 3$, so the period is $\frac{\pi}{3}$.

Step2: Identify the vertical - asymptotes

The vertical asymptotes of $y=\cot(x)$ occur at $x = n\pi$, $n\in\mathbb{Z}$. For $y = 3\cot(3x)$, the vertical asymptotes occur when $3x=n\pi$, or $x=\frac{n\pi}{3}$, $n\in\mathbb{Z}$. In the one - period interval, we can consider the interval $(0,\frac{\pi}{3})$. When $x = 0$, $\cot(3x)$ is not defined, and as $x$ approaches $0$ from the right, $\cot(3x)\to+\infty$, and as $x$ approaches $\frac{\pi}{3}$ from the left, $\cot(3x)\to-\infty$. Also, when $x=\frac{\pi}{6}$, $y = 3\cot(\frac{\pi}{2})=0$.

Step3: Analyze the behavior of the function

The coefficient $A = 3$ stretches the graph of $y=\cot(3x)$ vertically by a factor of 3. The graph of $y = 3\cot(3x)$ is a vertically stretched version of the basic cotangent function $y=\cot(3x)$ with a period of $\frac{\pi}{3}$ and vertical asymptotes at $x = 0$ and $x=\frac{\pi}{3}$ and passing through the point $(\frac{\pi}{6},0)$.

Answer:

C.