Introduction to Rational Equations 15 November 2010

Rational Equations Definition a n x n + a n-1 x n-1 + … + a 2 x 2 + a 1 n 1 + a 0 b n x n + b n-1 x n-1 + … + b 2 x 2 + b 1 n 1 + b 0 A polynomial divided by a polynomial Rational equations are fractions Polynomial

Polynomials vs. Rationals Polynomial EquationsRational Equations Continuous Smooth Discontinuous Has 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 DrawnCalculator 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

Horizontal Asymptotes, cont. Hand DrawnCalculator 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

Removable Discontinuities, cont. Hand DrawnCalculator 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:

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:

Example 2: Domain: Range:

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.