Rational Functions

Rational Function

Definition

A function where the numerator and denominator are both polynomials

Proper and Improper

Definition

A rational function is proper if the degree of the denominator is greater than the degree of the numerator, otherwise it is improper

Properties

Properties

For the function

  • eqn of vertical asymptote:
    • Think of it like means
  • eqn of horizontal asymptote:
  • domain:
  • range:

Graphing

Steps:

  1. Factor
  2. Determine holes
  3. Determine asymptotes or linear oblique asymptotes if required
    • Check function degrees:
      • if numerator degree > denominator degree:
        • asymptote degree = numerator degree - denominator degree
        • E.g numerator degree = 3 and denominator degree = 2, difference = 1, so asymptote type = linear oblique asymptote
        • E.g numerator degree = 3 and denominator degree = 1, difference = 2, so "asymptote" type is quadratic
      • else if numerator degree < denominator degree:
      • else:
  4. Determine x-intercepts and y-intercepts
  5. Graph

Examples

Determining Equations

State a possible equation of the rational function with the following features:

Example

Feature Value Notes
Vertical asymptote In the denominator
Horizontal asymptote Top and bottom will be the same degree
X-intercept Goes in the numerator

Example

Feature Value Notes
VA
HA
X-int 2 x-intercepts, means top and bottom must be degree 2

Example

Feature Value Notes
VA In the denominator like solving quadratic roots
HA None Must have a linear oblique asymptote
Degree of numerator must be greater than the degree of the denominator
X-int

Example

Feature Value Note
VA
HA Degree of numerator must be less than the degree of the denominator
X-int
Hole must exist on both the denominator and numerator.

Graph

Reciprocal 1.png
Using the Big Little Concept

Determining Sign Changes

The sign changes of a graph can be determined, in some instances, by the equation alone

E.g

We can determine that the equation will change signs at