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Chapter 4: Problem 5

Graph two periods of the given tangent function. $$y=3 \tan \frac{x}{4}$$

Short Answer

Expert verified
The graph of \(y=3 \tan \frac{x}{4}\) consists of vertically stretched tangent waves with a period of \(4\pi\), with asymptotes at \(x=2\pi + 4k\pi\). Two full periods of this function would extend from \(x=0\) to \(x=8\pi\).

Step by step solution

01

Determine the period

In the tangent function \(y = a \tan(bx)\), the standard period of function \(y = \tan(x)\) is \( \pi \). In given function \(y=3 \tan \frac{x}{4}\), the coefficient on x is 1/4, so the period of the function is \(4\pi\).
02

Mark period intervals

On the x-axis, mark intervals that represent the period of your tangent function. It will be most effective to start at x=0 and mark off intervals of \(4\pi\), continuing for two periods.
03

Plot key points

For the function \(y = \tan(x)\), \(y=\infty\) at \(x=\frac{\pi}{2} + k\pi\) and \(y=-\infty\) at \(x=-\frac{\pi}{2} + k\pi\), where k is an integer. Hence, for the function \(y=3 \tan \frac{x}{4}\), \(y=\infty\) at \(x=2\pi + 4k\pi\) and \(y=-\infty\) at \(x=-2\pi + 4k\pi\) . Plot these key points on your graph.
04

Draw the graph

Connect the points using smooth, continuous curves to form two periods of the tangent function. The tangent function has asymptotes at \(x=2\pi + 4k\pi\) which should also be included on the graph.

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Most popular questions from this chapter

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