Problem 54 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.