Problem 41 - Precalculus Essentials

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 \(\frac{1}{2}\) and its period is 6 units. The function is a reflected cosine function with a completed period at x=6.

Step 1: Find the Amplitude

The amplitude of the function is the absolute value of the coefficient in front of the cosine function. Therefore, the amplitude of the function \(y=-\frac{1}{2} \cos \frac{\pi}{3} x\) is \( |-\frac{1}{2}| = \frac{1}{2} \).

Step 2: Find the period

The period of the function is obtained by dividing \(2 \pi\) by the absolute value of the coefficient in front of the variable 'x'. Therefore, the period of the function \(y=-\frac{1}{2} \cos \frac{\pi}{3} x\) is \( \frac{2 \pi}{\frac{\pi}{3}} = 2\pi \cdot \frac{3}{\pi} = 6 \).

Step 3: Graph the function

This function behaves like the standard cosine function with some transformations. The amplitude modifies the height and the period modifies the length of the function's cycle. Because the coefficient of the cosine function is negative, the function is reflected over the x-axis. The completed period ends at x=6 instead of \(2 \pi\). This means the function starts at (0, -0.5), goes up to (1.5, 0.5), comes back down to (3, -0.5), goes up to (4.5, 0.5) and finally returns back to (6, -0.5) to complete the cycle.