Comprehensive Guide to Polynomial and Rational Functions

Function

A mathematical relationship mapping a set of inputs to a set of outputs where every input maps to exactly one output.

Domain

The set of all possible input values (independent variable x) for a function.

Range

The set of all possible output values (dependent variable y) for a function.

Rate of Change

The rate at which the output (y) changes with respect to the input (x).

Increasing Function

A function where as inputs increase, outputs also increase.

Decreasing Function

A function where as inputs increase, outputs decrease.

Concave Up

A function where the rate of change is increasing; the graph bends upward.

Concave Down

A function where the rate of change is decreasing; the graph bends downward.

Average Rate of Change (AROC)

The slope of the secant line connecting two points on a graph.

Linear Function

A function with a constant rate of change.

Quadratic Function

A function where the average rate of change is linear.

Cubic Function

A polynomial function where the rate of change changes at a changing rate.

Degree of a Polynomial

The highest exponent of a polynomial function.

Leading Coefficient

The coefficient of the term with the highest degree in a polynomial function.

End Behavior

The behavior of a polynomial function as x approaches infinity or negative infinity.

Vertical Asymptote (VA)

A vertical line where the function approaches infinity or negative infinity.

Hole (Removable Discontinuity)

A point where a function is not defined because both numerator and denominator equal zero.

X-Intercepts

Points where the numerator of a rational function equals zero, but the denominator does not.

Horizontal Asymptote

A horizontal line that the graph approaches as x approaches infinity.

Slant Asymptote

An asymptote found through polynomial long division when the degree of the numerator is greater than the degree of the denominator by one.

Fundamental Theorem of Algebra

A polynomial of degree n has exactly n complex zeros (counting multiplicity).

Extrema

Turning points of a polynomial where it changes from increasing to decreasing (max) or vice versa (min).

Point of Inflection

A point where the concavity of the function changes.

Polynomial Long Division

A method for dividing polynomial expressions to find slant asymptotes.

Binomial Theorem

A way to expand binomials without repeated multiplication, following Pascal's triangle.

Transformation of Functions

Operations including shifts and dilations applied to the parent function.

Vertical Shift

Moving a function up or down on the y-axis.

Horizontal Shift

Moving a function left or right along the x-axis.

Vertical Dilation

Stretching or compressing a function along the y-axis.

Horizontal Dilation

Stretching or compressing a function along the x-axis.

Contextual Domain Restrictions

Limitations based on the real-world context of a mathematical model.

Mathematical Domain Restrictions

Limitations based on mathematical properties, such as denominators not being zero.

Rounding in Models

Interpreting predictions of continuous data into integers based on context.

Constant First Differences

Indication that a linear model may be appropriate for a dataset.

Constant Second Differences

Indication that a quadratic model may be appropriate for a dataset.

Cubic/Quartic Polynomial Functions

Polynomial functions used for modeling complex curves with multiple turning points.

Rational Function

A function expressed as a ratio of two polynomial functions.

Asymptote Behavior

The anticipated behavior of a function in relation to its asymptotes.

Turning Points

Points on a graph where the direction of the curve changes.

Coefficient

A numerical factor in a term of a polynomial.

Graph Tangent

A graph touching but not crossing the x-axis at the root for even multiplicity.

Slope of Secant Line

The average rate of change between two points on a function's graph.

Increasing Rate of Change

Indicates a graph that is getting steeper upwards.

Decreasing Rate of Change

Indicates a graph that is getting less steep or downward.

Complex Zeros

Roots of a polynomial that may not be real numbers but occur in pairs.

Polynomial Function

A function defined as the sum of terms consisting of variables raised to non-negative integer exponents.