Introduction to Rational Equations

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Presentation transcript:

Introduction to Rational Equations 14 March 2011

Rational Equation

Definition Polynomial Rational = Rational equations are fractions in which both the numerator and the denominator are polynomials

Polynomials vs. Rationals Non-Example and Example Polynomial Equations Rational Equations y = x3 – 7x + 6

Your Turn: Complete problems 1 – 10 on the Introduction to Rational Equations handout

Discontinuities Discontinuity – a point or a line where the graph of an equation has as a hole, a jump, a break, or a gap Affect the shape, domain and range of an equation

Discontinuities, cont. Three major types of discontinuities: Vertical Asymptotes Horizontal Asymptotes Holes Asymptotic Point Discontinuity

Asymptotes Lines that the graph approaches but never crosses Represented by a dashed line Not part of the equation

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 Occur when the degree of the denominator is ≥ the degree of the numerator Ex. Can be crossed when |x| is very large Describes the end behavior of a rational eqn Always in the form y = Abbreviated HA

Horizontal Asymptotes, cont. Hand Drawn Calculator Drawn

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

Holes, cont. Hand Drawn Calculator Drawn

Graphing Calculators and Removable Discontinuities Graphing calculators have difficulty showing removable discontinuities Check the table for errors!

Example #1 Roots: y-int: VA: HA: Holes:

Example #2 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 Holes → 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.