Get started for free
Log In Start studying!
Get started for free Log out
Chapter 12: Problem 27
Plot the curve for \(f(x)=\frac{1}{x^2}\) and label any asymptotes.
Short Answer
Expert verified
Vertical asymptote: x = 0. Horizontal asymptote: y = 0.
Step by step solution
01
Understand the Function
The given function is a rational function: \[ f(x) = \frac{1}{x^2} \]This function has a denominator that can never be zero, so it has specific behaviors to identify such as asymptotes.
02
Identify Vertical Asymptote
A vertical asymptote occurs where the denominator is zero. For the function \( f(x) = \frac{1}{x^2} \), the denominator is zero when \( x = 0 \). Therefore, the vertical asymptote is at \( x = 0 \). Label this on the graph.
03
Identify Horizontal Asymptote
As \( x \) approaches infinity or negative infinity, the value of \( f(x) = \frac{1}{x^2} \) approaches 0. Thus, the horizontal asymptote is at \( y = 0 \). Label this on the graph.
04
Plot Key Points
Calculate and plot some key points to understand the behavior of the function. For example, when \( x = 1 \), \( f(1) = 1 \); when \( x = 2 \), \( f(2) = \frac{1}{4} \); when \( x = -1 \), \( f(-1) = 1 \); and when \( x = -2 \), \( f(-2) = \frac{1}{4} \). Plot these points on the graph.
05
Sketch the Curve
Using the asymptotes and key points, sketch the curve of the function. The curve will approach but never touch the asymptotes at \( x = 0 \) and \( y = 0 \), and it will be in the first and second quadrants because \( f(x) > 0 \) for all real values of \( x \) except at the vertical asymptote.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vertical Asymptotes
Understanding vertical asymptotes is crucial when plotting rational functions. A vertical asymptote represents a value of x where the function tends toward infinity. For the function, \( f(x) = \frac{1}{x^2} \), look at the denominator, \( x^2 \). The denominator becomes zero at \( x = 0 \). Therefore, we have a vertical asymptote at \( x = 0 \). This means the graph will approach this line but never touch or cross it. It's essential to recognize this behavior to understand how the function behaves near this point.
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of a function as x approaches very large positive or negative values. For \( f(x) = \frac{1}{x^2} \), as \( x \to \infty \) or \( x\to -\infty \), the value of \( f(x) \) tends to zero. Therefore, the horizontal asymptote is at \( y = 0 \). This tells us that no matter how large or small x gets, the function's value will approach 0, but never actually touch or reach zero. This is another characteristic you should mark on your graph to help sketch the curve accurately.
Key Points Plotting
Plotting key points helps to form a clearer picture of the function's behavior. With the function \( f(x) = \frac{1}{x^2} \), let's find some simple values:
- When \( x = 1 \), \( f(1) = 1 \).
- When \( x = 2 \), \( f(2) = \frac{1}{4} \).
- When \( x = -1 \), \( f(-1) = 1 \).
- When \( x = -2 \), \( f(-2) = \frac{1}{4} \).
Plotting these points gives us visuals on both sides of the asymptote at \( x = 0 \). These points highlight how quickly the function values drop as x moves away from zero, yet remain positive.
Curve Sketching
Curve sketching takes into account all information compiled from asymptotes and key points plotting. Begin by drawing your asymptotes: a vertical line at \( x = 0 \) and a horizontal line at \( y = 0 \).
Then, plot the key points you calculated earlier. Keep in mind:
- The curve will approach the x-axis as it extends to infinity in either direction.
- The function will never touch or cross the vertical asymptote at \( x = 0 \).
- The graph exists only in the first and second quadrants because all y-values are positive.
The careful plotting of points and understanding of asymptotes helps in drawing an accurate and informative graph.
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
Recommended explanations on Math Textbooks
Decision Maths
Read ExplanationDiscrete Mathematics
Read ExplanationCalculus
Read ExplanationProbability and Statistics
Read ExplanationMechanics Maths
Read ExplanationStatistics
Read ExplanationWhat do you think about this solution?
We value your feedback to improve our textbook solutions.
Study anywhere. Anytime. Across all devices.
Sign-up for free
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept
Privacy & Cookies Policy
Privacy Overview
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Always Enabled
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.