Chapter 3 – Polynomial and Rational Functions Rational Functions
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 Rational Functions
Basic Rational Function We want to identify the characteristics of rational functions Rational Functions DomainRange x-interceptsy-intercepts Asymptotes (VA, HA, SA)Directional Limits Max Min Increase Decrease
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 Rational Functions
Arrow Notation We will be using the following arrow notation for asymptotes: Rational Functions
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 Rational Functions
Vertical Asymptotes (VA) To find the VA 1. Set the denominator = 0 and solve for x. 2. Check using arrow notation Rational Functions
Horizontal Asymptotes The line y = b is a horizontal asymptote of the function y = f (x) if y approaches b as x approaches ∞ Rational Functions
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) Rational Functions
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 Rational Functions
Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function 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
Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function 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
Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function 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
Example Given the above equation, find the characteristics of rational functions and sketch a graph of the function Rational Functions DomainRange x-interceptsy-intercepts Asymptotes (VA, HA, SA)Directional Limits Max Min Increase Decrease