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Chapter 4: Problem 38

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

Short Answer

Expert verified
The graph of \( f(x) = -\cos(x) \) is a wave that starts from (0,-1), moves up to (\(\pi\),1), then down to \((2 \pi, -1)\). The graph of \( g(x) = -\cos(x - \pi) \) is the same wave, but shifted \(\pi\) units to the right.

Step by step solution

01

Understand Basic Cosine Graph

The basic graph of \( y = \cos(x) \) is a wave that oscillates between -1 and 1 on the y-axis. It starts at the maximum point (1,0), decreases to the minimum at (180 degrees or \(\pi\), -1), then returns to the maximum at (360 degrees or \(2 \pi\), 1). The period of the cosine function is \(2 \pi\) or 360 degrees. Coefficients can reflect, stretch, shift, or compress the graph.
02

Sketch - cos(x)

For the function \( f(x) = - \cos(x) \), the negative sign indicates reflection about the x-axis. That means, the graph of \( f(x) = - \cos(x) \) will start from the minimum point (0,-1), move to the maximum point at \((\pi, 1)\), then return to the minimum at \((2 \pi, -1)\). This forms a complete period and the process repeats for more periods.
03

Sketch -cos(x-\pi)

For the function \( g(x) = - \cos(x-\pi) \), besides reflecting as in step 2, \(\pi\) in \(x-\pi\) means the graph of this function will shift \(\pi\) units to the right compared to standard cosine function. So the first minimum point will be \((\pi ,-1)\); then move to the maximum at \((2\pi, 1)\), and back to the minimum at \((3\pi ,-1)\).

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