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Chapter 4: Problem 81

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\frac{1}{2} \tan \pi x$$

Short Answer

Expert verified
The function \(y = \frac{1}{2} \tan \pi x\) is a standard tangent function that oscillates between \(-\infty\) and \(+\infty\), with a period of 1. To capture two periods, graphically represent the x-range between -1 and 1, and a convenient y-range can be from -10 to 10. The function exhibits asymptotic behavior at odd multiples of \(\frac{1}{2}\).

Step by step solution

01

Understanding the Function

The function to be graphed is \(y = \frac{1}{2} \tan \pi x\). The tangent function has a period of \(\pi\), but the coefficient of \(x\) inside the function causes the function to 'speed up' by that factor, in effect reducing the period. Here, because \(\pi\) is multiplied by \(x\), the period remains \(1\) (since \(\pi / \pi = 1\)). Further, the amplitude of the function is not affected by a scalar multiple. So, the amplitude of \(y = \frac{1}{2} \tan \pi x\) is the same as that of the basic tan function, i.e., it extends from \(-\infty\) to \(+\infty\).
02

The Viewing Rectangle

Based on the period, to demonstrate at least two periods, the viewing rectangle's x-range should span at least two units (since each period is 1 unit). And in y-direction, as the range of the function is \(-\infty\) to \(+\infty\), one may choose the range to clearly show the function's periodic behavior, e.g., from -10 to 10.
03

Graphing the Function

Using a graphing tool, track the function \(y = \frac{1}{2} \tan \pi x\), with the viewing rectangle as determined in step 2. Our x-range is between -1 and 1, to clearly show two periods of the function, and the y-range is from -10 to 10 to capture the large range of the function. Note that the function has vertical asymptotes, or lines that the function approaches but never reaches, at the points where the cosine function (the denominator of the tangent function) equals zero, i.e., at \(x = \frac{1}{2}\), \(\frac{3}{2}\), etc.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. a. Graph \(y=\cos x\) for \(0 \leq x \leq \pi\) b. Based on your graph in part (a), does \(y=\cos x\) have an inverse function if the domain is restricted to \([0, \pi] ?\) Explain your answer. c. Determine the angle in the interval \([0, \pi]\) whose cosine is \(-\frac{\sqrt{3}}{2} .\) Identify this information as a point on your graph in part (a).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Determine the range of each of the following functions. Then give a viewing rectangle, or window, that shows two periods of the function's graph. a. \(f(x)=3 \sin \left(x+\frac{\pi}{6}\right)-2\) b. \(g(x)=\sin 3\left(x+\frac{\pi}{6}\right)-2\)

Use a graphing utility to graph two periods of the function. $$y=0.2 \sin \left(\frac{\pi}{10} x+\pi\right)$$

Use a graphing utility to graph two periods of the function. $$y=3 \sin (2 x-\pi)+5$$

Use a vertical shift to graph one period of the function. $$y=2 \sin \frac{1}{2} x+1$$
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