1. Functions AII.7 e 2009

2. Objectives: Find the Vertical Asymptotes Find the Horizontal Asymptotes

3. Rational Functions A rational function can have more than one vertical asymptote, but it can have at most one horizontal asymptote. A rational function f ( x ) is a function that can be written as where p ( x ) and q ( x ) are polynomial functions and q ( x ) 0.

4. Vertical Asymptotes If p ( x ) and q ( x ) have no common factors, then f ( x ) has vertical asymptote(s) when q ( x ) = 0. Thus the graph has vertical asymptotes at the zeros of the denominator.

5. Since the zeros are 1 and -1. Thus the vertical asymptotes are x = 1 and x = -1. Vertical Asymptotes Find the vertical asymptote of Example: V.A. is x = a, where a represents real zeros of q ( x ).

6. Horizontal Asymptotes The horizontal asymptote is determined by looking at the degrees of p ( x ) and q ( x ). A rational function f ( x ) is a function that can be written as where p ( x ) and q ( x ) are polynomial functions and q ( x ) 0.

7. Horizontal Asymptotes a.If the degree of p ( x ) is less than the degree of q ( x ), then the horizontal asymptote is y = 0. b. If the degree of p ( x ) is equal to the degree of q ( x ), then the horizontal asymptote is c. If the degree of p ( x ) is greater than the degree of q ( x ), then there is no horizontal asymptote.

8. Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Example : Find the horizontal asymptote: S ince the degree of the numerator is less than the degree of the denominator, horizontal asymptote is y = 0. Degree of numerator = 1 Degree of denominator = 2

9. Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Example : Find the horizontal asymptote: Degree of numerator = 1 Degree of denominator = 1 S ince the degree of the numerator is equal to the degree of the denominator, horizontal asymptote is.

10. S ince the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Example : Find the horizontal asymptote: Degree of numerator = 2 Degree of denominator = 1

11. Vertical & Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). H.A. : Answer Now

12. Vertical & Horizontal Asymptotes Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. H.A. : V.A. : x = H.A.: none

13. Vertical & Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). H.A. : Answer Now

14. Vertical & Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). H.A. : V.A. : none H.A.: y = 0 is not factorable and thus has no real roots.

15. Vertical & Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). H.A. : Answer Now

16. Vertical & Horizontal Asymptotes deg of p ( x ) < deg of q ( x ), then H.A. is y = 0 deg of p ( x ) = deg of q ( x ), then H.A. is deg of p ( x ) > deg of q ( x ), then no H.A. Practice : Find the vertical and horizontal asymptotes: V.A. : x = a, where a represents real zeros of q ( x ). H.A. : V.A. : x = -1 H.A.: y = 2