/*! 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 38 Sketch one full period of the gr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 38

Sketch one full period of the graph of each function. $$y=3 \csc \frac{\pi x}{2}$$

Short Answer

Expert verified
The graph of the function \(y = 3 \csc \frac{\pi x}{2}\) has a period of \(4\), with maximum and minimum values of \( \pm 3\), and asymptotes at \(x = 0\) and \(x = 2\).

Step by step solution

01

Determine the Period

The period of the function \(y = \csc Bx\) is given by the formula \(T = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) in the argument of the function. For this function, \(B = \frac{\pi}{2}\), so the period \(T = \frac{2\pi}{|\frac{\pi}{2}|} = 4\). The function completes one full cycle over this interval.
02

Identify the Asymptotes

Csc(x) is undefined where sine is zero. Therefore, there are vertical asymptotes at \(x=0\) and \(x = +T/2 = 4/2 = 2\), such that the function is undefined at these points as well as for any integer multiples of these points within the period.
03

Identify Maximum and Minimum Values

The maximum and minimum values of the cosecant function are the values farthest from zero, which occur at the same \(x\) values where sine is at its maximum or minimum. In this case, these occur at the midpoints between the asymptotes, which are \(x = T/4 = 1\) and \(x=3T/4 = 3\). At these points, \(y = 3 \csc(\frac{\pi x}{2}) = \pm 3\).
04

Sketch the Graph

Plot the asymptotes as dotted vertical lines at \(x = 0\) and \(x = 2\). Then, plot the maximum and minimum values at \(x = 1\) and \(x = 3\). Between these points, sketch the graph of the given 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

Simplify: \(\frac{2 \pi}{2 / 5}\).

Write an equation for a sine function using the given information. Amplitude \(=4 ;\) period \(=2\)

Use the fundamental trigonometric identities to write each expression in terms of a single trigonometric function or a constant. $$\sin ^{2} t\left(1+\cot ^{2} t\right)$$

Estimate, to the nearest tenth, \(\tan \frac{\pi}{3} .[4.2]\)

Graph at least one full period of the function defined by each equation. $$y=-\left|2 \cos \frac{\pi x}{2}\right|$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete 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.