Introduction to Rational Equations

15 November 2010

Rational EquationsDefinition

anxn + an-1xn-1 + … + a2x2 + a1n1 + a0

bnxn + bn-1xn-1 + … + b2x2 + b1n1 + b0

A polynomial divided bya polynomial

Rational equations are fractions

Polynomial

Polynomial

Polynomials vs. Rationals

Polynomial Equations

Rational Equations

ContinuousSmooth

DiscontinuousHas jumps, breaks, sharp bends, and/or holes

Discontinuities

Affect the shape, domain and range of an equation

Three major types for rational equations: Vertical Asymptotes Horizontal Asymptotes Removable Discontinuities (aka

Holes)

Asymptotes

A line that the equation approaches but can never reach

Not part of the equation

Represented by a dashed line

Vertical Asymptotes

Occur when the denominator equals zero

Can never be crossed Always in the form x = Abbreviated VA

Vertical Asymptotes, cont.Hand Drawn Calculator Drawn

Horizontal Asymptotes

Occurs when the degree of the numerator ≤ the degree of the denominator Ex.

Can be crossed Always in the form y = Abbreviated HA

3x

xy

Horizontal Asymptotes, cont.

Hand Drawn Calculator Drawn

Removable Discontinuities

aka Holes Gaps in the graph at a single point Occur when Always in the form x = Represented by an open circle (or hole) in

the graph

0

0y

Removable Discontinuities, cont.

Hand Drawn Calculator Drawn

Graphing Calculators and Removable Discontinuities Graphing calculators have difficulty

showing removable discontinuities Check the table for errors!

Example

Roots:

y-int:

VA:

HA:

Holes:

2

1

x

y

Your Turn:

On the “Identifying Features of Rational Equations” handout, answer problems 1 – 8.

Don’t answer the domain and range questions!

Discontinuities and Domain and Range

Discontinuities affect the domain and range of a rational equation

Vertical Asymptotes → Domain Horizontal Asymptotes → Range Removable Discontinuities →

Domain and Range

Example 1:

Domain:

Range:

3

2

x

xy

Example 2:

Domain:

Range:

2

42

x

xy

Your Turn:

Answer the domain and range questions for problems 1 – 8 on the “Identifying Features of Rational Equations” handout.

Homework

Complete problems 1 – 6 on the “Identifying the Features of Rational Equations Practice” handout.