/*! 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 24 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 24

Sketch the graph of the function. (Include two full periods.) $$y=-2 \sec 4 x+2$$

Short Answer

Expert verified
The graph of the function \(y=-2 \sec 4x + 2\) is the graph of a secant function that is vertically reflected, vertically stretched by a factor of 2, horizontally compressed by a factor of 1/4, and translated upward by 2 units. Note for sketching: the function will have vertical asymptotes at x=(n+0.5)蟺/2, maxima at -4, minima at 4 and the period of the function is 蟺/2.

Step by step solution

01

Vertical Reflection

Since the amplitude is negative, this function reflects vertically. That is, wherever the sec(x) graph is, the -sec(x) is the reflection across the x-axis.
02

Vertical Stretching

The '-2' in front of the secant reflects the function and also indicates a vertical stretch by a factor of 2. Remember that sec(x) has asymptotes at x=(n+0.5)蟺, where n is an integer, and maxima and minima at multiples of 蟺. After vertical stretching, the secant function will have maxima and minima at 卤2.
03

Horizontal Compression

The 4 in 4x indicates a horizontal compression by a factor of 1/4. That means, the period of the function is 蟺/2 instead of 2蟺.
04

Vertical Translation

The '+2' at the end denotes that the graph is shifted upward 2 units. Therefore, the new maxima and minima are at (卤2)+ 2 = 卤4, respectively.
05

Sketch the Graph

Now that all transformations are understood, the sketching of the graph can be done. Remember, in sketching secant functions, start with sketching the cosine function and then convert it to secant by removing pieces at the zeros of cosine and moving maximums and minimums to the new specified values.

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

Writing When the radius of a circle increases and the magnitude of a central angle is constant, how does the length of the intercepted arc change? Explain your reasoning.

Converting to \(\mathrm{D}^{\circ} \mathrm{M}^{\prime} \mathrm{S}^{\prime \prime}\) Form \(\quad\) Convert each angle measure to degrees, minutes, and seconds without using a calculator. Then check your answers using a calculator. (a) \(240.6^{\circ}\) (b) \(-145.8^{\circ}\)

The circular blade on a saw rotates at 5000 revolutions per minute. (a) Find the angular speed of the blade in radians per minute. (b) The blade has a diameter of \(7 \frac{1}{4}\) inches. Find the linear speed of a blade tip.

\(\quad\) A point on the end of a tuning fork moves in simple harmonic motion described by \(d=a \sin \omega t .\) Find \(\omega\) given that the tuning fork for middle C has a frequency of 264 vibrations per second.

Airplane Ascent During takeoff, an airplane's angle of ascent is \(18^{\circ}\) and its speed is 275 feet per second. (a) Find the plane's altitude after 1 minute. (b) How long will it take for the plane to climb to an altitude of \(10,000\) feet?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.