1. Introduction to Rational Equations 15 November 2010

2. 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

3. Polynomials vs. Rationals Polynomial EquationsRational Equations Continuous Smooth Discontinuous Has jumps, breaks, sharp bends, and/or holes

4. Discontinuities Affect the shape, domain and range of an equation Three major types for rational equations: Vertical Asymptotes Horizontal Asymptotes Removable Discontinuities (aka Holes)

5. Asymptotes A line that the equation approaches but can never reach Not part of the equation Represented by a dashed line

6. Vertical Asymptotes Occur when the denominator equals zero Can never be crossed Always in the form x = Abbreviated VA

7. Vertical Asymptotes, cont. Hand DrawnCalculator Drawn

8. 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

9. Horizontal Asymptotes, cont. Hand DrawnCalculator Drawn

10. 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

11. Removable Discontinuities, cont. Hand DrawnCalculator Drawn

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

13. Example Roots: y-int: VA: HA: Holes:

14. Your Turn: On the “Identifying Features of Rational Equations” handout, answer problems 1 – 8. Don’t answer the domain and range questions!

15. 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

16. Example 1: Domain: Range:

17. Example 2: Domain: Range:

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

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