**arithmetic sequence** a sequence in which, starting with the second term, each term is found by adding the same value, known as the common difference, to the previous term.

**arithmetic series** the sum of the terms of an arithmetic sequence with a definite number of terms.

**asymptote lines** dashed lines on a graph representing the limits of values where a rational function or hyperbola is defined; a graph may approach its asymptotes, but will never reach them.

**axis of symmetry (of an ellipse)** either of the two axes intersecting at its center; the longer is the major axis, the shorter, the minor one.

**axis of symmetry (of a parabola)** the line that passes through the vertex and focus.

**binomial** an expression containing two terms separated by a + or – sign.

**center** the point in a circle from which all points are equidistant; in an ellipse, the midpoint of the segment joining the two foci.

**circle** a conic section; the set of all points in a plane equidistant from one point.

**combination** similar to a permutation, but when the order is not important. The combination of 8 objects taken 3 at a time would be *C*(8,3) or _{8}*C*_{3}.

**common difference** can be found by taking any term in a sequence and subtracting its preceding term. See *arithmetic sequence*.

**common logarithm** understood to be base 10 when the base of a logarithm is not written. See *logarithm*.

**common ratio** found by taking any term in a sequence and dividing it by its preceding term. See *geometric sequence*.

**completely factored** incapable of being further simplified by division.

**completing the square** a technique for solving quadratic equations.

**complex conjugates** two binomials with the same two terms, but opposite signs, which represent the sum or difference of an imaginary number and a real number. For example, *a* + *bi* and *a* – *bi*. See *conjugate*.

**complex fraction** a fraction containing one or more additional fractions (in the numerator, denominator, or both).

**complex number** any expression that is a sum of a pure imaginary number and a real number, usually in the form *a* + *bi*.

**composite function** a function in which the variable name has been replaced by another function.

**compound inequality** a mathematical sentence with two inequality statements joined by "and" or "or."

**conic section** cross section formed by a plane slicing through a point-to-point pair of cones; see *circle*, *parabola, ellipse*, and *hyperbola*.

**conjugate axis** the axis that passes through the center of the hyperbola and is perpendicular to the transverse axis. See *hyperbola*.

**conjugates** two binomials with the same two terms but opposite signs between them. For example, 5*x* + 3 and 5*x* – 3.

**constant of proportionality** the multiplier of the independent variable in a variation relationship (usually represented by *k*). For example, *y* =; *kx*.

**coordinates of a point** the pair of numbers in the form (*x*,*y*) designating the location of any point on a plane.

**Cramer's rule** method of solving systems of equations by using determinants.

**dependent system** second version of the same equation, whose graphs coincides with each other.

**descending order** the general practice of writing polynomials in more than one variable so that the exponents decrease from right to left. For example:

**determinant** a square array of numerals or variables between vertical lines. A determinant differs from a matrix in that it has a numeric value.

**difference of cubes** an expression in the form of

**difference of squares** a special pattern that is the result of the product of conjugates. For example *x*^{2} – *y*^{2} is the product of conjugates (*x* + *y*)(*x* – *y*), *x*^{2} – 36 =; (*x* + 6)(*x* – 6), etc.

**direct variation** "*y* varies directly as *x*" means that as *x* gets larger, *y* also gets larger.

**directrix** the line from which the set of points in a parabola are equidistant. See *parabola*.

**dividend** in a division problem, the number being divided into. See *quotient*.

**divisor** in a division problem, the number being divided by. See *quotient*.

**domain** set of all the *x*-values (first number of each ordered pair) in a relation.

**ellipse** a conic section; the set of points in a plane such that the sum of the distances from two given points in that plane stays constant. Each of those two points is called a focus point. The line passing through the foci is the major axis; its endpoints (on the ellipse) are its major intercepts. The line crossing the ellipse perpendicular to the major axis through the vertex is the minor axis. Its endpoints are at the minor intercepts.

**equation** a statement that says two mathematical expressions are equal.

**exponential equation** an equation in which the variable appears as an exponent.

**exponential function** any function defined by

**extraneous solution** solution that does not make the original equation true. Extraneous solutions are most likely to appear in equations that have been raised to a power or multiplied by a variable term to solve.

**factor (n.)** a number that is multiplied by another number to make a product. The factors of 6, for example, are 2 and 3, as well as 1 and 6.

**factor, to (v.)** to divide a polynomial by a constant or variable common to all of its terms. To divide. To rewrite a polynomial as a product of polynomials or polynomials and monomials.

**factorial** a way of expressing a natural number multiplied by all its preceding natural numbers. 4! is read "4 factorial" and means (4)(3)(2)(1) =; 24.

**first degree equation** another name for a linear equation. See *linear equation*.

**focus** the point from which the set of points in a conic section are equidistant. In a circle, the focus is called the center. See *parabola, hyperbola*, and *ellipse*.

**formula** an algebraic equation that describes a rule, relationship, fact, principle, rule, etc. *I* =; *PRT*, for example, is the formula for finding simple interest.

**function** a relation in which none of the domain values is repeated.

**GCF (greatest common factor)** the largest expression that can be factored (divided perfectly) out of another expression. For 3*x*^{2} + 6*x* + 12, the GCF is 3, yielding 3(*x*^{2} + 2*x* + 4).

**general term** *n*th term of a sequence; a term of some order to be determined.

**geometric sequence** a sequence in which each term is found by multiplying the same value times the previous term. Taking any term in a geometric sequence and dividing it by its preceding term yields the common ratio.

**geometric series** the sum of the terms in a geometric sequence.

**graph** a pictorial display of solutions to mathematical equations. Also the point associated with an ordered pair.

**greatest common factor** see *GCF*.

**hyperbola** a conic section. The set of all points in a plane such that the absolute values of the difference of the distances between two given points stays constant; the two given points are the foci, and the midpoint of the segment joining the foci is the center. The transverse axis runs along the direction the hyperbola opens in. The conjugate axis passes through the center of the hyperbola and is perpendicular to the conjugate axis. The points of intersection of the hyperbola and the transverse axis are the vertices.

**identity function** *y* =; *x*, or *f*(*x*) =; *x* since for each replacement, the result is identical to *x*.

**imaginary value** *i*represents , which is an expression with no real value.

**inconsistent system** a system of non-intersecting equations. Their solution is the null set.

**index** in a radical expression (), the *n*, which is an integer greater than 1. If a radical expression has no index, the index is assumed to be 2. See *radical expression*.

**inequality** a mathematical sentence using a relational symbol other than the equal sign (=;).

**inverse function** a function in which the *x* and *y* variables have been switched; represented by *f* –1 (*x*). No domain element appears twice.

**inverse relation** the set of ordered pairs created when the ordered pairs of the original relation are reversed.

**inverse variation** "*y* varies inversely as *x*" means that as *x* gets larger, *y* gets smaller, and as *x* gets smaller, *y* gets larger.

**like radical expressions** radical expressions with identical index and radicand. See *radical expression*.

**linear equation** an equation with one variable whose exponent is 1. The graph of a linear equation is a straight line.

**linear inequality** a linear sentence not containing an equal sign (=;).

**logarithm** exponent expressing the power to which a fixed number (the base) must be raised in order to produce a given number. Abbreviated as *log*. It is usually computed to the base 10 (common logs, where the base is not written), or to the base *e* (known as natural logs and abbreviated ln); the purpose is to shorten mathematical calculations.

**logarithmic equation** an equation that involves the logarithm of an expression containing a variable.

**logarithmic function** a function of the form

**major axis** the line passing through the foci of an ellipse, having its endpoints on the ellipse. See *ellipse*.

**major intercepts** the points where the major axis of an ellipse touch the curve itself. See ellipse.

**matrix (pl. matrices)** a rectangular array of numerals or variables that can be used to represent systems of equations.

**minor axis** see ellipse.

**minor intercepts** see ellipse.

**monomial** a single term expression, not containing separate parts separated by + or – signs. For example: 5, *x*, 3*a*, 4 *x*^{2}*y*^{2}.

**multiplication principle for events** a principle used to determine how many different ways a particular event can occur. For example, if one event can occur in *p* different ways and another in *q* different ways and *p* and *q* are independent events, then together they can occur in *pq* different ways.

**natural logarithm** a term that represents log base *e* (also log_{e}), which is written as ln. See *logarithm*.

**ordered pair** represented as (*x*,*y*). The *x*-value always comes first, separated from the *y*-value by a comma. See *coordinates of a point*.

**origin** the point (0,0) where *x*-axis and *y*-axis intersect.

**parabola** a conic section. The set of points in a plane that are the same distance from a given point and a given line in that plane. The given line is called the *directrix*, and the given point is called the *focus*.

**Pascal's triangle** a graphical representation of binomial expansion, named after the French mathematician Blaise Pascal.

**permutation** the arrangement of objects in a certain order. For example, 8 objects arranged 3 at a time would be *P*(8,3) or 8 *P*_{3}.

**point-slope form (of a non-vertical line)** takes the following form, where (*x* – *x*_{1}) =; difference in *x*-coordinates, and (*y* – *y*_{1}) =; difference in *y*-coordinates; *m* is the slope.

**polynomial** an expression consisting of terms separated by some combination of + signs, – signs, or both.

**polynomial function** any function of the form

where the coefficients *a*_{0}, *a*_{1}, *a*_{2}, . . . , *a*_{n} are real numbers, and *n* is a whole number.

**proportion** an equation stating that 2 rational expressions are equal.

**pure imaginary number** any product of a real number and *i*. For example: 3*i*, 5*i*, etc. See *imaginary value*.

**quadrants** the four regions defined by the intersection of the *x*- and *y*-axes and designated by Roman numerals. Beginning in the top right and proceeding in a counterclockwise direction, quadrant I is the top right; quadrant II the top left; quadrant III the bottom left, and quadrant IV the bottom right.

**quadratic equation** any equation in the following form:

**quadratic form** any equation of the following form; such equations may be solved by quadratic formula:

**quadratic formula** a formula that may be used to solve any and all quadratic equations in standard quadratic form:

**quotient** the answer to a division problem. In 10 ÷ 5 =; 2, 10 is the dividend, 5 is the divisor, and 2 is the quotient.

**radical** the bracket also known as the "square root" sign (if its index is 2).

**radical equation** an equation in which the variable is under a radical sign.

**radical expression** the name given the following: The bracket is known as the radical sign; *a* is the radicand, and *n* is the index. If no *n* appears on the radical sign, the index is assumed to be 2. The above is read as "the *n*th root of *a*."

**radicand** the number under the radical. See *radical expression*.

**radius** the distance from the center of a circle to any point on the circle.

**range** set of all the *y*-values (second number of each ordered pair) in a relation.

**rational equation** an equation involving rational expressions.

**rational expression** the quotient of two polynomials, usually expressed as a fraction. The denominator may never be zero.

**rational function** If *f* (*x*) is a rational expression, then *y* =; *f* (*x*) is a rational function.

**rationalizing the denominator** a process used to remove radicals from denominators of rational expressions. To rationalize a denominator, multiply by the conjugate of the denominator over itself.

**relation** a set of ordered pairs.

**sequence** an ordered list of numbers.

**slope intercept form** *y* =; *mx* + *b*, where *x* and *y* are the coordinates of a point on the graph of the line, *m* is the line's slope, and *b* is some constant.

**slope of a line** the line's rise over its run (or its change in *y* divided by its change in *x*) as the graph of the line moves to the right. A line that descends as it moves right has a negative slope; a horizontal line has a slope of 0; a vertical line's slope is undefined.

**square trinomial** the expression produced by squaring a binomial:

**standard form of a line** the standard form for the equation of a line is

where *A, B*, and *C* are integers and *A* is positive.

**sum of cubes** an expression in the following form:

**synthetic division** a shortcut for dividing a polynomial by a binomial of the form *x* – *a*; only coefficients are used.

**term** any number in a sequence or piece of a polynomial separated by a + or – sign.

**transverse axis** the line along the direction the hyperbola opens in passing through its vertices. See *hyperbola*.

**trinomial** an expression containing three terms separated by + or – signs.

**varies directly** as one quantity increases or decreases, so does another quantity. See *direct variation*.

**varies inversely** as one quantity increases another decreases, and vice versa. See *inverse variation*.

**vertex (of hyperbola)** either of the two points of intersection of the hyperbola and the transverse axis. See *hyperbola*.

**vertex (of parabola)** the midpoint of the perpendicular segment from the focus to the directrix.

**vertical line test** a test for functions. If a vertical line passes through more than one point on a graph, then a domain point has been repeated, and the graphed relation is not a function.

*x*-axis the horizontal axis; all points with a *y*-coordinate of 0.

*x*-coordinate the number to the left of the comma in an ordered pair.

*x*-intercept the point at which a graph crosses the *x*-axis.

*y*-axis the vertical axis; all points with an *x*-coordinate of 0.

*y*-coordinate the number to the right of the comma in an ordered pair.

*y*-intercept the point at which a graph crosses the *y*-axis.

**zero of a function** any value for the variable that will produce a solution of 0.