| 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. |