Precalculus Glossary

absolute value (of a complex number) see modulus.

amplitude the value by which the graph of a trigonometric function such as sine or cosine is stretched; the amplitude is always a positive value.

argument the angle measured from the positive x-axis to the segment joining the origin and the point representing the graph of complex number c.

augmented matrix a matrix containing more than simply coefficients; it may contain a column of solutions or even an appended identity matrix, as in the method of calculating inverse matrices.

axis of symmetry the line passing through the vertex of a parabola about which the graph of the parabola is symmetric.

center (of a circle) the point from which all points on a given circle are equidistant.

center (of an ellipse) the midpoint of an ellipse's major axis.

center (of a hyperbola) the midpoint of the transverse axis.

circle a set of coplanar points equidistant from a fixed point called the center.

coefficient matrix a matrix whose entries are the coefficients for a system of equations.

cofactor the value Cij = (−1)i + j · Mij based upon some element aij in a square matrix, where Mij is the minor associated with aij.

cofunctions trigonometric function pairs which differ only in the presence or absence of the prefix "co," such as sine and cosine.

common logarithm a logarithm of base 10; if a logarithm is written without an explicit base (like log 3x), the base is understood to be 10.

complex numbers any number of the form a + bi, where a and b are real numbers and . If b = 0, the complex number is also a real number. If, however, a = 0, the number is said to be purely imaginary.

component form method of writing a vector's terminal point which presupposes that its initial point is the origin.

composition of functions the act of plugging one function into another, usually written as f(g(x)) or (f°g)(x).

conjugate axis the segment perpendicular to the transverse axis at a hyperbola's center.

constraints linear inequalities that bound the feasible region in a linear programming problem.

coterminal angles angles in standard position that share the same terminal ray.

counting numbers the most basic set of numbers, often learned when one is first taught to count: {1, 2, 3, 4, 5, 6, . . . }. They are also called the natural numbers.

Cramer's Rule a method for solving systems of equations with matrices.

critical number a value for which an expression is either undefined or is equal to zero.

degree (angle measurement) 1/360th of a ray's full rotation around the origin.

degree (of a polynomial) the greatest exponent within a polynomial.

DeMoivre's Theorem allows you to calculate powers of complex numbers written in trigonometric form.

dependent describes a system of equations that has infinitely many solutions.

Descartes' Rule of Signs a method used to determine the number of possible positive and negative real roots of a polynomial.

determinant a real number that is defined for any square matrix A, expressed either as det (A) or ∣A∣.

diagonal the elements a11, a22, a33, . . . , ann in the square matrix An × n.

directrix the fixed line used to define a parabola; all points on the parabola must be the same distance from the directrix as they are from the parabola's focus.

dot product of two vectors, v = <a,b> and w = <c,d>, is v · w = ac + bd.

eccentricity the value =  for an ellipse which describes whether the graph tends more toward an oval or circular shape.

ellipse the set of coplanar points such that the sum of the distances from each point to two distinct coplanar points (called the foci) is constant.

Euler's number the irrational mathematical constant written as e, which has a value approximately equal to 2.71828182845904523 . . . .

even functions functions such that f(−x) = −f(x).

exponential function has form f(x) = ax, for some real number a, as long as a > 0.

exponentiating the process of raising a constant to the power of both sides of the equation in order to cancel out a logarithm. The exponentiated form of .

factorial the product of a natural number, n, with all its preceding natural numbers, written "n!".

feasible solutions the region for the system of inequalities which act as constraints in linear programming.

foci (of an ellipse) the two fixed focus points which define an ellipse.

foci (of a hyperbola) the two fixed focus points which define a hyperbola.

focus (of a parabola) the fixed point used to define a parabola.

function a relation in which every input results in one and only one output.

Gaussian elimination the process used to put a matrix in row-echelon form.

Gauss-Jordan elimination the process used to put a matrix in reduced row-echelon form.

Heron's area formula used to calculate the area of an oblique triangle given the lengths of all its sides.

hyperbola set of points such that the difference of the distances from each point to two distinct, fixed points (called the foci) is a positive constant.

identity elements numbers that, when applied in specific operations, do not alter the values you begin with.

identity matrix a square matrix which contains all 0 elements except for its diagonal, which contains only 1 elements.

inconsistent describes a system of equations that has no solutions.

index the small number outside of a radical sign.

inverse function the function, labeled f−1(x), which contains all the ordered pair of f(x), with its coordinates reversed. In other words, if f(x) contains (a,b), then f−1(x) contains (b,a).

inverse matrix the unique n × n matrix A−1 corresponding to the n × n matrix A such that A−1 · A equals the n × n identity matrix.

irrational numbers any number that cannot be expressed as the quotient , where a and b are integers and b is nonzero.

leading coefficient the coefficient in the term of a polynomial containing the variable raised to its highest power.

Leading Coefficient Test describes what direction (either up or down) the graph is heading at the far right and left edges of the coordinate axes.

linear programming technique used to optimize a function whose solution set is subject to a set of linear inequality constraints.

logarithmic function function of form f(x) = logc x (read "the log base c of x").

magnitude the length of a vector; the magnitude of v is written ∣v∣.

major axis the line segment (whose ends are vertices) which passes through the foci of an ellipse.

matrix a rectangular collection of numbers, arranged in rows and columns, surrounded by a single set of brackets on either side.

minor notated Mij, and corresponding to a square matrix A, it is equal to the determinant of the matrix created by deleting the ith row and jth column of A.

minor axis the line segment, perpendicular to the major axis, which passes through the center of an ellipse and has endpoints on the ellipse.

modulus the distance from the origin to the point on the coordinate plane representing the graph of the complex number c = a + bi; also called the absolute value of c.

natural exponential function the exponential function with Euler's number as its base: f(x) = ex.

natural logarithm the logarithmic function of base e, written "ln x" and read either "natural log of x" or "L-N of x."

natural numbers the most basic set of numbers, often learned when one is first taught to count: {1, 2, 3, 4, 5, 6, . . . }. They are also called the counting numbers.

oblique triangles triangles which do not contain a right angle.

odd functions functions such that f (−x) = −f (x).

one-to-one a term used to describe a function for which every output has only one corresponding input. Only one-to-one functions have inverses.

optimal maximum or minimum values of a function.

order describes how many rows and columns are in a matrix.

orthagonal describes two vectors which are perpendicular to one another.

parabola a set of coplanar points equidistant from a fixed point (the focus) and a fixed line (the directrix).

parametric equations two equations (usually "x =" and "y =") defined in terms of a third variable, called the parameter.

partial sum sum of the terms of a series whose upper summation limit is finite.

Pascal's triangle the triangular arrangement of the coefficients of binomial expansions; the (n + 1)th row of the triangle gives the coefficients for the expression (a + b)n.

period the shortest length along the x-axis after which a periodic graph will repeat itself.

periodic describes a graph which will repeat itself infinitely after some fixed length of the x-axis, called the period.

polar axis the fixed ray in polar coordinates representing the initial side of the angle θ.

polar coordinates coordinates in the form (r, θ), where r is the distance from the pole and θ is the angle from the polar axis.

pole the fixed point in polar coordinates from which the distance r to the point is measured.

principal the initial investment in a compound interest problem.

quadrantal an angle in standard position whose terminal side falls upon a coordinate axis.

radian measurement of an angle in standard position that, when extended to a circle of radius r centered at the origin, will mark the endpoints of an arc whose length is also r.

radius the fixed distance between the center of a circle and any point on that circle.

rational numbers any number that can be expressed as a fraction , where a is an integer and b is a non-zero integer.

Rational Root Test a method used to determine all possible rational roots for a polynomial.

real numbers any number which is either rational or irrational is also a real number, because the real numbers are made up by combining those two, smaller groups.

rectangular coordinates coordinates in the form (x,y) in the Cartesian plane.

recursive sequence sequence whose terms are defined based on one or more preceding terms of the sequence.

reduced row-echelon form the form of a matrix in which the diagonal contains only 1s, all elements above and below the diagonal are 0s, and any rows containing only zeros are placed at the bottom of the matrix.

reference angle an acute angle that helps calculate trigonometric function values of an oblique angle.

row-echelon form the form of a matrix in which its diagonal contains only 1s, all elements to the left of the diagonal are 0s, and all rows made up entirely of zeros appear at the bottom of the matrix.

scalar term used to refer to a numeric, non-vector quantity when dealing with vectors.

sequence ordered list of numbers a1, a2, a3, . . . .

series the sum of the terms of a sequence.

singular describes a matrix that has no inverse.

slant asymptote a linear asymptote that is neither vertical nor horizontal.

square matrix a matrix that has the same number of rows and columns.

standard form (of a vector) describes a vector whose initial point lies on the origin.

standard position describes an angle whose initial side lies on the positive x-axis and whose vertex lies on the origin of the coordinate plane.

synthetic division a shortcut alternative to long division, which uses only the coefficients of the divisor and dividend; it is only applicable if the divisor is linear.

system of equations set of equations for which you are seeking coordinates that makes all of the equations in the set true.

test points points chosen based on the graph of an inequality to determine which regions of the graph (as defined by the inequality) make it true.

transverse axis segment passing through the foci of a hyperbola whose endpoints are the hyperbola's vertices.

unit circle a circle, centered at the origin with radius 1, which is used to calculate the sine and cosine values of certain angles.

unit vector a vector with magnitude 1.

vector quantity that possesses both magnitude and direction.

vertex (of an angle) the endpoint shared by the two rays forming an angle.

vertex (of linear programming) the point at which two constraints intersect.

vertex (of a parabola) the point at which the direction of a parabola changes.

vertical line test if a vertical line can be drawn through a graph, intersecting it in two or more places, then the graph cannot be that of a function.

vertices (of an ellipse) the endpoints of the major axis.

vertices (of a hyperbola) the endpoints of the transverse axis.

zero matrix a matrix of any order whose elements are all zeros.

zero vector written 0, it is the vector with component form <0,0>; it is orthagonal to all vectors by definition, although it is not actually perpendicular to anything because its magnitude is 0.