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Chapter 5: Problem 54

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

Short Answer

Expert verified
The graph of \(y = \sin x - 2\) is the same as the graph of \(y = \sin x\) but shifted 2 units downwards. This means that for any value of \(x\), the \(y\)-value on the shifted graph will be 2 units lower than on the normal sine graph.

Step by step solution

01

Identify the Basic Function and Vertical Shift

The given function is \(y= \sin x - 2\). It's clear here that the '-2' represents a vertical shift, indicating that the graph is moved down by 2 units.
02

Sketch the Basic sine Function

Draw the basic sine function \(y=\sin x\) on a graph spanning the x-values from 0 to \(2\pi\). The sine function starts at the origin (0,0) and cycles every \(2\pi\) units with a peak of 1 and a valley of -1.
03

Apply the Vertical Shift

Apply the vertical shift to the graph of \(y=\sin x\). In this case, you are shifting the graph down by 2 units. To do this, take each point on your original sine curve and move it down by 2 units. This shifted graph now represents \(y = \sin x - 2\). The period doesn't change because vertical shifts do not affect the span of the cycle.
04

Sketch the Final Graph

Finish the graph by maintaining the same periodic oscillations of the sine curve, but simply 2 units lower. The graph will still cycle every \(2\pi\), but now the peak will be at -1 and valley at -3, reflecting the downward shift of 2 units.

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