# Chapter 4 - Graphs of the Circular Functions - Section 4.3 Graphs of the Tangent and Cotangent Functions - 4.3 Exercises - Page 166: 4

$D$

#### 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 $D$.

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