A Rational Function is a quotient (fraction) where there the numerator and the denominator are both polynomials.
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
But what does this mean?
Why is this important?
Well, the one distinguishing characteristic of a Rational Function is that there is a variable in the denominator.
Rational Function with Vertical and Horizontal Asymptotes
And this is important because the graph of all Rational Functions have Asymptotes!
What’s an asymptote?
An asymptote is a line that a function either never touches or rarely touches, as Math is Fun so nicely states.
Think of a speed limit. Imagine you are driving on a road and the posted sign says 55 mph. Now, if we were perfect, law abiding citizens, we would only drive as fast as the sign says, without ever going faster. This is just like an asymptote. It is a curve’s speed limit. The graph will snuggle up to the asymptote, but never cross.
But why are there asymptotes in the first place?
Great question! Okay, so we know that a variable is a letter or symbol that represents a number. Which means, we can allow variables to be any number we want!
But, if the variable is in the denominator, then we must restrict the types of numbers we are allowed to use. Simply put, we can’t divide by zero!
Therefore, we need a way to identify these asymptotes, so we know how to restrict the variables.
Rational Function with an Oblique Asymptote
Don’t worry; the process is really quite simple!
First, we will talk about the three different types of asymptotes:
- Vertical Asymptotes
- Horizontal Asymptotes
- Oblique Asymptotes
Each of the first two types gives us a good picture of what they look like – vertical line, horizontal line.
What’s an Oblique Asymptote?
An oblique asymptote is anything that isn’t horizontal or vertical. It can be diagonal (slant), parabolic, cubic, etc.
Next, we will talk about a very important concept called Removable Discontinuity. These are special circumstances where we will be removing a vertical asymptote and replacing it with a hole.
We will rely heavily on our ability to solve equations as well as our knowledge of Synthetic Division and Long Division, and together we will be able to identify all asymptotes for various Rational Functions.
Finding Asymptotes (How-To) – Video
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