Answer
$D$
Work Step by Step
RECALL:
The function $y=\cot{x}$ has:
(1) a period of $\pi$;
(2) the vertical lines $x=n\pi$ (where $n$ is an integer) as its vertical asymptotes (e.g., $x=-\pi, 0, \pi, ...$).
(3) a graph that is decreasing from left to right.
Thus, the only possible graphs of $y=\cot{(x-\frac{\pi}{4})}$ are the ones in options C, D, and F.
RECALL:
The function $y=\cot{(x−d)}$ involves a horizontal (phase) shift of $|d|$ units of the parent function $y=\cot{x}$. The shift is to the right when $d\gt0$ and to the left when $d\lt0$.
This means that the given function, with $d=\frac{\pi}{4}$, involves a $\frac{\pi}{4}$-unit shift to the right.
Thus, two consecutive vertical asymptotes of the given function are:
$x=0+\frac{\pi}{4}=\frac{\pi}{4}$ and $x=\pi + \frac{\pi}{4}=\frac{5\pi}{4}$
The only graph that has these vertical asymptotes is the one in Option $D$.
