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Chapter 5: Problem 44

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

Short Answer

Expert verified
The amplitude of the function is 1. The period of the function is \(2\pi\). The phase shift is \(-\frac{\pi}{2}\). The graph starts at \(-\frac{\pi}{2}\), and ends at \(\frac{3\pi}{2}\), completing one whole period.

Step by step solution

01

Identify the amplitude

The amplitude of a trigonometric function is the absolute value of the coefficient of the function. In the function \(y=\cos\left(x+\frac{\pi}{2}\right)\), the coefficient is 1 (since it's not explicitly stated). Therefore, the amplitude of the function is \(|1|=1\).
02

Determine the period

The period of a standard cosine function is \(2\pi\). In the given function, there is no coefficient to \(x\), so the period is not affected and remains \(2\pi\).
03

Identify the phase shift

The phase shift of a function is determined by the value added or subtracted from \(x\) inside the function. For the function \(y=\cos\left(x+\frac{\pi}{2}\right)\), the value added is \(\frac{\pi}{2}\), which means there is a phase shift to the left by \(\frac{\pi}{2}\).
04

Graph the function

The graph of one period of the function would start at the point corresponding to \(-\frac{\pi}{2}\) (due to the phase shift), rise to 1 (the amplitude), fall back to -1, and then rise back to 1, completing the cycle at the point \(2\pi-\frac{\pi}{2} = \frac{3\pi}{2}\).
05

Repeat the cycle

Finally, to complete the graph for one period, the same cycle will be repeated starting from \(\frac{3\pi}{2}\).

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