1. Chapter 3 – Polynomial and Rational Functions 3.7 - Rational Functions

2. Example Rational functions are quotients of polynomials. For example, functions that can be expressed as where P(x) and Q(x) are polynomials and Q(x)  0. Note: We assume that P(x) and Q(x) have no factors in common. 3.7 - Rational Functions

3. Basic Rational Function We want to identify the characteristics of rational functions 3.7 - Rational Functions DomainRange x-interceptsy-intercepts Asymptotes (VA, HA, SA)Directional Limits Max Min Increase Decrease

4. Domain In order to find the domain of a rational function, we must set the denominator equal to zero. These values are where our function does not exist. Hint: If possible, always factor the denominator first before finding the domain. 3.7 - Rational Functions

5. Arrow Notation We will be using the following arrow notation for asymptotes: 3.7 - Rational Functions

6. Vertical Asymptotes The line x = a is a vertical asymptote of the function y = f (x) if y approaches  ∞ as x approaches a from the right or left. 3.7 - Rational Functions

7. Vertical Asymptotes (VA) To find the VA 1. Set the denominator = 0 and solve for x. 2. Check using arrow notation. 3.7 - Rational Functions

8. Horizontal Asymptotes The line y = b is a horizontal asymptote of the function y = f (x) if y approaches b as x approaches  ∞. 3.7 - Rational Functions

9. Horizontal Asymptotes (HA) To find the HA, we let r be the rational function 1. If n < m, then r has the horizontal asymptote y=0. 2. If n = m, then r has the horizontal asymptote. 3. If n > m, then r has no horizontal asymptotes. We need to check for a slant asymptote (SA). 3.7 - Rational Functions

10. Slant Asymptotes  To find the SA, we perform long division and get where R(x)/Q(x) is the remainder and the SA is y = ax + b. 3.7 - Rational Functions

11. Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function. 3.7 - Rational Functions DomainRange x-interceptsy-intercepts Asymptotes (VA, HA, SA) Directional Limits Max Min Increase Decrease DomainRange x-interceptsy-intercepts Asymptotes (VA, HA, SA)Directional Limits Max Min Increase Decrease

12. Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function. 3.7 - Rational Functions DomainRange x-interceptsy-intercepts Asymptotes (VA, HA, SA) Directional Limits Max Min Increase Decrease DomainRange x-interceptsy-intercepts Asymptotes (VA, HA, SA)Directional Limits Max Min Increase Decrease

13. Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function. 3.7 - Rational Functions DomainRange x-interceptsy-intercepts Asymptotes (VA, HA, SA) Directional Limits Max Min Increase Decrease DomainRange x-interceptsy-intercepts Asymptotes (VA, HA, SA)Directional Limits Max Min Increase Decrease

14. Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function. 3.7 - Rational Functions DomainRange x-interceptsy-intercepts Asymptotes (VA, HA, SA)Directional Limits Max Min Increase Decrease