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Chapter 4: Problem 77

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\tan \frac{x}{4}$$

Short Answer

Expert verified
The graph of the function \(y=\tan(x/4)\) shows the periodic nature of the tangent function with a period of \(4\pi\). It repeats after every \(4\pi\) interval when graphed using a viewing rectangle that spans at least two periods (from \(-4\pi\) to \(4\pi\) in x-values).

Step by step solution

01

Understanding the Tangent Function

The tangent function has a period of \(\pi\). So, it repeats after every \(\pi\) interval.
02

Adjusting for the Scale Factor

The given function is \(y=tan(x/4)\). The division by 4 adjusts the period of the function by a factor of 4, so the new period is \(4\pi\). This means the function repeats every \(4\pi\) interval.
03

Setting the Viewing Rectangle

To see at least two periods of the function, the viewing rectangle should cover at least \(8\pi\) in the x-dimension. Therefore, a reasonable window might be from \(-4\pi\) to \(4\pi\) for x-values and from -10 to 10 for y-values, as the tangent function approaches \(\pm\infty\) as x-approaches odd multiples of \(\pi/2\).
04

Using a Graphing Utility

Input the function \(y=tan(x/4)\) into the graphing utility using the viewing window determined in step 3 and sketch the graph.

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