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Chapter 4: Problem 33

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

Short Answer

Expert verified
The function \(f(x)=\cos x\) oscillates between 1 and -1 over the interval of \(2\pi\). \(g(x)=2+\cos x\), has the same period but vertically shifted by 2 units, oscillating between 1+2=3 and -1+2=1.

Step by step solution

01

Sketch the Graph of \(f(x)=\cos x\)

Start at the origin and sketch one full period \(2\pi\) of the cosine function. Cosine starts at the maximum value of 1, descends to a minimum of -1, and then returns to 1. Complete the second period with same pattern.
02

Plot the Transformed Function \(g(x)=2+\cos x\)

The graph of \(g(x) = 2 + \cos x\) is obtained by shifting the graph of \(f(x)=\cos x\) upward by 2 units. Start at \(y=2\) at \(x=0\) and follow the pattern of the cosine function. The value should oscillate from 1 to 3, ensuring the shift upwards is considered.
03

Compare the Graphs

Both the original and transformed function graphs should be shown together. This will give a clear visual representation of the vertical shift from \(f(x)\) to \(g(x)\).

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