Chapter 1: Introduction to Parabolic Asymptotes

During my time as a math tutor at a community college, I found great joy in teaching. Unfortunately, I lost that position due to the pandemic, a job I truly cherished. Such passion for one’s work is hard to come by. However, this discussion will focus on the topic of parabolic asymptotes.

In my tutoring experience, many students exhibited a distinct apprehension towards parabolic asymptotes. They were generally comfortable with horizontal asymptotes and even slant asymptotes, despite the occasional presence of vertical asymptotes. Yet, parabolic asymptotes posed a significant challenge for them.

To illustrate this, there once was a television series called "The Beastmaster," which symbolized a mastery that my students lacked when it came to understanding parabolic asymptotes.

Let’s explore the concept further. A rational function can be expressed as f(x) = g(x)/h(x), where both g(x) and h(x) are polynomials with rational coefficients. While we could introduce more complexity by using real coefficients, we will keep it simpler for now.

The expression f(x) = g(x)/h(x) can be simplified using polynomial long division, a technique well-known to students who have completed intermediate algebra. By setting y = q(x), we can identify our end-behavior asymptote. If q(x) is a polynomial of degree two, we refer to it as a parabolic asymptote. This is straightforward, as students recognize the graph of a second-degree polynomial as a parabola. The term "asymptote" signifies that the function f(x) approaches this parabola as the absolute value of x increases infinitely in both directions.

Section 1.1: Analyzing Rational Functions

Let’s consider a rational function, where g(x) is a fourth-degree polynomial in the numerator and h(x) is an irreducible quadratic in the denominator. By performing the long division, we can express it as a quotient q(x) along with a remainder r(x).

From our calculations, we find that q(x) = 3x² + x + 4 and r(x) = x + 1. We then graph y = q(x) as a dashed line to represent the parabolic asymptote. To determine the points where the function intersects its end-behavior asymptote, we solve r(x) = 0. In this case, solving x + 1 = 0 leads us to x = -1.

To further clarify why we resolve r(x) = 0 to find the intersections with the end-behavior asymptote, we can set up a system of equations: y = q(x) + r(x)/h(x) and y = q(x). The solution to this system will show where the original function intersects its end-behavior asymptote, which is evident at the points where r(x) = 0.

It’s important to note that the x-axis is not depicted, as it would interfere with the distinction between f and q.

Section 1.2: Homework and Practice

Note the location of point A. What must its x-coordinate be? The anticipated answer is negative one.

For additional practice, I encourage you to tackle the following problems. Each one should include a comprehensive graph with all pertinent details, including every asymptote associated with the function.

Chapter 2: Deepening Understanding of Asymptotes

This video titled "Rational Functions - Parabolic Asymptotes" explores how to identify and understand parabolic asymptotes in rational functions, enhancing your grasp of the concept.

The second video, "Calculus 2.3d - Rational Functions - Parabolic Asymptote," provides a detailed breakdown of parabolic asymptotes, further solidifying your knowledge and skills in this area.

Thank you for engaging with this material!