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Chapter 4: Problem 31

Sketch the graph of the function. (Include two full periods.) $$y=\tan \frac{\pi x}{4}$$

Short Answer

Expert verified
The tan function \(y=\tan \frac{\pi x}{4}\) has a period of 4, asymptotes at every odd multiple of 2, and key points at -2, 0, and 2. The function starts at origin (0,0), goes through key points, and approaches the asymptotes in the positive and negative directions.

Step by step solution

01

Identify the period of the function

The period of the standard y = tan x function is \(\pi\). If you multiply the x by a factor, it will change the period of the function. In the given function \(y=\tan \frac{\pi x}{4}\), x is multiplied by \(\frac{\pi}{4}\). This means the period of the function will be \(\frac{4\pi}{\pi}\) or 4.
02

Identify asymptotes

The tan function has vertical asymptotes where the function is not defined, in this case, on every odd multiple of \(\frac{\pi}{2}\). However, because of horizontal compression by \(\frac{1}{4}\), the new asymptotes will be at every odd multiple of \(\frac{4}{2}\), or every odd multiple of 2.
03

Identify Key Points

The key points for one cycle of standard y = tan x function occur at intervals of \(\frac{\pi}{2}\) and they are: (-\(\frac{\pi}{2}\),undefined), (0,0) and (\(\frac{\pi}{2}\), undefined). For the function \(y=\tan \frac{\pi x}{4}\), horizontal compression affects these points as well. They become (-2, undefined), (0,0) and (2, undefined).
04

Sketch the function

From the above information, begin plotting the function. First, draw the asymptotes at x = -2 and x = 2. Then draw the point at the origin (0,0). At points left of -2 and right of 2 (i.e., at -3 and 3), the function approaches 0. So, draw a tan curve between the two asymptotes. Now, since period is 4, the same pattern is repeated for every interval of 4. In this way, continue to extend the graph to sketch two full periods.

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