Answer
$\color{blue}{y=2 \cdot \csc{x} - 1}$.
Work Step by Step
The graph looks like that of the basic cosecant function $y=\csc{x}$.
The period of the given function is $2\pi$, the same as that of the basic cosecant function.
This means that the tentative equation of the function whose graph is given is $y=a\cdot csc{x}+c$
The vertices of the two U-shaped curves of the basic cosecant function $y=\csc{x}$, where $a=1$, are $(\frac{\pi}{2}, 1)$ and $(\frac{3\pi}{2}, -1)$. These vertices are equidistant from the line $y=0$.
The distance between these two vertices is 2 units.
In the given graph, the distance between the two vertices is:
$=1-(-3)
\\=1+3
\\=4$
This means that $a= \frac{4}{2}=2$. ($a$ is positive since the graph does not involve a reflection about the x-axis of the basic cosecant function.)
Notice that in the given graph, the vertices are equidistant from the line $y=-1$.
This means that the function involves a $1$-unit shift downwards of the parent function.
Therefore, the equation of the function whose graph is given must be $\color{blue}{y=2 \cdot \csc{x} - 1}$.
