Answer
Refer to the graph below.
Work Step by Step
RECALL:
The function $y=c+\csc{x}$ has:
period = $\pi$
vertical shift= $|c|$, (upward when $c \gt0$, downward when $c\lt0$)
consecutive vertical asymptotes: $x=0, \pi, $ and $2\pi$
The given function has $c=-1$.
Thus, it has:
period = $2\pi$
vertical shift = $|-1|=1$
One period of this function is in the interval $[0, 2\pi]$.
Divide this interval into four equal parts to obtain the key x-values $\frac{\pi}{2}, \pi, \frac{3\pi}{2}$.
Note that $\pi$ is a vertical asymptote of the function.
To graph the given function, perform the following steps:
(1) Create a table of values using the key x-values listed above. (Refer to the table below.)
(2) Graph the vertical asymptotes listed above.
(3) Draw a U-shaped curve between the asymptotes $x=0$ and $x=\pi$ whose vertex is at $(\frac{\pi}{2}, 0)$ . Draw an inverted-U-shaped curve between the asymptotes $x=\pi$ and $2\pi$ whose vertex is at $(\frac{3\pi}{2}, -2)$.
(Refer to the graph in the answer part above.)
