/*! 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 18 In Exercises \(5-30,\) find an a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 18

In Exercises \(5-30,\) find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. $$ y=1-\frac{1}{x+3} $$

Short Answer

Expert verified
Use x: [-10, 4] and y: [-5, 5] in graphing software to show key features.

Step by step solution

01

Identify Key Features of the Function

The function given is \( y=1 - \frac{1}{x+3} \). It has a vertical asymptote where the function is undefined, which occurs at \( x = -3 \). Additionally, since it contains a rational term \( -\frac{1}{x+3} \), you should expect the graph to change significantly near this point.
02

Determine Horizontal Asymptote

As \( x \to \infty \) or \( x \to -\infty \), the fraction \( \frac{1}{x+3} \) approaches zero, hence \( y \to 1 \). Thus, there is a horizontal asymptote at \( y = 1 \). This feature should be included in the graph to provide overall behavior of the function.
03

Choose Appropriate X-Values for Viewing Window

To capture the vertical asymptote at \( x = -3 \) and extend sufficiently to observe the horizontal behavior, you might choose \( x \) values from \( -10 \) to \( 4 \). This range should capture the essential changes in the function without extending unnecessarily.
04

Choose Appropriate Y-Values for Viewing Window

Since the function approaches \( y = 1 \) and becomes undefined at \( x = -3 \), observe a change below and above \( y = 1 \). It's reasonable to start with \( y \) values from \(-5 \) to \( 5 \) to ensure all interesting y-values are captured.
05

Verify with Graphing Software

Input the function into graphing software using the chosen windows (X: \[-10, 4\], Y: \[-5, 5\]) and observe the behavior. Check to ensure that the vertical asymptote at \( x = -3 \) and the horizontal asymptote are accurately represented.

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.

Vertical Asymptote
A vertical asymptote occurs in a function at a specific x-value where the function becomes undefined or approaches infinity. In the given problem, the function is defined as \( y = 1 - \frac{1}{x+3} \). Here, the rational part of the function \( -\frac{1}{x+3} \) becomes undefined when \( x = -3 \). This is because you cannot divide by zero, and when you substitute \( x = -3 \) into the function, the denominator becomes zero. This makes \( x = -3 \) the location of a vertical asymptote.
  • Vertical asymptotes signify points where the function will shoot up to positive or negative infinity.
  • As you move closer to these x-values, the function's value becomes very large (positively or negatively).
  • On a graph, vertical asymptotes look like lines that the curve approaches but never crosses.
Understanding vertical asymptotes is crucial, especially when graphing rational functions.
Horizontal Asymptote
Horizontal asymptotes describe the behavior of a graph as x approaches negative or positive infinity. In our function, \( y = 1 - \frac{1}{x+3} \), analyzing what happens as \( x \) becomes very large or very small is necessary. As \( x \to \infty \) or \( x \to -\infty \), the fraction \( \frac{1}{x+3} \) shrinks towards zero because dividing by a larger number makes the fraction smaller. Therefore, \( y \to 1 \) because \( 1 - 0 = 1 \). Hence, the horizontal asymptote occurs at \( y = 1 \).
  • Horizontal asymptotes indicate the value the function approaches but may never actually reach.
  • These lines are more about long-term behavior rather than immediate changes in the graph.
  • They are particularly important for understanding end behavior in graphs of rational functions.
Recognizing horizontal asymptotes helps estimate where the graph flattens out.
Rational Functions
Rational functions are formed by the ratio of two polynomials, often producing interesting behaviors such as asymptotes. The given problem, \( y = 1 - \frac{1}{x+3} \), is a rational function involving a linear polynomial divided by another linear polynomial. This functionality results in characteristics like vertical and horizontal asymptotes due to divisions in the form of \( \frac{1}{x+3} \).
  • Rational functions have at least one vertical asymptote, marked by points where the denominator equals zero.
  • They may possess horizontal asymptotes that offer insights into the function's end behavior.
  • Graphing rational functions involves careful consideration to capture critical changes, particularly around asymptotes.
Mastering rational functions aids in comprehending the complexity behind many algebraic expressions.
Graphing Software
Graphing software tools help visualize mathematical functions beyond manual plotting, especially beneficial for complex functions like rational functions. When dealing with functions such as \( y = 1 - \frac{1}{x+3} \), picking the right viewing window is essential for accurately representing the function's important traits, such as its asymptotes and behavior.
  • Choose x-value windows to include features like vertical asymptotes, e.g., \( x = -10 \) to \( 4 \) to observe both asymptotic boundaries in this instance.
  • Select y-values to capture the function's vertical range, like \( y = -5 \) to \( 5 \), highlighting where the graph settles around the horizontal asymptote.
  • Ensuring that these critical transformational features are covered gives a complete understanding of the function's overall behavior.
You gain a deeper comprehension of the scope of rational functions by leveraging graphing software.

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

A triangle has sides \(a=2\) and \(b=3\) and angle \(C=60^{\circ} .\) Find the length of side \(c .\)

A triangle has sides \(a=2\) and \(b=3\) and angle \(C=40^{\circ} .\) Find the length of side \(c .\)

In Exercises \(71-74,\) you will explore graphically the general sine function $$f(x)=A \sin \left(\frac{2 \pi}{B}(x-C)\right)+D$$ as you change the values of the constants \(A, B, C,\) and \(D .\) Use a CAS or computer grapher to perform the steps in the exercises. The vertical shift \(D\) Set the constants \(A=3, B=6, C=0\) . a. Plot \(f(x)\) for the values \(D=0,1,\) and 3 over the interval \(-4 \pi \leq x \leq 4 \pi .\) Describe what happens to the graph of the general sine function as \(D\) increases through positive values. b. What happens to the graph for negative values of \(D ?\)

For Exercises \(51-54,\) solve for the angle \(\theta,\) where \(0 \leq \theta \leq 2 \pi\) $$\cos 2 \theta+\cos \theta=0$$

Use graphing software to graph the functions specified in Exercises \(31-36\) . Select a viewing window that reveals the key features of the function. Graph two periods of the function \(f(x)=3 \cot \frac{x}{2}+1\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.