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Chapter 4: Problem 52

Sketch the graph of the function. (Include two full periods.) $$y=4 \cos \left(x+\frac{\pi}{4}\right)$$

Short Answer

Expert verified
The graph of the function \(y=4 \cos \left(x+\frac{\pi}{4}\right)\) is a cosine curve with amplitude 4 and phase shift \(-\frac{\pi}{4}\). The period of the function is \(2\pi\). Sketching two full periods of this function involves plotting key points at \(-\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}\), and \(\frac{7\pi}{4}\), and then connecting these with a cosine curve.

Step by step solution

01

Identify the amplitude

The amplitude of a function of the form \(y=A \cos (Bx+C)\) is given by the absolute value of \(A\). Here, \(A = 4\), so the amplitude of this function is \(4\).
02

Identify the period

The period of a function of the form \(y=A \cos (Bx+C)\) is given by \(\frac{2\pi}{|B|}\). Since there are no modifications to the \(x\) (i.e., \(B = 1\)), the period of this function is \(2\pi\).
03

Identify the phase shift

The phase shift of a function of the form \(y=A \cos (Bx+C)\) is \(-\frac{C}{B}\). For our function, this becomes \(-\frac{\pi/4}{1} = -\frac{\pi}{4}\). Thus, the function begins \(\frac{\pi}{4}\) units to the left of the standard cos function.
04

Set up the x-values for one period of the function

For the standard cos function, key x-values in one period are \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\), and \(2\pi\). Adjust these values by the phase shift to get x-values for our function: \(-\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}\), and \(\frac{7\pi}{4}\).
05

Evaluate the function at these x-values

To obtain y-values, substitute each x-value into our function: \(4 \cos \left(-\frac{\pi}{4}\right), 4 \cos \left(\frac{\pi}{4}\right), 4 \cos \left(\frac{3\pi}{4}\right),\), and so on. These are key points for sketching the graph.
06

Sketch the graph

Plot the key points obtained in Step 5 on a grid, and sketch a cosine curve connecting these points. Repeat this pattern to obtain two full periods of the function.

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