Problem 37 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The amplitude of the function \(y=4 cos 2 \pi x\) is 4 and its period is \(\frac{1}{2}\).

Step 1: Determine the amplitude

The amplitude of the function is given by the absolute value of the coefficient of the cosine function. Here, the coefficient is 4, so the amplitude is \(|4|=4\).

Step 2: Determine the period

The period can be found by taking the reciprocal of the absolute value of the coefficient of x inside the cosine function, which is \(2 \pi\). So the period is \(\frac{1}{|2 \pi|} = \frac{1}{2}\).

Step 3: Sketch the graph

The standard cosine function starts at a maximum, decreases to a minimum, and then returns to the maximum, which is one complete cycle or period. The amplitude tells us the maximum and minimum values. Here, the maximum is 4 and the minimum is -4. The period tells us where the maximum, minimum, and end of one cycle occur. Here, one cycle ends at \(x = \frac{1}{2}\). Sketch these keypoints and connect them smoothly, reproducing the characteristic 'S'-shape of the cosine graph between the maximum, minimum, and endpoint of the period.