1. Introduction to Rational Equations
14 March 2011

2. Rational Equation

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

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

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

6. 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

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

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

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

10. Vertical Asymptotes, cont.
Hand Drawn
Calculator Drawn

11. 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

12. Horizontal Asymptotes, cont.
Hand Drawn
Calculator Drawn

13. 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

14. Holes, cont.
Hand Drawn
Calculator Drawn

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

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

17. Example #2
Roots:
y-int:
VA:
HA:
Holes:

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

19. 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

20. Example 1:
Domain:
Range:

21. Example 2:
Domain:
Range:

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

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