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Chapter 5: Problem 49

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

Short Answer

Expert verified
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 by step solution

01

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 \(|-3| = 3\).
02

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 \( \frac {2\pi} {2} = \pi\).
03

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.
04

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.

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