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Chapter 4: Problem 37

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{array}{l} f(x)=2 \cos x \\ g(x)=2 \cos (x+\pi) \end{array}$$

Short Answer

Expert verified
Both functions, \(f(x) = 2cos(x)\) and \(g(x) = 2cos(x + \pi)\), have the same amplitude and frequency but differ by a phase shift of \(\pi\). This phase shift means their peaks and valleys occur at different points along the x-axis despite having the same shape.

Step by step solution

01

Analyzing function f(x) = 2cos(x)

The function \(f(x)=2 \cos x\) is a cosine function with amplitude 2 and no phase shift. A full period of the cosine function is \(2\pi\), and its graph repeats every \(2\pi\). The function will start at the point (0, 2), and one full period will end at \(2\pi\). Over one period, the function will hit the major points: (0, 2), \(\pi/2, 0\), \(\pi, -2\), \((3\pi)/2, 0\), \(2\pi, 2\). Then, this pattern repeats for the second period.
02

Analyzing function g(x) = 2cos(x+蟺)

The function \(g(x)=2 \cos (x+\pi)\) is also a cosine function with amplitude 2 but with a phase shift of \(\pi\). This is equivalent to a horizontal shift of the graph by \(\pi\) to the left. The graph will start at the point (\(-\pi\), 2), and one period ends at \(\pi\). Over one period, the function hits the major points: (\(-\pi\), 2), \(-\pi/2, 0\), \(0, -2\), \(\pi/2, 0\), \(\pi, 2\). The graph repeats this pattern for the second period.
03

Plot the graphs

Using the points determined in previous steps, plot both \(f(x)\) and \(g(x)\) on the same coordinate plane. Begin with the function \(f(x)\) and then plot \(g(x)\). Each function should clearly complete two full periods. The function \(f(x) = 2 cos x\) will start at the point (0, 2), peak at (0, 2), touch the x-axis at (\(\pi/2\), 0), bottom out at (\(\pi\), -2), touch the x-axis again at (\((3\pi)/2\), 0), and then peak again at (\(2\pi\), 2). Meanwhile, the function \(g(x) = 2 cos(x + \pi)\) initiates each period at a minimum at \(-\pi, pi, 3pi\), etc. Both functions are similar in shape but are essentially 'out of sync' due to the phase shift of \(g(x)\).

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