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Chapter 4: Problem 49

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

Short Answer

Expert verified
The graph of the function \(y=3 \cos(x+\pi)\) is a standard cosine wave that has been stretched vertically by a factor of \(3\) and shifted \(\pi\) units to the left. Two periods of this function span from \(-\pi\) to \(3\pi\).

Step by step solution

01

Identify Key Characteristics of the Function

First, identify the main aspects of the function. The amplitude \(a\) is \(3\), so the output values will extend from \(3\) to \(-3\). As \(b\), which affects the period \(\frac{2\pi}{b}\), is \(1\), the period remains \(2\pi\). The function is shifted horizontally by \(-\pi\), and there is no vertical shift since \(k=0\). Thus, the maximum value is \(3\), the minimum value is \(-3\), and the period is \(2\pi\). Note also the phase shift of \(-\pi\). Drawing these values on the x and y axes will give the framework for the graph.
02

Sketch the cosine function

Start by sketching the standard cosine function \(\cos(x)\) within the period \(0\) to \(2\pi\). The cosine function starts at the maximum, then reaches the equilibrium, descends to the minimum, returns to the equilibrium, and finally ascends back to the maximum.
03

Apply the transformation

Now apply the phase shift \(-\pi\). This moves the graph of the standard cosine function \(\pi\) units to the left. Apply the amplitude scaling which stretches the graph vertically by a factor of \(3\). This completes the graph for \(y=3 \cos(x+\pi)\), in the interval \(-\pi\) to \(\pi\).
04

Plot a second period

To sketch a second period, simply copy the pattern from the first period to the right, for the interval \(\pi\) to \(3\pi\).

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