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Chapter 4: Problem 50

Sketch the graph of the function. (Include two full periods.) $$y=\sin (x-2 \pi)$$

Short Answer

Expert verified
The graph of the function \(y=\sin (x-2\pi)\) is identical to the graph of the standard sine function. It crosses the x-axis at \(x=0, \pi, 2\pi, ...\), reaches maximum at \(x=\pi/2, 3\pi/2, ...\), and minimum at \(x=3\pi/2, 5\pi/2, ...\). It repeats every \(2\pi\).

Step by step solution

01

Understanding the sine function

The sine function, \(y=\sin x\), is a periodic function with a period of \(2\pi\). This means the function repeats its values every \(2\pi\). It oscillates between -1 and 1, reaching these extremums at \(-\pi/2\) and \(\pi/2\) correspondingly, and crosses the x-axis at \(x=0\) and \(x=\pi\).
02

Shifts of the sine function

Shifting the graph of a function means that each point of the graph is moved to a different location. If the shift is a horizontal shift to the right, then the graph is shifted in the positive direction along the x-axis by whatever the shift factor is. In this case the shift factor is \(2\pi\). So each point of the standard sine graph will be shifted to the right by \(2\pi\).
03

Sketch the graph

Because sine is periodic and repeats every \(2\pi\), the graph of \(y=\sin (x-2\pi)\) is exactly the same as that of \(y=\sin x\). The x-intercepts are \(0, \pi, 2\pi\), etc. The maximum point in this period is at \(\pi/2,1\) and the minimum is at \(3\pi/2, -1\). The plot is symmetric about the origin.

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