Problem 49 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function \(y=-3 \cos \left(2 x-\frac{\pi}{2}\right)\) has an amplitude of 3, a period of \(\pi\) and a phase shift of \(-\frac {\pi} {4}\) to the right. The graph will be a flipped cosine curve starting from -3, moving upward and repeating after an interval of \(\pi\).

Step 1: Determine the Amplitude

The amplitude of the function is given by the absolute value of coefficient 'a'. Here, the coefficient in front of the cosine function is '-3', therefore, the amplitude is \(|-3| = 3\).

Step 2: Calculate the Period

The period of the cosine function is given by \( \frac {2\pi} {b}\), in this case 'b' is 2. Hence the period of the function is \( \frac {2\pi} {2} = \pi\).

Step 3: Find the Phase Shift

The phase shift can be obtained by dividing 'c' by 'b'. In this case, 'c' is \(-\frac {\pi} {2}\) and 'b' is 2. The phase shift will be \(\frac {c} {b} = -\frac {\frac {\pi} {2}} {2} = -\frac {\pi} {4}\). The negative sign indicates a shift to right.

Step 4: Graph the Function

To graph the function, first plot the amplitude which is 3. The period is \(\pi\) means the function will compete a full cycle in \(\pi\) units along x-axis instead of \(2\pi\) which is normal for 'cos' function. The phase shift is \(-\frac {\pi} {4}\) which means function will start from \(-\frac {\pi} {4}\) instead of 0, moving right along the x-axis. Also, as our function is '-cos' instead of 'cos', the graph will be flipped over the x-axis, starting from -3 instead of 3 and moving upward.