91Ó°ÊÓ

The given cosine function is \(y = 3 \cos(2x + \pi)\). Here, the amplitude \(A\) is 3 (which is the absolute value of the coefficient of \(\cos\)), the number \(B\) is 2 (which determines the period), and the phase shift \(-C\) is \(\pi\) (inside the parentheses, opposite the x).

Normally, the period of a cosine function is \(2\pi\). However, the period \(P\) of \(y = A \cos(Bx - C) + D\) is determined by dividing \(2\pi\) by the absolute value of \(B\). So, in this case, the period is \(2\pi / |2| = \pi\).

The amplitude is the absolute value of \(A\). So, in this case, the amplitude is \(|3| = 3\). The amplitude represents the maximum displacement from the equilibrium position, or the distance from the maximum to the equilibrium. This will be the maximum and minimum value of the y-coordinate.

The phase shift is given by \(C = -\pi\). This is the amount by which the function is shifted horizontally. In this case, this means the graph is shifted by \(\pi\) to the left.

First mark down the phase shift on the x-axis, this is the starting point of the graph. From this point, mark down the period \(\pi\), which will be the width of one full cycle of the cosine wave. Make sure you also mark down the halfway point, which will indicate the minimum peak of the cosine wave. Then indicate the amplitude on the y-axis, this will give the height of the maximum and minimum peaks of the cosine wave. Sketch your wave starting from the phase shift, going up to the maximum peak at the halfway point of the phase shift and period, down to the minimum peak at the period, and back to the equilibrium at the end of the period. Your graph should look like an 'M' with the middle peak reaching a maximum height of 3 and the ends reaching the equilibrium.