Mastery Guide: Polynomial Functions for AP Precalculus

Linear Functions

Functions characterized by a constant rate of change.

Polynomial Functions

Functions characterized by varying rates of change.

Average Rate of Change (AROC)

Measures the slope of the secant line connecting two points on a curve.

Concave Up

When the rate of change is increasing, graph creates a cup shape.

Concave Down

When the rate of change is decreasing, graph creates a frown shape.

Standard Form of a Polynomial

p(x) = an x^n + a{n-1} x^{n-1} + … + a1 x + a0.

Degree of a Polynomial

The highest exponent in a polynomial.

Leading Coefficient

The coefficient in front of the leading term in a polynomial.

End Behavior

Describes the behavior of the function as x approaches positive or negative infinity.

Turning Points

Points where a polynomial changes direction, at most n-1 for degree n.

Zeros of a Polynomial

x-values where f(x) = 0, corresponding to the x-intercepts of the graph.

Multiplicity of a Zero

The exponent on a factor indicating how many times that root occurs.

Single Zero

A zero with multiplicity 1 where the graph crosses the x-axis.

Odd Multiplicity

A multiplicity greater than 1 where the graph crosses the x-axis but flattens out.

Even Multiplicity

A multiplicity where the graph touches the x-axis and turns around.

Polynomial Inequalities

Inequalities like P(x) > 0 or P(x) < 0 that require analyzing intervals.

Sign Chart Method

A method used to determine where a polynomial is positive or negative.

Test Values in Intervals

Values selected within each interval to determine the sign of P(x).

Leading Term Test

A method to determine the sign of the polynomial based on its leading term.

Limit Notation

Describes behavior of a function as its input approaches a value.

Strict Inequalities

Inequalities that do not include the endpoints, denoted with parentheses.

Inclusive Inequalities

Inequalities that include the endpoints, denoted with brackets.

Bounce Factor

The behavior at roots with even multiplicity where signs do not change.

Inflection Point

A point on the graph where the curve changes concavity.

Secant Line

A line that connects two points on a curve, used to calculate AROC.

Second Difference

Used to determine concavity in polynomial graphs.

Real Roots

The solutions to a polynomial equation that correspond to x-intercepts.