Problem 52 - Precalculus Essentials

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \cos (2 \pi x+4 \pi)$$

Step-by-Step Solution

Answer

The amplitude of the function is 3, the period is 1, and the phase shift is 2 units to the left. By employing these values, one period of the function can be graphed.

Step 1: Determine the Amplitude

The amplitude of the function is given by the absolute value of the coefficient of the cosine function. In this case it is \(|3|\), which is just 3.

Step 2: Calculate the Period

The period of the function is determined by \(2\pi / |B|\). In this case, the 'B' value is \(2\pi\), so period = \(2\pi / |2\pi|\) = 1.

Step 3: Find the Phase Shift

Phase shift is determined by C/B. In this function, 'C' is \(4\pi\) and 'B' is \(2\pi\). Thus, phase shift = \(4\pi / 2\pi\) = 2. This means the function will be shifted 2 units to the left.

Step 4: Graph the Function

To graph one period of the function, a wave with amplitude 3 and period 1 is drawn on a normal x,y graph. The wave is then shifted 2 units to the left to indicate the phase shift. In graphing, note that one period of the wave includes all the points from a crest to the next or from a trough to the next, or from any point to the next corresponding point along the wave.