Applied Algebra: Modeling and functions

A.

absolute value

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 \(\abs{-7} = 7\text{.}\)

absolute value equation

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

absolute value inequality

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

algebraic expression

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

algebraic fraction

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

algebraic solution

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

allometric equation

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 \(y = k (\text{mass})^p\text{.}\)

altitude

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.

amortization

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

amount

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

amplitude

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

annuity

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

approximation

n, an inexact result.

area

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.

ascending powers

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

associative law of addition

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 \((a + b) + c = a + (b + c)\text{.}\)

associative law of multiplication

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 \((a \cdot b) \cdot c = a \cdot (b \cdot c)\text{.}\)

asymptote

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.

augmented matrix (for a linear systterm with \(n\) variables in standard form),

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.

axis

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

axis of symmetry

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

B.

back substitution

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

bar graph

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

base

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.

base angles

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

binomial

n, a polynomial with exactly two terms.

binomial expression

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

build (a fraction)

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

building factor

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

cartesian coordinate systterm

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

C.

cartesian plane

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

change in (a variable)

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

change of variables

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 \(y = ax + b\text{.}\)

circle

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

circumference

n, the distance around a circle.

closed interval

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 \(a\text{.}\)

coefficient

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 \(7\text{.}\)

coefficient matrix (for a linear systterm with n variables in standard form)

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).

common factor (of two or more expressions)

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

common log or common logarithm (of a given positive number x)

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

commutative law of addition

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 \(a + b = b + a\text{.}\)

commutative law of multiplication

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 \(a\cdot b = b\cdot a\text{.}\)

compltermentary angles

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

complete the square

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 \(a(x + k)^2\text{.}\)

complex conjugate (of a complex number)

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

complex fraction

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

complex plane

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.

complex number

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

component

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

compound inequality

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 \(2\text{.}\)"

compound interest (or compounded interest)

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.

compounding period

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

concave down (of a graph)

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.

concave up (of a graph)

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.

concavity

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

concentric (of circles or spheres)

adj, having the same center.

conditional equation

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

cone

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.

congruent

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.

conjugate

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.

conjugate pair

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

consistent (of a systterm of equations)

adj, having at least one solution.

consistent and independent (of a systterm of linear equations)

adj, having exactly one solution.

constant

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.

constant

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

constant of proportionality

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

constant of variation

see constant of proportionality.

constraint

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

continuous

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.

continuous compounding

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.

conversion factor

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

coordinate

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

coordinate axis

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

coordinate plane

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

corollary

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

costs

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.

counting number

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

cube

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

cube

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

cube root

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

cubic

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

cylinder

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.

D.

decay factor

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

decimal

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

decimal place

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.

decimal point

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 \(1.3\text{.}\)

decreasing

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.

degree

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

degree

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.

demand equation

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

denominator

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

dependent

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

dependent variable

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

descending powers

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

diagonal

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

diameter

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.

difference

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

difference of squares

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

dimension

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.

direct variation

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.

directed distance

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 \(2 - 5 = -3\text{.}\)

directly proportional

adj, describing variables related by direct variation.

discriminant

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

distributive law

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

divisor

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

domain

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

doubling time

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

E.

elementary row operation

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.

elimination

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.

empirical model

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

entry

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

equation

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

equation in two variables

n, an equation that involves two variables.

equilateral

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

equilibrium point

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

equivalent

adj, representing the same value.

equivalent equations

n, equations that have the same solutions.

equivalent expressions

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

error tolerance

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

evaluate

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

exact

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

exact solution

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

exponent

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 \(3^5 = 3\times 3\times 3\times 3\times 3\text{.}\)

exponential decay

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 \(0 \lt b\lt 1\text{.}\)

exponential equation

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

exponential function

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.

exponential growth

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 \(b\gt 1\text{.}\)

exponential notation

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 \(3^{1/2}\text{.}\)

expression

see algebraic expression.

extraction of roots

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

extraneous solution

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

extrapolate

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

F.

factor

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

factor

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

factored form

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 \(y = a(x - r_1)(x - r_2)\text{.}\)

feasible solution

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

FOIL

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.

formula

n, an equation involving two or more variables.

fraction bar

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 \(2\text{.}\)

function

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

function of two variables

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.

function value

n, an output value of a function.

fundamental principle of fractions

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 \(\displaystyle{\frac{a\cdot c}{b\cdot c}=\frac{a}{b}} \text{.}\)

G.

Gaussian reduction

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

geometrically similar

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

graph

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

graph

v, to draw a graph.

graph of an equation (or inequality)

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

graphical solution

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

greatest common factor (GCF) of two or more expressions

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

growth factor

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

guidepoints

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

H.

half-life

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

half-plane

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

height

see altitude.

htermisphere

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

horizontal asymptote

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.

horizontal axis

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

horizontal intercept

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

horizontal line test

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.

horizontal translation (of a graph)

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

hypotenuse

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

I.

identity

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

imaginary axis

n, the vertical axis in the complex plane.

imaginary number

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

imaginary part

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 \(- 7\text{.}\)

imaginary unit

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 \(-1\text{.}\)

inconsistent

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

increasing

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.

independent

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.

independent variable

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

index

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.

inequality

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

inflation

n, a persistent increase over time of consumer prices.

inflection point

n, a point where a graph changes concavity.

initial value

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

input

n, value of the independent variable.

integer

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

intercept

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

intercept method

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

interest

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.

interest rate

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.

interpolate

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

intersection point

n, a point in common to two graphs.

interval

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 \(a\)).

interval notation

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 \(3\text{.}\)

inverse function

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

inverse square law

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.

inverse variation

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.

inversely proportional

adj, describing variables related by inverse variation.

irrational number

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

isolate

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).

isosceles triangle

n, a triangle with two sides of equal length.

J.

joint variation

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.

L.

law of exponents

n, a basic property about powers and exponents.

lead coefficient

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

leading entry

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

leg

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

like fractions

n, fractions with equivalent denominators.

like terms

n, terms with equivalent variable parts.

line segment

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

linear combination

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.

linear combinations

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

linear equation

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

linear programming

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

linear regression

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

linear system

n, a set of linear equations.

linear term

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

log

see logarithm.

log scale

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

log-log paper

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

logarithm

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

logarithmic equation

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

logarithmic function

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

lowest common denominator (LCD)

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

lowest common multiple (LCM)

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

M.

mathematical model

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

matrix

n, a rectangular array of numbers.

maximum

adj, largest or greatest.

maximum

n, largest value.

maximum value

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

mean

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 \(\frac{5+2+11}{3}= 6\text{.}\)

mechanistic model

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

median

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 \(\frac{6+9}{2}= 7.5\text{.}\)

minimum

adj, least or smallest.

minimum

n, smallest value.

minimum value

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

mode

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 \(1\text{.}\)

model

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.

model

v, to create a model.

monomial

n, an algebraic expression with only one term.

monotonic

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

multiplicative property (of absolute values)

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

multiplicity

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.

natural base

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

natural exponential function

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

natural log or natural logarithm

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

natural number

n, a counting number.

negative number

n, a number that is less than zero.

negative of

n, the opposite of.

net change

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.

nonstrict inequality

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

normal

adj, perpendicular.

\(n\)th root

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 \(a\text{.}\)

number line

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

numerator

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

numerical solution

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

O.

objective function

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

one-to-one

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

open interval

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 \(a\text{.}\)

operation

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

opposite

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.

order

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.

order of operations

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

order symbol

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

ordered pair

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.

ordered triple

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.

origin

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

output

n, value of the dependent variable.

P.

parabola

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

parallel lines

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

parameter

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.

percent

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."

percent increase

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

perfect square

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

perimeter

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

perpendicular lines

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

piecewise defined function

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

point-slope form

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

polygon

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.

polynomial

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.

polynomial function

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.

positive number

n, a number greater than zero.

power

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

power function

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

prime (or prime number)

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

principal

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

principal root

see principal square root.

principal square root

n, the nonnegative square root.

product

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

profit

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

proportion

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

proportional

see directly proportional, inversely proportional.

pyramid

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

Pythagorean theorem:

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

Q.

quadrant

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.

quadratic

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

quadratic equation

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

quadratic formula

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

quadratic function

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

quadratic polynomial

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

quadratic regression

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

quadratic term

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

quadratic trinomial

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

quadrilateral

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

quartic

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

quotient

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

R.

radical

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

radical expression

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

radical equation

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

radical notation

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

radical sign

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 \(2\text{.}\)

radicand

n, the expression under a radical sign.

radius

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.

raise to a power

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

range

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.

rate

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

rate of change

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.

ratio

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 \(\frac{1}{2} \text{.}\)

rational

adj, having to do with ratios.

rational exponent

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

rational expression

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

rational function

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

rational number

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

rationalize the denominator

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 \(\frac{\sqrt{2}}{2} \text{.}\)

real axis

n, the horizontal axis in the complex plane.

real line

see number line.

real part

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 \(4\text{.}\)

real number

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

reciprocal (of a number)

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 \(1\text{.}\)

rectangle

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

reduce a fraction

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

reduced row echelon form

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.

reflection (of a point or graph across a line)

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

regression line

n, the line used for linear regression.

regular polygon

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

restricted domain

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

revenue

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

right angle

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

right triangle

n, a triangle that includes one right angle.

root

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

round

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 \(3.14\text{.}\)

row echelon form

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.

S.

satisfy

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

scale

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.

scale

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

scale factor

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

scaling exponent

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

scatterplot

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.

scientific notation

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 \(1.2\times 10^4\text{.}\)

semicircle

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

signed number

n, a positive or negative number.

significant digit

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.

significant figure

see significant digit.

similar

see geometrically similar.

simplify

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

sinusoidal

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

slope

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

slope-intercept form

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

slope-intercept method

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

solution

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.

solve

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.

sphere

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.

square

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

square

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

square matrix

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

square root

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

standard form

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.

strict inequality

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

subscript

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 \(1\text{.}\)

substitution method

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

sum

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

surface area

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

supplementary angles

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

supply equation

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

symmetry

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

system of equations

n, two or more equations involving the same variables.

T.

term

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.

test point

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.

transform

v, to apply a transformation.

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.

translation

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

trapezoid

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

triangle

n, a three-sided figure in the plane.

triangle inequality

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

triangular form

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).

trinomial

n, a polynomial with exactly three terms.

turning point

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

U.

union

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

unit circle

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

unlike fractions

n, fractions whose denominators are not equivalent.

unlike terms

n, terms with variable parts that are not equivalent.

upper triangular form

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 \(i \gt j\text{.}\)

V.

variable

adj, not constant, subject to change.

variable

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

variation

see direct variation; inverse variation.

verify

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

vertex

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.

vertex angle

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

vertex form

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

vertical asymptote

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

vertical axis

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

vertical compression

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 \(1\)).

vertical intercept

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

vertical line test

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.

vertical stretch

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 \(1\)).

vertical translation

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

vertices

n, the plural of vertex.

volume

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.

W.

whole number

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

X.

\(x\)-axis

see horizontal axis.

\(x\)-intercept

see horizontal intercept.

\(xy\)-plane

see coordinate plane.

Y.

\(y\)-axis

see vertical axis.

\(y\)-intercept

see vertical intercept.

Z.

zero

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 \(0\text{.}\)

zero-factor principle

see Zero-Factor Principle  below.

Properties of Numbers.