Overview
This lecture covers the basics of rational functions and explains how to determine their end behavior by examining the leading terms of the numerator and denominator. It introduces scenarios for different outcomes and how they affect horizontal and slant asymptotes.
Rational Functions: Definition and Requirements
- A rational function is a polynomial divided by another polynomial.
- The denominator cannot be zero.
- The denominator must have degree one or higher (must include at least an x term).
- Standard form (not factored) is preferred for analyzing end behavior.
Combining and Simplifying Rational Functions
- All analysis should be done with a single fraction (one numerator, one denominator).
- Combine fractions by finding a common denominator and adjusting numerators.
End Behavior of Rational Functions
- End behavior examines function outputs as x approaches positive or negative infinity (very large magnitude values).
- Focus only on the leading term (highest power) of numerator and denominator.
- Divide leading terms to analyze end behavior; three outcomes are possible:
Scenario 1: Polynomial Result
- If dividing leading terms gives a polynomial (no x left in denominator), the rational function's end behavior mirrors that polynomial.
- If the result is linear, there is a slant (oblique) asymptote with the same slope.
- No horizontal asymptote if the result is a polynomial.
- Examples:
- Even degree, negative leading coefficient: both ends go to negative infinity.
- Odd degree, positive leading coefficient: left end negative infinity, right end positive infinity.
Scenario 2: Constant Result
- If the division yields a constant, there is a horizontal asymptote at y = that constant.
- End behavior (left and right) approaches the constant value.
Scenario 3: Rational Function Result
- If an x remains in the denominator, there is a horizontal asymptote at y = 0.
- Both end behaviors approach zero.
Factored Form Reminder
- For end behavior, you only need the leading term from each factor; multiply them if needed.
- Always convert to standard form for end behavior analysis.
End Behavior on Graphs
- Horizontal asymptotes act as "vacuums": function values approach the asymptote as x approaches ±∞.
- Slant asymptotes: function end behavior follows the slant line (goes to infinity in the same direction as the asymptote).
Key Terms & Definitions
- Rational Function — a function that is a ratio of two polynomials.
- Leading Term — term with the highest degree in a polynomial.
- End Behavior — the output values of a function as input x approaches ±∞.
- Horizontal Asymptote — a horizontal line that the graph approaches at extreme x-values.
- Slant (Oblique) Asymptote — a diagonal line the graph approaches when the degree of the numerator is one more than the denominator.
Action Items / Next Steps
- Practice combining rational expressions into a single fraction.
- For each rational function, identify and divide the leading terms to determine end behavior.
- Convert factored-form rational functions to standard form (just leading terms) for end behavior analysis.
- Review graphical examples and relate asymptotes to end behavior outcomes.