| 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. |
| Term | Definition |
|---|---|
| base | The number b in exponential functions b^x or logarithmic functions log_b x, where b > 0 and b ≠ 1. |
| exponential function | A function of the form f(x) = ab^x where a ≠ 0 is the initial value and b > 0, b ≠ 1 is the base. |
| exponential growth | A pattern of change where output values increase multiplicatively as input values increase additively. |
| identity function | The function h(x) = x, which serves as the line of reflection between inverse exponential and logarithmic functions. |
| inverse function | A function that reverses the mapping of another function, such that if f(x) = y, then f⁻¹(y) = x. |
| inverse relationship | A relationship between two functions where the input and output values are reversed, such that one function undoes the other. |
| logarithmic function | A function of the form f(x) = a log_b x where b > 0, b ≠ 1, and a ≠ 0, characterized by output values changing additively as input values change multiplicatively. |
| logarithmic growth | A pattern of change where output values increase additively as input values increase multiplicatively. |
| ordered pair | A pair of values (x, y) representing a point on a graph or a relationship between input and output values. |
| reflection | A transformation that flips a graph over a line, such as the line y = x. |
| Term | Definition |
|---|---|
| arithmetic sequence | A sequence where each term after the first is found by adding a fixed number called the common difference to the previous term. |
| common difference | The constant difference between successive terms in an arithmetic sequence, denoted by d. |
| common ratio | The constant factor by which each term in a geometric sequence is multiplied to obtain the next term. |
| constant proportional change | A relationship where successive terms change by the same multiplicative factor, characteristic of geometric sequences. |
| constant rate of change | The uniform change between successive terms in an arithmetic sequence. |
| general term | A formula that represents any term in a sequence based on its position, such as g_n = g_0 r^n for geometric sequences. |
| geometric sequence | A sequence where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. |
| initial value | The starting value of a function, represented by b in linear functions and a in exponential functions. |
| sequence | A function from the whole numbers to the real numbers, producing a list of ordered values. |
| whole numbers | The set of non-negative integers {0, 1, 2, 3, ...} used as the domain for a sequence function. |
| Term | Definition |
|---|---|
| additive transformation | A transformation of a function involving addition or subtraction, resulting in vertical and horizontal translations. |
| 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. |
| 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. |
| exponential function | A function of the form f(x) = ab^x where a ≠ 0 is the initial value and b > 0, b ≠ 1 is the base. |
| extremum | Maximum or minimum points on a function; logarithmic functions do not have extrema except on closed intervals. |
| increasing function | A function over an interval where output values always increase as input values increase. |
| inverse function | A function that reverses the mapping of another function, such that if f(x) = y, then f⁻¹(y) = x. |
| logarithmic function | A function of the form f(x) = a log_b x where b > 0, b ≠ 1, and a ≠ 0, characterized by output values changing additively as input values change multiplicatively. |
| 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. |
| range | The set of all possible output values that a function can produce. |
| vertical asymptote | A vertical line x = a where the graph of a rational function approaches infinity or negative infinity as the input approaches a. |
| Term | Definition |
|---|---|
| change of base property | The logarithmic property stating that log_b x = (log_a x)/(log_a b), which allows conversion between logarithms of different bases. |
| 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). |
| logarithmic expressions | Mathematical expressions involving logarithms that can be rewritten in different equivalent forms. |
| natural logarithm | The logarithmic function with base e, denoted ln(x), that is particularly useful for modeling real-world phenomena. |
| power property for logarithms | The logarithmic property stating that log_b x^n = n log_b x, which allows exponents inside a logarithm to be written as coefficients. |
| product property for logarithms | The logarithmic property stating that log_b(xy) = log_b x + log_b y, which allows products inside a logarithm to be written as sums. |
| 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 |
|---|---|
| additive transformations | Shifts of a function's graph vertically or horizontally, represented by the parameters h and k in transformed function forms. |
| exponential equations | Equations in which the variable appears in the exponent, solved using properties of exponents and logarithms. |
| exponential function | A function of the form f(x) = ab^x where a ≠ 0 is the initial value and b > 0, b ≠ 1 is the base. |
| exponential inequalities | Inequalities in which the variable appears in the exponent, solved using properties of exponents and logarithms. |
| extraneous solutions | Solutions obtained through algebraic manipulation that do not satisfy the original equation or are excluded by mathematical or contextual limitations. |
| inverse function | A function that reverses the mapping of another function, such that if f(x) = y, then f⁻¹(y) = x. |
| inverse operations | Operations that undo each other, such as addition and subtraction or exponentiation and logarithms, used to reverse a function's mapping. |
| inverse relationship between exponential and logarithmic functions | The mathematical relationship where exponential and logarithmic functions undo each other, allowing conversion between exponential and logarithmic forms. |
| logarithmic equations | Equations involving logarithms, solved using properties of logarithms and the inverse relationship with exponential functions. |
| logarithmic function | A function of the form f(x) = a log_b x where b > 0, b ≠ 1, and a ≠ 0, characterized by output values changing additively as input values change multiplicatively. |
| logarithmic inequalities | Inequalities involving logarithms, solved using properties of logarithms and the inverse relationship with exponential functions. |
| properties of exponents | Rules governing operations with exponential expressions, used to simplify and solve exponential equations. |
| properties of logarithms | Rules governing operations with logarithmic expressions, including product, quotient, and power properties, used to solve logarithmic equations. |
| Term | Definition |
|---|---|
| exponential function | A function of the form f(x) = ab^x where a ≠ 0 is the initial value and b > 0, b ≠ 1 is the base. |
| logarithmic function | A function of the form f(x) = a log_b x where b > 0, b ≠ 1, and a ≠ 0, characterized by output values changing additively as input values change multiplicatively. |
| logarithmic regression | A statistical method using technology to construct a logarithmic function model that best fits a given data set. |
| natural logarithm | The logarithmic function with base e, denoted ln(x), that is particularly useful for modeling real-world phenomena. |
| proportional growth | Growth where input values change proportionally over equal-length output-value intervals, often modeled by logarithmic functions. |
| real zero | A real number value that makes a polynomial function equal to zero, corresponding to an x-intercept on the graph. |
| repeated multiplication | A process where an initial value is multiplied by the same proportion multiple times, which can be modeled using logarithmic functions. |
| transformation | Changes applied to a parent function such as translations, reflections, stretches, or compressions. |
| Term | Definition |
|---|---|
| dependent variable | The variable representing output values in a function. |
| exponential characteristics | Properties of data or functions that follow an exponential pattern, such as constant multiplicative rates of change. |
| exponential data | Data that follows an exponential model of the form y = ab^x, where the rate of change increases or decreases exponentially. |
| exponential model | A mathematical model of the form y = ab^x that describes data or phenomena exhibiting exponential growth or decay. |
| initial linear value | The y-intercept of a linear model on a semi-log plot, which equals log_n a for an exponential model y = ab^x. |
| linear model | A mathematical representation of a linear relationship, expressed in the form y = mx + b, where m is the slope and b is the y-intercept. |
| linear rate of change | The slope of a linear model, representing the constant change in y for each unit change in x; in a semi-log plot of exponential data, this equals log_n b. |
| linearization | The process of transforming exponential data into a linear form so that it can be modeled with a linear function. |
| logarithmically scaled | An axis on a graph where values are marked according to their logarithms, compressing large ranges of data and making exponential functions appear linear. |
| semi-log graph | A graph with a logarithmic scale on one axis and a linear scale on the other axis, used to linearize exponential relationships. |
| semi-log plot | A graph where one axis (typically the y-axis) uses a logarithmic scale while the other axis uses a linear scale, used to visualize exponential relationships. |
| Term | Definition |
|---|---|
| arithmetic sequence | A sequence where each term after the first is found by adding a fixed number called the common difference to the previous term. |
| common difference | The constant difference between successive terms in an arithmetic sequence, denoted by d. |
| common ratio | The constant factor by which each term in a geometric sequence is multiplied to obtain the next term. |
| 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. |
| domain | The set of all possible input values for which a function is defined. |
| exponential function | A function of the form f(x) = ab^x where a ≠ 0 is the initial value and b > 0, b ≠ 1 is the base. |
| geometric sequence | A sequence where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. |
| initial value | The starting value of a function, represented by b in linear functions and a in exponential functions. |
| linear function | A polynomial function of degree 1 with the form f(x) = mx + b, representing a constant rate of change. |
| point-slope form | A way of expressing linear functions as f(x) = y_i + m(x - x_i) based on a known slope and a point on the line. |
| proportional change | Output values that change by a constant factor or ratio over equal-length input intervals, characteristic of exponential functions. |
| slope | The rate of change of a line, representing how much the output changes for each unit change in the input. |
| Term | Definition |
|---|---|
| additive transformation | A transformation of a function involving addition or subtraction, resulting in vertical and horizontal translations. |
| base | The number b in exponential functions b^x or logarithmic functions log_b x, where b > 0 and b ≠ 1. |
| 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. |
| domain | The set of all possible input values for which a function is defined. |
| exponential decay | An exponential function where a > 0 and 0 < b < 1, resulting in output values that decrease as input values increase. |
| exponential function | A function of the form f(x) = ab^x where a ≠ 0 is the initial value and b > 0, b ≠ 1 is the base. |
| exponential growth | A pattern of change where output values increase multiplicatively as input values increase additively. |
| extremum | Maximum or minimum points on a function; logarithmic functions do not have extrema except on closed intervals. |
| initial value | The starting value of a function, represented by b in linear functions and a in exponential functions. |
| 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. |
| proportional over equal-length input-value intervals | A characteristic where the ratio of output values remains constant for equal changes in input values, which identifies exponential functions. |
| Term | Definition |
|---|---|
| exponential expressions | Mathematical expressions of the form b^x where b is a base and x is an exponent, which can be rewritten in multiple equivalent forms. |
| exponential unit fraction | An exponent in the form of a unit fraction 1/k where k is a natural number, representing a root of the base. |
| 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). |
| kth root | The value that, when raised to the power k, equals the base b, represented as b^(1/k). |
| negative exponent property | The rule stating that b^(-n) = 1/b^n, which expresses negative exponents as reciprocals. |
| power property for exponents | The rule stating that (b^m)^n = b^(mn), allowing an exponential expression raised to a power to be simplified. |
| product property for exponents | The rule stating that b^m · b^n = b^(m+n), allowing products of exponential expressions with the same base to be combined. |
| 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). |
| Term | Definition |
|---|---|
| base | The number b in exponential functions b^x or logarithmic functions log_b x, where b > 0 and b ≠ 1. |
| base of the exponent | The value b in an exponential function f(x) = ab^x that determines the rate at which the function grows or decays. |
| dependent variable | The variable representing output values in a function. |
| domain | The set of all possible input values for which a function is defined. |
| equivalent forms | Different ways of writing the same mathematical expression that have equal values for all valid inputs. |
| exponential function | A function of the form f(x) = ab^x where a ≠ 0 is the initial value and b > 0, b ≠ 1 is the base. |
| exponential models | Mathematical functions of the form f(x) = ab^x used to represent situations where quantities grow or decay by a constant factor over equal intervals. |
| exponential regression | A statistical method using technology to fit an exponential function model to a data set by finding the best-fitting values for the parameters. |
| growth factor | The base b in an exponential function f(x) = ab^x, representing the multiplicative change in the output for each unit increase in the input. |
| initial value | The starting value of a function, represented by b in linear functions and a in exponential functions. |
| input | The independent variable or value that is entered into a function. |
| natural base e | The mathematical constant approximately equal to 2.718, commonly used as the base in exponential functions that model real-world scenarios. |
| percent change | The relative change in a quantity expressed as a percentage, which is related to the growth factor in an exponential model. |
| proportional growth | Growth where input values change proportionally over equal-length output-value intervals, often modeled by logarithmic functions. |
| repeated multiplication | A process where an initial value is multiplied by the same proportion multiple times, which can be modeled using logarithmic functions. |
| transformation | Changes applied to a parent function such as translations, reflections, stretches, or compressions. |
| Term | Definition |
|---|---|
| data set | A collection of numerical values or observations that represent measurements or information about variables. |
| error | The difference between a model's predicted value and the actual observed value in a data set. |
| exponential model | A mathematical model of the form y = ab^x that describes data or phenomena exhibiting exponential growth or decay. |
| linear model | A mathematical representation of a linear relationship, expressed in the form y = mx + b, where m is the slope and b is the y-intercept. |
| model | A mathematical representation constructed from a data set to describe relationships or predict values. |
| overestimate | A predicted value that is greater than the actual observed value. |
| quadratic model | A mathematical representation of a relationship between two variables using a quadratic function of the form f(x) = ax² + bx + c. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| regression | A statistical method for fitting a function to a data set to model the relationship between variables. |
| residual plot | A graph displaying the residuals of a regression model, used to assess whether a model is appropriate for a data set. |
| residuals | The differences between the predicted values from a regression model and the actual observed values in a data set. |
| underestimate | A predicted value that is less than the actual observed value. |
| Term | Definition |
|---|---|
| additive transformation | A transformation of a function involving addition or subtraction, resulting in vertical and horizontal translations. |
| analytic representation | An algebraic formula or equation that explicitly defines a function. |
| commutative | A property where the order of operations does not affect the result; composition of functions is not commutative, meaning f ∘ g typically produces a different result than g ∘ f. |
| composite function | A function formed by combining two or more functions, where the output of one function becomes the input of another function, denoted as f ∘ g or f(g(x)). |
| composition of functions | A function operation where one function is applied to the output of another function, written as (f ∘ g)(x) = f(g(x)). |
| decomposed | The process of breaking down a complex function into simpler component functions that can be composed together. |
| domain | The set of all possible input values for which a function is defined. |
| function composition | The process of combining two or more functions where the output of one function becomes the input of another function. |
| graphical representation | A visual representation of a function displayed on a coordinate plane. |
| 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). |
| identity function | The function h(x) = x, which serves as the line of reflection between inverse exponential and logarithmic functions. |
| input value | The x-values or independent variable values used as inputs to a function. |
| multiplicative transformation | A transformation of a function involving multiplication, resulting in vertical and horizontal dilations. |
| numerical representation | A representation of a function using tables of values or ordered pairs (x, y). |
| output value | The y-values or results produced by a function for given input values. |
| 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 |
|---|---|
| composite function | A function formed by combining two or more functions, where the output of one function becomes the input of another function, denoted as f ∘ g or f(g(x)). |
| composition of functions | A function operation where one function is applied to the output of another function, written as (f ∘ g)(x) = f(g(x)). |
| contextual restrictions | Limitations on a function's domain or range based on the real-world context or practical applicability of the function. |
| domain | The set of all possible input values for which a function is defined. |
| identity function | The function h(x) = x, which serves as the line of reflection between inverse exponential and logarithmic functions. |
| 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. |
| inverse function | A function that reverses the mapping of another function, such that if f(x) = y, then f⁻¹(y) = x. |
| inverse operations | Operations that undo each other, such as addition and subtraction or exponentiation and logarithms, used to reverse a function's mapping. |
| invertible domain | The domain of a function on which the function is one-to-one and therefore has an inverse function. |
| invertible function | A function that has an inverse function; a one-to-one function where each output corresponds to exactly one input. |
| output value | The y-values or results produced by a function for given input values. |
| reflection over the line y = x | A transformation that reverses the roles of x- and y-coordinates, used to graph an inverse function. |
| reverse mapping | The process by which an inverse function exchanges the roles of inputs and outputs from the original function. |
| Term | Definition |
|---|---|
| base | The number b in exponential functions b^x or logarithmic functions log_b x, where b > 0 and b ≠ 1. |
| common logarithm | A logarithm with base 10, used when the base of a logarithmic expression is not specified. |
| exponential form | The representation of a logarithmic equation in the form b^a = c, equivalent to the logarithmic form log_b c = a. |
| logarithm | The exponent or power to which a base must be raised to obtain a given number. |
| logarithmic expression | A mathematical expression of the form log_b c, where b is the base and c is the argument, representing the exponent to which the base must be raised to obtain the value c. |
| logarithmic scale | A scale where each unit represents a multiplicative change equal to the base of the logarithm, such as powers of 10 on a base-10 logarithmic scale. |
| Term | Definition |
|---|---|
| concavity | The curvature of a function, describing whether the graph curves upward (concave up) or downward (concave down). |
| cycle | A single complete repetition of a periodic pattern that can be used to construct the entire graph of a periodic relationship. |
| intervals of decrease | Sections of a function's domain where the output values are getting smaller as the input increases. |
| intervals of increase | Sections of a function's domain where the output values are getting larger as the input increases. |
| period | The smallest positive value k such that a periodic function repeats its pattern, meaning f(x+k) = f(x) for all x in the domain. |
| periodic function | A function that repeats its values at regular intervals, where f(x+k) = f(x) for all x in the domain, with k being the period. |
| periodic relationship | A relationship between two variables where output values demonstrate a repeating pattern over successive equal-length intervals as input values increase. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| repeating pattern | A sequence of values or behaviors that recurs at regular intervals in a periodic relationship. |
| Term | Definition |
|---|---|
| domain restrictions | Limitations on the input values of a function based on mathematical validity, contextual meaning, or extreme values in the data set. |
| inverse trigonometric functions | Functions that reverse the operation of trigonometric functions, taking a trigonometric ratio as input and returning an angle measure as output. |
| periodic | A property of trigonometric functions where they repeat their values at regular intervals. |
| trigonometric equations | Equations that contain trigonometric functions and require finding the values of the variable that satisfy the equation. |
| trigonometric inequalities | Inequalities that contain trigonometric functions and require finding the values of the variable that satisfy the inequality. |
| Term | Definition |
|---|---|
| cosecant function | The reciprocal of the sine function, denoted f(θ) = csc θ, defined where sin θ ≠ 0. |
| cotangent function | The reciprocal of the tangent function, denoted f(θ) = cot θ, equivalent to cos θ/sin θ where sin θ ≠ 0. |
| range | The set of all possible output values that a function can produce. |
| reciprocal function | A function formed by taking the reciprocal (1/f) of another function. |
| secant function | The reciprocal of the cosine function, denoted f(θ) = sec θ, defined where cos θ ≠ 0. |
| vertical asymptotes | Lines where a function approaches infinity; for secant and cosecant functions, these occur where cosine and sine equal zero, respectively. |
| Term | Definition |
|---|---|
| algebraic manipulation | The process of rewriting expressions using algebraic operations to transform them into equivalent forms. |
| cosine sum identity | The trigonometric identity cos(α + β) = cos α cos β - sin α sin β, which expresses the cosine of a sum of two angles. |
| difference identities | Trigonometric identities derived from sum identities by substituting negative angles, used to express trigonometric functions of angle differences. |
| domain restrictions | Limitations on the input values of a function based on mathematical validity, contextual meaning, or extreme values in the data set. |
| double-angle identities | Trigonometric identities that express trigonometric functions of twice an angle in terms of functions of the original angle. |
| equivalent analytic representations | Different algebraic forms of trigonometric expressions that are mathematically equal and can reveal different properties or simplify problem-solving. |
| equivalent forms | Different ways of writing the same mathematical expression that have equal values for all valid inputs. |
| Pythagorean identity | The fundamental trigonometric identity sin² θ + cos² θ = 1, derived from the Pythagorean Theorem applied to the unit circle. |
| sine sum identity | The trigonometric identity sin(α + β) = sin α cos β + cos α sin β, which expresses the sine of a sum of two angles. |
| trigonometric equations | Equations that contain trigonometric functions and require finding the values of the variable that satisfy the equation. |
| trigonometric expressions | Mathematical expressions involving trigonometric functions such as sine, cosine, tangent, and their reciprocals. |
| trigonometric identity | An equation involving trigonometric functions that is true for all values in the domain of the functions. |
| trigonometric inequalities | Inequalities that contain trigonometric functions and require finding the values of the variable that satisfy the inequality. |
| unit circle | A circle with radius 1 centered at the origin, used to define trigonometric functions where a point on the circle has coordinates (cos θ, sin θ). |
| Term | Definition |
|---|---|
| angle in standard position | An angle positioned in the coordinate plane with its vertex at the origin and one ray coinciding with the positive x-axis. |
| complex number | A number of the form a + bi, where a and b are real numbers and i is the imaginary unit. |
| complex plane | A coordinate system where complex numbers are represented as points, with the real part on the horizontal axis and the imaginary part on the vertical axis. |
| origin | The central point in a polar coordinate system from which all distances (radii) are measured. |
| polar coordinate | A coordinate system in which points are located by their distance from the origin (radius r) and their angle measure (θ) from the positive x-axis. |
| polar coordinate system | A coordinate system based on circles centered at the origin and lines through the origin, where points are located using an ordered pair (r, θ). |
| radius | In polar coordinates, the distance from the origin to a point, represented by |r|. |
| rectangular coordinate | An ordered pair (x, y) representing the horizontal and vertical position of a point in the plane. |
| rectangular coordinate system | A coordinate system where points are located using an ordered pair (x, y) representing horizontal and vertical distances from the origin. |
| terminal ray | The ray that forms the final side of an angle in standard position. |
| Term | Definition |
|---|---|
| angle measure | The input value in a polar function that represents the direction from the positive x-axis, typically measured in radians or degrees. |
| domain | The set of all possible input values for which a function is defined. |
| origin | The central point in a polar coordinate system from which all distances (radii) are measured. |
| polar coordinate | A coordinate system in which points are located by their distance from the origin (radius r) and their angle measure (θ) from the positive x-axis. |
| polar functions | Functions of the form r = f(θ) where the input is an angle measure and the output is a radius, used to create graphs in polar coordinates. |
| positive x-axis | The reference direction in a polar coordinate system from which angle measures are taken. |
| radius | In polar coordinates, the distance from the origin to a point, represented by |r|. |
| 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]. |
| polar function | A function of the form r = f(θ) that describes a curve in the polar coordinate system, where r is the distance from the origin and θ is the angle. |
| relative extremum | A point on a function where the function changes from increasing to decreasing (relative maximum) or from decreasing to increasing (relative minimum). |
| Term | Definition |
|---|---|
| cosine function | A trigonometric function that gives the x-coordinate (horizontal displacement from the y-axis) of a point on the unit circle corresponding to a given angle. |
| radian measure | The measure of an angle defined as the ratio of the arc length subtended by the angle to the radius of the circle. |
| sine function | A trigonometric function that gives the y-coordinate (vertical displacement from the x-axis) of a point on the unit circle corresponding to a given angle. |
| standard position | The position of an angle with its vertex at the origin and its initial side along the positive x-axis. |
| tangent function | A trigonometric function, denoted f(θ) = tan θ, that gives the slope of the terminal ray of an angle in standard position on the unit circle. |
| terminal ray | The ray that forms the final side of an angle in standard position. |
| unit circle | A circle with radius 1 centered at the origin, used to define trigonometric functions where a point on the circle has coordinates (cos θ, sin θ). |
| Term | Definition |
|---|---|
| angle in standard position | An angle positioned in the coordinate plane with its vertex at the origin and one ray coinciding with the positive x-axis. |
| circle centered at the origin | A circle whose center is at the point (0, 0) on the coordinate plane. |
| coordinates | The ordered pair (x, y) that specifies the location of a point on a coordinate plane. |
| cosine function | A trigonometric function that gives the x-coordinate (horizontal displacement from the y-axis) of a point on the unit circle corresponding to a given angle. |
| equilateral triangle | A triangle with all three sides equal in length and all angles measuring 60°. |
| isosceles right triangle | A right triangle with two equal sides and angles of 45°-45°-90°. |
| multiples of π/4 | Angles that are integer multiples of π/4 radians (45°), including π/4, π/2, 3π/4, π, etc. |
| multiples of π/6 | Angles that are integer multiples of π/6 radians (30°), including π/6, π/3, π/2, 2π/3, etc. |
| quadrant | One of the four regions of the coordinate plane divided by the x-axis and y-axis. |
| radian | A unit of angle measure where one radian is the angle formed when the arc length equals the radius of the circle. |
| radius | In polar coordinates, the distance from the origin to a point, represented by |r|. |
| sine function | A trigonometric function that gives the y-coordinate (vertical displacement from the x-axis) of a point on the unit circle corresponding to a given angle. |
| terminal ray | The ray that forms the final side of an angle in standard position. |
| Term | Definition |
|---|---|
| cosine function | A trigonometric function that gives the x-coordinate (horizontal displacement from the y-axis) of a point on the unit circle corresponding to a given angle. |
| domain | The set of all possible input values for which a function is defined. |
| oscillate | To move back and forth between two values in a regular, repeating pattern. |
| sine function | A trigonometric function that gives the y-coordinate (vertical displacement from the x-axis) of a point on the unit circle corresponding to a given angle. |
| standard position | The position of an angle with its vertex at the origin and its initial side along the positive x-axis. |
| terminal ray | The ray that forms the final side of an angle in standard position. |
| unit circle | A circle with radius 1 centered at the origin, used to define trigonometric functions where a point on the circle has coordinates (cos θ, sin θ). |
| x-coordinate | The horizontal position of a point, representing its distance from the y-axis. |
| y-coordinate | The vertical position of a point, representing its distance from the x-axis. |
| Term | Definition |
|---|---|
| amplitude | The absolute value of the coefficient a in a sinusoidal function, representing the maximum distance from the midline to the peak or trough of the graph. |
| 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. |
| cosine function | A trigonometric function that gives the x-coordinate (horizontal displacement from the y-axis) of a point on the unit circle corresponding to a given angle. |
| even function | A function that is graphically symmetric over the line x = 0 and satisfies the property f(−x) = f(x). |
| frequency | The number of complete cycles of a sinusoidal function that occur over a unit interval of input values. |
| midline | The horizontal line around which a sinusoidal function oscillates, located at y = d in the function a sin(b(θ + c)) + d. |
| odd function | A function that is graphically symmetric about the point (0,0) and satisfies the property f(−x) = −f(x). |
| period | The smallest positive value k such that a periodic function repeats its pattern, meaning f(x+k) = f(x) for all x in the domain. |
| reflective symmetry | A property of a graph that is mirror-symmetric across a line; the cosine function has reflective symmetry over the y-axis. |
| rotational symmetry | A property of a graph that looks the same when rotated 180 degrees about a point; the sine function has rotational symmetry about the origin. |
| sine function | A trigonometric function that gives the y-coordinate (vertical displacement from the x-axis) of a point on the unit circle corresponding to a given angle. |
| sinusoidal function | A function of the form f(θ) = a sin(b(θ + c)) + d or g(θ) = a cos(b(θ + c)) + d, where a, b, c, and d are real numbers and a ≠ 0. |
| Term | Definition |
|---|---|
| amplitude | The absolute value of the coefficient a in a sinusoidal function, representing the maximum distance from the midline to the peak or trough of the graph. |
| 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). |
| midline | The horizontal line around which a sinusoidal function oscillates, located at y = d in the function a sin(b(θ + c)) + d. |
| period | The smallest positive value k such that a periodic function repeats its pattern, meaning f(x+k) = f(x) for all x in the domain. |
| phase shift | A horizontal translation of a sinusoidal function represented by the constant c, which shifts the graph left or right by -c units. |
| sinusoidal function | A function of the form f(θ) = a sin(b(θ + c)) + d or g(θ) = a cos(b(θ + c)) + d, where a, b, c, and d are real numbers and a ≠ 0. |
| 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 shift | A vertical translation of a sinusoidal function represented by the additive constant d, which moves the entire graph up or down and shifts the midline. |
| 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 |
|---|---|
| amplitude | The absolute value of the coefficient a in a sinusoidal function, representing the maximum distance from the midline to the peak or trough of the graph. |
| contextual domain | The range of input values for which a sinusoidal function model is meaningful and applicable within a real-world context. |
| frequency | The number of complete cycles of a sinusoidal function that occur over a unit interval of input values. |
| period | The smallest positive value k such that a periodic function repeats its pattern, meaning f(x+k) = f(x) for all x in the domain. |
| periodic phenomena | Events or patterns that repeat at regular intervals over time or space. |
| phase shift | A horizontal translation of a sinusoidal function represented by the constant c, which shifts the graph left or right by -c units. |
| sinusoidal function | A function of the form f(θ) = a sin(b(θ + c)) + d or g(θ) = a cos(b(θ + c)) + d, where a, b, c, and d are real numbers and a ≠ 0. |
| sinusoidal regression | A statistical method using technology to fit a sinusoidal function to a data set by estimating the best-fit parameters. |
| vertical shift | A vertical translation of a sinusoidal function represented by the additive constant d, which moves the entire graph up or down and shifts the midline. |
| Term | Definition |
|---|---|
| additive transformation | A transformation of a function involving addition or subtraction, resulting in vertical and horizontal translations. |
| asymptote | Lines that a graph approaches but never reaches, indicating behavior at infinity or at points of discontinuity. |
| 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. |
| cosine function | A trigonometric function that gives the x-coordinate (horizontal displacement from the y-axis) of a point on the unit circle corresponding to a given angle. |
| 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. |
| period | The smallest positive value k such that a periodic function repeats its pattern, meaning f(x+k) = f(x) for all x in the domain. |
| periodic | A property of trigonometric functions where they repeat their values at regular intervals. |
| phase shift | A horizontal translation of a sinusoidal function represented by the constant c, which shifts the graph left or right by -c units. |
| 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. |
| 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. |
| sine function | A trigonometric function that gives the y-coordinate (vertical displacement from the x-axis) of a point on the unit circle corresponding to a given angle. |
| slope | The rate of change of a line, representing how much the output changes for each unit change in the input. |
| standard position | The position of an angle with its vertex at the origin and its initial side along the positive x-axis. |
| tangent function | A trigonometric function, denoted f(θ) = tan θ, that gives the slope of the terminal ray of an angle in standard position on the unit circle. |
| terminal ray | The ray that forms the final side of an angle in standard position. |
| unit circle | A circle with radius 1 centered at the origin, used to define trigonometric functions where a point on the circle has coordinates (cos θ, sin θ). |
| 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 |
|---|---|
| arccosine | The inverse function of cosine, denoted cos⁻¹(x), that returns an angle whose cosine equals the input value. |
| arcsine | The inverse function of sine, denoted sin⁻¹(x), that returns an angle whose sine equals the input value. |
| arctangent | The inverse function of tangent, denoted tan⁻¹(x), that returns an angle whose tangent equals the input value. |
| inverse trigonometric functions | Functions that reverse the operation of trigonometric functions, taking a trigonometric ratio as input and returning an angle measure as output. |
| invertible function | A function that has an inverse function; a one-to-one function where each output corresponds to exactly one input. |
| periodic | A property of trigonometric functions where they repeat their values at regular intervals. |
| restricted domain | A limited interval of input values for a trigonometric function that makes it one-to-one and therefore invertible. |
| Term | Definition |
|---|---|
| column | The vertical lines of elements in a matrix. |
| dot product | A scalar quantity obtained by multiplying the magnitudes of two vectors and the cosine of the angle between them; equals zero when vectors are perpendicular. |
| matrix | A rectangular array of numbers arranged in rows and columns that represents a linear transformation. |
| matrix product | The product of two matrices that represents the composition of their corresponding linear transformations. |
| row | The horizontal lines of elements in a matrix. |
| Term | Definition |
|---|---|
| domain | The set of all possible input values for which a function is defined. |
| parameter | An independent variable (often denoted t) used to express the coordinates of points on a curve in parametric form. |
| parametric equations | A pair of equations that express x and y coordinates as functions of a parameter, typically written as x(t) and y(t). |
| parametric function | A function that expresses the coordinates of a point as functions of a parameter, typically time t, written as f(t) = (x(t), y(t)). |
| Term | Definition |
|---|---|
| 2 × 2 matrix | A square matrix with 2 rows and 2 columns. |
| column vector | Vectors represented as columns in a matrix; when two column vectors form a 2×2 matrix, the absolute value of the determinant gives the area of the parallelogram they span. |
| determinant | A scalar value calculated from a square matrix that determines whether the matrix is invertible; for a 2×2 matrix [a b; c d], the determinant equals ad - bc. |
| identity matrix | A square matrix with 1s on the main diagonal (from top left to bottom right) and 0s everywhere else. |
| invertibility | The property of a square matrix that has an inverse; a matrix is invertible if and only if its determinant is nonzero. |
| matrix inverse | A matrix that, when multiplied by the original matrix, produces the identity matrix; a square matrix has an inverse if and only if its determinant is nonzero. |
| parallel vectors | Vectors that have the same or opposite direction, resulting from scalar multiplication of a vector by a constant. |
| parallelogram | A quadrilateral formed by two vectors; the area of the parallelogram spanned by two vectors equals the absolute value of the determinant of the matrix formed by those vectors. |
| row vector | Vectors represented as rows in a matrix; when two row vectors form a 2×2 matrix, the absolute value of the determinant gives the area of the parallelogram they span. |
| square matrix | A matrix with the same number of rows and columns; only square matrices can have determinants and inverses. |
| Term | Definition |
|---|---|
| input vector | The vector that is mapped or transformed by a linear transformation. |
| linear transformation | A function that maps vectors to vectors while preserving vector addition and scalar multiplication, represented by a matrix. |
| matrix multiplication | The operation of multiplying a transformation matrix by a vector or matrix to produce output vectors. |
| output vector | The resulting vector produced by applying a linear transformation to an input vector. |
| ℝ² | The two-dimensional real vector space consisting of all ordered pairs of real numbers. |
| transformation matrix | A 2 × 2 matrix A that represents a linear transformation, where L(v) = Av for vectors v in ℝ². |
| zero vector | A vector with components ⟨0, 0⟩ that occurs when the tail and head are at the same point. |
| Term | Definition |
|---|---|
| composition of functions | A function operation where one function is applied to the output of another function, written as (f ∘ g)(x) = f(g(x)). |
| composition of linear transformations | The result of applying one linear transformation followed by another linear transformation. |
| determinant | A scalar value calculated from a square matrix that determines whether the matrix is invertible; for a 2×2 matrix [a b; c d], the determinant equals ad - bc. |
| dilation | A linear transformation that scales regions by a constant factor, with the magnitude determined by the absolute value of the determinant. |
| inverse transformations | Two linear transformations that are inverses if their composition maps any vector to itself, effectively undoing each other's effects. |
| linear transformation | A function that maps vectors to vectors while preserving vector addition and scalar multiplication, represented by a matrix. |
| matrix | A rectangular array of numbers arranged in rows and columns that represents a linear transformation. |
| matrix inverse | A matrix that, when multiplied by the original matrix, produces the identity matrix; a square matrix has an inverse if and only if its determinant is nonzero. |
| matrix product | The product of two matrices that represents the composition of their corresponding linear transformations. |
| rotation | A linear transformation that rotates every vector by a fixed angle about the origin without changing its length. |
| unit vector | A vector with a magnitude of 1, often used to indicate direction. |
| vector | A mathematical object with both magnitude and direction, represented as an ordered pair of components in ℝ². |
| Term | Definition |
|---|---|
| discrete intervals | Separate, distinct time periods or steps used to measure changes in a system, rather than continuous time. |
| future states | The predicted conditions or distributions of a system at subsequent time steps using matrix multiplication. |
| matrix inverse | A matrix that, when multiplied by the original matrix, produces the identity matrix; a square matrix has an inverse if and only if its determinant is nonzero. |
| matrix models | Mathematical representations using matrices to represent transitions or changes between different states in a system. |
| past states | The predicted conditions or distributions of a system at previous time steps using the inverse of a transition matrix. |
| percent change | The relative change in a quantity expressed as a percentage, which is related to the growth factor in an exponential model. |
| repeated multiplication | A process where an initial value is multiplied by the same proportion multiple times, which can be modeled using logarithmic functions. |
| state vector | A column vector that represents the distribution or values across different states at a particular point in time. |
| steady state | A distribution between states that remains unchanged from one step to the next after repeated matrix multiplication. |
| transition matrix | A matrix that models the probabilities or rates of moving from one state to another in a system. |
| transitions between states | Changes or movements from one condition or situation to another in a system being modeled. |
| Term | Definition |
|---|---|
| horizontal extrema | The maximum and minimum x-coordinates reached by a particle during its motion, found by identifying extrema of x(t). |
| parametric function | A function that expresses the coordinates of a point as functions of a parameter, typically time t, written as f(t) = (x(t), y(t)). |
| particle motion | The path and position of a particle as it moves through space over time, modeled using parametric equations. |
| planar motion | The movement of a particle or object in a two-dimensional plane. |
| real zero | A real number value that makes a polynomial function equal to zero, corresponding to an x-intercept on the graph. |
| vertical extrema | The maximum and minimum y-coordinates reached by a particle during its motion, found by identifying extrema of y(t). |
| 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. |
| y-intercepts | The points where the particle's path crosses the y-axis, corresponding to the real zeros of x(t). |
| 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]. |
| direction of motion | The path direction a particle follows in the plane, determined by whether x(t) and y(t) are increasing or decreasing. |
| parametric planar motion function | A function that describes the motion of a particle in a plane using a parameter (typically time) to define both x and y coordinates independently. |
| parametrization | A representation of a curve using a pair of functions (x(t), y(t)) where both x and y are expressed in terms of a parameter t. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| slope of the graph | The ratio of the average rate of change of y to the average rate of change of x between two points on a parametric curve. |
| Term | Definition |
|---|---|
| circular path | A curve traced in the plane that forms a circle, defined by parametric equations. |
| counterclockwise revolution | Motion around a circle in the counterclockwise direction, completing a full 360-degree rotation. |
| line segment | The portion of a line between two endpoints, characterized by a starting point and an ending point. |
| linear path | A straight line segment connecting two points in the coordinate plane. |
| parametric function | A function that expresses the coordinates of a point as functions of a parameter, typically time t, written as f(t) = (x(t), y(t)). |
| parametrically | Expressed using parametric equations where x and y coordinates are defined as functions of a parameter, typically time (t). |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| transformation | Changes applied to a parent function such as translations, reflections, stretches, or compressions. |
| unit circle | A circle with radius 1 centered at the origin, used to define trigonometric functions where a point on the circle has coordinates (cos θ, sin θ). |
| Term | Definition |
|---|---|
| equation involving two variables | A mathematical statement with an equals sign containing two different variables, which can describe one or more functions or be graphed in a coordinate plane. |
| implicitly defined function | A function defined by an equation relating x and y, rather than explicitly solving for y in terms of x. |
| ordered pairs | Points on a graph represented as (x, y) coordinates that satisfy the equation of a function. |
| rate of change | The measure of how quickly a function's output changes relative to changes in its input. |
| solutions to an equation | The ordered pairs of values that satisfy an equation involving two variables and can be plotted as points on a graph. |
| Term | Definition |
|---|---|
| asymptote | Lines that a graph approaches but never reaches, indicating behavior at infinity or at points of discontinuity. |
| circle | A special case of an ellipse where the horizontal and vertical radii are equal (a = b). |
| conic sections | Curves formed by the intersection of a plane with a cone, including parabolas, ellipses, circles, and hyperbolas. |
| ellipse | A conic section formed when a plane intersects a cone at an angle, or the set of points where the sum of distances to two foci is constant. |
| horizontal radius | The distance from the center of an ellipse to its edge along the horizontal axis, represented by the value a. |
| hyperbola | A conic section formed when a plane intersects both nappes of a cone, or the set of points where the difference of distances to two foci is constant. |
| parabola | A conic section formed when a plane intersects a cone parallel to its side, or the set of points equidistant from a focus and directrix. |
| vertex | The point (h, k) that represents the center or turning point of a parabola. |
| vertical radius | The distance from the center of an ellipse to its edge along the vertical axis, represented by the value b. |
| Term | Definition |
|---|---|
| conic sections | Curves formed by the intersection of a plane with a cone, including parabolas, ellipses, circles, and hyperbolas. |
| domain | The set of all possible input values for which a function is defined. |
| ellipse | A conic section formed when a plane intersects a cone at an angle, or the set of points where the sum of distances to two foci is constant. |
| hyperbola | A conic section formed when a plane intersects both nappes of a cone, or the set of points where the difference of distances to two foci is constant. |
| implicitly defined function | A function defined by an equation relating x and y, rather than explicitly solving for y in terms of x. |
| inverse function | A function that reverses the mapping of another function, such that if f(x) = y, then f⁻¹(y) = x. |
| invertible function | A function that has an inverse function; a one-to-one function where each output corresponds to exactly one input. |
| parabola | A conic section formed when a plane intersects a cone parallel to its side, or the set of points equidistant from a focus and directrix. |
| parameter | An independent variable (often denoted t) used to express the coordinates of points on a curve in parametric form. |
| parametric equations | A pair of equations that express x and y coordinates as functions of a parameter, typically written as x(t) and y(t). |
| parametrization | A representation of a curve using a pair of functions (x(t), y(t)) where both x and y are expressed in terms of a parameter t. |
| trigonometric parametrization | A method of representing curves using trigonometric functions (such as sine, cosine, secant, and tangent) as the parametric equations. |
| Term | Definition |
|---|---|
| angle measure | The input value in a polar function that represents the direction from the positive x-axis, typically measured in radians or degrees. |
| components | The horizontal (a) and vertical (b) values of a vector, where a = x₂ - x₁ and b = y₂ - y₁. |
| directed line segment | A line segment with a specified direction from a starting point to an ending point. |
| direction | The orientation of a vector, which is parallel to the line segment from the origin to the point with coordinates (a, b). |
| dot product | A scalar quantity obtained by multiplying the magnitudes of two vectors and the cosine of the angle between them; equals zero when vectors are perpendicular. |
| head | The ending point or tip of a vector. |
| Law of Cosines | A relationship used to find side lengths or angle measures in a triangle when given other side lengths and angles. |
| Law of Sines | A relationship stating that the ratio of a side length to the sine of its opposite angle is constant for all sides and angles in a triangle. |
| magnitude | The length of a vector, calculated as the square root of the sum of the squares of its components. |
| parallel vectors | Vectors that have the same or opposite direction, resulting from scalar multiplication of a vector by a constant. |
| perpendicular | Two vectors are perpendicular when the angle between them is 90 degrees, indicated by a dot product of zero. |
| scalar multiplication | The multiplication of a constant (scalar) by a vector, resulting in a new vector whose components are each multiplied by that constant. |
| standard basis vectors | The unit vectors →i = ⟨1, 0⟩ and →j = ⟨0, 1⟩ that point in the positive x and y directions, respectively, in ℝ². |
| tail | The starting point or beginning of a vector. |
| unit vector | A vector with a magnitude of 1, often used to indicate direction. |
| vector | A mathematical object with both magnitude and direction, represented as an ordered pair of components in ℝ². |
| vector addition | The process of combining two or more vectors to produce a resultant vector. |
| vector sum | The addition of two vectors by adding their corresponding components to produce a new vector. |
| zero vector | A vector with components ⟨0, 0⟩ that occurs when the tail and head are at the same point. |
| Term | Definition |
|---|---|
| parametric function | A function that expresses the coordinates of a point as functions of a parameter, typically time t, written as f(t) = (x(t), y(t)). |
| planar motion | The movement of a particle or object in a two-dimensional plane. |
| position vector | A vector that represents the location of a particle relative to the origin, with magnitude equal to the distance from the origin. |
| speed | The magnitude of the velocity vector, representing the rate at which a particle is moving regardless of direction. |
| vector-valued function | A function that outputs vectors, typically expressed as p(t) = ⟨x(t), y(t)⟩ or p(t) = x(t)i + y(t)j, where each input t produces a vector output. |
| velocity vector | A vector-valued function v(t) = ⟨x'(t), y'(t)⟩ that represents the rate of change of position with respect to time, indicating both direction and speed of motion. |
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