/*! 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 45 Use a graphing utility to graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 45

Use a graphing utility to graph the function. (Include two full periods.) $$y=-\csc (4 x-\pi)$$

Short Answer

Expert verified
The graph of the function \(y = -\csc(4x - \pi)\) will have vertical asymptotes at \(x=\frac{(n\pi + \pi)}{4}\) and period of \(\pi/4\). It will reflect about x-axis and shift \(\pi\) units to right from original cosecant function.

Step by step solution

01

Understanding the Original Function

The original function before transformation can be written as \(y = \csc(x)\). The cosecant function \(y = \csc(x)\) is basically the reciprocal of \(y = \sin(x)\) function. It has vertical asymptotes at \(x = n\pi\) where \(n\) is an integer. The period of the function is \(\pi\). The function is positive in quadrants I and II, while negative in quadrants III and IV. Consequently, the \(y=\csc(x)\) graph goes to positive infinity at \(n\pi\) and negative infinity at \((n+0.5)\pi\), with a switch in direction of 'infinity' each time.
02

Modify the function

The given function is \(y=-\csc (4x-\pi)\). This represents the transformations from original function. The negative sign inverts the graph, a factor of 4 inside the function compresses the graph along the x-axis by a factor of 1/4, and finally, subtraction by \(-\pi\) shifts the graph \(\pi\) units to the right.
03

Draw the modified function

By taking into account the aforementioned transformations into original graph, the new graph can be plotted. Vertical asymptotes are now at \( x=\frac{(n\pi+ \pi)}{4}\). The period of the function is now changed to \(\pi/4\). The function is still positive in quadrants I and II, and negative in quadrants III and IV, but because of the negative sign in front the graph will reflect about x-axis. Finally plot at least two full periods of this 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

True or False? Determine whether the statement is true or false. Justify your answer. The difference between the measures of two coterminal angles is always a multiple of \(360^{\circ}\) when expressed in degrees and is always a multiple of \(2 \pi\) radians when expressed in radians.

Find the length of the sides of a regular hexagon inscribed in a circle of radius 25 inches.

Sketch a graph of the function. $$f(x)=\arctan 2 x$$

A carousel with a 50 -foot diameter makes 4 revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed (in feet per minute) of the platform rim of the carousel.

Use a graphing utility to graph the function. $$f(x)=2 \arccos (2 x)$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

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.