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Chapter 4: Problem 44

Use a graphing utility to graph the function. (Include two full periods.) $$y=\frac{1}{4} \cot \left(x-\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The graph of the function \(y=\frac{1}{4} \cot (x-\frac{\pi}{2})\) consists of vertically compressed cotangent curves, shifted \(\frac{\pi}{2}\) units to the right.

Step by step solution

01

Identify the Basic Function

The function given is a transformation of the basic cotangent function. So first sketch a basic function \(y=\cot(x)\), which has a period of \( \pi\) and asymptotes at \( x=0, \pi, 2\pi, \ldots\)
02

Apply the Vertical Scaling

Next, apply the vertical scaling factor of \(\frac{1}{4}\) which compresses the basic cotangent function towards the x-axis. Now, maximum and minimum values are 1/4 of the basic cotangent function.
03

Apply the Phase Shift

Finally, apply the phase shift of \( -\frac{\pi}{2}\). This means you will move each point on the previous graph to the right by \(\frac{\pi}{2}\). The asymptotes shift to \( x=\frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}\), etc. Hence, the final graph of \( y=\frac{1}{4} \cot (x-\frac{\pi}{2}) \) is obtained.

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Most popular questions from this chapter

Write the function in terms of the sine function by using the identity $$A \cos \omega t+B \sin \omega t=\sqrt{A^{2}+B^{2}} \sin \left(\omega t+\arctan \frac{A}{B}\right).$$ Use a graphing utility to graph both forms of the function. What does the graph imply? $$f(t)=4 \cos \pi t+3 \sin \pi t$$

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