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Chapter 4: Problem 34

Sketch the graph of the function. (Include two full periods.) $$y=\csc (2 x-\pi)$$

Short Answer

Expert verified
The sketch of the given function \(y=\csc (2 x-\pi)\) will include two full periods where the graph has vertical asymptotes at \(x=\pi/2 + n*\pi\) where \(n=0,卤1,卤2,...\), and the maxima and minima appear at y=-1 and y=1 respectively.

Step by step solution

01

Identify Period, Shift, and Asymptotes

First, recognize the period and phase shift of the function. The period of the parent function \(\csc(x)\) is \(2\pi\), broader than the sine or cosine functions. However, in this case, b=2, which adjusts the period to \(2\pi/|2|= \pi\). The function is phase-shifted to the right by \(c/b = \pi/2\). This means that there will be vertical asymptotes at \(x=\pi/2 + n*\pi\) where \(n=0,卤1,卤2,...\). The period is important for completing two full periods of the function and the phase shift and asymptotes will help in sketching it.
02

Sketch Vertical Asymptotes

Start sketching the function by drawing the vertical asymptotes at \(x=\pi/2 + n*\pi\) where \(n=0,卤1,卤2,...\). These are vertical lines where the function will be undefined.
03

Plot the Maximum and Minimum Values

Next, calculate and plot the maximum and minimum values for two full periods of the function. Since the range of the function is \(-\infty \lt y \leq -1\) and \(1 \leq y < \infty\), those y-values of -1 and 1 will be the maximum and minimum values. Those points are located halfway between the asymptotes.
04

Connect the points with Curves

Now, connect the points with smooth curves that approach but do not touch the asymptotes.
05

Repeat for the Second Period

Then, repeat the steps for the second period of the function, maintaining the same pattern of curves between asymptotes.

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