Problem 35 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The amplitude of the function \(y = \cos 2x\) is 1 and the period is \(\pi\). The graph of one period of the function starts at its maximum value of 1 at x=0, goes to its minimum value of -1 at x=\(\pi/2\), then returns to 1 at x=\(\pi\).

Step 1: Identify the Amplitude

Amplitude is the distance from the function's maximum or minimum value to its midline. For the cosine function, the amplitude is usually 1, as the function oscillates between -1 and 1. In our specific function, which is \(y = \cos 2x\), there is no coefficient before the cosine, and therefore, the amplitude is 1.

Step 2: Identify the Period

The period of a function is the distance over which the function repeats itself. The standard period of the cosine function, \(y = cos(x)\), is \(2\pi\). Since the function in this exercise is \(y = \cos 2x\), the frequency of cosine has doubled, and hence the period is \(\pi\).

Step 3: Draw the Graph

To draw one period of the function, mark the x-axis from 0 to \(\pi\). Since the amplitude is 1, the function will oscillate between -1 and 1 on the y-axis. Using the periodic and oscillatory properties of the cosine function, draw a curve starting at its maximum value (1) at x=0, fully going down to its minimum value (-1) at x=\(\pi/2\), and back at 1 at x=\(\pi\).