- Use arrow notation to describe how rational functions behave at different points
- Apply rational functions to solve real-world problems
- Find out which values are allowed for rational functions and understand what limits these values set
- Spot where rational functions shoot up to infinity (vertical asymptotes) and explore why these points are important
- Find where rational functions level off at infinity (horizontal asymptotes) and see how these affect the graph
- Draw graphs of rational functions carefully, including places where the graph breaks or levels out, and where it’s not defined
Solve Applied Problems Involving Rational Functions
A rational function is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.
- A horizontal line of the form [latex]y=c[/latex]
- The constant [latex]c[/latex] represents a number that the function value (output) approaches in the long run, either as the input grows very small or very large.
- Horizontal asymptotes represent the long-run behavior (the end behavior) of the graph of the funtion.
- A function’s graph may cross a horizontal asymptote briefly, even more than once, but will eventually settle down near it, as the value of the function approaches the constant [latex]c[latex].
Vertical Asymptote
- A vertical line of the form [latex]x=a[/latex]
- The constant [latex]a[/latex] represents an input for which the function value (output) is undefined.
- Substituting the value of [latex]a[/latex] into the function will result in a zero in the function's denominator.
- The graph of the function "bends around", either increasing or decreasing without bound as the input nears [latex]a[/latex]
- A function's graph will never cross a vertical asymptote.