Problem 51 - Algebra and Trigonometry

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 function \(y=2 \cos (2 \pi x+8 \pi)\) has an amplitude of 2, a period of 1, and a phase shift of -4.

Step 1: Identify the Coefficients A, B, and C

In the given function \(y=2 \cos (2 \pi x+8 \pi)\), we can identify that the coefficient \(A=2\), \(B=2\pi\), and \(C=8\pi\).

Step 2: Calculate the Amplitude

The amplitude is determined by the absolute value of A. So, for this function, the amplitude is \(|A|=|2|=2\).

Step 3: Calculate the Period

The period is determined by calculating \((2\pi)/|B|\). So, for this function, the period is \((2\pi)/|2\pi|=1\).

Step 4: Calculate the Phase Shift

The phase shift is determined by calculating \(-C/B\). So, for this function, the phase shift is \(-8\pi/(2\pi)=-4\).

Step 5: Graph One Period of the Function

To graph the function, start at the phase shift (-4) on the x-axis, this is where the period starts. Mark the amplitude (2) on the y-axis. The graph will have a cycle of up-down-up (since it is cosine) spanning 1 unit (the period) along the x-axis, with maximum and minimum values of 2 and -2 (based on the amplitude) on the y-axis. Repeat this graph to visualize more periods.