How do you graph $y=4\csc 2x$?

Hint: First compare the function $y=4\csc 2x$ with the base function. Find the period and frequency of the function. Then take some different ‘y’ values for corresponding ‘x’ values and plot the graph.

Complete step-by-step solution:
$y=4\csc 2x$ is a trigonometric function of the base function $y=\csc x$.
As we know $\csc x$ is the reciprocal of $\sin x$, so $\csc x$ will not be defined at the points where $\sin x$ is 0. Hence, the domain of $\csc x$ will be $R-n\pi $, where ‘R’ is the set of real numbers and ‘n’ is an Integers. Similarly, the range of cosec x will be $R-\left( -1,1 \right)$. Since, $\sin x$ lies between $\left( -1,1 \right)$, so $\csc x$ can never lie in the region of $\left( -1,1 \right)$.
Now, considering our equation $y=4\csc 2x$
Here the range of the function will be 4 times from that of $y=\csc x$
Again as we know the period of $\csc x=2\pi $, so the period of $y=4\csc 2x$ will be$=\dfrac{2\pi }{\left| 2 \right|}=\pi $
And the frequency$=\dfrac{1}{\pi }$ (as frequency is the reciprocal of time period)
For the graph we have to take some different values of ‘y’ for corresponding ‘x’ values

Taking these values of ‘x’ and ‘y’ the graph can be drawn as


Note: The base function of $y=4\csc 2x$ is $y=\csc x$. From the above graph, we can conclude that the graph of the function $\csc x$ does not have a maximum or a minimum value. The function goes to infinity periodically and is symmetric with the origin.