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Chapter 4: Problem 34

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{aligned} &f(x)=2 \cos 2 x\\\ &g(x)=-\cos 4 x \end{aligned}$$

Short Answer

Expert verified
The graph of function f(x) will be a cos wave of amplitude 2, peaking at y=2 and dropping to y=-2, completing a cycle every \(\pi\) units, while the graph of function g(x) will also be a cos wave but flipped vertically with amplitude 1, peaking at y=1 and dropping to y=-1, completing a cycle every \(\pi/2\) units.

Step by step solution

01

Understand the Cosine Function

The cosine function has a wave-like shape. Its maximum value is 1, and its minimum value is -1. However, the maximum and minimum values will be higher or lower depending on the amplitude. The period of the function is usually \(2\pi\), unless there's a coefficient to x, which multiplies the original period by \(1/\)(the coefficient).
02

Plot f(x)

The function f(x) is a scaled cosine function with amplitude 2 and frequency 2. Start by plotting the points for one period (from \(x=0\) to \(x=\pi\)) and then repeat for the second period (from \(x=\pi\) to \(x=2\pi\)). The maximum point will be at \(y=2\) and the minimum point will be at \(y=-2\). The complete cycle will repeat every \(\pi\) units due to the presence of 2x in the function.
03

Plot g(x)

Function g(x) is \(-\cos 4x\), which is a vertically flipped cosine graph with frequency 4 (due to 4x) and an amplitude of 1 (since there is no scale factor other than -1). Therefore, make the plotting for one period and then repeat that second period. The cycle will complete every \(\pi/2\) units due to the presence of 4x in the function.
04

Align both plots

With both plots on the same graph, it will be clear that f(x) completes one full cycle in the same interval where g(x) completes two cycles due to the difference in frequencies.

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