The first coordinate in an ordered pair.  For the point (8, –2) the abscissa is 8.

Absolute Value

The matrix obtained by changing the sign of every matrix element. The additive inverse of matrix A is written –A.

Note: The sum of a matrix and its additive inverse is the zero matrix.

The study of geometric figures using the coordinate plane or coordinates in space. Formulas from analytic geometry include the distance formula, midpoint formula, point of division formula, centroid formula, area of a convex polygon.

The change in the value of a quantity divided by the elapsed time. For a function, this is the change in the y-value divided by the change in the x-value for two distinct points on the graph.

Any of the following formulas can be used

Arithmetic Sequence
Arithmetic Progression

A matrix form of a linear system of equations obtained from the coefficient matrix as shown below. It is created by adding an additional column for the constants on the right of the equal signs. The new column is set apart by a vertical line.

Binomial Coefficients in Pascal's Triangle

Numbers written in any of the ways shown below. Each notation is read aloud "n choose r".

Binomial Theorem

A method for distributing powers of binomials as shown below.

(x, y) or (x, y, z) coordinates.

A formula that allows you to rewrite a logarithm in terms of logs written with another base. This is especially helpful when using a calculator to evaluate a log to any base other than 10 or e.

The number multiplied times a product of variables or powers of variables in a term. For example, 123 is the coefficient in the term 123x³y.

Coefficient Matrix

The matrix formed by the coefficients in a linear system of equations.

Column of a Matrix

A vertical set of numbers in a matrix.

A selection of objects from a collection. Order is irrelevant.

Combination Formula

A formula for the number of possible combinations of r objects from a set of nobjects. This is written in any of the ways shown below.

The mathematics of counting, especially counting how many elements are in very large sets.

Common Logarithm

The logarithm base 10 of a number. That is, the power of 10 necessary to equal a given number. The common logarithm of x is written log x. For example, log 100 is 2 since 10² = 100.

Common Ratio

For a geometric sequence or geometric series, the common ratio is the ratio of a term to the previous term. This ratio is usually indicated by the variable r.

Compatible Matrices

Two matrices with dimensions arranged so that they may be multiplied. The number of columns of the first matrix must equal the number of rows of the second.

Complex Conjugate

The complex conjugate of a + bi  is a – bi, and similarly the complex conjugate of a – bi  is a + bi. This consists of changing the sign of the imaginary part of a complex number. The real part is left unchanged.

Compound Fraction
Complex Fraction

A fraction which has, as part of its numerator and/or denominator, at least one other fraction.

Complex Number Formulas

Algebra rules and formulas for complex numbers are listed below.

Complex Numbers

Numbers like 3 – 2i or [image] that can be written as the sum or difference of a real number and an imaginary number. Complex numbers are indicated by the symbol [image].

Built from more than one thing.

Combining two functions by substituting one function's formula in place of each x in the other function's formula. The composition of functions f and g is written f ° g, and is read aloud "f composed with g." The formula for f ° g is written (f °g)(x). This is read aloud "f composed with g of x."

Compound Inequality

Two or more inequalities taken together. Often this refers to a connected chain of inequalities, such as 3 < x < 5.

Compound Interest

A method of computing interest in which interest is computed from the up-to-date balance. That is, interest is earned on the interest and not just on original balance.

Continuously Compounded Interest

Interest that is, hypothetically, computed and added to the balance of an account every instant. This is not actually possible, but continuous compounding is well-defined nevertheless as the upper bound of "regular" compound interest. The formula, given below, is sometimes called the shampoo formula (Pert^®).

A transformation in which all distances on the coordinate plane are shortened by multiplying either all x-coordinates (horizontal compression) or all y-coordinates (vertical compression) of a graph by a common factor less than 1.

To figure out or evaluate. For example, "compute 2 + 3" means to figure out that the answer is 5.

Conditional Equation

An equation that is true for some value(s) of the variable(s) and not true for others.

Conditional Inequality

An inequality that is true for some value(s) of the variable(s) and not true for others.

Conic Sections

The family of curves including circles, ellipses, parabolas, and hyperbolas. All of these geometric figures may be obtained by the intersection a double conewith a plane, hence the name conic section. All conic sections have equations of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0.

The result of writing sum of two terms as a difference or vice-versa. Note: Conjugates are similar to, but not the same as, complex conjugates.

Conjugate Pair Theorem

An assertion about the complex zeros of any polynomial which has real numbers as coefficients.

Consistent System of Equations

A system of equations that has at least one solution.

Constant

As a noun, a term or expression with no variables. Also, a term or expression for which any variables cancel out. For example, –42 is a constant. So is 3x + 5 – 3x, which simplifies to just 5.

Constant Function

A function of the form y = constant or f(x) = constant, such as y = –2.

Continued Sum

A notation using the Greek letter sigma (Σ) that allows a long sum to be written compactly.

Contrapositive

Switching the hypothesis and conclusion of a conditional statement and negating both. For example, the contrapositive of "If it is raining then the grass is wet" is "If the grass is not wet then it is not raining."

To approach a finite limit. There are convergent limits, convergent series, convergent sequences, and convergent improper integrals.

Convergent Sequence

A sequence with a limit that is a real number. For example, the sequence 2.1, 2.01, 2.001, 2.0001, . . . has limit 2, so the sequence converges to 2. On the other hand, the sequence 1, 2, 3, 4, 5, 6, . . . has a limit of infinity (∞). This is not a real number, so the sequence does not converge. It is a divergent sequence.

Convergent Series

An infinite series for which the sequence of partial sums converges. For example, the sequence of partial sums of the series0.9 + 0.09 + 0.009 + 0.0009 + ··· is 0.9, 0.99, 0.999, 0.9999, .... This sequence converges to 1, so the series 0.9 + 0.09 + 0.009 + 0.0009 + ··· is convergent.

On the coordinate plane, the pair of numbers giving the location of a point (ordered pair). In three-dimensional coordinates, the triple of numbers giving the location of a point (ordered triple). In n-dimensional space, a sequence of n numbers written in parentheses.

Cramer’s Rule

A method for solving a linear system of equations using determinants. Cramer’s rule may only be used when the system is square and the coefficient matrix is invertible.

Cube Root

A number that must be multiplied times itself three times to equal a given number. The cube root of x is written [image]or [image].

Cubic Polynomial

A polynomial of degree 3. For example, x³ - 1, 4a³ - 100a² + a - 6, and m²n + mn² are all cubic polynomials.

Decreasing Function

A function with a graph that moves downward as it is followed from left to right. For example, any line with a negative slope is decreasing.

A degenerate triangle is the "triangle" formed by three collinear points. It doesn’t look like a triangle, it looks like a line segment.

Degree of a Polynomial

The highest degree of any term in the polynomial.

Degree of a Term

For a term with one variable, the degree is the variable's exponent. With more than one variable, the degree is the sum of the exponents of the variables.

Greek Alphabet

The letters of ancient Greece, which are frequently used in math and science.

Dependent Variable

A variable that depends on one or more other variables. For equations such as y= 3x – 2, the dependent variable is y. The value of y depends on the value chosen for x. Usually the dependent variable is isolated on one side of an equation. Formally, a dependent variable is a variable in an expression, equation, or function that has its value determined by the choice of value(s) of other variable(s).

Descartes' Rule of Signs

A method for determining the maximum number of positive zeros for a polynomial. This maximum is the number of sign changes in the polynomial when written as shown below.

A single number obtained from a matrix that reveals a variety of the matrix's properties. Determinants of small matrices are written and evaluated as shown below. Determinants may also be found using expansion by cofactors.

Diagonal Matrix

A square matrix which has zeros everywhere other than the main diagonal. Entries on the main diagonal may be any number, including 0.

Difference Quotient

For a function f, the formula [image]. This formula computes the slope of the secant line through two points on the graph of f. These are the points with x-coordinates x and x + h. The difference quotient is used in the definition the derivative.

Dilation

A transformation in which a figure grows larger. Dilations may be with respect to a point (dilation of a geometric figure) or with respect to the axis of a graph(dilation of a graph).

Dilation of a Graph

A transformation in which all distances on the coordinate plane are lengthened by multiplying either all x-coordinates (horizontal dilation) or all y-coordinates (vertical dilation) by a common factor greater than 1.

On the most basic level, this term refers to the measurements describing the size of an object. For example, length and width are the dimensions of a rectangle.

Dimensions of a Matrix

The number of rows and columns of a matrix, written in the form rows×columns. The matrix below has 2 rows and 3 columns, so its dimensions are 2×3. This is read aloud, "two by three."

A relationship between two variables in which one is a constant multiple of the other. In particular, when one variable changes the other changes in proportion to the first.

Directrix of a Parabola

A line perpendicular to the axis of symmetry used in the definition of a parabola. A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is thelocus of points such that the distance to the focus equals the distance to the directrix.

Discriminant of a Quadratic

The number D = b² – 4ac determined from the coefficients of the equation ax² +bx + c = 0. The discriminant reveals what type of roots the equation has.

Distance Formula

The formula [image] is the distance between points (x₁, y₁) and (x₂, y₂).

Distinct

Different. Not identical.

To multiply out the parts of an expression. Distributing is the opposite of factoring.

Distributing Rules

Algebra rules for distributing expressions. See factoring rules as well.

To fail to approach a finite limit. There are divergent limits, divergent series, divergent sequences, and divergent improper integrals.

Divergent Sequence

A sequence that does not converge. For example, the sequence 1, 2, 3, 4, 5, 6, 7, ... diverges since its limit is infinity (∞). The limit of a convergent sequence must be a real number.

Divergent Series

A series that does not converge. For example, the series 1 + 2 + 3 + 4 + 5 + ··· diverges. Its sequence of partial sums 1, 1 + 2, 1 + 2 + 3 , 1 + 2 + 3 + 4 , 1 + 2 + 3 + 4 + 5, ... diverges.

The set of values of the independent variable(s) for which a function or relationis defined. Typically, this is the set of x-values that give rise to real y-values.

Alternate terms for domain used to make it clear that the domain being referred to is not a restricted domain.

Double Cone

A geometric figure made up of two right circular cones placed apex to apex as shown below. Typically a double cone is considered to extend infinitely far in both directions, especially when working with conic sections and degenerate conic sections.

Double Root

A root of a polynomial equation with multiplicity 2. Also refers to a zero of a polynomial function with multiplicity 2.

Doubling Time

For a substance growing exponentially, the time it takes for the amount of the substance to double.

A matrix form used when solving linear systems of equations.

Element of a Matrix

One of the entries in a matrix. The address of an element is given by listing the row number then the column number.

Formally, an ellipse can be defined as follows: For two given points, the foci, an ellipse is the locus of points such that the sum of the distance to each focus is constant. The standard form for the equation of an ellipse is given below.

End Behavior

The appearance of a graph as it is followed farther and farther in either direction. For polynomials, the end behavior is indicated by drawing the positions of the arms of the graph, which may be pointed up or down. Other graphs may also have end behavior indicated in terms of the arms, or in terms of asymptotes or limits.

Equation

A mathematical sentence built from expressions using one or more equal signs (=).

Equation of a Line

The various common forms for the equation of a line are listed below. In all forms, slope is represented by m, the x-intercept by a, and the y-intercept by b.

Equivalent Systems of Equations

Systems of equations that have the same solution set.

To figure out or compute. For example, "evaluate [image]" means to figure out that the expression simplifies to 17.

Even Function

A function with a graph that is symmetric with respect to the y-axis. A function is even if and only if f(–x) = f(x).

Even Function

A function with a graph that is symmetric with respect to the y-axis. A function is even if and only if f(–x) = f(x).

To multiply out the parts of an expression. Distributing is the opposite of factoring.

Explicit Formula of a Sequence

A formula that allows direct computation of any term for a sequence a₁, a₂, a₃, . . . , a_n, . . . .

Exponent

x in the expression a^x. For example, 3 is the exponent in 2³.

Exponent Rules

Algebra rules and formulas for exponents are listed below.

Exponential Decay

A model for decay of a quantity for which the rate of decay is directly proportional to the amount present. The equation for the model is A = A₀b^t(where 0 < b < 1 ) or A = A₀e^kt (where k is a negative number representing the rate of decay). In both formulas A₀ is the original amount present at time t = 0.

A function of the form y = a·b^x where a > 0 and either 0 < b < 1 or b > 1. The variables do not have to be x and y. For example, A = 3.2·(1.02)^t  is an exponential function.

Exponential Growth

A model for growth of a quantity for which the rate of growth is directly proportional to the amount present. The equation for the model is A = A₀b^t(where b > 1 ) or A = A₀e^kt (where k is a positive number representing the rate of growth). In both formulas A₀ is the original amount present at time t = 0.

The use of exponents.

Expression

Any mathematical calculation or formula combining numbers and/or variables using sums, differences,products, quotients (including fractions), exponents, roots, logarithms, trig functions, parentheses,brackets, functions, or other mathematical operations. Expressions may not contain the equal sign (=) or any type of inequality.

A solution of a simplified version of an equation that does not satisfy the original equation. Watch out for extraneous solutions when solving equations with a variable in the denominator of a rational expression, with a variable in the argument of a logarithm, or a variable as the radicand in an nth root when n is an even number.

Extreme Values of a Polynomial

The graph of a polynomial of degree n has at most n – 1 extreme values(minima and/or maxima). The total number of extreme values could be n – 1 or n – 3 or n – 5 etc.

Extremum

An extreme value of a function. In other words, the minima and maxima of a function. Extrema may be either relative (local) or absolute (global).

A factor of polynomial P(x) is any polynomial which divides evenly into P(x). For example, x + 2 is a factor of the polynomial x² – 4.

Factor Theorem

The theorem that establishes the connection between the zeros and factors of a polynomial.

Factorial

The product of a given integer and all smaller positive integers. The factorial of n is written n! and is read aloud "n factorial".

Factoring Rules

Algebra formulas for factoring.

Falling Bodies

A formula used to model the vertical motion of an object that is dropped, thrown straight up, or thrown straight down.

Fibonacci Sequence

The sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . for which the next termis found by adding the previous two terms. This sequence is encountered in many settings, from population models to botany.

A transformation in which a geometric figure is reflected across a line, creating a mirror image. That line is called the axis of reflection.

A step function of x which is the greatest integer less than or equal to x. The floor function is written a number of different ways: with special brackets [image]or [image], or by using either boldface brackets [x] or plain brackets [x].

Focal Radius

This term has distinctly different definitions for different authors.

Foci of an Ellipse

Two fixed points on the interior of an ellipse used in the formal definition of the curve. An ellipse is defined as follows: For two given points, the foci, an ellipse is the locus of points such that the sum of the distance to each focus is constant.

Foci of a Hyperbola

Two fixed points located inside each curve of a hyperbola that are used in the curve's formal definition. A hyperbola is defined as follows: For two given points, the foci, a hyperbola is the locus of points such that the difference between the distance to each focus is constant.

Focus (conic section)

A special point used to construct and define a conic section. A parabola has one focus. An ellipse has two, and so does a hyperbola. A circle can be thought of as having one focus at its center.

Focus of a Parabola

The focus of a parabola is a fixed point on the interior of a parabola used in the formal definition of the curve.

FOIL Method

A technique for distributing two binomials. The letters FOIL stand for First, Outer, Inner, Last. First means multiply the terms which occur first in each binomial. Then Outer means multiply the outermost terms in the product. Inner means multiply the innermost two terms. Last means multiply the terms which occur last in each binomial. Then simplify the products and combine any like terms which may occur.

Fractional Equation

An equation which has a rational expression on one or both sides of the equal sign. Sometimes rational equations have extraneous solution.

Fractional Exponents

The use of rational numbers as exponents. A rational exponent represents both an integer exponent and an nth root. The root is found in the denominator (like a tree, the root is at the bottom), and the integer exponent is found in the numerator.

Fractional Expression

An expression that can be written as a polynomial divided by a polynomial.

Function

A relation for which each element of the domain corresponds to exactly one element of the range. For example, [image] is a function because each number x in the domain has only one possible square root. On the other hand, [image] is not a function because there are two possible values for any positive value of x.

Function Operations

Definitions for combining functions by adding, subtracting, multiplying, dividing, and composing them.

Fundamental Theorem of Algebra

The theorem that establishes that, using complex numbers, all polynomials can be factored. A generalization of the theorem asserts that any polynomial ofdegree n has exactly n zeros, counting multiplicity.

Fundamental Theorem of Arithmetic

The assertion that prime factorizations are unique. That is, if you have found a prime factorization for a positive integer then you have found the only such factorization. There is no different factorization lurking out there somewhere.

Odds in Gambling

A way of representing gambling payoffs of an event by a method similar to odds against. If the gambling odds are m:n (read aloud "m to n"), then a bet of ndollars pays m dollars profit if the bettor wins.

Gauss-Jordan Elimination

A method of solving a linear system of equations. This is done by transforming the system's augmented matrix into reduced row-echelon form by means of row operations.

Gaussian Elimination

A method of solving a linear system of equations. This is done by transforming the system's augmented matrix into row-echelon form by means of row operations. Then the system is solved by back-substitution.

GCF

The largest integer that divides evenly into each of a given set of numbers. Often abbreviated GCF or gcf. For example, 6 is the gcf of 30 and 18. Sometimes GCF is written using parentheses: (30, 18) = 6.

General Form for the Equation of a Line

Ax + By = C, where A > 0 and, if possible, A, B, and C are relatively primeintegers. The standard form is used in some algebra classes for practice in manipulating equations. Otherwise it is used far less often than other forms for the equation of a line.

Geometric Mean

A kind of average. To find the geometric mean of a set of n numbers, multiply the numbers and then take the nth root of the product.

Geometric Progression

A sequence such as 2, 6, 18, 54, 162 or [image] which has a constant ratiobetween terms. The first term is a₁, the common ratio is r, and the number of terms is n.

Geometric Series

A series such as 2 + 6 + 18 + 54 + 162 or [image] which has a constantratio between terms. The first term is a₁, the common ratio is r, and the number of terms is n.

A transformation in which a graph or geometric figure is picked up and moved to another location without any change in size or orientation.

The number [image], or about 1.61803. The Golden Mean arises in many settings, particularly in connection with the Fibonacci sequence. Note: The reciprocal of the Golden Mean is about 0.61803, so the Golden Mean equals its reciprocal plus one. It is also a root of x² – x – 1 = 0.

Golden Spiral

A spiral that can be drawn in a golden rectangle as shown below. The figure forming the structure for the spiral is made up entirely of squares and golden rectangles.

Graph of an Equation or Inequality

The picture obtained by plotting all the points of an equation or inequality.

Graphic Methods

The use of graphs and/or pictures as the main technique for solving a math problem. When a problem is solved graphically, it is common to use a graphing calculator.

An interval that contains one endpoint but not the other.

Half-Life

For a substance decaying exponentially, the amount of time it takes for the amount of the substance to diminish by half.

Harmonic Mean

A kind of average. To find the harmonic mean of a set of n numbers, add the reciprocals of the numbers in the set, divide the sum by n, then take the reciprocal of the result. The harmonic mean of {a₁, a₂, a₃, a₄, . . ., a_n} is given below.

Horizontal Dilation

A stretch in which a plane figure is distorted horizontally.

A conic section that can be thought of as an inside-out ellipse.

Horizontal Line Equation

y = b, where b is the y-intercept.

Horizontal Line Test

A test use to determine if a function is one-to-one. If a horizontal line intersects a function's graph more than once, then the function is not one-to-one.

Horizontal Translation

A shift in which a plane figure moves horizontally.

Identity (Equation or Inequality)

An equation which is true regardless of what values are substituted for any variables (if there are any variables at all).

Identity Matrix

A square matrix which has a 1 for each element on the main diagonal and 0 for all other elements.

Identity of an Operation

The quantity which, when combined with another quantity using an operation, leaves the quantity unchanged.

Complex numbers with no real part, such as 5i.

Imaginary Part

The coefficient of i in a complex number. For a complex number a + bi, the imaginary part is b.

Improper Rational Expression

A rational expression in which the degree of the numerator polynomial is greater than or equal to the degree of the denominator polynomial.

Inconsistent System of Equations

A system of equations which has no solutions.

Increasing Function

A function with a graph that goes up as it is followed from left to right. For example, any line with a positive slope is increasing.

Independent Variable

A variable in an equation that may have its value freely chosen without considering values of any other variable. For equations such as y = 3x – 2, the independent variable is x. The variable y is not independent since it depends on the number chosen for x.

Inequality

Definition 1: Any of the symbols <, >, ≤, or ≥.

Inequality

Definition 1: Any of the symbols <, >, ≤, or ≥.

Inequality Rules

Algebra rules for manipulating inequalities are listed below.

Infinite Geometric Series

An infinite series that is geometric. An infinite geometric series converges if its common ratio r satisfies –1 < r < 1. Otherwise it diverges.

Infinite Series

A series that has no last term, such as [image]. The sum of an infinite series is defined as the limit of the sequence of partial sums.

Interest

The process by which an amount of money increases over time. With interest, a fixed percentage of the money is added at regular time intervals.

Interval

The set of all real numbers between two given numbers. The two numbers on the ends are the endpoints. The endpoints might or might not be included in the interval depending whether the interval is open, closed, or half-open (same as half-closed).

Interval Notation

A notation for representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included. For example, [3, 8) is the interval of real numbers between 3 and 8, including 3 and excluding 8.

The quantity which cancels out the a given quantity. There are different kinds of inverses for different operations.

Inverse Function

The function obtained by switching the x- and y-variables in a function. The inverse of function f is written f ^-1.

For a square matrix A, the inverse is written A^-1. When A is multiplied by A^-1 the result is the identity matrix I. Non-square matrices do not have inverses.

A relationship between two variables in which the product is a constant. When one variable increases the other decreases in proportion so that the product is unchanged.

A square matrix which has an inverse. A matrix is nonsingular if and only if its determinant does not equal zero.

When we say z is jointly proportional to a set of variables, it means that z is directly proportional to each variable taken one at a time.

Latus Rectum

The line segment through a focus of a conic section, perpendicular to the major axis, which has both endpoints on the curve.

The smallest positive integer into which two or more integers divide evenly. For example, 24 is the LCM of 8 and 12. Sometimes the LCM is written using brackets: [8, 12] = 24.

Leading Coefficient

The coefficient of a polynomial's leading term. For example, 5 is the leading coefficient of 5x⁴ – 6x³ + 4x – 12.

Leading Term

The term in a polynomial which contains the highest power of the variable. For example, 5x⁴ is the leading term of 5x⁴ – 6x³ + 4x – 12.

Least Common Denominator

The smallest whole number that can be used as a denominator for two or more fractions. The least common denominator is the least common multiple of the original denominators.

Like Terms

Terms which have the same variables and corresponding powers and/or roots. Like terms can be combined using addition an subtraction. Terms that are not like cannot be combined using addition or subtraction.

Like a line. A description of any graph or data that can be modeled by a linear polynomial.

Linear Combination

A sum of multiples of each variable in a set.

Linear Factorization

A factored form of a polynomial in which each factor is a linear polynomial.

Linear Equation

An equation that can be written in the form "linear polynomial = linear polynomial" or "linear polynomial = constant".

Linear Inequality

An inequality that can be written in the form "linear polynomial > linear polynomial" or "linear polynomial > constant". The > sign may be replaced by <, ≤, or ≥.

Linear Polynomial

A polynomial with degree 1. For example, the following are all linear polynomials: 3x + 5, y – ½, and a.

Linear Programming

An algorithm for solving problems asking the largest or smallest possible value of a linear polynomial. Any restrictions on the problem must be expressed as a system of inequalities; in particular, all equations and/or inequalities must be linear.