1. Rational Expressions, Vertical Asymptotes, and Holes

2. Rational Expression It is the quotient of two polynomials. A rational function is a function defined by a rational expression. Examples: Not Rational:

3. Find the domain: Graph it:

4. Find the domain: Graph it:

5. Find the domain: Graph it:

6. Find the domain: Graph it:

7. Vertical Asymptote If x – a is a factor of the denominator of a rational function but not a factor of the numerator, then x = a is a vertical asymptote of the graph of the function.

8. Find the domain: Graph it using the graphing calculator. What do you see?

9. Find the domain: Graph it using the graphing calculator. What do you see?

10. Hole (in the graph) If x – b is a factor of both the numerator and denominator of a rational function, then there is a hole in the graph of the function where x = b, unless x = b is a vertical asymptote. The exact point of the hole can be found by plugging b into the function after it has been simplified.

11. Find the domain and identify vertical asymptotes & holes.

19. Horizontal Asymptotes & Graphing

20. Horizontal Asymptotes Degree of numerator = Degree of denominator Degree of numerator < Degree of denominator Degree of numerator > Degree of denominator Horizontal Asymptote:

21. Find all asymptotes & holes & then graph: