Problem 54 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.