Problem 51 - Precalculus Essentials

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=2 \cos (2 \pi x+8 \pi)$$

Step-by-Step Solution

Answer

The amplitude of the function \(y = 2 \cos(2 \pi x + 8 \pi)\) is 2, the period is 1, and the phase shift is -4.

Step 1: Determine the Amplitude

The amplitude of a trigonometric function is the absolute value of the coefficient of the trig function. Given \( y = 2 \cos(2 \pi x + 8 \pi) \), the coefficient of the cosine function is 2. Therefore, the amplitude of the function is \( |2| = 2 \).

Step 2: Determine the Period

The period of a cosine function is determined by the coefficient B of x inside the cosine function. It is calculated as \((2\pi) / |B|\). For the given function \( y = 2 \cos(2 \pi x + 8 \pi) \), B is equal to \(2\pi\). Hence, the period is \( (2\pi) / |2\pi| = 1 \).

Step 3: Determine the Phase Shift

The phase shift is determined by the constant C inside the cosine function, and is calculated as -C/B. For the function \( y = 2 \cos(2 \pi x + 8 \pi) \), C is equal to \(8\pi\), and B is equal to \(2\pi\). Hence, the phase shift is -C/B = -(8\pi / 2\pi) = -4.

Step 4: Graph the function

To graph one period of the function \( y = 2 \cos(2 \pi x + 8 \pi) \), we will start from the phase shift. The graph has a maximum value equal to the amplitude at the phase shifts, and repeats every period. Use these values to graph a rough sketch of the function.