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Chapter 4: Problem 42

Determine the amplitude and period of each function. Then graph one period of the function. $$y=-\frac{1}{2} \cos \frac{\pi}{4} x$$

Short Answer

Expert verified
The amplitude of the function is \( \frac{1}{2} \) and the period is 8.

Step by step solution

01

Determining the Amplitude

The amplitude of function \(y=-\frac{1}{2} \cos \frac{\pi}{4} x\) is obtained from vertical stretching or compressing factor. In this case it is \(-\frac{1}{2}\). The amplitude is always positive, hence the amplitude is \(\frac{1}{2}\).
02

Determining the Period

The period of a cosine function is \( \frac{2\pi}{|B|} \) where B is the horizontal stretch or compression factor. In our function, it is \(\frac{\pi}{4}\). So, the period is \( \frac{2\pi}{|\frac{\pi}{4}|} = 8 \).
03

Graphing the function

When graphing, it's key to remember that the period will determine the width of the wave, and the amplitude its height. The Cosine wave will start at the maximum value (amplitude) since there's no phase shift, but this value is negative due to the multiplying factor. The wave will then drop down to the minimum value (negative amplitude), oscillating from top downwards, over the interval of its period, which is 8 in this case.

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Most popular questions from this chapter

will help you prepare for the material covered in the next section. a. Graph \(y=-3 \cos \frac{x}{2}\) for \(-\pi \leq x \leq 5 \pi\) b. Consider the reciprocal function of \(y=-3 \cos \frac{x}{2}\) namely, \(y=-3 \sec \frac{x}{2} .\) What does your graph from part (a) indicate about this reciprocal function for \(x=-\pi, \pi, 3 \pi,\) and \(5 \pi ?\)

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=-2.5 \sin \frac{\pi}{3} x \text { and } y=-2.5 \csc \frac{\pi}{3} x$$

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=0.8 \sin \frac{x}{2} \text { and } y=0.8 \csc \frac{x}{2}$$

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=2 \cos (2 \pi x+8 \pi)$$

Use a graphing utility to graph two periods of the function. $$y=-2 \cos \left(2 \pi x-\frac{\pi}{2}\right)$$
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