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Chapter 4: Problem 45

Sketch the graph of the function. (Include two full periods.) $$y=\cos 2 \pi x$$

Short Answer

Expert verified
In order to sketch the function \(y = \cos 2 \pi x\), first find the period of the function which in this case is 1. The function repeats every one unit length on the x-axis. Once the period is clear, key points for the function for one period can be identified and marked on the graph. Lastly, connect these points with a smooth curve to get the shape of one full period of the function. Repeat this for the second period.

Step by step solution

01

Identify the period of the function

The period of the function \(y=\cos 2 \pi x\) is given by the formula \(Period = \frac{2\pi}{|B|}\) where B is the coefficient of x. Here, B is \(2\pi\). Therefore, the period of the function is \(Period = \frac{2\pi}{|2\pi|} = 1\). This implies that the function repeats itself for every unit length on the x-axis.
02

Identify key points for one period

For cosine function, starting from zero, the key points are: at \(x=0\), \(y=\cos(0)=1\); at \(x=\frac{1}{4}\), \(y=\cos2\pi \times 1/4 = 0\); at \(x=\frac{1}{2}\), \(y=\cos2\pi \times 1/2 = -1\); at \(x= \frac{3}{4}\), \(y=\cos2\pi \times 3/4 = 0\); and at \(x=1\), \(y=\cos2\pi \times 1 = 1\).
03

Plot the graph for one period

Start by marking the points identified above on the graph. They are (0, 1), (\(\frac{1}{4}\), 0), (\(\frac{1}{2}\), -1), (\(\frac{3}{4}\), 0), and (1, 1). Once the points are marked, sketch a smooth curve that passes through these points with the typical 'wave' shape of the cosine function.
04

Sketch the second period of the graph

Repeat the pattern for the next unit interval on the x-axis. For cosine function, the pattern repeats itself after every period. Thus, the key points for the second unit length on the x-axis will be (1, 1), (\(1\frac{1}{4}\), 0), (\(1\frac{1}{2}\), -1), (\(1\frac{3}{4}\), 0), and (2, 1). Now, sketch a smooth curve that passes through these points. This completes two full periods of the function \(y=\cos 2\pi x\).

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