/*! 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 57 (a) Draw the graphs of the famil... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 57

(a) Draw the graphs of the family of functions $$ f(x)=\frac{a}{2}\left(e^{x / a}+e^{-x / a}\right) $$ (b) How does a larger value of \(a\) affect the graph?

Short Answer

Expert verified
A larger value of \( a \) makes the graph wider and flatter.

Step by step solution

01

Understand the Function

The function \( f(x) = \frac{a}{2} \left( e^{x/a} + e^{-x/a} \right) \) is a hyperbolic cosine function scaled by \( a \). It can be rewritten as \( f(x) = a \cdot \cosh(\frac{x}{a}) \), where \( \cosh \) is the hyperbolic cosine function.
02

Graph the Core Function

Plotting \( \cosh(x) \) shows a symmetrical shape about the \( y \)-axis. \( \cosh(x) \) is always positive, with its minimum value 1 at \( x=0 \) and increasing exponentially as \( |x| \) increases.
03

Scale the Graph Vertically

For a general \( a \), the function \( f(x) = a \cdot \cosh(\frac{x}{a}) \) scales the graph vertically by a factor of \( a \). This means the minimum value at \( x=0 \) will be \( a \) instead of 1.
04

Scale the Graph Horizontally

The term \( \cosh(\frac{x}{a}) \) compresses the function horizontally by a factor of \( a \). Wider values of \( a \) elongate the graph horizontally, making it less steep as compared to narrow ones.
05

Explore Different Values of \(a\)

Graph functions for different values of \( a \), for example, \( a = 1, 2, 3 \). Observe how changes in \( a \) adjust the width (spread) of the graph.
06

Analyze the Effect of Larger \(a\)

Larger values of \( a \) result in a graph that is wider and flatter. The exponential growth/decline on either side of \( x=0 \) occurs more slowly, indicating less steepness in the curve.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions
Understanding how to graph functions is a vital math skill. It helps visualize equations and provides insight into their behavior. At its core, graphing a function involves plotting its values within a coordinate system. This means you assign points on a graph that correspond to functional output values for particular inputs. In this way, you observe how a function behaves across a specified domain. Graphing involves several key steps:
  • Identify the function type. Here, we are dealing with a hyperbolic cosine function.
  • Calculate key points. For instance, find where the function reaches its minimum, maximum, intersect, etc.
  • Draw the curve. This involves smoothly connecting these points to reflect the function's continuity.
In this example, you graph the function by transforming a basic hyperbolic cosine function, resulting in symmetrically curved lines which offer a broad insight into its exponential behavior as it stretches along the graph.
Hyperbolic Cosine
The hyperbolic cosine function is often abbreviated as cosh. Formally, it is expressed as \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). It's a critical mathematical function used in various applications, including physics, engineering, and hyperbolic geometry. Here are its fundamental characteristics:
  • Symmetry: Hyperbolic cosine is even, meaning \( \cosh(x) = \cosh(-x) \).
  • Minimum Value: At \( x=0 \), \( \cosh(x) \) reaches its minimum value of 1.
  • Exponential Growth: As \( x \) moves away from zero in either direction, \( \cosh(x) \) increases rapidly.
Due to its symmetrical nature, the graph of the hyperbolic cosine rises identically on both sides of the y-axis. This unique trait makes it distinct from circular trigonometric functions, offering a different form of exponential growth profile.
Function Transformations
Function transformations alter the basic appearance of a graph, allowing it to stretch, compress, or shift within a coordinate plane. They are central concepts in understanding changes in graph behavior based on variable adjustments. For the given hyperbolic cosine function, the expression \( f(x) = a \cdot \cosh(\frac{x}{a}) \) applies both vertical and horizontal transformations:
  • Vertical Stretch: Multiplying by \( a \) scales the graph vertically, meaning the peak at \( x=0 \) changes from 1 to \( a \).
  • Horizontal Compression: Using \( \frac{x}{a} \) shortens the graph horizontally. Increasing \( a \) makes the graph wider and less steep.
Combined, these transformations not only change the appearance but also the steepness and spread of the graph. Larger \( a \) values result in broader, flatter curves, illustrating that function transformations are powerful tools in graph manipulation.

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 population of the world was 5.7 billion in 1995 and the observed relative growth rate was 2% per year. (a) By what year will the population have doubled? (b) By what year will the population have tripled?

Solve the inequality. $$ 3 \leq \log _{2} x \leq 4 $$

An infectious strain of bacteria increases in number at a relative growth rate of 200% per hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 24 hours. How long will it take for the critical level to be reached if the same person is infected with 10 bacteria?

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ e^{x}=-x $$

Compound Interest Nancy wants to invest \(\$ 4000\) in saving certificates that bear an interest rate of 9.75\(\%\) per year, compounded semiannully. How long a time period should she choose in order to save an amount of \(\$ 5000 ?\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.