Rational Functions

Major objectives:
- Rational Functions and Asymptotes
- Transformations of ${y = 1/x }$
- Asymptotes of Rational Functions
- Graphing Rational Functions
- Slant Asymptotes and End Behavior
- Applications
Rational Functions and Asymptotes
A rational function is a ratio of polynomials:

${r(x)=\frac{P(x)}{Q(x)}}$
What can go wrong? We've got a denominator: perhaps it's zero, which would be bad....
Transformations of
This function is interesting: it is odd (possesses symmetry), it is its own inverse, and its graph is made up of special functions called hyperbolas:
- Consider rational functions of the form
  
  ${r(x)=\frac{ax+b}{cx+d}}$
  Each one of their graphs can be obtained from that of 1/x by transformations. Let's see how -- by long-division.
- Mathematica Demonstration of these linear fractional transformations

Asymptotes of Rational Functions

Here's two of the most important things about rational functions:

They have vertical asymptotes (where the denominator polynomial is zero but the numerator polynomial is not zero), and
They have end-behavior asymptotes (when x gets big in size, tending towards ${\pm\infty}$ )

So it's easy to find the vertical asymptotes: for ${r(x)=\frac{P(x)}{Q(x)}}$ just solve the equation ${Q(x)=0}$ :

The end-behavior asymptote is the polynomial that the function ${r(x)}$ approaches as ${x\longrightarrow\pm\infty}$ .

This might be a horizontal asymptote (which means that ${r(x)\longrightarrow{c}}$ , a constant function):

This might be a slant asymptote (which means that ${r(x)\longrightarrow{mx+b}}$ , a linear function):

Or this may be more complicated (e.g. a quadratic asymptote, where ${r(x)\longrightarrow{ax^2+bx+c}}$ , a quadratic function):

It turns out that every rational function can be expressed as the sum of a polynomial ${e(x)}$ and rational function ${z(x)}$ where the numerator degree is less than the denominator degree:

${r(x)=e(x)+z(x)}$

I use this notation for the functions ${e(x)}$ and ${z(x)}$ because ${e(x)}$ controls the end behavior, whereas ${z(x)}$ controls the behavior near the origin (the "near" behavior).

${r(x)=\frac{x^3-2x^2+1}{x-2}=x^2+\frac{1}{x-2}}$

The inner behaviour (near zero) The end behavior (as ${x\to\pm\infty}$ The transition zone, where both terms are clearly visible

In this example, ${e(x)=x^2}$ and ${z(x)=\frac{1}{x-2}}$ . (degree 0 over degree 1).
${e(x)}$ controls the end behavior, and ${z(x)}$ controls the near behavior.

The way we find the end-behavior asymptote is via long-division (I hear you groan!). Let's try one....

Graphing Rational Functions
The graphing strategy for rational functions is the same as for polynomials, only we include the asymptotes.
Links:
- Mathematica Demonstration of rational functions

Website maintained by Andy Long. Comments appreciated.


${r(x)=\frac{x^3-2x^2+1}{x-2}=x^2+\frac{1}{x-2}}$
The inner behaviour (near zero)	The end behavior (as ${x\to\pm\infty}$	The transition zone, where both terms are clearly visible
In this example, ${e(x)=x^2}$ and ${z(x)=\frac{1}{x-2}}$ . (degree 0 over degree 1). ${e(x)}$ controls the end behavior, and ${z(x)}$ controls the near behavior.