/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Graph two periods of the given c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 39

Graph two periods of the given cosecant or secant function. $$y=-\frac{1}{2} \sec \pi x$$

Short Answer

Expert verified
The given secant function after analyzing will have a period of 2, vertical asymptotes at \(x = \frac{1}{2} + k\) and \(x = \frac{3}{2} + k\), where \(k\) is an integer, and will be reflected about the x-axis due to the negative sign.

Step by step solution

01

Identify the Properties of the Function

The given function is \(y = - \frac{1}{2} \sec \pi x\). The secant function is the reciprocal of the cosine function and it is undefined when cosine is equal to zero. Hence, it has vertical asymptotes at these points. Additionally, the secant function has a period of \(2 \pi\). In the given function, \(\pi\) is being used to compress the function horizontally. This means the new period of the function will be \( \frac {2 \pi}{|\pi|} = 2 \). Finally, the negative sign and the \(\frac {1}{2}\) will stretch the function vertically by a factor of -1/2, which will lead to a reflection of the function about the x-axis.
02

Determine the Asymptotes

The cosine function equals zero at \(\frac{\pi}{2}\) and \(\frac{3 \pi}{2}\). Since the period has been compressed by a factor of \(\pi\), we need to adjust these points accordingly. Therefore, these asymptotes when modified due to the horizontal compression become \(x = \frac{1}{2}\) and \(x = \frac{3}{2}\). Since the period is 2, the asymptotes will be \(x = \frac{1}{2} + k\) and \(x = \frac{3}{2} + k\), where \(k\) is an integer.
03

Sketch the Function

For \(x = 0\), calculate the value of \(y\) by substituting \(x = 0\) in the given function which will give \(y = - \frac{1}{2}\). The function will be reflected about the x-axis due to the negative sign. Using these asymptotes and points, sketch the secant function.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. If \(\theta=\frac{3}{2},\) is this angle larger or smaller than a right angle?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made an error because the angle I drew in standard position exceeded a straight angle.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using the equation \(y=A \sin B x,\) if I replace either \(A\) or \(B\) with its opposite, the graph of the resulting equation is a reflection of the graph of the original equation about the \(x\) -axis.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing one complete cycle of \(y=A \sin (B x-C)\) I find it easiest to begin my graph on the \(x\) -axis.

In Chapter \(5,\) we will prove the following identities: $$ \begin{aligned} \sin ^{2} x &=\frac{1}{2}-\frac{1}{2} \cos 2 x \\ \cos ^{2} x &=\frac{1}{2}+\frac{1}{2} \cos 2 x \end{aligned} $$ Use these identities to solve. Use the identity for \(\sin ^{2} x\) to graph one period of \(y=\sin ^{2} x\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.