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Chapter 4: Problem 68

Use a graphing utility to graph the function. (Include two full periods.) Be sure to choose an appropriate viewing window. $$y=-4 \sin \left(\frac{2}{3} x-\frac{\pi}{3}\right)$$

Short Answer

Expert verified
The function \(y=-4 \sin \left(\frac{2}{3} x-\frac{\pi}{3}\right)\) is graphed on the viewing window \([-2\pi, 8\pi] \times [-5,5]\). There are two complete periods visible in this graph.

Step by step solution

01

Identify Key Aspects

The given function is \(y=-4 \sin \left(\frac{2}{3} x-\frac{\pi}{3}\right)\). Here, the amplitude is |-4| = 4, the period is \(\frac{2\pi}{|2/3|} = 3\pi\), the phase shift is \(\frac{\pi}{3}\) to the right, and there is no vertical shift since nothing is added or subtracted outside the sine function.
02

Choose an Appropriate Viewing Window

An appropriate window will include two full periods. With a period of \(3\pi\), two periods will have a total length of \(6\pi\). For the y-values, use the amplitude, which is 4, for the maximum and minimum values since the function ranges from -4 to 4. Also, include some space above and below this range for clarity. Thus, choose the viewing window as \([-2\pi, 8\pi] \times [-5,5]\). This includes the phase shift to the right of \(\frac{\pi}{3}\) and two complete periods.
03

Graph the Function

Now, using the graphing utility, graph the function \(y=-4 \sin \left(\frac{2}{3} x-\frac{\pi}{3}\right)\) in the viewing window chosen in the previous step.

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