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Chapter 4: Problem 30

Graph two periods of the given cosecant or secant function. $$y=2 \csc x$$

Short Answer

Expert verified
The graph of the function \(y = 2 \csc{x}\) for two periods consists of a series of 'U'-shaped curves, centered at \(y = 2\) and \(y = -2\), with vertical asymptotes at integer multiples of \(\pi\).

Step by step solution

01

Identify the Key Features

For \(y=csc(x)\), the period is \(2\pi\), the amplitude is undefined, and it has vertical asymptotes at \(x = 0\), \(x = \pi\), \(x = 2\pi\), etc. There are maximum and minimum values at \(x = \pi/2 + n\pi\), where \(n\) is any integer. The modifications due to the 2 coefficient will only affect the maximum and minimum values of the function.
02

Draw the Vertical Asymptotes

Draw vertical dotted lines (representing the asymptotes) at \(x = 0\), \(x = \pi\), \(x = 2\pi\), \(x = -\pi\), and \(x = -2\pi\) to cover two periods of the function.
03

Plot the Maximum and Minimum Values

Mark points at \(x = \pi/2\), \(x = 3\pi/2\), \(x = 5\pi/2\), \(x = -\pi/2\), and \(x = -3\pi/2\). Due to the 2 coefficient, these points will be at \(y = 2\) and \(y = -2\) instead of \(y = 1\) and \(y = -1\).
04

Draw the Graph

Draw a curve passing through each of the plotted points and 'approaching' the vertical asymptotes, being careful to remember that the function is positive for \(0 < x < \pi\) and negative for \(-\pi < x < 0\). Repeat this for the second period of the function.

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Most popular questions from this chapter

Rounded to the nearest hour, Los Angeles averages 14 hours of daylight in June, 10 hours in December, and 12 hours in March and September. Let \(x\) represent the number of months after June and let \(y\) represent the number of hours of daylight in month \(x .\) Make a graph that displays the information from June of one year to June of the following year.

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=0.8 \sin \frac{x}{2} \text { and } y=0.8 \csc \frac{x}{2}$$

will help you prepare for the material covered in the next section. $$\text { Simplify: } \frac{-\frac{3 \pi}{4}+\frac{\pi}{4}}{2}$$

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \cos (2 \pi x+4 \pi)$$

Will help you prepare for the material covered in the next section. a. Graph \(y=\sin x\) for \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\) b. Based on your graph in part (a), does \(y=\sin x\) have an inverse function if the domain is restricted to \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] ?\) Explain your answer. c. Determine the angle in the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) whose sine is \(-\frac{1}{2} .\) Identify this information as a point on your graph in part (a).
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