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Chapter 4: Problem 54

Use a vertical shift to graph one period of the function. $$y=\sin x-2$$

Short Answer

Expert verified
The one period of the function \(y=\sin x - 2\) is a sine wave starting at \((0, -2)\) and ending at \((2\pi,-2)\), and it's shifted 2 units downwards from a typical sine wave.

Step by step solution

01

Understand the concept

A 'vertical shift' is a transformation that shifts the graph of a function up or down in the coordinate plane. Given the function \(y=\sin x-2\), the '-2' is the vertical shift that moves the graph 2 units down. The 'period' of a function is the distance required for the function to start repeating. The sin function repeats every \(2\pi\) units.
02

Identify key points

For sine function, some key points within one period (0 to \(2\pi\)) are: at \(x=0\), \(\sin x = 0\); at \(x=\pi/2\), \(\sin x = 1\); at \(x=\pi\), \(\sin x = 0\); at \(x=3\pi/2\), \(\sin x = -1\); finally at \(x=2\pi\), \(\sin x = 0\). But remember our function is vertically shifted, thus each y-coordinate should be subtracted by 2.
03

Plot the function

Plot these points: \((0,-2), (\pi/2,-1), (\pi,-2), (3\pi/2,-3)\) and \((2\pi,-2)\) on the coordinate plane. Connect these points using smooth curves. You should notice the wave-like form of the sin function, but instead it starts and ends 2 units below the x-axis.

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Most popular questions from this chapter

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