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Chapter 4: Problem 57

Sketch the graph of the function. (Include two full periods.) $$y=3 \cos (x+\pi)-3$$

Short Answer

Expert verified
The graph of the function \(y=3 \cos (x + \pi) -3\) is a cosine wave of amplitude 3, phase shift of \(\pi\) units to the right, and a shift of 3 units down with its midline being y=-3. The graph can be plotted by constructing one period of the cosine wave, then applying the phase shift and vertical shift, and finally extending the graph to include two full periods, which includes -2\(\pi\) and 4\(\pi\).

Step by step solution

01

Identify the amplitude, period, phase shift, vertical shift and sketch basic cosine wave

The amplitude of the given function is 3, which means the wave goes 3 units above and below its midline. The period is \(2\pi\) as there is no frequency change. The phase shift is \(\pi\) units. This means the entire graph of the cosine function is shifted \(\pi\) units right. The vertical shift is 3 units down, so the midline is at y=-3. First, sketch the basic cosine wave without phase and vertical shifts taking into account the amplitude and period.
02

Apply the phase and vertical shifts

Once the basic cosine wave is plotted, we apply the phase and vertical shifts. The phase shift is \(\pi\), which means the graph will shift to the right by \(\pi\). For the vertical shift, the whole graph should move down by 3 units. Adjust the midline, maximum and minimum points accordingly to reflect the phase and vertical shifts.
03

Plot the full graph

Now you are ready to sketch the final graph of \(y=3 \cos (x+\pi) -3\). It should start at \((-\pi,-3)\), go through points such as \((0,0)\) and \((\pi,-3)\), and continue based on the cosine wave pattern. Make sure to include two full periods of the function.

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