Problem 45 - Algebra and Trigonometry

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 of the function is 3, the period is \(\pi\), and the phase shift is \(\pi/2\) units to the right. To sketch the graph, make these transformations to a standard \(\cos\) graph.

Step 1: Amplitude of the Function

The amplitude of the function is given by the absolute value of the coefficient of the \(\cos\) term. In this case, it is \(|3|\), which results in an amplitude of 3.

Step 2: Period of the Function

The period of a cosine function is normally \(2\pi\). However, this period gets divided by the absolute value of the coefficient of \(x\) in the argument of the cosine function. Thus, the period here is \(2\pi / |2|\), which simplifies to \(\pi\).

Step 3: Phase Shift of the Function

The phase shift of the function is given by the constant term in the argument of the \(\cos\) function divided by the coefficient of \(x\), with the sign reversed. Thus, the phase shift is \(\pi / 2 = \pi/2\). Since the sign in front of the phase shift is negative, the graph is shifted \(\pi/2\) units to the right.

Step 4: Graphing of the Function

To graph the function, start with the basic \(\cos\) graph. Then, stretch it vertically by a factor of 3, compress it horizontally by a factor of 2, and shift it right by \(\pi/2\) units. Mark the important points (peak, trough, and zero crossings) according to these transformations.