1. Rational Functions and Conics Advanced Math Chapter 4

2. Rational Functions and Asymptotes Advanced Math Section 4.1

3. Advanced Math 4.13 Rational functions N(x) and D(x) are polynomialsN(x) and D(x) are polynomials D(x) is not zeroD(x) is not zero

4. Advanced Math 4.14 Domain of a rational function All real numbers except x-values that make the denominator zeroAll real numbers except x-values that make the denominator zero Examples: find the domains of the following functionsExamples: find the domains of the following functions

5. Advanced Math 4.15 Vertical asymptotes The line x = a is a vertical asymptote of the graph of f ifThe line x = a is a vertical asymptote of the graph of f if f(x)  ∞f(x)  ∞ or f(x)  - ∞or f(x)  - ∞ –as x  a from either the right or the left

6. Advanced Math 4.16 Horizontal asymptotes The line y = b is a horizontal asymptote of the graph of f ifThe line y = b is a horizontal asymptote of the graph of f if –f(x)  b as x  ∞as x  ∞ or x  -∞or x  -∞

7. Advanced Math 4.17 Example Find any horizontal or vertical asymptotes of the function below:Find any horizontal or vertical asymptotes of the function below:

8. Advanced Math 4.18 Finding vertical asymptotes The graph has vertical asymptotes at the zeros of D(x)The graph has vertical asymptotes at the zeros of D(x)

9. Advanced Math 4.19 Finding horizontal asymptotes The graph has one or zero horizontal asymptote determined by comparing the degrees of N(x) and D(x)The graph has one or zero horizontal asymptote determined by comparing the degrees of N(x) and D(x) If the degree of N < the degree of D, then the line y = 0 (the x-axis) is a horizontal asymptoteIf the degree of N < the degree of D, then the line y = 0 (the x-axis) is a horizontal asymptote If the degree of N > the degree of D, then there is no horizontal asymptoteIf the degree of N > the degree of D, then there is no horizontal asymptote If the degrees are the same, then find the horizontal asymptote by taking the ratio of the leading coefficientsIf the degrees are the same, then find the horizontal asymptote by taking the ratio of the leading coefficients

10. Advanced Math 4.110 Examples Find all vertical and horizontal asymptotes of the graphs of the following functions:Find all vertical and horizontal asymptotes of the graphs of the following functions: