Problem 38 - Precalculus Essentials

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=5 \cos 2 \pi x$$

Step-by-Step Solution

Answer

The amplitude is 5 and the period is 1. The plot of one period of the function \(y=5\cos(2\pi x)\) is a wave starting at (0,5) and ending up at (1,5) with the lowest point at (0.5, -5).

Step 1: Identify the amplitude

The amplitude of the function is the absolute value of \(A\) in the standard form. In this function, the amplitude \(A=5\). Amplitude represents the maximum distance from the midline or the highest peak or the lowest point of the wave, so in this function, this distance is 5 units.

Step 2: Identify the period

The period of the function is given by \(2\pi /B\) in the standard form. In this function, \(B=2\pi\). Therefore, the period is calculated as follows: \(2\pi/2\pi = 1\). So, the period of this function is 1 unit.

Step 3: Plot the function

When plotting the function, you would start at (0,5) because the cosine of 0 is 1, and the amplitude is 5. Familiar points of a cosine function should be labeled: (0,5), (0.25,0), (0.5,-5), (0.75,0), and (1,5). These points complete one period of the function.