| Term | Definition |
|---|---|
| denominator | The polynomial expression in the bottom part of a rational function. |
| holes | Points where a rational function is undefined due to common factors in the numerator and denominator that cancel out, creating a gap in the graph. |
| limit | The value that a function approaches as the input approaches a specific value. |
| multiplicity | The number of times a linear factor appears in the complete factorization of a polynomial; determines how the graph behaves at that zero. |
| numerator | The polynomial expression in the top part of a rational function. |
| rational function | A function expressed as the ratio of two polynomials, where the denominator is not equal to zero. |
| real zero | A real number value that makes a polynomial function equal to zero, corresponding to an x-intercept on the graph. |
| Term | Definition |
|---|---|
| concave down | A characteristic of a graph where the rate of change is decreasing, creating a curve that opens downward. |
| concave up | A characteristic of a graph where the rate of change is increasing, creating a curve that opens upward. |
| decreasing function | A function over an interval where output values always decrease as input values increase. |
| dependent variable | The variable representing output values in a function. |
| domain | The set of all possible input values for which a function is defined. |
| function | A mathematical relation that maps each input value to exactly one output value. |
| function rule | The mathematical relationship that determines how input values map to output values, which can be expressed graphically, numerically, analytically, or verbally. |
| increasing function | A function over an interval where output values always increase as input values increase. |
| independent variable | The variable representing input values in a function. |
| input | The independent variable or value that is entered into a function. |
| input value | The x-values or independent variable values used as inputs to a function. |
| output | The dependent variable or value that results from applying a function to an input. |
| output value | The y-values or results produced by a function for given input values. |
| range | The set of all possible output values that a function can produce. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| x-axis | The horizontal axis on a coordinate plane representing input values. |
| zero | A value of the input for which a polynomial function equals zero; also called a root of the equation p(x)=0. |
| Term | Definition |
|---|---|
| analytic representations | Different mathematical forms or expressions used to represent the same function, such as factored form or standard form. |
| asymptote | Lines that a graph approaches but never reaches, indicating behavior at infinity or at points of discontinuity. |
| binomial | A polynomial expression consisting of exactly two terms, such as (a + b). |
| binomial coefficients | The numerical coefficients that appear in the expansion of (a + b)^n, found in the rows of Pascal's Triangle. |
| binomial theorem | A mathematical theorem that provides a formula for expanding expressions of the form (a + b)^n using binomial coefficients. |
| domain | The set of all possible input values for which a function is defined. |
| end behavior | The behavior of a function as the input values approach positive or negative infinity. |
| equivalent forms | Different ways of writing the same mathematical expression that have equal values for all valid inputs. |
| factored form | A representation of a polynomial or rational expression written as a product of its factors, which reveals the real zeros and x-intercepts. |
| holes | Points where a rational function is undefined due to common factors in the numerator and denominator that cancel out, creating a gap in the graph. |
| Pascal's Triangle | A triangular array of numbers where each row contains the binomial coefficients used in the binomial expansion of (a + b)^n. |
| polynomial expressions | Mathematical expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. |
| polynomial long division | An algebraic process similar to numerical long division used to divide one polynomial by another, producing a quotient and remainder. |
| quotient | The result obtained when one polynomial is divided by another polynomial in polynomial long division. |
| range | The set of all possible output values that a function can produce. |
| rational expressions | Mathematical expressions that represent the ratio of two polynomials, written as a fraction with a polynomial numerator and polynomial denominator. |
| rational function | A function expressed as the ratio of two polynomials, where the denominator is not equal to zero. |
| real zero | A real number value that makes a polynomial function equal to zero, corresponding to an x-intercept on the graph. |
| remainder | The polynomial left over after polynomial long division, which has a degree less than the divisor polynomial. |
| slant asymptote | A linear asymptote that occurs when the numerator polynomial of a rational function has a degree one greater than the denominator polynomial. |
| standard form | A representation of a polynomial or rational expression in expanded form, which reveals information about end behavior. |
| x-intercept | The point where a graph crosses or touches the x-axis, occurring at (a, 0) when a is a real zero of the function. |
| Term | Definition |
|---|---|
| additive transformation | A transformation of a function involving addition or subtraction, resulting in vertical and horizontal translations. |
| domain | The set of all possible input values for which a function is defined. |
| horizontal dilation | A transformation that stretches or compresses the graph of a function horizontally by multiplying the input by a constant factor b, written as g(x) = f(bx). |
| horizontal translation | A transformation that shifts the graph of a function left or right by adding a constant to the input, written as g(x) = f(x + h). |
| multiplicative transformation | A transformation of a function involving multiplication, resulting in vertical and horizontal dilations. |
| parent function | The simplest form of a family of functions, used as a base for creating transformed functions. |
| range | The set of all possible output values that a function can produce. |
| reflection over the x-axis | A transformation that flips the graph of a function across the x-axis, occurring when the multiplicative factor is negative in a vertical dilation. |
| reflection over the y-axis | A transformation that flips the graph of a function across the y-axis, occurring when the multiplicative factor is negative in a horizontal dilation. |
| vertical dilation | A transformation that stretches or compresses the graph of a function vertically by multiplying the function by a constant factor a, written as g(x) = af(x). |
| vertical translation | A transformation that shifts the graph of a function up or down by adding a constant k to the function, written as g(x) = f(x) + k. |
| Term | Definition |
|---|---|
| assumptions | Underlying conditions or beliefs about what remains consistent or how quantities behave in a function model. |
| cubic function | A polynomial function of degree 3 with the form f(x) = ax³ + bx² + cx + d. |
| degree | The highest power of the variable in a polynomial function, which determines the number of differences needed to reach a constant value. |
| domain restrictions | Limitations on the input values of a function based on mathematical validity, contextual meaning, or extreme values in the data set. |
| function model | A mathematical function used to represent and analyze relationships in a data set or real-world scenario. |
| linear function | A polynomial function of degree 1 with the form f(x) = mx + b, representing a constant rate of change. |
| maximum | The highest points or local maximum values on a function's graph. |
| minimum | The lowest points or local minimum values on a function's graph. |
| nth differences | The differences calculated by repeatedly subtracting consecutive terms in a sequence, used to identify polynomial degree. |
| polynomial function | A function that can be expressed in the form p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a positive integer and a_n is nonzero. |
| quadratic function | A polynomial function of degree 2 with the form f(x) = ax² + bx + c, creating a parabolic graph. |
| range restrictions | Limitations on the output values of a function, such as rounding values, based on mathematical validity, contextual meaning, or extreme values in the data set. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| real zero | A real number value that makes a polynomial function equal to zero, corresponding to an x-intercept on the graph. |
| restrictions | Limitations placed on the domain or range of a function model based on mathematical, contextual, or data-based considerations. |
| Term | Definition |
|---|---|
| average rate of change | The change in the output of a function divided by the change in the input over a specified interval, calculated as (f(b) - f(a))/(b - a) for the interval [a, b]. |
| changing rates of change | The variation in how quickly a quantity changes over different intervals, indicating acceleration or deceleration in the rate of change. |
| contextual scenario | A real-world situation or problem context in which a mathematical model is applied. |
| cubic function | A polynomial function of degree 3 with the form f(x) = ax³ + bx² + cx + d. |
| cubic regression | A regression technique that fits a cubic function to a data set. |
| electromagnetic force | The force between charged objects or magnetic poles, which is inversely proportional to the square of the distance between them. |
| function model | A mathematical function used to represent and analyze relationships in a data set or real-world scenario. |
| gravitational force | The attractive force between two objects due to their masses, which is inversely proportional to the square of the distance between them. |
| inversely proportional | A relationship between two quantities where one quantity increases as the other decreases by a constant factor, typically expressed as y = k/x. |
| linear function | A polynomial function of degree 1 with the form f(x) = mx + b, representing a constant rate of change. |
| linear regression | A regression technique that fits a linear function to a data set. |
| parent function | The simplest form of a family of functions, used as a base for creating transformed functions. |
| piecewise-defined function | A function defined by different expressions over different intervals of its domain. |
| polynomial function | A function that can be expressed in the form p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a positive integer and a_n is nonzero. |
| quadratic function | A polynomial function of degree 2 with the form f(x) = ax² + bx + c, creating a parabolic graph. |
| quadratic regression | A regression technique that fits a quadratic function to a data set. |
| quartic function | A polynomial function of degree 4 with the form f(x) = ax⁴ + bx³ + cx² + dx + e. |
| quartic regression | A regression technique that fits a quartic (fourth-degree polynomial) function to a data set. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| rational function | A function expressed as the ratio of two polynomials, where the denominator is not equal to zero. |
| regression | A statistical method for fitting a function to a data set to model the relationship between variables. |
| regression analysis | A statistical method used to fit a function to a set of data points to model the relationship between variables. |
| restrictions | Limitations placed on the domain or range of a function model based on mathematical, contextual, or data-based considerations. |
| transformation | Changes applied to a parent function such as translations, reflections, stretches, or compressions. |
| Term | Definition |
|---|---|
| average rate of change | The change in the output of a function divided by the change in the input over a specified interval, calculated as (f(b) - f(a))/(b - a) for the interval [a, b]. |
| domain | The set of all possible input values for which a function is defined. |
| input value | The x-values or independent variable values used as inputs to a function. |
| interval | A connected subset of the domain over which a function's behavior is analyzed. |
| negative rate of change | A rate of change where one quantity increases while the other decreases, or vice versa. |
| output value | The y-values or results produced by a function for given input values. |
| positive rate of change | A rate of change where both quantities increase together or both decrease together. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| rate of change at a point | The instantaneous rate at which output values change with respect to input values at a specific point on a function. |
| Term | Definition |
|---|---|
| average rate of change | The change in the output of a function divided by the change in the input over a specified interval, calculated as (f(b) - f(a))/(b - a) for the interval [a, b]. |
| concave down | A characteristic of a graph where the rate of change is decreasing, creating a curve that opens downward. |
| concave up | A characteristic of a graph where the rate of change is increasing, creating a curve that opens upward. |
| constant rate | A rate of change that remains the same across all intervals; for quadratic functions, the rate at which average rates of change are changing. |
| equal-length input-value intervals | Consecutive intervals along the input axis that have the same width, used to compare average rates of change. |
| linear function | A polynomial function of degree 1 with the form f(x) = mx + b, representing a constant rate of change. |
| quadratic function | A polynomial function of degree 2 with the form f(x) = ax² + bx + c, creating a parabolic graph. |
| secant line | A line that intersects a curve at two points, used to represent the average rate of change between those points. |
| sequence | A function from the whole numbers to the real numbers, producing a list of ordered values. |
| slope | The rate of change of a line, representing how much the output changes for each unit change in the input. |
| Term | Definition |
|---|---|
| concave down | A characteristic of a graph where the rate of change is decreasing, creating a curve that opens downward. |
| concave up | A characteristic of a graph where the rate of change is increasing, creating a curve that opens upward. |
| decreasing | A characteristic of a function where output values fall as input values increase over an interval. |
| degree | The highest power of the variable in a polynomial function, which determines the number of differences needed to reach a constant value. |
| even degree | A polynomial function where the highest power of the variable is an even number. |
| global maximum | The greatest of all local maximum values of a polynomial function. |
| global minimum | The least of all local minimum values of a polynomial function. |
| increasing | A characteristic of a function where output values rise as input values increase over an interval. |
| leading coefficient | The coefficient a_n of the leading term in a polynomial function. |
| leading term | The term in a polynomial with the highest degree, which dominates the function's behavior as input values increase or decrease without bound. |
| local maximum | A point where a polynomial function switches from increasing to decreasing, producing a relative highest output value in that region. |
| local minimum | A point where a polynomial function switches from decreasing to increasing, producing a relative lowest output value in that region. |
| point of inflection | A point on the graph of a polynomial where the concavity changes from concave up to concave down or vice versa, occurring where the rate of change changes from increasing to decreasing or decreasing to increasing. |
| polynomial function | A function that can be expressed in the form p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a positive integer and a_n is nonzero. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| real zero | A real number value that makes a polynomial function equal to zero, corresponding to an x-intercept on the graph. |
| Term | Definition |
|---|---|
| complex conjugate | For a non-real complex number a+bi, its conjugate is a-bi; non-real zeros of polynomials with real coefficients always occur in conjugate pairs. |
| complex zero | A zero of a polynomial function that is a complex number (including non-real complex numbers of the form a+bi). |
| even function | A function that is graphically symmetric over the line x = 0 and satisfies the property f(−x) = f(x). |
| even multiplicity | When a zero of a polynomial has an even multiplicity, the graph is tangent to the x-axis at that point and does not cross it. |
| graphically symmetric | A property of a function's graph where it mirrors itself across a line or point. |
| linear factor | An expression of the form (x-a) that divides evenly into a polynomial function, where a is a zero of the polynomial. |
| multiplicity | The number of times a linear factor appears in the complete factorization of a polynomial; determines how the graph behaves at that zero. |
| non-real zero | A zero of a polynomial function that is not a real number; a complex number with a non-zero imaginary part. |
| odd function | A function that is graphically symmetric about the point (0,0) and satisfies the property f(−x) = −f(x). |
| odd multiplicity | When a zero of a polynomial has an odd multiplicity, the graph crosses the x-axis at that point. |
| polynomial function | A function that can be expressed in the form p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a positive integer and a_n is nonzero. |
| polynomial inequality | An inequality involving a polynomial function, where real zeros serve as endpoints for intervals that satisfy the inequality. |
| real zero | A real number value that makes a polynomial function equal to zero, corresponding to an x-intercept on the graph. |
| root | A solution to the equation p(x)=0; a value where the polynomial function equals zero. |
| x-intercept | The point where a graph crosses or touches the x-axis, occurring at (a, 0) when a is a real zero of the function. |
| zero | A value of the input for which a polynomial function equals zero; also called a root of the equation p(x)=0. |
| Term | Definition |
|---|---|
| degree | The highest power of the variable in a polynomial function, which determines the number of differences needed to reach a constant value. |
| end behavior | The behavior of a function as the input values approach positive or negative infinity. |
| leading term | The term in a polynomial with the highest degree, which dominates the function's behavior as input values increase or decrease without bound. |
| nonconstant polynomial function | A polynomial function with degree greater than zero. |
| polynomial function | A function that can be expressed in the form p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a positive integer and a_n is nonzero. |
| Term | Definition |
|---|---|
| degree | The highest power of the variable in a polynomial function, which determines the number of differences needed to reach a constant value. |
| end behavior | The behavior of a function as the input values approach positive or negative infinity. |
| horizontal asymptote | A horizontal line that a rational function's graph approaches as input values increase or decrease without bound. |
| leading term | The term in a polynomial with the highest degree, which dominates the function's behavior as input values increase or decrease without bound. |
| limit notation | Mathematical notation using lim to describe the value that a function approaches as the input approaches a specific value or infinity. |
| polynomial function | A function that can be expressed in the form p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a positive integer and a_n is nonzero. |
| rational function | A function expressed as the ratio of two polynomials, where the denominator is not equal to zero. |
| slant asymptote | A linear asymptote that occurs when the numerator polynomial of a rational function has a degree one greater than the denominator polynomial. |
| Term | Definition |
|---|---|
| asymptote | Lines that a graph approaches but never reaches, indicating behavior at infinity or at points of discontinuity. |
| domain | The set of all possible input values for which a function is defined. |
| numerator | The polynomial expression in the top part of a rational function. |
| rational function inequalities | Inequalities of the form r(x) ≥ 0 or r(x) ≤ 0 where r is a rational function, used to determine intervals where the function is non-negative or non-positive. |
| zeros of rational functions | The real values of x for which a rational function equals zero, which correspond to the real zeros of the numerator when those values are in the domain of the function. |
| Term | Definition |
|---|---|
| denominator | The polynomial expression in the bottom part of a rational function. |
| multiplicity | The number of times a linear factor appears in the complete factorization of a polynomial; determines how the graph behaves at that zero. |
| numerator | The polynomial expression in the top part of a rational function. |
| polynomial function | A function that can be expressed in the form p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a positive integer and a_n is nonzero. |
| rational function | A function expressed as the ratio of two polynomials, where the denominator is not equal to zero. |
| real zero | A real number value that makes a polynomial function equal to zero, corresponding to an x-intercept on the graph. |
| vertical asymptote | A vertical line x = a where the graph of a rational function approaches infinity or negative infinity as the input approaches a. |