Problem 45 - Precalculus Essentials

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \cos (2 x-\pi)$$

Step-by-Step Solution

Answer

The amplitude is 3, the period is \(\pi\), and the phase shift is \(\pi / 2\) to the right.

Step 1: Determine Amplitude

The amplitude is given by the absolute value of the coefficient of the cosine function, which in this case is 3.

Step 2: Determine Period

The period of a cosine function is given by \(2\pi\) divided by the absolute value of the coefficient of \(x\). In this case, the coefficient of \(x\) is 2, hence the period of the function is \(\pi\).

Step 3: Determine Phase Shift

The cosine function is shifted by \(\phi\) units to the right if it is of the form \(y = \cos (x - \phi)\). In this case, it is \(\cos (2x - \pi)\), so it is shifted by \(\pi / 2\) to the right. The phase shift is positive as the direction of shift is to right.

Step 4: Plot the Function

Plot the cosine function taking into account the amplitude of 3, the period of \(\pi\), and the phase shift of \(\pi / 2\). This would result in a cosine wave that starts from \(\pi / 2\), goes up till 3, comes down through 0 at \(3\pi/2\), goes down till -3 at \(2\pi\), then goes back to 0 at \(5\pi/2\) completing one cycle or period of \(\pi\).