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Chapter 4: Problem 32

Sketch the graph of the function. (Include two full periods.) $$y=\tan (x+\pi)$$

Short Answer

Expert verified
The graph of the function \(y = \tan(x + \pi)\) is equivalent to that of the standard tangent function \(y = \tan(x)\), but shifted \(-\pi\) units to the left, with the vertical asymptotes at \(x = n\pi - \frac{\pi}{2}\), where n is an integer. This includes two full periods of the function.

Step by step solution

01

Understand the Standard Tangent Function

Start by recognizing the standard tangent function, \(y = \tan(x)\). The tangent function is periodic with a period of \(\pi\). Its graph consists of a series of vertical asymptotes, located at \(x = n\pi + \frac{\pi}{2}\), where n is an integer, between which the graph oscillates between negative and positive infinity.
02

Identify the Phase Shift

Then, identify the phase shift in the given function. In a general function of the form \(y = \tan(ax+b)\), the term \(b\) represents the phase shift. In this case, the function is \(y = \tan(x + \pi)\), so there is a phase shift of \(-\pi\) to the left.
03

Sketch the Tan Function with Phase Shift

Lastly, sketch the graph of the function. Start by sketching the standard \(y = \tan(x)\) function for reference. Then apply the phase shift. Move every point on the standard graph and the vertical asymptotes \(-\pi\) units to the left. This results in the vertical asymptotes being at \(x = n\pi - \frac{\pi}{2}\), where n is an integer.

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