Question

graph the function over a two - period interval.
y = cot(9x - π)
which graph below shows two periods of the function?
○ a.
○ b.
○ c.
○ d.

Explanation:

Step1: Recall the period formula for cotangent function

The general form of a cotangent function is $y = A\cot(Bx - C)+D$, and its period is given by $T=\frac{\pi}{|B|}$. For the function $y = \cot(9x-\pi)$, $B = 9$, so the period $T=\frac{\pi}{9}$.

Step2: Find the two - period interval

The two - period interval has a length of $2T$. Since $T=\frac{\pi}{9}$, the length of the two - period interval is $\frac{2\pi}{9}$.
We can find the vertical asymptotes of $y=\cot(9x - \pi)$ by setting $9x-\pi=k\pi$, where $k\in\mathbb{Z}$. Solving for $x$ gives $x=\frac{k\pi+\pi}{9}=\frac{(k + 1)\pi}{9}$.
When $k = 0$, $x=\frac{\pi}{9}$; when $k=- 1$, $x = 0$.
The cotangent function $y=\cot(9x-\pi)$ has the same shape as the basic cotangent function $y = \cot(x)$ but is horizontally compressed by a factor of $\frac{1}{9}$ and shifted to the right by $\frac{\pi}{9}$ units.
The basic cotangent function $y=\cot(x)$ has a graph that decreases from $0$ to $\pi$ between consecutive vertical asymptotes. For $y=\cot(9x - \pi)$, between consecutive vertical asymptotes $x=\frac{k\pi}{9}$ and $x=\frac{(k + 1)\pi}{9}$, the function is decreasing.

Answer:

We need to analyze the key features such as vertical asymptotes and the behavior of the function (increasing or decreasing) between them. Without seeing the actual graphs A, B, C, D in detail, we know that the vertical asymptotes of $y=\cot(9x-\pi)$ are at $x=\frac{k\pi}{9},k\in\mathbb{Z}$ and the function is decreasing between consecutive vertical asymptotes. You would choose the graph that has vertical asymptotes at $x = 0,x=\frac{\pi}{9},x=\frac{2\pi}{9}$ and is decreasing between these asymptotes for a two - period interval.