AP Precalculus Unit 1: Polynomial Functions, Rates of Change, and Behavior

Vocabulary flashcards covering core AP Precalculus concepts from Lessons 1.1 to 1.6, including rates of change, concavity, polynomial characteristics, zeros with multiplicity, complex roots, function symmetry, and end behavior.

Positive Function

A function whose graph lies entirely above the $x$-axis over an interval, meaning $f(x) > 0$.

Negative Function

A function whose graph lies entirely below the $x$-axis over an interval, meaning $f(x) < 0$.

Increasing Function

A function whose output values increase as input values increase over an interval ($\Delta y > 0$ as $\Delta x > 0$).

Decreasing Function

A function whose output values decrease as input values increase over an interval ($\Delta y < 0$ as $\Delta x > 0$).

Average Rate of Change (AROC)

The slope of the secant line between two points on a function, calculated by $AROC = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{f(b) - f(a)}{b - a}$.

Secant Line

A line connecting two distinct points on the graph of a function; its slope represents the average rate of change between those two points.

Tangent Line

A line that touches a curve at a single point of tangency; its slope indicates whether the instantaneous rate of change is positive (rising), negative (falling), or zero (horizontal at a turning point).

Concave Up (CCU)

A graphical characteristic where the rate of change (slope of the tangent or secant lines) is increasing over an interval, and secant lines lie above the graph.

Concave Down (CCD)

A graphical characteristic where the rate of change (slope of the tangent or secant lines) is decreasing over an interval, and secant lines lie below the graph.

Point of Inflection (POI)

A point on a graph where the function changes concavity from concave up to concave down, or vice versa; points of inflection of $f(x)$ correspond to relative extrema of the rate of change function.

Linear Function (Rate of Change Property)

A function whose average rate of change (first difference, or $1D$) is constant over equal-length consecutive input intervals.

Quadratic Function (Rate of Change Property)

A function whose first differences ($1D$) change at a constant rate, meaning the second differences ($2D$) of the outputs are constant over equal-length consecutive input intervals.

Leading Term

The term in a polynomial function that contains the highest power of the variable ($a_n x^n$).

Leading Coefficient

The numerical coefficient ($a_n$) associated with the leading term of a polynomial function.

Degree of a Polynomial

The greatest exponent of the variable among all terms in the polynomial.

Relative (Local) Extrema

All maximum and minimum output values within an open interval of a function, including the endpoints on a restricted domain.

Absolute (Global) Extrema

The single greatest output value (global maximum) or least output value (global minimum) attained by a function across its entire domain.

Multiplicity of a Zero

The exponent or power of the linear factor $(x - a)$ corresponding to the zero $x = a$ in a completely factored polynomial function.

Odd Multiplicity

A property of a polynomial zero where the factor exponent is odd, causing the graph to cross through the $x$-axis at that zero.

Even Multiplicity

A property of a polynomial zero where the factor exponent is even, causing the graph to touch and bounce off (turn around tangent to) the $x$-axis at that zero.

Complex Conjugate Pairs

Pairs of complex roots of the form $a + bi$ and $a - bi$; for polynomials with real coefficients, all non-real imaginary zeros always occur in these pairs.

Fundamental Theorem of Algebra

A theorem stating that a polynomial function of degree $n$ has exactly $n$ complex zeros (including real and non-real imaginary zeros) when counting multiplicities.

Successive Differences Property

A property stating that for a table of polynomial values with equal-width input intervals, the number of successive finite differences required to produce a constant row is equal to the degree $n$ of the polynomial.

Even Function

A function symmetric with respect to the $y$-axis satisfying $f(-x) = f(x)$, meaning opposite $x$-values yield identical $y$-values (and all polynomial terms have even exponents).

Odd Function

A function symmetric with respect to the origin satisfying $f(-x) = -f(x)$, meaning opposite $x$-values yield opposite $y$-values (and all polynomial terms have odd exponents, with no non-zero constants).

End Behavior

The behavior of the output values of a function as the input values increase without bound ($\lim_{x \rightarrow \infty} f(x)$) or decrease without bound ($\lim_{x \rightarrow -\infty} f(x)$).

Right End Behavior Rule

The mathematical principle stating that the right end behavior of a polynomial function is determined entirely by the sign of its leading coefficient: if positive, $\lim_{x \rightarrow \infty} f(x) = \infty$; if negative, $\lim_{x \rightarrow \infty} f(x) = -\infty$.

Left End Behavior Rule

The mathematical principle stating that if a polynomial function has an even degree, its left end behavior matches its right end behavior; if it has an odd degree, its left end behavior goes in the opposite direction.

Sign Chart

A number line partitioned by the real zeros of a polynomial that uses test values within each interval to determine whether the polynomial is positive ($> 0$) or negative ($< 0$) when solving non-linear inequalities.