Modeling, Functions, and Graphs

n, the distance on the number line from a number to \(0\text{.}\) For example, the absolute value of \(-7\) is \(7\text{.}\) This fact is expressed by the equation \(\abs{-7} = 7\text{.}\)

n, an equation in which the variable occurs between the absolute value bars.

n, an inequality in which the variable occurs between the absolute value bars.

n, a meaningful combination of numbers, variables, and operation symbols. Also called an expression.

n, a fraction whose numerator and denominator are polynomials. Also called a rational expression.

n, a method for solving equations (or inequalities) by manipulating the equations (or inequalities). Compare with graphical solution and numerical solution.

n, an equation showing the (approximate) relationship between a living organism’s body mass and another of the organism’s properties or processes, usually given in the form \(y = k (\text{mass})^p\text{.}\)

n, (i) the distance above the ground or above sea level; (ii) the vertical distance between the base and the opposite vertex of a triangle, pyramid, or cone; (iii) the distance between parallel sides of a parallelogram, trapezoid, or rectangle. Also called height.

n, the payment of a debt through regular installments over a period of time.

(in an interest-bearing account), n, the sum of the principal that was invested and all the interest earned.

n, the vertical distance between the midline and the maximum value of a sinusoidal function.

n, sequence of equal payments or deposits made at equal time intervals.

n, an inexact result.

n, a measure of the two-dimensional space enclosed by a polygon or curve, typically expressed in terms of square units, such as square meters or square feet, etc.

n, an ordering of the terms of a polynomial so that the exponents on the variable are increasing, such as in the polynomial \(1 + x + x^2\text{.}\)

n, the property that when adding three or more terms, the grouping of terms does not affect the sum. We express this formally by saying that if \(a\text{,}\) \(b\text{,}\) and \(c\) are any numbers, then \((a + b) + c = a + (b + c)\text{.}\)

n, the property that when multiplying three or more factors, the grouping of factors does not affect the product. We express this formally by saying that if \(a\text{,}\) \(b\text{,}\) and \(c\) are any numbers, then \((a \cdot b) \cdot c = a \cdot (b \cdot c)\text{.}\)

n, a reference line (or curve) towards which the graph of an equation tends as the value of x and/or y grows or diminishes without bound.

n, the matrix obtained by making each row of the matrix correspond to an equation of the systterm, with the coefficients of the variables filling the first \(n\) columns, and the last (that is, the \(n + 1\)) column having the constants.

n, (plural axes), a line used as a reference for position and/or orientation.

n, a line that cuts a plane figure into two parts, each a mirror image of the other.

n, a technique for solving a triangular systterm of linear equations.

n, a picture of numerical information in which the lengths or heights of bars are used to represent the values of variables.

n, (i) a number or algebraic expression that is used as a repeated factor, where an exponent indicates how many times the base is used as a factor. For example, when we write \(3^5\text{,}\) the base is \(3\text{.}\) (ii) The bottom side of a polygon. (iii) The bottom face of a solid.

n, the angles opposite the equal sides in an isosceles triangle.

n, a polynomial with exactly two terms.

n, a sum of two unlike terms, such as \(\sqrt{3}+\sqrt{2} \text{.}\)

v, to find an equivalent fraction by multiplying numerator and denominator by the same nonzero expression.

n, an expression by which both numerator and denominator of a given fraction are multiplied (in order to build the fraction).

n, the grid that associates points in the coordinate plane to ordered pairs of numbers.

n, a plane with a pair of coordinate axes. Also called a coordinate plane.

n, the final value (of the variable) minus the starting value.

n, (i) a transformation of data, (ii) substitution of a new variable for a variable expression, for example, replacing \(t^2\) with \(x\) so that the equation \(y = at^2 + b\) becomes \(y = ax + b\text{.}\)

n, the set of all points in a plane at a fixed distance (the radius) from the center.

n, the distance around a circle.

n, a set of numbers, denoted by \([a, b]\text{,}\) which includes all the numbers between \(a\) and \(b\) as well as the numbers \(a\) and \(b\) thtermselves, where \(a\) and \(b\) are real numbers and \(a \lt b\text{.}\) Or the set of numbers denoted by \((-\infty, b]\text{,}\) which includes the real number \(b\) and all numbers less than \(b\text{,}\) or the set of numbers denoted by \([a, \infty)\text{,}\) which includes the real number \(a\) and all numbers greater than \(a\text{.}\)

n, the numerical factor in a term. For example, in the expression \(32a + 7b\text{,}\) the coefficient of \(a\) is \(32\) and the coefficient of \(b\) is \(7\text{.}\)

n, the matrix of \(n\) columns obtained by making each row of the matrix correspond to an equation of the systterm, with the coefficients of the variables filling the \(n\) columns (and the constants are not represented in the matrix).

n, a quantity that divides evenly into each of the given expressions.

n, the exponent, denoted by \(\log x\) (or by \(\log_{10} x\)) for the number \(10\) to obtain the value \(x\text{,}\) that is, \(10^{\log x} = x\text{.}\)

n, the property that when adding terms, the order of the terms does not affect the sum. We express this formally by saying that if \(a\) and \(b\) are any numbers, then \(a + b = b + a\text{.}\)

n, the property that when multiplying factors, the order of the factors does not affect the product. We express this formally by saying that if \(a\) and \(b\) are any numbers, then \(a\cdot b = b\cdot a\text{.}\)

n, two angles whose measures add up to \(90\degree\text{.}\)

v, to determine the appropriate constant to add to a binomial of the form \(ax^2 + bx\) so that the result can be written in the form \(a(x + k)^2\text{.}\)

n, the complex number with the same real part and opposite imaginary part; for example, the complex conjugate of \(1 + i\) is \(1 - i\text{.}\)

n, a fraction that contains one or more fractions in its numerator and/or in its denominator.

n, a coordinate plane representing complex numbers, with the real parts corresponding to the values on the horizontal axis and imaginary parts corresponding to values on the vertical axis.

n, a number that can be written in the form \(a + bi\text{,}\) where \(a\) and \(b\) are real numbers and \(i^2=-1\text{.}\)

n, one of the values of an ordered pair or ordered triple.

n, a mathtermatical statterment involving two order symbols. For example, the compound inequality \(1\lt x\lt 2\) says that "\(1\) is less than \(x\text{,}\) and \(x\) is less than \(2\text{.}\)"

n, an interest earning agreterment in which the interest payment at a given time is computed based on the sum of the original principal and any interest money already accrued.

n, the time interval between consecutive interest payments to an account that earns interest.

adj, curving so that the ends of a flexible rod would need to be bent downward (compared with a straight rod) to lie along the graph. Or equivalently, curving so that a line segment tangent to the curve will lie above the curve.

adj, curving so that the ends of a flexible rod would need to be bent upward (compared with a straight rod) to lie along the graph. Or equivalently, curving so that a line segment tangent to the curve will lie below the curve.

n, a description of a curve as either concave up or concave down.

adj, having the same center.

n, an equation that is true for some (but not all) values of the variable(s).

n, a three-dimensional object whose base is a circle and whose vertex is a point above the circle. The points on the segments joining the circle to the vertex make up the cone.

adj, having all measure(s) matching exactly. For example, two line segments are congruent when they have the same length; two triangles are congruent if all three sides and all three angles of one match exactly with the corresponding parts of the other triangle.

n, (i) (of a complex number) the complex number with the same real part and opposite imaginary part; (ii) (of a binomial expression) the binomial expression with the same first term and opposite second term.

n, (i) (of a complex number) a complex number and its conjugate; (ii) (of a binomial expression) the binomial expression and its conjugate.

adj, having at least one solution.

adj, having exactly one solution.

adj, unchanging, not variable. For example, we say that the product of two variables is constant if the product is always the same number, for any values of the variables.

n, a number (as opposed to a variable).

n, the quotient of two directly proportional variables, or the product of two inversely proportional variables. Also called the constant of variation.

see constant of proportionality.

n, an equation or inequality involving one or more variables, typically specifying a condition that must be true in the given context.

adj, without holes or gaps. For example, a curve is continuous if it can be drawn without lifting the pencil from the page, and a function is continuous if its graph can be drawn without lifting the pencil from the page.

n, an interest earning agreterment in which the amount in the account is \(Pe^{rt}\text{,}\) where \(P\) is the initial principal, \(r\) is the annual interest rate, and \(e\approx 2.71828\) is the base of the natural logarithm.

n, a ratio used to convert from one unit of measure to another.

n, a number used with a number line or an axis to designate position.

n, one of the two perpendicular number lines used to define the coordinates of points in the plane.

n, a plane with a pair of coordinate axes. Also called the Cartesian plane or \(xy\)-plane.

n, a mathtermatical fact that is a consequence of a previously known fact.

n, money that an individual or group must pay out. For example, the costs of a company might include payments for wages, supplies, and rent.

n, one of the numbers \(1, 2, 3, 4, \ldots\text{.}\)

n, (i) a three-dimensional box whose six faces all consist of squares; (ii) an expression raised to the power \(3\text{.}\)

v, to raise an expression to the power \(3\text{.}\) For example, to cube \(2\) means to form the product of three \(2\)s: \(2^3 = 2 \times 2 \times 2 = 8\text{.}\)

n, a number that when raised to the power \(3\) gives a desired value. For example, \(2\) is the cube root of \(8\) because \(2^3 = 8\text{.}\)

adj, having to do with the third degree of a variable or with a polynomial of degree \(3\text{.}\)

n, a three-dimensional figure in the shape of a soft drink can. The top and base are circles of identical size, and the line segments joining the two circles are perpendicular to the planes containing the two circles.

n, the factor by which an initial value of a diminishing quantity is multiplied to obtain the final value.

adj, having to do with a base-\(10\) numeration systterm.

n, the position of a digit relative to the decimal point. For example, in the number \(3.14159\text{,}\) the digit \(4\) is in the second decimal place, or hundredths place.

n, the mark "." that is written between the whole number part and the fractional part of a decimal number. For example, the decimal form of \(1 \frac{3}{10}\) is \(1.3\text{.}\)

adj, (i) (of numbers) moving to the left on a number line: Positive numbers are decreasing when getting closer to zero, and negative numbers are decreasing when they move farther from \(0\text{;}\) (ii) (of a graph) having decreasing values of \(y\) when moving along the graph from left to right; (iii) (of a function) having a decreasing graph.

n, a measure of angle equal to \(\frac{1}{360}\) of a complete revolution.

n, (i) (of a monomial) the exponent on the variable, or if there are more than one variable, the sum of the exponents of all the variables; (ii) (of a polynomial) the largest degree of the monomials in the polynomial.

n, an equation that gives the quantity of some product that consumers are willing to purchase in terms of the price of that product.

n, the expression below the fraction bar in a fraction.

adj, (of a systterm of equations) having infinitely many solutions.

n, a variable whose value is determined by specifying the value of the independent variable.

n, expressed with the term with the highest degree written first, then the term with the second highest degree, etc.

n, (i) a line segment joining one vertex of a quadrilateral to the opposite vertex; (ii) a line segment joining opposite corners of a box; (iii) the entries of a matrix whose row number match the column number, that is, the \((1, 1), (2, 2), \ldots , (n, n)\) entries

n, (i) a line segment passing through the center of a circle (or sphere) with endpoints on the circle (sphere); (ii) the length of that line segment.

n, the result of a subtraction. For example, the expression \(a - b\) represents the difference between \(a\) and \(b\text{.}\)

n, an expression of the form \(a^2 - b^2\text{.}\)

n, (i) (of a matrix) the numbers of rows and columns respectively of the matrix, also called the order of the matrix. For example, a matrix with dimension \(2\) by \(3\) (or \(2\times 3\)) has two rows and three columns; (ii) a measurterment defining a geometric figure, for example, the length and width are dimensions of a rectangle.

n, a relation between two variables in which one is a constant multiple of the other (so that the ratio between the two variables is the constant), or in which one is a constant multiple of a positive exponent power of the other variable.

n, the difference between the ending coordinate and the starting coordinate of points on a number line; the directed distance is negative if the ending value is smaller than the starting value. For example, the directed distance from \(5\) to \(2\) is \(2 - 5 = -3\text{.}\)

adj, describing variables related by direct variation.

n, (for the quadratic polynomial \(ax^2 + bx + c\)) the quantity \(b^2 - 4ac\text{.}\)

n, the property that for any numbers \(a\text{,}\) \(b\text{,}\) and \(c\text{,}\) \(a(b + c) = ab + ac\text{.}\)

n, a quantity that is divided into another quantity. For example, in the expression \(a\div b\text{,}\) the divisor is \(b\text{.}\)

n, the set of all acceptable inputs for a function or equation.

n, (of exponential growth) the time required for a quantity to double in size.

n, one of the three following operations: (1) an exchange of two rows, (2) multiplying all entries of a row by a nonzero constant, (3) adding a multiple of any row to another row.

n, a method for solving a system of equations that involves adding together the equations of the systterm or multiples of the equations of the systterm.

n, an equation whose graph (approximately) fits a given set of data (but gives no information about the physical processes involved).

n, a value in a matrix, often identified by specifying location by row and column.

n, a mathtermatical statterment that two expressions are equal, for example, \(1 + 1 = 2\text{.}\)

n, an equation that involves two variables.

adj, (of a polygon) having all sides of equal length.

n, the point where the graphs of the supply and demand equations intersect

adj, representing the same value.

n, equations that have the same solutions.

n, expressions that have the same value for all permissible values of their variables.

n, the allowable difference between an estimate and the actual value.

v, to determine the value of an expression when the variable in the expression is replaced by a number.

adj, not simply close, but with absolutely no deviation from an intended value.

n, the exact value of a solution, i.e., not an approximation.

n, the expression that indicates how many times the base is used as a factor. For example, when we write \(3^5\text{,}\) the exponent is \(5\text{,}\) and \(3^5 = 3\times 3\times 3\times 3\times 3\text{.}\)

n, a manner of decreasing characterized by a constant decay factor for any fixed specified interval of time, or equivalently, modeled by a function \(f\) with the form \(f(t) = ab^t\text{,}\) where \(a\) and \(b\) are positive constants and \(0 \lt b\lt 1\text{.}\)

n, an equation containing a variable expression as an exponent.

n, a function \(f\) which can be put in the form \(f(x) = ab^x\text{,}\) where \(a\) is a nonzero constant and \(b\ne 1\) is a positive constant.

n, growth characterized by a constant growth factor for any fixed specified interval of time, or equivalently, modeled by a function \(f\) with the form \(f(t) = ab^t\text{,}\) where \(a\) and \(b\) are positive constants and \(b\gt 1\text{.}\)

n, a way of writing an expression that involves radicals and/or reciprocals in terms of powers that have fractional and/or negative exponents. For example, the exponential notation for \(\sqrt{3}\) is \(3^{1/2}\text{.}\)

see algebraic expression.

n, a method used to solve (quadratic) equations.

n, a value that is not a solution to a given equation but is a solution to an equation derived from the original.

v, to estimate the value of a dependent variable for a value of the independent variable that is outside the range of the data.

n, an expression that divides evenly into another expression. For example, \(2\) is a factor of \(6\text{.}\)

v, to write as a product. For example, to factor \(6\) we write \(6 = 2\times 3\text{.}\)

n, (i) (of a polynomial or algebraic expression) an expression written as a product of two or more factors, where the algebraic factors cannot be further factored; (ii) (of an equation of a parabola) the form \(y = a(x - r_1)(x - r_2)\text{.}\)

n, an ordered pair which satisfies the constraints of a linear programming problterm.

n, an acronym for a method for computing the product of two binomials: F stands for First terms, O for Outer terms, I for Inner terms, and L stands for Last terms.

n, an equation involving two or more variables.

n, the line segment separating the numerator and denominator of a fraction. In the fraction \(\frac{1}{2} \text{,}\) the fraction bar is the short segment between the \(1\) and the \(2\text{.}\)

n, a relationship between two variables in which each value of the input variable determines a unique value of the output variable.

n, a relationship between an output variable and an ordered pair of input variables in which each ordered pair of the input variables determines a unique value of the output variable.

n, an output value of a function.

n, the property that the value of a fraction is unchanged when both its numerator and denominator are multiplied by the same nonzero value. We express this formally by saying if \(a\) is any number, and \(b\) and \(c\) are nonzero numbers, then \(\displaystyle{\frac{a\cdot c}{b\cdot c}=\frac{a}{b}} \text{.}\)

n, the process of performing eltermentary row operations on a matrix to obtain a matrix in echelon form.

adj, having the same shape (but possibly different size).

n, a visual representation of the values of a variable or variables, typically drawn on a number line or on the Cartesian plane.

v, to draw a graph.

n, a picture of the solutions of an equation (or inequality) using a number line or coordinate plane.

n, a method for solving equations (or inequalities) by reading values off an appropriate graph. Compare with algebraic solution and numerical solution.

n, the largest factor that divides evenly into each expression.

n, the factor by which an initial value of a growing quantity is multiplied to obtain the final value.

n, individual points that are plotted to help draw a graph (by hand).

n, (of exponential decay) the time required for a quantity to diminish to half its original size.

n, either of the two regions of a plane that has been divided into two regions by a straight line

see altitude.

n, half a sphere (on one side or the other of a plane passing through the center).

n, a line parallel to the \(x\)-axis toward which the graph of an equation tends as the value of \(x\) grows or diminishes without bound.

n, the horizontal coordinate axis. Often called the \(x\)-axis.

n, where the graph meets the horizontal axis. Also called \(x\)-intercept.

n, a test to determine if a function has an inverse function: If no horizontal line intersects the graph of a function more than once, then the inverse is also a function.

n, the result of moving all points of the graph straight left (or all straight right) by the same distance.

n, the longest side of a right triangle. (It is always the side opposite the right angle.)

n, an equation that is true for all permissible values of the variable(s).

n, the vertical axis in the complex plane.

n, a complex number of the form \(bi\text{,}\) where \(b\) is a real number and \(i^2 = -1\text{.}\)

n, (of a complex number) the coefficient of \(i\) when the complex number is written in the form \(a + bi\text{,}\) where \(a\) and \(b\) are real numbers. For example, the imaginary part of \(4 - 7i\) is \(- 7\text{.}\)

n, a nonreal number denoted by \(i\) and which satisfies \(i^2= -1\text{,}\) that is, \(i\) is defined to be a square root of \(-1\text{.}\)

adj, (of a system of equations) having no solution.

adj, (i) (of numbers) moving to the right on a number line: Positive numbers are increasing when moving farther from zero, and negative numbers are increasing when they move closer to \(0\text{;}\) (ii) (of a graph) having increasing values of y when moving along the graph from left to right; (iii) (of a function) having an increasing graph.

adj, (i) (of a system of \(2\) linear equations in \(2\) variables) having different graphs for the two equations; (ii) (of a system of \(n\) linear equations in \(n\) variables) having no one equation equal to a linear combination of the others.

n, a variable whose value determines the value of the dependent variable.

n, (of a radical) the number at the left of the radical symbol that indicates the type of root involved; for example, the index of \(3\) in the expression \(\sqrt[3]{x}\) indicates a cube root.

n, a mathematical statement of the form \(a\lt b\text{,}\) \(a\le b\text{,}\) \(a\gt b\text{,}\) \(a\ge b\text{,}\) or \(a\ne b\text{.}\)

n, a persistent increase over time of consumer prices.

n, a point where a graph changes concavity.

n, the starting value of a variable, often when \(t = 0\text{.}\)

n, value of the independent variable.

n, a whole number or the negative of a whole number.

n, a point where a graph meets a coordinate axis.

n, a method for graphing a line by finding its horizontal and vertical intercepts.

n, money paid for the use of money. For example, after borrowing money, the borrower must pay the lender not only the original amount of money borrowed (known as the principal) but also the interest on the principal.

n, the fraction of the principal that is paid as interest for one year. For example, an interest rate of \(10\%\) means that the interest for one year will be \(10\%\) of the principal.

v, to estimate the value of a dependent variable based on data that include both larger and smaller values of the independent variable.

n, a point in common to two graphs.

n, a set of numbers that includes all the numbers between \(a\) and \(b\) (possibly but not necessarily including \(a\) and/or \(b\)), where \(a\) and \(b\) are real numbers. Or the set of all numbers less than \(b\) (and possibly including \(b\)), or the set all numbers greater than \(a\) (and possibly including \(a\)).

n, notation used to designate an interval. For example, \([2, 3]\) is the interval notation to designate all the real numbers from \(2\) to \(3\text{,}\) including both \(2\) and \(3\text{.}\)

n, a function whose inputs are outputs of a given function \(f\text{,}\) and whose outputs are the corresponding inputs of \(f\text{.}\)

n, a physical law that states that the magnitude of some quantity is inversely proportional to the square of the distance to the source of that quantity.

n, a relation between two variables in which one is a constant divided by the other (so that the product of the two variables is the constant), or in which one is a constant divided by a positive exponent power of the other.

adj, describing variables related by inverse variation.

n, a number that is not rational but does correspond to a point on the number line.

v, (a variable or expression) to create an equivalent equation (or inequality) in which the variable or expression is by itself on one side of the equation (or inequality).

n, a triangle with two sides of equal length.

n, a relationship among three or more variables in which whenever all but two variables are held constant, those remaining two variables vary directly or inversely with each other.

n, a basic property about powers and exponents.

n, (of a polynomial) the coefficient of term with highest degree.

n, (of a row in a matrix) the first nonzero entry of the row, when read from left to right.

n, one of the two shorter sides of a right triangle, or the length of that side.

n, fractions with equivalent denominators.

n, terms with equivalent variable parts.

n, the points on a single line that join two specified points (the endpoints) on that line.

n, (i) the sum of a nonzero constant multiple of one equation and a nonzero constant multiple of a second equation; (ii) the sum of constant multiples of quantities.

n, a procedure for solving a linear system of equations which requires taking one or more linear combination of equations.

n, an equation such as \(2x + 3y = 4\) or \(x - 3y = 7\) in which each term has degree \(0\) or \(1\text{.}\)

n, the study of optimizing functions with constraint equations and/or constraint inequalities.

n, the process of using a line to predict values of a (dependent) variable.

n, a set of linear equations.

n, a term that consists of a constant times a variable.

see logarithm.

n, a scale of measurement that uses the logarithm of a physical quantity rather than the quantity itself.

n, a type of graph paper in which both horizontal and vertical axes use log scales.

n, (i) an exponent; (ii) a function whose outputs are exponents associated with a given base.

n, an equation involving the logarithm of a variable expression.

n, a function of the form \(f (x) = \log_b(x)\text{,}\) where \(b\) is a positive constant different from \(1\text{.}\)

n, (of two or more fractions) the smallest denominator that is a multiple of the denominators in the given fractions.

n, (of two or more counting numbers) the smallest counting number that the given numbers divide into evenly.

n, a representation of relationships among quantities using equations, tables, and/or graphs.

n, a rectangular array of numbers.

adj, largest or greatest.

n, largest value.

n, (of a variable expression) the largest value that the expression can equal when the variable is allowed to assume all possible values.

n, the average of a set of numbers, computed by adding the numbers and dividing by how many are in the set. For example, the mean of \(5\text{,}\) \(2\text{,}\) and \(11\) is \(\frac{5+2+11}{3}= 6\text{.}\)

n, an equation whose graph (approximately) fits a given set of data and whose parameters are estimates about the physical properties involved.

n, the middle number in a set of numbers when written in increasing order. For example, the median of \(5\text{,}\) \(2\text{,}\) and \(11\) is \(5\text{.}\) If the set has two numbers in the middle when written in order, then the median of the set is the mean of those middle numbers. For example, the median of \(6\text{,}\) \(1\text{,}\) \(9\text{,}\) and \(27\) is \(\frac{6+9}{2}= 7.5\text{.}\)

adj, least or smallest.

n, smallest value.

n, (of a variable expression) the smallest value that the expression can equal when the variable is allowed to assume all possible values.

n, the number that occurs most frequently in a set of numbers. For example, the mode of \(1\text{,}\) \(1\text{,}\) \(2\text{,}\) and \(3\) is \(1\text{.}\)

n, a mathematical equation or graph or table used to represent a situation in the world or a situation described in words. For example, the equation \(P = R - C\) is a model for the relationship among the variables of profit, revenue, and cost.

v, to create a model.

n, an algebraic expression with only one term.

adj, (of a function or graph) either never increasing or never decreasing.

n, the property that \(\abs{a\cdot b} = \abs{a}\cdot\abs{b}\) for any real numbers \(a\) and \(b\text{.}\)

n, (i) (of a zero of a polynomial) the number of times the corresponding linear factor appears as a factor of the polynomial. For example, \(-9\) is a zero of multiplicity one and \(7\) is a zero of multiplicity two for the polynomial \(p(x) =x^3 - 5x^2 - 77x + 441\) because \(p(x)\) factors as \(p(x) =(x + 9) (x - 7)^2\text{;}\) (ii) (of a solution to a polynomial equation) the multiplicity of the zero of the corresponding polynomial. For example, \(-9\) is a solution of multiplicity one and \(7\) is a solution of multiplicity two for the polynomial equation \(x^3 = 5x^2 + 77x - 441\) because the equation can be written in the standard form \(p(x) = 0\text{,}\) where \(p(x)\) factors as \(p(x) =(x + 9)(x - 7)^2\text{.}\)

n, the irrational number \(e\approx 2.71828182846\text{,}\) which is useful in calculus, statistics, and other mathematical topics.

n, the function \(f(x) = e^x\text{,}\) where \(e\) is the natural base.

n, the logarithm with base \(e\text{,}\) where \(e\) is the natural base.

n, a counting number.

n, a number that is less than zero.

n, the opposite of.

n, the final value of a variable minus the initial value. For example, if an object’s weight decreases from \(15\) pounds to \(13\) pounds, the net change in weight is \(-2\) pounds.

n, a mathematical statement of the form \(a \le b\) or \(a \ge b\text{.}\)

adj, perpendicular.

n, a number which when raised to the power \(n\) gives a desired value. When \(b^n = a\text{,}\) then \(b\) is an \(n\)th root of \(a\text{.}\)

n, a line with coordinates marked on it representing the real numbers.

n, the expression in a fraction that is above the fraction bar.

n, a method for solving equations by reading values from an appropriate table of values. Compare with algebraic solution and graphical solution.

n, (in linear programming) the function that is to be optimized.

adj, (pertaining to a function) having the property that every output comes from one and only one input.

n, a set of numbers denoted by \((a, b)\text{,}\) which includes all the numbers between \(a\) and \(b\) but not the numbers \(a\) and \(b\) themselves, where \(a\) and \(b\) are real numbers and \(a \ne b\text{.}\) Or the set of numbers denoted by \((-\infty b)\text{,}\) which includes all numbers less than \(b\text{,}\) or the set of numbers denoted by \((a, \infty)\text{,}\) which includes all numbers greater than \(a\text{.}\)

n, addition, subtraction, multiplication, or division (or raising to a power or taking a root).

n, the number on the number line that is on the other side of \(0\) and at the same distance. For example, \(5\) and \(-5\) are opposites.

n, (of a matrix) the numbers of rows and columns respectively of the matrix, also called the dimension of the matrix. For example, a matrix with order \(2\) by \(3\) (or \(2 \times 3\)) has two rows and three columns.

n, rules that prescribe the order in which to carry out the operations in an expression.

n, one of the four symbols \(\lt\text{,}\) or \(\le\text{,}\) or \(\gt\text{,}\) or \(\ge\text{.}\)

n, a pair of numbers enclosed in parentheses, like this: \((x, y )\text{.}\) Often used to specify a point or a location on the coordinate plane.

n, three numbers enclosed in parentheses, like this: \((x, y, z)\text{.}\) Often used to specify a solution to a system of equations in three variables or a point in three-dimensional space.

n, the point where the coordinate axes meet. It has coordinates (0, 0).

n, value of the dependent variable.

n, a curve with the shape of the graph of \(y = ax^2\text{,}\) where \(a\ne 0\text{.}\)

n, lines that lie in the same plane but do not intersect, even if extended indefinitely.

n, a constant in an equation that varies in other equations of the same form. For example, in the slope-intercept formula \(y = b + mx\text{,}\) the constants \(b\) and \(m\) are parameters.

n, a fraction with (an understood) denominator of \(100\text{.}\) For example, to express the fraction \(\frac{51}{100}\) as a percent, we write \(51\%\) or say "\(51\) percent."

n, the change in some quantity, expressed as a percentage of the starting amount.

n, the square of an integer. For example, \(9\) is a perfect square because \(9 = 3^2\text{.}\)

n, the distance around the edge or boundary of a two-dimensional figure.

n, lines that meet and form right angles with each other.

n, a function defined by multiple expressions, one expression for each specified interval of the independent variable.

n, one way of writing the equation for a line: \(y-y_1=m(x-x_1)\) or \(\frac{y-y_1}{x-x_1}= m\text{.}\)

n, a simple closed geometric figure in the plane consisting of line segments (called sides) that meet only at their endpoints. For example, triangles are polygons with three sides.

n, a sum of terms, where each term is either a constant or a constant times a power of a variable, and the exponent is a positive integer.

n, a function that can be written in the form \(f (x) = a_nx^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} +\cdots + a_2x_2 + a_1x + a_0\) where \(a_0, a_1, a_2, \ldots a_n\) are constants.

n, a number greater than zero.

n, an expression that consists of a base and an exponent.

n, a function of the form \(f(x) = ax^p\text{,}\) where \(a\) and \(p\) are constants.

n, an integer greater than \(1\) whose only whole number factors are itself and \(1\text{.}\)

n, the original amount of money deposited in an account or borrowed from a lender. (Compare with interest.)

see principal square root.

n, the nonnegative square root.

n, the result of a multiplication. For example, the expression \(a\cdot b\) represents the product of \(a\) and \(b\text{.}\)

n, the money left after counting all the revenue that came in and subtracting the costs that had to be paid out.

n, an equation in which each side is a ratio.

see directly proportional, inversely proportional.

n, a three-dimensional object like a cone except that the base is a polygon instead of a circle.

If the legs of a right triangle are \(a\) and \(b\) and the hypotenuse is \(c\text{,}\) then \(a^2 + b^2 = c^2\text{.}\)

n, any of the four regions into which the coordinate axes divide the plane. The first quadrant consists of the points where both coordinates are positive; the second quadrant where the first coordinate is negative and the second coordinate positive; the third quadrant consists of points where both coordinates are negative; and the fourth quadrant contains the points where the first coordinate is positive and the second coordinate is negative.

adj, relating to the square of a variable (or of an expression).

n, an equation that equates zero to a polynomial of degree \(2\) (or an equivalent equation).

n, the formula that gives the solutions of the quadratic equation \(ax^2 + bx + c = 0\text{,}\) namely \(x = \frac{-b±\sqrt{b^2-4ac}}{2a}\text{.}\)

n, a function of the form \(f (x) = ax^2 + bx + c\text{.}\)

n, a polynomial whose degree is \(2\text{.}\)

n, the process of using a quadratic function to predict values of a (dependent) variable.

n, a term whose degree is \(2\text{.}\)

n, a polynomial of degree \(2\) and having exactly \(3\) terms.

n, a polygon with exactly \(4\) sides.

adj, (pertaining to a polynomial) having degree \(4\text{.}\)

n, the result of a division. For example, the expression \(a \div b\) represents the quotient of \(a\) and \(b\text{.}\)

n, a root of a number, such as a square root or a cube root.

n, a square root, a cube root, or an \(n\)th root.

n, an equation in which the variable appears under a radical sign.

n, notation using the radical sign to indicate a root.

n, the symbol \(\sqrt{~}\text{,}\) which is used to indicate the principal square root, or the symbol \(\sqrt[3]{~}\text{,}\) which is used to indicate cube root, or the symbol \(\sqrt[n]{~}\text{,}\) which is used to indicate \(n\)th root, where \(n\) is a counting number greater than \(2\text{.}\)

n, the expression under a radical sign.

n, (i) a line segment from the center of a circle (or sphere) to a point on the circle (sphere), (ii) the length of that line segment.

v, use as a repeated factor, for example, to raise \(x\) to the power \(2\) is the same as multiplying \(x\cdot x\text{.}\)

n, (i) the set of all output values for a function; (ii) the difference between the largest and smallest values in a set of data.

n, a ratio that compares two quantities (typically) with different units.

n, the ratio of change in the dependent variable to the corresponding change in the independent variable, measuring the change in the dependent variable per unit change in the independent variable.

n, (i) a way to compare two quantities by division, (ii) a fraction. For example, "the ratio of \(1\) to \(2\)" can be written as \(\frac{1}{2} \text{.}\)

adj, having to do with ratios.

n, an exponent that is a rational number. For example, the expression \(x^{1/3}\) has a rational exponent of \(1/3\text{,}\) and \(x^{1/3} = \sqrt[3]{x}\text{.}\)

n, a ratio of two polynomials. Also called an algebraic fraction.

n, a function of the form \(f (x) = \frac{p(x)}{q(x)}\text{,}\) where \(p\) and \(q\) are polynomial functions.

n, a number that can be expressed as the ratio of two integers.

v, to find an equivalent fraction that contains no radical in the denominator. For example, when we rationalize \(\frac{1}{\sqrt{2}} \text{,}\) we obtain \(\frac{\sqrt{2}}{2} \text{.}\)

n, the horizontal axis in the complex plane.

see number line.

n, (of a complex number) the term which does not include \(i\) when the complex number is written in the form \(a + bi\text{,}\) where \(a\) and \(b\) are real numbers. For example, the real part of \(4 - 7i\) is \(4\text{.}\)

n, a number that corresponds to a point on a number line.

n, the result of dividing \(1\) by the given number. For example, the reciprocal of \(2\) is \(\frac{1}{2} \text{.}\) Two numbers are reciprocals of each other when their product is \(1\text{.}\)

n, a four-sided figure (in the plane) with four right angles. The opposite sides are equal in length and parallel.

v, to find an equivalent fraction whose numerator and denominator share no common factors (other than \(1\) and \(-1\)).

n, (of a matrix) a row echelon form matrix that also satisfies (1) the leading entry in each nonzero row is a \(1\text{,}\) (2) each leading \(1\) is the only nonzero entry in its column.

n, the transformation that replaces each point of a graph with its mirror image on the other side of the line.

n, the line used for linear regression.

n, a polygon all of whose sides have equal length and all of whose angles are congruent.

n, a domain of a function that does not include all real numbers.

n, money that an individual or group receives. For example, a person might have revenues from both a salary and from earnings on investments.

n, an angle of \(90\degree\text{.}\)

n, a triangle that includes one right angle.

n, the solution to an equation. See also cube root; \(n\)th root; principal square root; square root.

v, to give an approximate value of a number by choosing the nearest number of a specified form. For example, to round \(3.14159\) to two decimal places, we use \(3.14\text{.}\)

n, (of a matrix) a matrix in which (1) only zeros occur below each nonzero leading entry, (2) the leading entry in any row is to the right of any leading entry above it, and (3) any row consisting entirely of zeros is below all rows with any nonzero entry.

v, to make an equation true (when substituted for the variable or variables). For example, the number \(5\) satisfies the equation \(x - 2 = 3\text{.}\)

n, marked values on a number line or axes to establish how wide an interval of numbers is represented by a physical distance on the number line.

v, (i) to determine the scale on an axis or axes; (ii) to multiply (measurements) by a fixed number (the scale factor).

n, a fixed number by which measurements or values are multiplied.

n, the exponent defining direct variation or a power function. For example, if \(y = 3x^4\text{,}\) then the scaling exponent is \(4\text{.}\)

n, a type of graph used to represent pairs of data values. Each pair of data values provides the coordinates for one point on the scatterplot. Also called a scatter diagram.

n, a standard method for writing very large or very small numbers that uses powers of \(10\text{.}\) For example, the scientific notation for \(12,000\) is \(1.2\times 10^4\text{.}\)

n, half a circle (on one side or the other of a diameter).

n, a positive or negative number.

n, (in the decimal form of a number) a digit warranted by the accuracy of the measuring device. When the decimal point is present, the significant digits are all those from the leftmost nonzero digit to the rightmost digit after the decimal point. When there is no decimal point, the significant digits are all those from the leftmost nonzero digit to the rightmost nonzero digit. For example, \(123.40\) has five significant digits, but \(12,340\) has only four significant digits. Also called significant figure.

see significant digit.

see geometrically similar.

v, to write in an equivalent but simpler or more convenient form. For example, we can simplify the expression \(\sqrt{16}\) to \(4\text{.}\)

adj, having the shape of a sine or cosine graph.

n, a measure of the steepness of a line or of the rate of change of one variable with respect to another.

n, a standard form for the equation of a nonvertical line: \(y = b + mx\text{.}\)

n, a method for graphing a line that uses the slope and the \(y\)-intercept.

n, a value for the variable that makes an equation or an inequality true. A solution to an equation in two variables is an ordered pair that satisfies the equation. A solution to a system is an ordered pair that satisfies each equation of the system.

v, (i) (an equation) to find any and all solutions to an equation, inequality, or system; (ii) (a formula) to write an equation for one variable in terms of any other variables, for example, when we solve \(5x + y = 3\) for \(y\) to get \(y = -5x + 3\text{;}\) (iii) (a triangle) to find the measures of all three sides and of all three angles.

n, a three-dimensional object in the shape of a ball. A sphere consists of all the points in space at a fixed distance (the radius) from the center of the sphere.

n, (i) any expression times itself; (ii) a rectangle whose sides are all the same length.

v, to multiply by itself, that is, to raise to the power \(2\text{.}\)

n, a matrix with the same number of rows as columns.

n, a number that when squared gives a desired value. For example, \(7\) is a square root of \(49\) because \(7^2 = 49\text{.}\)

n, (i) (of a linear, quadratic, or other polynomial equation) an equation in which the right side is 0, so the equation has the form \(p(x) = 0\text{;}\) (ii) (of a system of linear equations) a system in which the variables occur only on the left side of each equation and in alphabetic order.

n, a mathematical statement of the form \(a \lt b\) or \(a \gt b\text{.}\)

n, a small number written below and to the right of a variable. For example, in the equation \(x_1 = 3\text{,}\) the variable \(x\) has the subscript \(1\text{.}\)

n, a method for solving a system of equations that begins by expressing one variable in terms of the other.

n, the result of an addition. For example, the expression \(a + b\) represents the sum of \(a\) and \(b\text{.}\)

n, the total area of the faces or surfaces of a three-dimensional object.

n, two angles whose measures add up to \(180\degree\text{.}\)

n, an equation that gives the quantity of some product that producers are willing to produce in terms of the price of that product.

n, a geometric property of having sameness on opposite sides of a line (or plane) or about a point.

n, two or more equations involving the same variables.

n, (i) (in a sum) a quantity that is added to another. For example, in the expression \(x + y - 4\text{,}\) \(x\text{,}\) \(y\text{,}\) and \(-4\) are the terms; (ii) an algebraic expression that is not a sum or difference, for example, 4x is one term.

n, (for an inequality) a point in the plane (or on a number line) used to determine which side of the plane (or number line) is included in the solution.

v, to apply a transformation.

n, (i) (of data) applying a function to one or both components in a set of data, typically so that the resulting data becomes approximately linear; (ii) (of a graph) a change that occurs in the graph of an equation when one or more of the parameters defining that equation are altered.

n, (of a graph or geometric figure) sliding horizontally and/or vertically without rotating or changing any shapes.

n, a four-sided figure in the plane with one pair of parallel sides.

n, a three-sided figure in the plane.

n, the inequality \(\abs{a + b} \le \abs{a} +\abs{b}\text{,}\) which is true for any two real numbers \(a\) and \(b\text{.}\)

n, a system of linear equations in which the first variable does not occur in the second equation, the first two variables do not occur in the third equation (and the first three variables do not occur in the fourth equation if there are more than three variables, and so on).

n, an equation involving at least one of the trigonometric functions or ratios.

n, a polynomial with exactly three terms.

n, (of a graph) where the graph either changes from increasing to decreasing or vice versa.

n, the set obtained by collecting all the elements of one set along with all the elements of another set.

n, a circle of radius \(1\) unit (usually centered at the origin).

n, fractions whose denominators are not equivalent.

n, terms with variable parts that are not equivalent.

n, (of a matrix) a matrix with all zeros in the lower left corner. More precisely, the entry in the \(i\)th row and \(j\)th column is \(0\) whenever \(i \gt j\text{.}\)

adj, not constant, subject to change.

n, a numerical quantity that changes over time or in different situations.

see direct variation; inverse variation.

v, to prove the truth or validity of an assertion.

n, (plural vertices), (i) a point where two sides of a polygon meet; (ii) a corner or extreme point of a geometric object; (iii) the highest or lowest point on a parabola.

n, the angle between the equal sides in an isosceles triangle.

n, one way of writing a quadratic equation, \(y = a(x - x_v)2 + y_v\text{,}\) which displays the vertex, \((x_v, y_v )\text{.}\)

n, a line \(x = a\) parallel to the \(y\)-axis toward which the graph of an equation tends as the value of \(x\) approaches \(a\text{.}\)

n, the vertical coordinate axis. Often called the \(y\)-axis.

n, (of a graph) the result of replacing each point of the graph with the point obtained by scaling the \(y\)-coordinate by a fixed factor (when that factor is between \(0\) and \(1\)).

n, where the graph meets the vertical axis. Also called the \(y\)-intercept.

n, a test to decide whether a graph defines a function: A graph represents a function if and only if every vertical line intersects the graph in at most one point.

n, (of a graph) the result of replacing each point of the graph with the point obtained by scaling the \(y\)-coordinate by a fixed factor (when that factor is greater than \(1\)).

n, (of a graph) the result of moving all points of the graph straight up (or all straight down) by the same distance.

n, the plural of vertex.

n, a measure of the three-dimensional space enclosed by a three-dimensional object, typically expressed in terms of cubic units, such as cubic meters or cubic feet.

n, one of the numbers \(0, 1, 2, 3, \ldots\text{.}\)

see horizontal axis.

see horizontal intercept.

see coordinate plane.

see vertical axis.

see vertical intercept.

n, (i) the number \(0\text{,}\) with the property that when it is added to any other number, the resulting sum is equal to that second number; (ii) an input to a function which yields an output of \(0\text{.}\)

n, The statement that, if the product of two numbers is zero, then one (or both) of the numbers must be zero.