Problem 41 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify parameters of the function

From the given function \(y=−\frac{1}{2} \cos \frac{\pi}{3}x\), the amplitude A is the coefficient of cosine, so |A| = |−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.

Step 2: 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.

Step 3: 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.