/*! 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 32 Graph two periods of the given c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 32

Graph two periods of the given cosecant or secant function. $$y=\frac{3}{2} \csc \frac{x}{4}$$

Short Answer

Expert verified
The graph of the function \(y=\frac{3}{2} \csc \left(\frac{x}{4}\right)\) has amplitude of 3/2, period of \(8\pi\), and asymptotes at \(x = 0, \pm 4n\pi\) (for all integer values of n).

Step by step solution

01

Identity the Function鈥檚 Form

The function \(y=\frac{3}{2} \csc \left(\frac{x}{4}\right)\) is in the form of \(y=A \csc(Bx)\), where A is the amplitude and B affects the period. In this function, A = 3/2 and B = 1/4.
02

Calculate Amplitude

The amplitude of the function is |A|. In our case, it is \(\left|\frac{3}{2}\right| = \frac{3}{2}\). Amplitude is a measure of how high and low the graph goes from its midline.
03

Calculate Period

The period of a csc function is \(T=2\pi|\frac{1}{B}|\). Here, B = 1/4, so the period is \(T = 2\pi|\frac{1}{1/4}|= 2\pi*4 = 8\pi\). This is the width of one complete cycle on the graph.
04

Identify the Asymptotes

The function \(y=csc(x)\) is undefined wherever \(sin(x) = 0\). Thus, the function \(y=\frac{3}{2} \csc \left(\frac{x}{4}\right)\) will have a vertical asymptote wherever \(sin(\frac{x}{4}) = 0\), which occurs at \(x = 0, \pm 4n\pi\), for all integer values of n.
05

Draw the Graph

Start at \(x=0\) (where the first asymptote is). Mark the period intervals along the x-axis (each \(8\pi\)). Draw an upward curve reaching its max at the central point between two asymptotes where the y value is 3/2 and downward curve where the y value is -3/2. Repeat these steps for two periods. This way, the graph of the function is drawn.

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

The following figure shows the depth of water at the end of a boat dock. The depth is 6 feet at low tide and 12 feet at high tide. On a certain day, low tide occurs at 6 A.M. and high tide at noon. If \(y\) represents the depth of the water \(x\) hours after midnight, use a cosine function of the form \(y=A \cos B x+D\) to model the water's depth.

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\tan \frac{x}{4}$$

Use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. Convert each angle to a decimal in degrees. Round your answer to two decimal places. $$30^{\circ} 15^{\prime} 10^{\prime \prime}$$

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=2 \cos (2 \pi x+8 \pi)$$

Describe a relationship between the graphs of \(y=\sin x\) and \(y=\cos x\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.