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Chapter 4: Problem 39

Use a graphing utility to graph the function. (Include two full periods.) $$y=\tan \frac{x}{3}$$

Short Answer

Expert verified
The graph of \(y = \tan(\frac{x}{3})\) will look like the basic tangent graph but stretched horizontally by a factor of 3, meaning it takes longer to complete a full oscillation. The asymptotes will be at \(x = 3(\frac{\pi}{2} + n\pi)\) and the period will be \(3\pi\).

Step by step solution

01

Identify Features of Basic Tangent

Identify key features of the basic tangent graph. Here, it's noted that the tangent function has a period of \(\pi\) with asymptotes at the multiples of \(\frac{\pi}{2}\). Also, it's range is \(-\infty\) to \(+\infty\) and there is no amplitude as such because the tangent function never settles or repeats vertically.
02

Apply Modification to The Tangent Function

For \(y = \tan(\frac{x}{3})\), the \(\frac{x}{3}\) element modifies the standard period of the tangent function. Instead of a period of \(\pi\), the new period would be \(3\pi\), because dividing the variable \(x\) by 3 will stretch the basic tangent function horizontally by a factor of 3. Identify that asymptotes now occur at \(x = 3(\frac{\pi}{2} + n\pi)\) where \(n\) is any integer.
03

Graph the Function Using Graphing Tool

Use a graphing tool to graph \(y = \tan(\frac{x}{3})\). Make sure to include at least two full periods in your graph, which, for the modified function, would be between \(-3\pi\) to \(3\pi\). Carefully draw the tangent lines at the asymptotes that are multiples of \(3(\frac{\pi}{2} + n\pi)\), and sketch the curve between those lines, following the basic behavior of the tangent function.

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Most popular questions from this chapter

The normal monthly high temperatures \(H\) (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by $$H(t)=56.94-20.86 \cos \left(\frac{\pi t}{6}\right)-11.58 \sin \left(\frac{\pi t}{6}\right)$$ and the normal monthly low temperatures \(L\) are approximated by $$L(t)=41.80-17.13 \cos \left(\frac{\pi t}{6}\right)-13.39 \sin \left(\frac{\pi t}{6}\right)$$ where \(t\) is the time (in months), with \(t=1\) corresponding to January (see figure). (Source: National Climatic Data Center (GRAPH CANNOT COPY). (a) What is the period of each function? (b) During what part of the year is the difference between the normal high and normal low temperatures greatest? When is it smallest? (c) The sun is northernmost in the sky around June \(21,\) but the graph shows the warmest temperatures at a later date. Approximate the lag time of the temperatures relative to the position of the sun.

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