AP Precalculus Summary Notes

Average Rate of Change

• The average rate of change of a function on a specific interval equals the slope of the secant line connecting endpoints of that interval.

Binomial Theorem

• Used to expand polynomial functions.

• Coefficients in expansions are related to entries in Pascal's Triangle.

• The coefficient of a specific term can be found using rows of Pascal's Triangle.

Geometric Sequences

• Identified by the common ratio, derived from consecutive terms.

• Terms outcomes are determined by multiplication of the previous term by the common ratio.

Function Behavior

• Concavity identifies if a function is concave up or down based on the rate of change.

  • If a function's rate of change is decreasing, it is concave down.

Function Composition

• Understanding compositions of functions and their domains.

• Domain of the composed function may differ based on division by zero conditions.

Polynomial Characteristics

• Polynomials can reveal maximum and minimum values through their leading coefficients and degree.

• The multiplicity of roots affects the graph's local maxima and minima.

Asymptotic Behavior

• Rational functions can have horizontal or vertical asymptotes based on polynomial degrees.

• The behavior of a function at infinity helps determine end behavior and the presence of asymptotes.

Transformations

• Transformations can include dilations or translations in horizontal or vertical direction, impacting the function’s graph.

Residuals and Model Appropriateness

• Analyzing residuals from regression models can indicate model fit and appropriateness.