Problem 40 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Determining the Amplitude

The amplitude is given by the absolute value of the coefficient of the cosine function. Hence, the amplitude for the function \(y=-3 \cos \frac{1}{3} x\) is \(|-3|\), so the amplitude is 3.

Step 2: Finding the Period

The period can be found using the formula \(2\pi / B\). Here, \(B = \frac{1}{3}\). Thus, the period of the function is \(2\pi / B = 2\pi / (\frac{1}{3}) = 6\pi\).

Step 3: Graphing the Function

When creating the graph, the amplitude determines the peak and trough of the wave (in this case it will reach 3 and -3 units). The distance between two peaks (or troughs) can be determined by the period, which is found to be \(6\pi\) for this function. Since it is a negative cosine function, it will start from the bottom of the wave (-3), rather than the top. Plot these details on a graph to represent the function \(y=-3 \cos \frac{1}{3} x\).