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Chapter 4: Problem 38

Sketch the graph of the function. (Include two full periods.) $$y=2 \cot \left(x+\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The graph of the function \(y=2 \cot \left(x+\frac{\pi}{2}\right)\) is a periodic cotangent wave with period \(\pi\), shifted \(\frac{\pi}{2}\) units to the left, and an amplitude of 2.

Step by step solution

01

Understand the Function

Firstly, recognize that \(y = 2 \cot \left(x+\frac{\pi}{2}\right)\) is a cotangent function. Cotangent function is periodic with a period of \(\pi\), that is, the values repeat every \(\pi\) units.
02

Determine the shift

Notice that the function has a phase shift. It is shifted \(-\frac{\pi}{2}\) units to the left because of the term \(\left(x+\frac{\pi}{2}\right)\) inside the cotangent function.
03

Identify the amplitude

The amplitude of the function is 2. The original cotangent function has an amplitude of 1, so this function is twice as steep.
04

Sketch the Graph

Use the information from the previous two steps to sketch the graph. The graph will start from \(x = -\frac{\pi}{2}\), reach infinity at \(x = 0\), and then descend to negative infinity at \(x = \frac{\pi}{2}\). This constitutes one period of the function. For two full periods, repeat these key points at \(x = \frac{\pi}{2}\), \(x = \pi\), and \(x = \frac{3\pi}{2}\). The graph will be twice as steep as the original cotangent function because of the amplitude 2.

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