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Chapter 4: Problem 28

Sketch one full period of the graph of each function. $$y=2 \csc x$$

Short Answer

Expert verified
The graph of the function \(y=2 \csc x\) is undefined at integer multiples of \(\pi\), has maximum values of 2 at every integer multiple of \(2\pi\), and minimum values of -2 at every odd multiple of \(\pi\). It repeats its steep, U-shaped curve between each of these points in a cycle of \(2\pi\). Due to being the reciprocal of the sine function, its behavior is opposite, with extrema where the sine function crosses the x-axis, and asymptotes where the sine function has its maximum and minimum.

Step by step solution

01

Plot the points of \(y=2 \sin x\)

The cosecant function is the reciprocal of the sine function. So, we first need to plot the graph of the function \(y=2 \sin x\). This will help in plotting the graph of the cosecant function. The sine function has a period of \(2\pi\), crosses the x-axis at integer multiples of \(\pi\), has a maximum value of 2 at \(\pi/2\) and a minimum value of -2 at \(3\pi/2\). We plot these points.
02

Plot the graph of \(y=2 \csc x\) using the reciprocal values

Next, we plot the graph of the function \(y=2 \csc x\) by taking the reciprocal of the plotted sine values. This means the graph will be undefined at places where the sine function crosses the x-axis, i.e., at integer multiples of \(\pi\). The minimum values from the sine graph will become maximum values in the cosecant graph and vice versa. Plot these points on the graph.
03

Sketch the Graph

Finally, sketch the graph by connecting the plotted points for the function \(y=2 \csc x\). Remember to include vertical asymptotes at integer multiples of \(\pi\) where the function is undefined.

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