Problem 46 - Precalculus Essentials

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=4 \cos (2 x-\pi)$$

Step-by-Step Solution

Answer

The amplitude of the function \(y = 4 \cos (2 x-\pi)\) is 4, the period is \(\pi\), and it has a phase shift of \(\pi\) to the right.

Step 1: Determine the amplitude

The amplitude of a cosine function is the absolute value of A. Here it is |4| which equals 4.

Step 2: Determine the period

The period is given by the formula \(Period = 2\pi / B\). For the given cosine function \(B = 2\), so the Period = \(2\pi / 2 = \pi\).

Step 3: Determine the phase shift

The phase shift for the cosine function is given by the value of C. Here \(C = -\pi\). But as it moves the graph to the right or to the left according to the sign of C, for negative value of C, the phase will shift to the right by \(\pi\) units.

Step 4: Graph the function

The graph of the function \(y = 4\cos(2x - \pi)\) will be a wave starting from \(x = \pi\), going up to 4, down to -4, completing a full cycle at \(x = 2\pi\), and then repeating itself.