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Chapter 4: Problem 39

Sketch one full period of the graph of each function. $$y=3 \tan 2 \pi x$$

Short Answer

Expert verified
The graph of \(y = 3 \tan 2 \pi x\) is a vertically stretched tangent function with a period of 0.5. Its graph has asymptotes at the beginning and end of each period.

Step by step solution

01

Identify the amplitude

The amplitude for a tangent function isn't clearly defined as it ranges from \(-\infty\) to \(\infty\). However, the coefficient of the function is a vertical stretch factor. In this case, that factor is 3. This means that the graph will reach higher and lower, at a quicker rate, than the standard tangent function.
02

Identify the period

The period of a standard tangent function is \(\pi\). The period is affected by horizontal stretches or compressions. In this equation, the horizontal compression factor is \(2 \pi\). To find the new period, divide the standard period by this factor. Hence, the period of \(y = 3 \tan 2 \pi x\) is \(\pi / (2 \pi) = 0.5\). This means the function completes one full cycle over an interval of 0.5.
03

Identify asymptotes

Asymptotes for the tangent function occur where the function is undefined. For a standard tangent function, this is at odd multiples of \(\pi / 2\). This is altered by the horizontal stretch/compression, resulting in asymptotes at \(0.5 \times \)odd integer multiples in this case.
04

Sketch the Graph

Begin by drawing the asymptotes at \(x = 0\) and \(x = 0.5\) as the boundaries of the period. Next, plot the middle of these to be an x-intercept, at \(x = 0.25\). Finally, because of the vertical stretch factor of 3, the graph will reach its peak and valley more quickly than the standard function. Connect the points with a curve that approaches but never reaches the asymptotes.

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