Problem 59 - Precalculus Essentials

Question

Use a vertical shift to graph one period of the function. $$y=-3 \cos 2 \pi x+2$$

Step-by-Step Solution

Answer

The graph is a vertically shifted cosine curve which starts from bottom because of the negative amplitude at \(y=-1\), peaks at \(y=5\), and has one complete wave in the interval of \(1/2 \pi\) along the x axis.

Step 1: Identify the amplitude, frequency, and vertical shift

In the function \(y=-3 \cos 2 \pi x+2\), the amplitude is \(-3\), which means the wave will have a height of 3. The frequency is \(2 \pi\), which means one period of the wave will be completed in an interval of \(1/2 \pi\). The vertical shift is \(+2\), which means the whole wave will be moved up by 2 units from the x-axis.

Step 2: Determine the range

The range of the function is determined by the amplitude and the vertical shift. Because the amplitude is 3, the wave extends 3 units above and below its central position. The graph has been moved up by 2 units, so the graph will range from \(2-3=-1\) up to \(2+3=5\).

Step 3: Plot the function

First, draw a dashed line \((y=2)\) to represent the vertical shift of the wave. Then, remember that the cosine function starts at the peak of a wave. Because of the negative amplitude, these waves will instead start at the bottom. Draw a period of the cosine curve, making sure it has a peak at \(y=5\) and a trough at \(y=-1\). Also note that the period of one wave should span \(1/2 \pi\) along the x-axis.