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Chapter 4: Problem 102

Use a graphing utility to graph two periods of the function. $$y=-2 \cos \left(2 \pi x-\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The graph of the function \(y=-2 \cos \left(2 \pi x-\frac{\pi}{2}\right)\) has an amplitude of 2, a period of 1, a phase shift to the right by \(\frac{\pi}{2}\) and is reflected across the x-axis. The graph repeats its cycle every 1 unit and reaches its peak and trough at one quarter of the period away from each x-intercept, the maximum distance between the wave and x-axis is 2 units.

Step by step solution

01

Identify Amplitude and Period

The amplitude of the graph can be found from the coefficient in front of the cosine function, which in this case is 2. The period of the function is \(\frac{2\pi}{|2\pi|}=1\). Therefore, the graph will repeat every interval of length 1.
02

Identify Phase Shift

The phase shift, or horizontal shift, can be found from the term inside the parentheses. In this case it's \(\frac{\pi}{2}\), but since it's subtracted in the function it's actually a shift to the right by \(\frac{\pi}{2}\) units.
03

Identify Reflection

A negative sign in front of the function represents a reflection across the x-axis. Here, the negative sign in the function \(y=-2 \cos \left(2 \pi x-\frac{\pi}{2}\right)\) indicates that the usual cosine waveform is reflected.
04

Graph the Function

Taking all these values into account, it is possible to graph the function on a graphing utility. The graph should begin at the phase shift, reach its peak and trough at one quarter of the period away from each x-intercept, and repeat this cycle every period (in this case, every 1 unit). The maximum height of the wave from the x-axis is twice (due to the amplitude 2). As the function is negative, the pattern starts from its minimum instead of the maximum.

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