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Chapter 5: Problem 41

Determine the amplitude and period of each function. Then graph one period of the function. $$y=-\frac{1}{2} \cos \frac{\pi}{3} x$$

Short Answer

Expert verified
The amplitude of the function \(y=鈭抃frac{1}{2} \cos \frac{\pi}{3}x\) is 1/2 and the period is 6.

Step by step solution

01

Identify parameters of the function

From the given function \(y=鈭抃frac{1}{2} \cos \frac{\pi}{3}x\), the amplitude A is the coefficient of cosine, so |A| = |鈭�1/2| = 1/2. The period parameter B is \(\frac{\pi}{3}\), which affects the period of the function. The function has no shifts, since there is no constant being added or subtracted inside or outside the cosine function.
02

Calculate the period

The period of a cosine function in general form is given by \(T = \frac{2\pi}{|B|}\). For our function, the absolute value of B is \(\frac{\pi}{3}\). Substituting this into the formula, we get the period T = \(\frac{2\pi}{|\frac{\pi}{3}|} = 3 * 2 = 6\). So the period of the function is 6.
03

Plot the graph

To graph one period of the function, use the calculated period and amplitude. The function is a cosine function so it starts at its maximum value. The maximum and minimum values will be |A| and -|A| respectively. Thus, the function starts at 1/2 (because A = -1/2), decreases to -1/2, and then increases back to 1/2, completing one period from x = 0 to x = 6. Because A is negative, the cosine function is reflected about the x-axis.

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Most popular questions from this chapter

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