# Chapter 4 - Graphs of the Circular Functions - Section 4.3 Graphs of the Tangent and Cotangent Functions - 4.3 Exercises - Page 171: 30 RECALL: (1) The function $y=c+ a\cdot \tan{x}$ has: period = $\pi$ vertical shift = $|c|$, (upward when $c\gt 0$, downward when $c\lt0$) If $a \lt 0$, th graph of the tangent function will be a reflection about the x-axis of the parent function $y=\tan{x}$ (2) The consecutive vertical asymptotes of the function $y=\tan{x}$, are $x=-\frac{\pi}{2}$ and $x=\frac{\pi}{2}$. Thus, with $a=-1$, the graph of the given function involves a refection about the x-axis of the parent function. With $c=1$, there will be a $1$-unit upward shift of the reflected graph of the parent function $y=\tan{x}$. One period of the given function is in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$, so the next period is in the interval $[\frac{\pi}{2}, \frac{3\pi}{2}]$. Divide each of these intervals into four equal parts to obtain the following key x-values: $-\frac{\pi}{4}, 0, \frac{\pi}{4}, \frac{3\pi}{4}, \pi, \text{ and } \frac{5\pi}{4}$ To graph the given function, perform the following steps: (1) Create a table of values for the given function using the key x-values listed above. (Refer to the attached image table below.) (2) Graph the consecutive vertical asymptotes. (3) Plot each point from the table then connect them using a smooth curve, making sure that the curves are asymptotic with the lines in Step (2) above. Refer to the graph in the answer part above. 