rational-functions-plus-asymptotes

1. Section 5.2 – Properties of Rational Functions Defn: Rational Function • A function in the form: • The functions p and q are polynomials. • The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero.

2. Section 5.2 – Properties of Rational Functions Domain of a Rational Function • {x | x –4} • or • (-, -4)  (-4, )

3. graph

4. Section 5.2 – Properties of Rational Functions Domain of a Rational Function • {x | x 2} • or • (-, 2)  (2, )

5. graph

6. Section 5.2 – Properties of Rational Functions Domain of a Rational Function • {x | x –3, 3} • or • (-, -3)  (-3, 3)  (3, )

7. graph

8. Section 5.2 – Properties of Rational Functions Domain of a Rational Function • {x | x –3, 5} • or • (-, -3)  (-3, 5)  (5, )

9. Section 5.2 – Properties of Rational Functions Linear Asymptotes (vertical, horizontal, or oblique) Lines in which a graph of a function will approach. By approach we mean each successive value of X puts the graph closer to the asymptote than the previous value. Vertical Asymptote • A vertical asymptote exists for any value of x that makes the denominator zero AND is not a value that makes the numerator zero, in this case the factors would cancel. • Example • A vertical asymptotes exists at x = -5. • VA:

10. graph

11. Section 5.2 – Properties of Rational Functions Asymptotes Vertical Asymptote • Example • A vertical asymptote exists at x = 4. VA: • A vertical asymptote does not exist at x = 3 as it is a value that also makes the numerator zero. • A hole exists in the graph at x = 3.

12. graph

13. Section 5.2 – Properties of Rational Functions Asymptotes Horizontal Asymptote • A horizontal asymptote exists if the largest exponents in the numerator and the denominator are equal, • or • if the largest exponent in the denominator is larger than the largest exponent in the numerator. • If the largest exponent in the denominator is equal to the largest exponent in the numerator, then the horizontal asymptote is equal to the ratio of the coefficients. • If the largest exponent in the denominator is larger than the largest exponent in the numerator, then the horizontal asymptote is .

14. Section 5.2 – Properties of Rational Functions Asymptotes Horizontal Asymptote • Example • HA: • A horizontal asymptote exists at y = 5/2. • A horizontal asymptote exists at y = 0. • HA:

15. graph

16. Section 5.2 – Properties of Rational Functions Asymptotes Oblique (slant) Asymptote • An oblique asymptote exists if the largest exponent in the numerator is one degree larger than the largest exponent in the denominator. **Note** • Other non-linear asymptotes can exist for a rational function.

17. Section 5.2 – Properties of Rational Functions Asymptotes Oblique Asymptote • Example • An oblique asymptote exists. • Long division is required. • We ignore the remainder if it exists • An oblique asymptote exists at y = x. OA:

18. graph

19. Section 5.2 – Properties of Rational Functions Asymptotes Oblique Asymptote • Example • An oblique asymptote exists. • Long division is required. • An oblique asymptote exists at y = 2x • OA:

20. graph