Rational Functions
An overview of properties of rational functions with an emphasis on connections to limits.
Table of Contents
- Defining rational functions
- A review of polynomial functions
- Incorporating limit notation
- Defining rational functions
- Discontinuities
- A close look at zeros of rational functions
- Holes
- Vertical asymptotes
- Limits at discontinuities
- End behavior
- BOB0 - BOTN - EATS DC
- A close look at end behavior
- Putting it all together
A review of polynomial functions
A [b]rational function[/b] [math]r[/math] is a function that can be expressed in the form [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math], where [math]p[/math] is a polynomial and [math]q[/math] is a non-zero polynomial. [br][br]A [b]polynomial function[/b] [math]f[/math] is a function of the form [math]f\left(x\right)=c_nx^n+c_{n-1}x^{n-1}+...+c_1x+c_0[/math] where each of the [math]c_i[/math]s are real numbers and [math]n[/math] is a non-negative integer.[br][br]Before we talk too much about rational functions, it's important that you're first comfortable with polynomial functions.
Enter an example of a polynomial function below.
Let's make sure you understand the notation above. Suppose [math]f\left(x\right)=c_nx^n+c_{n-1}x^{n-1}+...+c_1x+c_0[/math] is a polynomial. Which of the following symbols represents [math]f[/math]'s leading coefficient?
If [math]f\left(x\right)=c_nx^n+c_{n-1}x^{n-1}+...+c_1x+c_0[/math] is a polynomial, which of the following symbols represents [math]f[/math]'s degree?
If [math]f\left(x\right)=c_nx^n+c_{n-1}x^{n-1}+...+c_1x+c_0[/math] is a polynomial, which of the following symbols represents [math]f[/math]'s [math]y[/math]-intercept?
If [math]f\left(x\right)=c_nx^n+c_{n-1}x^{n-1}+...+c_1x+c_0[/math] is a polynomial, what is the maximum number of real zeros [math]f[/math] can have?
Give an example of such a polynomial below.
If [math]f\left(x\right)=c_nx^n+c_{n-1}x^{n-1}+...+c_1x+c_0[/math] is a polynomial, what is the minimum number of real zeros [math]f[/math] can have?
Give an example of such a polynomial below.
If you're asked to find the real zeros of a given polynomial [math]f[/math], what would you do?
Plot a polynomial below with zeros at 0, 1, and 3.
A close look at zeros of rational functions
Plot a rational function with a zero at x=1.
If [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] is a rational function with a zero at [math]x=3[/math], what can you conclude?
The converse, however, is not true. That is, if [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] is a rational function with [math]p\left(3\right)=0[/math], it is not necessarily the case that [math]r[/math] has a zero at [math]x=3[/math]. Plot an example of such a rational function below.
Plot a rational function [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] below such that [math]p\left(-1\right)=p\left(2\right)=0[/math], and [math]r[/math]'s only zero occurs at [math]x=2[/math].
Correctly complete the claim below
[math]x=a[/math] is a zero of the rational function [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] if and only if:
BOB0 - BOTN - EATS DC
BOB0 - BOTN - EATS DC is one of those horrible mnemonic devices that pride-less students love. (Other such monstrosities include cross multiplication, FOIL, SOH-CAH-TOA, and that stupid quadratic formula song.) Let's unpack this mnemonic, I guess. I am not excited about this, but, also being pride-less, I will reluctantly do this if it means you will understand non-vertical asymptotes of rational functions.
"BOB0" means "Bigger On Bottom: 0"
Or, with slightly more detail, if a rational function is "Bigger On Bottom", then it has a horizontal aymptote of [math]y=0[/math]. Or, if you want to say it in a (prideful) way that won't make me cringe:[br][br]Let [math]r[/math] be a rational function that we write as [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math], where [math]p[/math] and [math]q[/math] are polynomials, and let the degree of [math]q[/math] be greater than the degree of [math]p[/math]. Then [math]r[/math] has a horizontal asymptote of [math]y=0[/math].[br][br]
Plot a "BOB0" rational function.
Let [math]f(x)[/math] be a BOB0 rational function, so therefore we can write a limit to describe its end behavior. We can write [math]\lim_{x\to A}f\left(x\right)=B[/math], where...
Which of the following could be [math]A[/math]?
Which of the following could be [math]B[/math]?
Explain in your own words why BOB0 works. (Meaning, why is it the case that Bigger On Bottom results in a horizontal asymptote of [math]y=0[/math]?)
"BOTN" means...
Well, what do you think BOTN is meant to convey? Try to explain it in a way that won't make me cringe. If you need to experiment before you answer, skip ahead to the next question.
Plot a "BOTN" rational function.
Let [math]f(x)[/math] be a BOTN rational function, so therefore we can write a limit to describe its end behavior. We can write [math]\lim_{x\to A}f\left(x\right)=B[/math], where...
Which of the following could be [math]A[/math]?
Which of the following could be [math]B[/math]?
Explain in your own words why BOTN works.
"EATS DC" means "Exponents Are The Same: Divide Coefficients"
Okay, so the way I would articulate EATS DC this way:[br][br]Let [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] be a rational function where [math]p[/math] is a polynomial with leading coefficient [math]a[/math], and [math]q[/math] is a polynomial with leading coefficient [math]b[/math]. Let the degree of [math]p[/math] be equal the degree of [math]q[/math]. Then [math]r[/math] has a horizontal asymptote of [math]y=\frac{a}{b}[/math].
Plot an "EATS DC" rational function.
Let [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] be an EATS DC rational function where [math]p[/math] is a polynomial with leading coefficient [math]a[/math], and [math]q[/math] is a polynomial with leading coefficient [math]b[/math]. We can therefore write [math]\lim_{x\to A}r\left(x\right)=B[/math] to describe [math]r[/math]'s end behavior, where...
Which of the following could be [math]A[/math]?
Which of the following could be [math]B[/math]?