Answer
Refer to the graph below.
Work Step by Step
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.
