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Chapter 4: Problem 43

Use a graphing utility to graph the function. (Include two full periods.) $$y=\tan \left(x-\frac{\pi}{4}\right)$$

Short Answer

Expert verified
The graph of the function \(y=\tan \left(x-\frac{\pi}{4}\right)\) is a graph of the tangent function shifted to the right by \(\frac{\pi}{4}\) units. The function has a period of \(\pi\) and the graph includes two full periods from \(x = 0\) to \(x = 2\pi\).

Step by step solution

01

Understanding the Given Function

The given function is \(y=\tan \left(x-\frac{\pi}{4}\right)\), which is a transformation of the tangent function. The \(-\frac{\pi}{4}\) part translates our original function to the side by \(\frac{\pi}{4}\) units. This results in shifting the function to the right by \(\frac{\pi}{4}\) units. A positive shift to \(x\) in the argument of the function shifts the graph to the left, whereas a negative shift to \(x\) in the argument of the function shifts it to the right.
02

Identifying the Period

The standard period of the tangent function is \(\pi\). In the given function, the coefficient of the angle is 1, which means the period remains the same, that is \(\pi\). Since the exercise asks for two full periods, this will be twice the standard period, i.e., \(2\pi\).
03

Plotting the Function

Use your graphing utility to graph the function. Start at \(x = 0\) and end at \(x = 2\pi\). Because of our shift, the vertical asymptotes of our function will now be at \(x=\frac{\pi}{4}\) and \(x=\frac{5\pi}{4}\). The graph should show two periods of the tangent function, displaced to the right by \(\frac{\pi}{4}\) units.
04

Checking the Function

To check whether the graph has been drawn correctly, plug some values of \(x\) into the function and check if they match with your graphed function. For instance for \(x=\frac{\pi}{4}\), \(y\) would be \(\tan{(0)} = 0\) and for \(x=\frac{5\pi}{4}\), \(y\) would be \(\tan{(\pi)} = 0\).

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