/*! 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 46 Use a graphing utility to graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 46

Use a graphing utility to graph the function. (Include two full periods.) $$y=2 \sec (2 x-\pi)$$

Short Answer

Expert verified
The graph of the function \(y = 2 sec(2x - \pi)\) is a vertically stretched and horizontally compressed version of the basic secant graph, also shifted to the right by \(\pi\) units. It has vertical asymptotes at \(x = \pi/2 + 2k\pi\), \(k\) being any integer, and this new period is \(\pi\). Two full periods of the function are captured in the interval \(-\pi \leq x \leq \pi\).

Step by step solution

01

Understand the Basic Secant Graph

The secant function, \(y = sec(x)\), is the reciprocal of the cosine function, \(y = cos(x)\). Its graph has a series of vertical asymptotes and curves, with each curve spanning the distance between two vertical asymptotes. It has a period of \(2\pi\) and undefined values at \(\pi/2 + k\pi\), where \(k\) is any integer.
02

Apply the Transformations to the Graph

The function to graph is \(y = 2 sec(2x - \pi)\). The multiplicative factor of 2 outside the secant function will vertically stretch the graph by a factor of 2. The term inside the secant function affects the graph in two ways: the \(-\pi\) shifts the graph to the right by \(\pi\) units, and the multiplier of 2 on the \(x\) term compresses the graph horizontally by a factor of 2. Hence, the period of the function is half of that of the basic secant function. Identify the new locations for the vertical asymptotes (at \(\pi/2 + 2k\pi\)) and sketch the vertically stretched, compressed, and shifted secant curve.
03

Sketching two full periods

To sketch two complete periods of this modified secant function, select a range for \(x\) that includes two instances of the periodicity. For instance, since the period has been halved due to the horizontal compression, the new period is \(\pi\) instead of \(2\pi\). Thus, you can sketch two full periods over the range \(-\pi \leq x \leq \pi\).

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

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$y=\frac{4}{x}+\sin 2 x, \quad x>0$$

Use a graphing utility to graph the functions \(f(x)=\sqrt{x}\) and \(g(x)=6\) arctan \(x .\) For \(x>0,\) it appears that \(g>f .\) Explain why you know that there exists a positive real number \(a\) such that \(ga .\) Approximate the number \(a\).

A sprinkler on a golf green sprays water over a distance of 15 meters and rotates through an angle of \(140^{\circ} .\) Draw a diagram that shows the region that the sprinkler can irrigate. Find the area of the region.

Finding Arc Length Find the length of the are on a circle of radius \(r\) intercepted by a central angle \(\boldsymbol{\theta}\). $$r=15 \text { inches, } \theta=120^{\circ}$$

Consider the function \(f(x)=x-\cos x\) (a) Use a graphing utility to graph the function and verify that there exists a zero between 0 and \(1 .\) Use the graph to approximate the zero. (b) Starting with \(x_{0}=1,\) generate a sequence \(x_{1}, x_{2}\) \(x_{3}, \ldots,\) where \(x_{n}=\cos \left(x_{n-1}\right) .\) For example \(x_{0}=1\) \(x_{1}=\cos \left(x_{0}\right)\) \(x_{2}=\cos \left(x_{1}\right)\) \(x_{3}=\cos \left(x_{2}\right)\) What value does the sequence approach?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Mechanics 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.