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Chapter 4: Problem 25

Sketch the graph of the function. (Include two full periods.) $$y=\csc \frac{x}{2}$$

Short Answer

Expert verified
The graph of the function y = csc(x/2) is a series of U-shaped curves flip, with asymptotes at \( x = 2n\pi \). The graph repeats every 4pi, doesn't intersect the x-axis and has its maximum and minimum where the sine function has respectively.

Step by step solution

01

Identify the Function

The function that needs to be graphed is \( y = \csc \frac{x}{2} \) . This is the cosecant function which is the reciprocal of sine function, \( y = \csc (x) = \frac{1}{\sin (x)} \). The graph will have similar shape to sine function but it will be undefined wherever sine function equals to zero.
02

Identify the Period

The period of basic cosecant function \( \csc(x) \) is \( 2\pi \). However, for \( \csc(\frac{x}{2}) \), the period is stretched by a factor of 2. Therefore, the new period for graphing the function \( y = \csc \frac{x}{2} \) is \( 2 * 2\pi = 4\pi \).
03

Identify the Asymptotes

Since the function \( \csc(x) \) is undefined for \( x = n\pi \), where \( n \) is any integer, the graph has vertical asymptotes at these points. However, the function is \( \csc \frac{x}{2} \), so the function is undefined at \( x = 2n\pi \), which means the vertical asymptotes occur at \( x = 2n\pi \).
04

Sketch the Graph

The graph of \( y = \csc \frac{x}{2} \) would be a series of U-shaped curves, alternating direction, repeating every \( 4\pi \) with asymptotes at \( x = 2n\pi \). Remember that the cosecant function is undefined when the sine function equals zero so it will not intersect the x-axis. The graph will have minimums and maximums where the sine function has maximums and minimums respectively.

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