Problem 36 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The amplitude of the function \(y=\cos 4x\) is 1, and the period is \(\pi/2\). The function is graphed over one period from 0 to \(\pi/2\), starting at the maximum amplitude, crossing the x-axis at \(\pi/8\) and reaching the minimum amplitude at \(\pi/4\).

Step 1: Determine the Amplitude

The amplitude of a cosine function is given by the absolute value of A, in the general form of the function \(y=A\cos(Bx+C)+D\). From the function \(y=\cos 4x\), the coefficient of the cosine function (A) is 1. Thus, the amplitude is \(|1|=1\).

Step 2: Determine the Period

To find the period of the cosine function, we use the formula \(T=2\pi/B\), where B is the coefficient of the argument of the cosine function. Here in the function \(y=\cos 4x\), B is 4. Therefore, substituting B = 4 into the formula, the period is \(T=2\pi/4=\pi/2\).

Step 3: Graph the Function

Plotting the function over one period: Start by drawing the x-axis from 0 to \(\pi/2\). Then, plot the points (0,1), \(\pi/8\), 0 and \(\pi/4\), -1, denoting the maximum, zero, and minimum points of the cosine function in a period respectively. Finally, connect the points with a smooth curve, noting that the cosine function starts from the peak, goes through zero at \(\pi/8\), and ends at the minimum point at \(\pi/4\).