Problem 41 - Precalculus Essentials

Question

Graph two periods of the given cosecant or secant function. $$y=\csc (x-\pi)$$

Step-by-Step Solution

Answer

The graph of the function \(y = \csc{(x-\pi)}\) starts at \(x=\pi\) and ends at \(x=\pi + 2\pi\) for the first period, and it continues to the second period, repeating the pattern. Vertical asymptotes are observed where the function is undefined, that is at the points where the sine function is 0.

Step 1: Understanding the Shifted Cosecant Function

The given function is \(y = \csc{(x-\pi)}\). This is a cosecant function that has been shifted \(\pi\) units to the right from the original cosecant function. After the shift, the period still remains the same which is \(2\pi\), but the function now starts at \(x = \pi\).

Step 2: Plotting the Sine Function

To graph the shifted cosecant function, it helps to initially sketch the graph of \(y=\sin(x-\pi)\). The sine function has a period of \(2\pi\) and it starts at \(x=\pi\), reaches its maximum of 1 at \(x=\pi + \pi/2\), then goes back to 0 at \(x=\pi + \pi\), reaches its minimum of -1 at \(x=\pi + 3\pi/2\), and returns to 0 at \(x=\pi + 2\pi\). Repeat this pattern to get two periods for this function.

Step 3: Graphing the Cosecant Function

Now, to get the graph of the cosecant function, take the reciprocal of the sine function at each x value, excluding the x values where the sine function equals 0. The cosecant function reaches positive and negative infinity as the sine function approaches 0. Also, since \(\csc{x}\) is undefined where \(\sin{x} = 0\), draw vertical asymptotes at these points.