Asymptotes of rational functions | Polynomial and rational functions | Algebra II | Khan Academy

TL;DR

This video explains how to graph a rational function and understand the behavior of the graph as x approaches infinity or certain values.

Transcript

In this video, we're going to see if we can graph a rational function. A rational function is just a function that has an expression on the numerator and the denominator. It has a polynomial in the numerator-- Let's see, we have x squared over-- and another polynomial in the denominator --x squared minus 16. We could obviously graph this by just tr... Read More

Key Insights

• 😑 Rational functions have a polynomial expression in the numerator and denominator.
• ♾️ As x approaches infinity or negative infinity, the graph approaches a horizontal asymptote.
• 🚦 Vertical asymptotes occur when the denominator becomes zero.
• 📈 The behavior of the graph near the asymptotes helps determine the shape of the graph.
• ☺️ Plugging in values of x helps understand how the function behaves as x approaches certain values.
• 🍉 The coefficients of the terms in the numerator and denominator determine the values of the asymptotes.
• ❓ Understanding asymptotes is crucial for accurate graphing and analysis of rational functions.

Questions & Answers

Q: What is a rational function?

A rational function is a function that has a polynomial in the numerator and the denominator.

Q: How can we graph a rational function?

To graph a rational function, it is helpful to understand the behavior of the graph as x approaches infinity and certain values. This helps us identify horizontal and vertical asymptotes.

Q: What is an asymptote?

An asymptote is a line that the graph of a function approaches but never touches. In the case of a rational function, there can be both horizontal and vertical asymptotes.

Q: How can we determine the behavior of a rational function as x approaches certain values?

By plugging in values of x and finding the corresponding y-values, we can see how the function behaves as x gets larger or approaches specific numbers.

Q: What happens to the graph of a rational function as x approaches infinity?

As x approaches infinity, the graph of a rational function approaches a horizontal asymptote. The y-values get closer and closer to a certain value.

Q: How can we identify vertical asymptotes?

Vertical asymptotes occur when the denominator of the rational function becomes zero. By factoring the denominator and identifying the values that make it zero, we can determine the vertical asymptotes.

Q: Can the horizontal asymptote be different for different rational functions?

Yes, the value of the horizontal asymptote depends on the coefficients of the terms in the numerator and denominator. It can be any real number or negative infinity.

Q: Why is it important to understand the asymptotes of a rational function?

Understanding the asymptotes helps us visualize the overall shape of the graph. It also tells us where the function is undefined and how it behaves as x approaches certain values.

Summary & Key Takeaways

• A rational function is a function with a polynomial expression in the numerator and denominator.

• To graph a rational function, it is important to understand its basic structure and the behavior as x approaches infinity.

• The graph of a rational function can approach a horizontal asymptote, a vertical asymptote, or both.