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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 function \(y=\csc \frac{x}{2}\) can be graphed by noting it has asymptotes every \(2\pi\) units, and two periods of the function occur between \(x=0\) and \(x=4\pi\). Given it's a cosecant function, it decreases from positive infinity to negative infinity for each period.

Step by step solution

01

Identify the parent function

Here, the parent function is \(\csc(x)\). The graph of \(\csc(x)\) has vertical asymptotes at every multiple of \(\pi\) where \(\sin(x) = 0\) and a period of \(2\pi\). The maximum and minimum points of \(\sin(x)\) become the points on the graph of \(\csc(x)\).
02

Determine the change in the period

Given function is \(\csc \frac{x}{2}\). The normal period of \(\csc(x)\) function is \(2\pi\) but because we have \(\frac{x}{2}\) in function, it will take twice as long to complete one period. Hence, the period of given function will be \(4\pi\).
03

Plot the asymptotes

Asymptotes occur wherever \(\sin(x)\) equals zero, and for \(\csc(\frac{x}{2})\), they will occur twice as often, meaning at every \(2\pi\) units. Drawing vertical dotted lines on these points will help guide the sketching of the graph.
04

Plot the graph

Using the period and asymptotes, then considering that the cosecant function is decreasing from a maximum value to a minimum value, sketch the graph of the function by drawing arcs from one asymptote to another, with a decreasing slope towards the x-axis. Sketch two full periods of the function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Period Change in Cosecant Functions
When dealing with the cosecant function, the "period" refers to the length of the interval required for the function to complete one full cycle. For the parent function \(\csc(x)\), this period is \(2\pi\). However, when the input to the cosecant function changes, the period often adjusts. In the function \(\csc \frac{x}{2}\), the expression \(\frac{x}{2}\) inside the function means the angle progresses at half the rate. Therefore, the period extends, taking twice as long for the function to repeat its cycle. This changes the period to \(4\pi\).
  • Original period of \(\csc(x)\): \(2\pi\)
  • New period of \(\csc \frac{x}{2}\): \(4\pi\)
Understanding how these period changes work can be crucial for sketching accurate graphs.
Understanding Asymptotes
Asymptotes play a vital role in sketching the graph of trigonometric functions like cosecant. In the case of \(\csc(x)\), vertical asymptotes appear wherever the sine function is zero because cosecant is undefined at these points. Typically, for \(\csc(x)\), these occur at every integer multiple of \(\pi\).
For \(\csc \frac{x}{2}\), these asymptotes become even more frequent, occurring at every \(2\pi\) units due to the halved rate of progress in the angle. When sketching, draw vertical dotted lines to represent these asymptotes, which help guide the graphing of arcs between them.
Graphing Trigonometric Functions
Graphing trigonometric functions, such as the cosecant function, involves several key steps. First, identify the characteristics like period and asymptotes. For \(\csc \frac{x}{2}\), the extended period and more frequent asymptotes must be considered.
  • Start by plotting the asymptotes.
  • Understand that between these asymptotes, the graph consists of arcs.
  • These arcs peak and dip, reflecting the maximum and minimum points of the corresponding sine function.
Since you need to represent two full periods, your graph should show consistent repetition, carefully following the characteristic shape of the cosecant's wave-like pattern.
The Role of the Parent Function
In trigonometry, understanding the parent function is essential. The parent function for cosecant is \(\csc(x)\), which serves as the foundational model. Knowing its period, asymptotes, and general shape helps when transformations like scaling and period changes are applied.
Every modification, whether to the period or amplitude, stems from the parent function. For instance, \(\csc \frac{x}{2}\) is a transformed version of \(\csc(x)\), with only the period modified due to the input adjustment.
  • Parent function: guides the shape and behavior of complex functions.
  • Transformed function: results from changes made to the parent function.
Familiarity with the parent function aids in predicting the effects of transformations and plotting accurate graphs.

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. $$\tan \frac{5 \pi}{4}=1 \rightarrow \arctan 1=\frac{5 \pi}{4}$$

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow \infty, \text { the value of } \arctan x \rightarrow$$ \(\square\).

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) \(x \rightarrow\left(\frac{\pi}{2}\right)^{+}\) (b) \(x \rightarrow\left(\frac{\pi}{2}\right)^{-}\) (c) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{+}\) (d) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{-}\) $$f(x)=\tan x$$

An airplane flying at 600 miles per hour has a bearing of \(52^{\circ} .\) After flying for 1.5 hours, how far north and how far east will the plane have traveled from its point of departure?

Use a graphing utility to graph the function. $$f(x)=\frac{\pi}{2}+\cos ^{-1}\left(\frac{1}{\pi}\right)$$
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