absolute maximum

the largest output value over the entire domain of a function

absolute minimum

the smallest output value over the entire domain of a function

absolute value of a complex number

The distance from the origin to the point representing the complex number in the complex plane; for [latex]z = a + bi[/latex], it equals [latex]\sqrt{a^2 + b^2}[/latex].

addition method (elimination method)

An algebraic technique where equations are added (after multiplying by constants if needed) to eliminate one variable.

Addition Principle

States that if one event can occur in A ways and a second event can occur in B ways, and both events cannot occur at the same time, then there are A+B ways for the first event OR the second event to occur.

adjacent side

In a right triangle, the side next to a given acute angle that is not the hypotenuse; the side between the angle and the right angle.

altitude

A perpendicular line segment from a vertex of a triangle to the opposite side (or the line containing the opposite side).

ambiguous case

A situation in triangle solving (SSA configuration) where the given information may result in zero, one, or two valid triangles.

angle

The union of two rays having a common endpoint called the vertex.

angle of elevation

The angle formed between a horizontal line and the line of sight looking upward to an object.

angle of rotation

An acute angle [latex]\theta[/latex] formed by a set of axes rotated from the Cartesian plane where: if [latex]\cot(2\theta)>0[/latex], then [latex]\theta[/latex] is between [latex]0°[/latex] and [latex]45°[/latex]; if [latex]\cot(2\theta)<0[/latex], then [latex]\theta[/latex] is between [latex]45°[/latex] and [latex]90°[/latex]; and if [latex]\cot(2\theta)=0[/latex], then [latex]\theta=45°[/latex].

angular speed

The angular rotation [latex]\theta[/latex] per unit time [latex]t[/latex], denoted [latex]\omega = \frac{\theta}{t}[/latex].

arc length

The length of the arc [latex]s[/latex] along a circle subtended by a central angle, calculated as [latex]s = r\theta[/latex] where [latex]\theta[/latex] is in radians.

Archimedes’ spiral

A polar curve with equation [latex]r = \theta[/latex] that spirals outward from the pole with distance increasing at a constant rate.

argument

The angle [latex]\theta[/latex] in the polar form of a complex number, measured from the positive real axis.

arithmetic sequence

A sequence where the difference between consecutive terms is always the same (constant difference d).

arrow notation a

way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value

augmented matrix

a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets

average rate of change

the change in output divided by the change in input over a specified interval

back-substitution

The process of substituting known variable values into previous equations to find remaining unknown variables.

base

The constant [latex]b[/latex] in an exponential function [latex]f(x) = ab^x[/latex] or in a logarithmic function [latex]f(x) = \log_b(x)[/latex]; must be positive and not equal to 1.

break-even point

The point where a cost function intersects a revenue function; where profit is zero.

cardioid

A heart-shaped polar curve with equation [latex]r = a \pm b\cos\theta[/latex] or [latex]r = a \pm b\sin\theta[/latex] where [latex]\frac{a}{b} = 1[/latex].

carrying capacity

The limiting value [latex]c[/latex] in a logistic model; represents the maximum sustainable population or quantity.

center of a hyperbola

The midpoint of both the transverse and conjugate axes of a hyperbola, where the axes intersect.

center of an ellipse

The midpoint of both the major and minor axes of an ellipse.

central rectangle

A rectangle centered at the hyperbola’s center with sides passing through each vertex and co-vertex; the diagonals of this rectangle are the asymptotes of the hyperbola.

change-of-base formula

A formula that allows evaluation of logarithms with any base using logarithms of another base: [latex]\log_b(M) = \frac{\log_n(M)}{\log_n(b)}[/latex].

coefficient matrix

a matrix that contains only the coefficients from a system of equations

cofunction identities

Trigonometric identities showing the relationship between a trigonometric function and the cofunction of its complementary angle; for example, [latex]\sin(t) = \cos(\frac{\pi}{2} - t)[/latex].

column

a set of numbers aligned vertically in a matrix

combination

A selection of objects where the order does not matter; the number of ways to choose objects without regard to arrangement.

common difference

The constant value d added to each term to get the next term in an arithmetic sequence, calculated as [latex]d=a_n−a_{n−1}[/latex].

common logarithm

A logarithm with base 10, written as [latex]\log(x)[/latex] instead of [latex]\log_{10}(x)[/latex]; used to measure phenomena like earthquakes (Richter scale), star brightness, and pH levels.

common ratio

The constant value r by which each term is multiplied to get the next term in a geometric sequence, calculated as [latex]r=a_n/a_{n−1}[/latex].

complement of an event

The complement E′ is the set of all outcomes in the sample space that are not in event E.

complementary angles

Two angles whose measures add up to 90° (or [latex]\frac{\pi}{2}[/latex] radians).

complex plane

A coordinate plane where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers; used to plot complex numbers.

composite function

the new function formed by function composition, when the output of one function is used as the input of another

concavity

The direction a curve bends; concave up curves bend upward (can hold water), concave down curves bend downward (cannot hold water).

conic section (conic)

Any shape resulting from the intersection of a right circular cone with a plane; includes circles, ellipses, parabolas, and hyperbolas.

conjugate axis

The axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints.

consistent system

A system that has at least one solution (either unique or infinitely many); can be independent or dependent.

constant of variation

the nonzero value [latex]k[/latex] that helps define the relationship between variables in direct or inverse variation

continuous function

A function that has no holes or breaks in its graph; a function [latex]f(x)[/latex] is continuous at [latex]x=a[/latex] if [latex]f(a)[/latex] exists, [latex]\underset{x\to a}{\mathrm{lim}}f(x)[/latex] exists, and [latex]\underset{x\to a}{\mathrm{lim}}f(x)=f(a)[/latex].

continuous growth/decay model

A model of the form [latex]A(t) = A_0 e^{kt}[/latex] where [latex]k > 0[/latex] represents growth and [latex]k < 0[/latex] represents decay.

cosecant

The reciprocal of the sine function: [latex]\csc t = \frac{1}{\sin t}[/latex].

cosine

For a point [latex](x, y)[/latex] on the unit circle corresponding to angle [latex]t[/latex], [latex]\cos t = x[/latex].

cost function

A function describing the costs of doing business; typically has fixed costs and variable costs.

cotangent

The reciprocal of the tangent function: [latex]\cot t = \frac{\cos t}{\sin t}[/latex].

coterminal angles

Two angles in standard position that have the same terminal side.

co-vertex

An endpoint of the minor axis of an ellipse or the conjugate axis of a hyperbola.

Cramer’s Rule

a method for solving systems of equations that have the same number of equations as variables using determinants

decreasing function

a function where output values decrease as input values increase

degree

A unit of angle measurement where one degree is [latex]\frac{1}{360}[/latex] of a circular rotation.

De Moivre’s Theorem

A formula for raising a complex number in polar form to a power: [latex]z^n = r^n[\cos(n\theta) + i\sin(n\theta)][/latex].

degenerate conic sections

Any of the possible shapes formed when a plane intersects a double cone through the apex; types include a point, a line, and intersecting lines.

dependent system

A system with infinitely many solutions; for two variables, the lines are coincident (the same line); for three variables, the planes intersect along a line or are identical.

dependent variable

the output variable in a function, often represented by y or [latex]f(x)[/latex]

derivative

The instantaneous rate of change of a function at a point; also represents the slope of the tangent line to the graph of the function at that point.

determinant

a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations

direct variation

a relationship where one quantity is a constant multiplied by another quantity; as one quantity increases, so does the other

directrix

A fixed line used in the definition of a conic section; for a parabola, all points on the curve are equidistant from the focus and the directrix. More generally, a line such that the ratio of the distance from points on the conic to the focus to the distance to the directrix is constant (the eccentricity).

discontinuous function

A function that is not continuous at [latex]x=a[/latex]; a function that has one or more holes or breaks in its graph.

diverges

A series diverges when its sum is not a real number or does not approach a finite value.

DMS form

Notation for expressing angles using degrees, minutes, and seconds (D° M’ S”), where 1° = 60′ and 1′ = 60″.

domain

the set of all possible input values for a function

double-angle formulas

Formulas derived from sum formulas by setting both angles equal, used to find trigonometric values of twice an angle

eccentricity

The ratio of the distances from a point [latex]P[/latex] on the conic to the focus [latex]F[/latex] and to the directrix [latex]D[/latex], represented by [latex]e=\frac{PF}{PD}[/latex], where [latex]e[/latex] is a positive real number. The value of [latex]e[/latex] determines the type of conic.

element

a number in a matrix

ellipse

The set of all points [latex](x,y)[/latex] in a plane such that the sum of their distances from two fixed points (foci) is constant.

ellipsis

The symbol (…) used to indicate that a sequence or pattern continues indefinitely.

event

Any subset of the sample space; a collection of outcomes from an experiment.

even function

a function whose graph is unchanged by horizontal reflection, satisfying [latex]f(x) = f(-x)[/latex], and is symmetric about the y-axis

even function

A function where [latex]f(-x) = f(x)[/latex]. Cosine and secant are even trigonometric functions.

even-odd identities

Identities that relate the value of a trigonometric function at an angle to its value at the negative of that angle; cosine and secant are even functions, while sine, tangent, cotangent, and cosecant are odd functions

explicit formula

A formula that defines the nth term of a sequence using the position n in the sequence.

exponential decay

A decrease based on a constant multiplicative rate of change over equal increments of time; occurs when [latex]0 < b < 1[/latex] in [latex]f(x) = ab^x[/latex].

exponential function

A function of the form [latex]f(x) = ab^x[/latex] where [latex]a[/latex] is any nonzero number, [latex]b > 0[/latex], and [latex]b \ne 1[/latex].

exponential growth

An increase based on a constant multiplicative rate of change over equal increments of time; occurs when [latex]b > 1[/latex] in [latex]f(x) = ab^x[/latex].

extrapolation

Using a model to make predictions outside the range of original data; less reliable and requires careful reasoning.

extraneous solution

A solution that emerges algebraically but doesn’t satisfy the original equation; common when logarithms of negative numbers or zero would be required.

extraneous solution

A solution that emerges from the algebraic process but doesn’t satisfy the original equation; common when squaring both sides or solving nonlinear systems.

finite sequence

A sequence with a limited number of terms.

focus (plural: foci)

A fixed point used in the definition of a conic section. For an ellipse, one of two fixed points on the major axis such that the sum of distances from these points to any point on the ellipse is constant. For a hyperbola, one of two fixed points such that the difference of distances is constant. For a parabola, a fixed point in the interior that lies on the axis of symmetry.

function

a relation where each input corresponds to exactly one output

Gaussian elimination

A systematic method for solving systems of linear equations using row operations to achieve upper triangular form.

Gaussian elimination

using elementary row operations to obtain a matrix in row-echelon form

geometric sequence

A sequence where the ratio between consecutive terms is always the same (constant ratio r).

growth factor

The value [latex]b[/latex] in an exponential growth function [latex]f(x) = ab^x[/latex] where [latex]b > 1[/latex]; also called the base or growth multiplier.

half-angle formulas

Formulas used to find exact values of trigonometric functions when the angle is half of a special angle

half-life

The time required for an exponentially decaying quantity to reduce to half its original amount; calculated as [latex]t = -\frac{\ln(2)}{k}[/latex].

Heron’s formula

A formula for finding the area of a triangle when all three side lengths are known: [latex]\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}[/latex] where [latex]s[/latex] is the semi-perimeter.

horizontal asymptote

a horizontal line [latex]y = b[/latex] where the graph approaches the line as the inputs increase or decrease without bound

horizontal compression

a transformation that compresses a function’s graph horizontally by multiplying the input by a constant [latex]b > 1[/latex]

horizontal line test

a test to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, it’s not one-to-one

horizontal reflection

a transformation that reflects a function’s graph across the y-axis by multiplying the input by [latex]-1[/latex]

horizontal shift

a transformation that shifts a function’s graph left or right by adding a constant to the input

horizontal stretch

a transformation that stretches a function’s graph horizontally by multiplying the input by a constant [latex]0 < b < 1[/latex]

hyperbola

The set of all points [latex](x,y)[/latex] in a plane such that the difference of the distances between latex[/latex] and two fixed points (foci) is a positive constant.

hypotenuse

The longest side of a right triangle; the side opposite the right angle.

identity matrix

a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra

identity property of logarithms

The logarithm of the base to itself equals 1: [latex]\log_b(b) = 1[/latex].

inconsistent system

A system with no solution; for two variables, the lines are parallel; for three variables, the planes don’t meet at a common point.

increasing function

a function where output values increase as input values increase

independent system

A system with exactly one solution; graphically, the lines or planes intersect at exactly one point.

independent variable

the input variable in a function, often represented by x

index of summation

The variable (often k, i, or n) used in summation notation to represent the position of terms being summed.

infinite discontinuity

A point of discontinuity in a function [latex]f(x)[/latex] at [latex]x=a[/latex] where one or both of the one-sided limits approaches positive or negative infinity; the graph typically has a vertical asymptote at [latex]x=a[/latex].

infinite sequence

A sequence that continues indefinitely without ending.

infinite series

The sum of all terms in an infinite sequence.

initial side

The fixed ray from which angle measurement begins, extending along the positive x-axis in standard position.

initial value

The value [latex]a[/latex] in an exponential function [latex]f(x) = ab^x[/latex]; represents the output when [latex]x = 0[/latex], so [latex]f(0) = a[/latex].

input

a value from the domain that is entered into a function

instantaneous rate of change

The rate at which a function is changing at a specific point, measured by the derivative; describes how quickly the output changes relative to the input at an exact moment.

interpolation

Using a model to make predictions within the range of original data; generally more reliable than extrapolation.

intersection of two events

The event [latex]E\cap F[/latex] that occurs if both event E and event F occur simultaneously. Mathematically represents “and.”

interval notation

a way of writing sets of numbers using brackets and parentheses to show included and excluded endpoints

invariant

A property or expression that remains unchanged after a transformation, such as rotation. The discriminant [latex]B^{2}-4AC[/latex] is invariant under rotation.

inverse function

for any one-to-one function [latex]f(x)[/latex], the inverse is a function [latex]f^{-1}(x)[/latex] such that [latex]f^{-1}(f(x)) = x[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]; this also implies that [latex]f(f^{-1}(x)) = x[/latex] for all [latex]x[/latex] in the domain of [latex]f^{-1}[/latex]

inverse property of logarithms

Logarithms and exponentials undo each other: [latex]b^{\log_b(x)} = x[/latex] and [latex]\log_b(b^x) = x[/latex].

inverse variation

a relationship where one quantity is a constant divided by another quantity; as one quantity increases, the other decreases

inversely proportional

a relationship where one quantity is a constant divided by the other quantity

irreducible quadratic factor

A quadratic expression that cannot be factored into linear factors with real coefficients (the discriminant [latex]b^2 - 4ac < 0[/latex]).

joint variation

a relationship where a variable varies directly or inversely with multiple variables

jump discontinuity

A point of discontinuity in a function [latex]f(x)[/latex] at [latex]x=a[/latex] where both the left-hand and right-hand limits exist, but [latex]\underset{x\to a^{-}}{\mathrm{lim}}f(x)\neq\underset{x\to a^{+}}{\mathrm{lim}}f(x)[/latex]; the graph “jumps” from one value to another.

latus rectum

The line segment that passes through the focus of a parabola, parallel to the directrix, with endpoints on the parabola; its length is [latex]|4p|[/latex].

Law of Cosines

A formula relating the sides and angles of any triangle: [latex]c^2 = a^2 + b^2 - 2ab\cos\gamma[/latex]. Generalizes the Pythagorean theorem to oblique triangles.

Law of Sines

A formula stating that in any triangle, the ratio of each side length to the sine of its opposite angle is constant: [latex]\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}[/latex].

left-hand limit

The limit of values of [latex]f(x)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] from the left, denoted [latex]\underset{x\to a^{-}}{\mathrm{lim}}f(x)=L[/latex]. The values of [latex]f(x)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] such that [latex]x

lemniscate

A figure-eight shaped polar curve with equation [latex]r^2 = a^2\cos(2\theta)[/latex] or [latex]r^2 = a^2\sin(2\theta)[/latex].

limaçon

A snail-shaped family of polar curves with equation [latex]r = a \pm b\cos\theta[/latex] or [latex]r = a \pm b\sin\theta[/latex]; includes dimpled, convex, and inner-loop varieties depending on the ratio [latex]\frac{a}{b}[/latex].

limit

When it exists, the value [latex]L[/latex] that the output of a function [latex]f(x)[/latex] approaches as the input [latex]x[/latex] gets closer and closer to [latex]a[/latex] but does not equal [latex]a[/latex]. The value of the output [latex]f(x)[/latex] can get as close to [latex]L[/latex] as we choose by using input values of [latex]x[/latex] sufficiently near to [latex]x=a[/latex], but not necessarily at [latex]x=a[/latex]. Denoted [latex]\underset{x\to a}{\mathrm{lim}}f(x)=L[/latex].

linear speed

local maximum

a point where the function value is greater than all nearby points

local minimum

a point where the function value is less than all nearby points

logarithm

The exponent to which a base must be raised to produce a given number; if [latex]b^y = x[/latex], then [latex]\log_b(x) = y[/latex].

logarithmic function

The inverse of an exponential function; a function of the form [latex]f(x) = \log_b(x)[/latex] where [latex]b > 0[/latex] and [latex]b \ne 1[/latex].

logistic growth

Growth that is exponential at first but slows as it approaches a maximum value (carrying capacity); modeled by [latex]f(x) = \frac{c}{1 + ae^{-bx}}[/latex].

lower limit of summation

The starting value of the index in summation notation.

major axis

The longer of the two axes of an ellipse that passes through the center, both foci, and both vertices.

matrix

a rectangular array of numbers

minor axis

The shorter of the two axes of an ellipse that passes through the center and both co-vertices, perpendicular to the major axis.

modulus

The absolute value or magnitude of a complex number; the distance [latex]r[/latex] from the origin in polar form.

multiplicative inverse of a matrix

a matrix that, when multiplied by the original, equals the identity matrix

Multiplication Principle

States that if one event can occur in A ways and a second event can occur in B ways after the first event has occurred, then the two events can occur in [latex]A\cdot B[/latex] ways. Also known as the Fundamental Counting Principle.

mutually exclusive events

Events that have no outcomes in common; events that cannot occur at the same time.

natural exponential function

The exponential function with base [latex]e[/latex]; written as [latex]f(x) = e^x[/latex] where [latex]e \approx 2.718282[/latex].

natural logarithm

A logarithm with base [latex]e[/latex], written as [latex]\ln(x)[/latex] instead of [latex]\log_e(x)[/latex]; commonly used in calculus and scientific applications.

Newton’s Law of Cooling

A model describing how an object’s temperature changes over time to equalize with surrounding temperature: [latex]T(t) = Ae^{kt} + T_s[/latex].

nondegenerate conic section

A shape formed by the intersection of a plane with a double right cone such that the plane does not pass through the apex; nondegenerate conics include circles, ellipses, hyperbolas, and parabolas.

nonlinear inequality

An inequality containing a nonlinear expression (such as [latex]x^2[/latex], [latex]xy[/latex], etc.).

nth term of the sequence

The general term of a sequence, denoted an, representing any term based on its position n.

oblique triangle

Any triangle that does not contain a right angle; can be acute (all angles less than 90°) or obtuse (one angle greater than 90°).

odd function

a function whose graph is unchanged by combined horizontal and vertical reflection, satisfying [latex]f(x) = -f(-x)[/latex], and is symmetric about the origin

odd function

A function where [latex]f(-x) = -f(x)[/latex]. Sine, tangent, cosecant, and cotangent are odd trigonometric functions.

one-to-one function

a function where each output value corresponds to exactly one input value

one-to-one property of exponential functions

If [latex]b^S = b^T[/latex], then [latex]S = T[/latex] for any positive real number [latex]b \neq 1[/latex].

one-to-one property of logarithmic functions

If [latex]\log_b(S) = \log_b(T)[/latex], then [latex]S = T[/latex] for positive real numbers [latex]S[/latex], [latex]T[/latex], and base [latex]b > 0[/latex], [latex]b \neq 1[/latex].

opposite side

In a right triangle, the side across from a given acute angle; the side not touching the angle.

ordered triple

A solution [latex](x, y, z)[/latex] to a system of three equations in three variables.

output

the value produced when an input is entered into a function

parabola

The set of all points [latex](x,y)[/latex] in a plane that are the same distance from a fixed point (the focus) and a fixed line (the directrix).

partial fraction decomposition

The process of breaking down a simplified rational expression into a sum or difference of simpler rational expressions.

partial fractions

The individual simpler fractions that make up the sum or difference of a rational expression before combining them.

partial sum

The sum of the first n terms of a series, denoted [latex]S_n[/latex].

period

The shortest interval [latex]P[/latex] over which a function completes one full cycle, where [latex]f(x + P) = f(x)[/latex].

permutation

An ordering of objects where the order matters; the number of ways to arrange objects in a specific sequence.

piecewise function

a function defined by different formulas over different parts of the domain

polar axis

The reference direction in the polar coordinate system, corresponding to the positive x-axis; the starting line from which angles are measured.

polar coordinates

An ordered pair [latex](r, \theta)[/latex] where [latex]r[/latex] is the distance from the pole and [latex]\theta[/latex] is the angle from the polar axis.

polar equation

An equation of a curve in polar coordinates [latex]r[/latex] and [latex]\theta[/latex].

polar form of a complex number

The representation [latex]z = r(\cos\theta + i\sin\theta)[/latex] or [latex]r\text{ cis }\theta[/latex], where [latex]r[/latex] is the modulus and [latex]\theta[/latex] is the argument.

pole

The origin of the polar coordinate system, corresponding to the point [latex](0, 0)[/latex] in rectangular coordinates.

power rule for logarithms

The logarithm of a power equals the exponent times the logarithm: [latex]\log_b(M^n) = n\log_b(M)[/latex].

probability

A number between 0 and 1 (or 0 and 100) that describes the likelihood that an event will occur.

probability model

A mathematical description of an experiment listing all possible outcomes and their associated probabilities. The sum of all probabilities must equal 1.

product rule for logarithms

The logarithm of a product equals the sum of the logarithms: [latex]\log_b(MN) = \log_b(M) + \log_b(N)[/latex].

product-to-sum formulas

Formulas that express products of trigonometric functions as sums or differences

profit function

The difference between revenue and cost: [latex]P(x) = R(x) - C(x)[/latex].

properties of limits

A collection of theorems for finding limits of functions by performing mathematical operations on the limits.

property of logarithmic equality

If [latex]\log_b(M) = \log_b(N)[/latex], then [latex]M = N[/latex] for any positive real numbers [latex]M[/latex], [latex]N[/latex], and base [latex]b > 0[/latex], [latex]b \neq 1[/latex].

Pythagorean identities

Three identities based on the Pythagorean theorem and the unit circle: [latex]\sin^2\theta + \cos^2\theta = 1[/latex], [latex]1 + \tan^2\theta = \sec^2\theta[/latex], and [latex]1 + \cot^2\theta = \csc^2\theta[/latex]

quadrantal angles

Angles whose terminal side lies on an axis, including 0°, 90°, 180°, 270°, and 360°.

quotient identities

Identities that express tangent and cotangent as quotients: [latex]\tan\theta = \frac{\sin\theta}{\cos\theta}[/latex] and [latex]\cot\theta = \frac{\cos\theta}{\sin\theta}[/latex]

quotient rule for logarithms

The logarithm of a quotient equals the difference of the logarithms: [latex]\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)[/latex].

radian

The measure of a central angle that intercepts an arc equal in length to the radius of the circle. A full revolution equals [latex]2\pi[/latex] radians.

radiocarbon dating

A method for determining the age of organic materials.

range

the set of all possible output values for a function

rate of change

a measure of how one quantity changes relative to another

rational equation

an equation that contains at least one rational expression with a variable in the denominator

rational function

a function that can be written as the ratio of two polynomials

rational inequality

an inequality that contains at least one rational expression with a variable in the denominator

reciprocal identities

Identities that relate trigonometric functions to their reciprocals, such as [latex]\sin\theta = \frac{1}{\csc\theta}[/latex]

recursive formula

A formula that defines each term of a sequence using one or more preceding terms. Must include initial term(s).

reduction formulas

Formulas that reduce even powers of sine or cosine to expressions involving the first power of cosine; also called power-reducing formulas

reference angle

The measure of the smallest positive acute angle formed by the terminal side of an angle and the horizontal axis.

regression

A method of fitting an algebraic model to data

relation

a relationship between two sets of numbers

removable discontinuity

a single point at which a function is undefined that, if filled in, would make the function continuous; appears as a hole on the graph

repeated factor

A factor in the denominator that appears more than once, written as [latex](ax + b)^n[/latex] where [latex]n > 1[/latex].

revenue function

A function calculating revenue, written as [latex]R = xp[/latex] where [latex]x[/latex] is quantity and [latex]p[/latex] is price.

right-hand limit

The limit of values of [latex]f(x)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] from the right, denoted [latex]\underset{x\to a^{+}}{\mathrm{lim}}f(x)=L[/latex]. The values of [latex]f(x)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] where [latex]x>a[/latex] and [latex]x\neq a[/latex].

rose curve

A petal-shaped polar curve with equation [latex]r = a\cos(n\theta)[/latex] or [latex]r = a\sin(n\theta)[/latex]; has [latex]n[/latex] petals if [latex]n[/latex] is odd, [latex]2n[/latex] petals if [latex]n[/latex] is even.

row

a set of numbers aligned horizontally in a matrix

row-echelon form

after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal

row-equivalent

two matrices A and B are row-equivalent if one can be obtained from the other by performing basic row operations

row operations

adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form

sample space

The set of all possible outcomes of an experiment, denoted S.

scalar multiple

an entry of a matrix that has been multiplied by a scalar

secant

The reciprocal of the cosine function: [latex]\sec t = \frac{1}{\cos t}[/latex].

sector

A region of a circle bounded by two radii and the intercepted arc, with area [latex]A = \frac{1}{2}\theta r^2[/latex].

semi-perimeter

One-half of the perimeter of a triangle, denoted [latex]s = \frac{a+b+c}{2}[/latex]; used in Heron’s formula.

sequence

A function whose domain is the set of positive integers; an ordered list of numbers following a specific pattern.

series

The sum of the terms in a sequence.

sigma

The Greek letter Σ used in summation notation to indicate a sum.

sine

For a point [latex](x, y)[/latex] on the unit circle corresponding to angle [latex]t[/latex], [latex]\sin t = y[/latex].

SOH-CAH-TOA

A mnemonic device for remembering the three primary trigonometric ratios in right triangles: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

solution set

The set of all ordered pairs or triples that satisfy all equations in a system.

standard position

An angle positioned with its vertex at the origin and its initial side extending along the positive x-axis.

stepwise function

A function that remains constant over intervals and then jumps instantaneously to different values; an example of a function with jump discontinuities.

substitution method

An algebraic technique where one equation is solved for one variable, and that expression is substituted into the other equation(s).

sum and difference formulas

Formulas that express trigonometric functions of sums or differences of angles in terms of functions of the individual angles

summation notation

A notation using the sigma symbol (Σ) to represent the sum of terms in a series.

sum-to-product formulas

Formulas that express sums or differences of trigonometric functions as products

supplementary angles

Two angles whose measures add up to 180° (or [latex]\pi[/latex] radians).

symmetry with respect to the polar axis

A graph is symmetric about the x-axis if replacing [latex](r, \theta)[/latex] with [latex](r, -\theta)[/latex] yields an equivalent equation.

symmetry with respect to the pole

A graph is symmetric about the origin if replacing [latex](r, \theta)[/latex] with [latex](-r, \theta)[/latex] yields an equivalent equation.

symmetry with respect to [latex]\theta = \frac{\pi}{2}[/latex]

A graph is symmetric about the y-axis if replacing [latex](r, \theta)[/latex] with [latex](-r, -\theta)[/latex] yields an equivalent equation.

system of linear equations

A set of two or more linear equations with two or more variables that must be considered simultaneously.

system of nonlinear equations

A system containing at least one equation of degree larger than one (not linear).

system of nonlinear inequalities

A system of two or more inequalities in two or more variables with at least one nonlinear inequality.

tangent

The ratio of sine to cosine: [latex]\tan t = \frac{\sin t}{\cos t}[/latex].

tangent line

A line that touches the graph of a function at exactly one point and has the same instantaneous rate of change (slope) as the function at that point.

term

An individual number in a sequence.

terminal side

The rotated ray of an angle after rotation from the initial side.

transverse axis

The axis of a hyperbola that passes through the center and has the vertices as its endpoints; the foci lie on this axis.

trigonometric equation

An equation that contains at least one trigonometric function and may have an infinite number of solutions due to the periodic nature of trigonometric functions

trigonometric identity

An equation involving trigonometric functions that is true for all values in the domain of the variable

two-sided limit

The limit of a function [latex]f(x)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] is equal to [latex]L[/latex], denoted [latex]\underset{x\to a}{\mathrm{lim}}f(x)=L[/latex], if and only if [latex]\underset{x\to a^{-}}{\mathrm{lim}}f(x)=\underset{x\to a^{+}}{\mathrm{lim}}f(x)[/latex].

union of two events

The event E∪F that occurs if either event E or event F (or both) occurs. Mathematically represents “or.”

unit circle

A circle centered at the origin with radius 1, where [latex]x^2 + y^2 = 1[/latex].

upper limit of summation

The ending value of the index in summation notation.

varies directly

a relationship where one quantity is a constant multiplied by another quantity

varies inversely

a relationship where one quantity is a constant divided by another quantity

verifying an identity

The process of showing that both sides of an equation are equal by transforming one side (usually the more complicated side) into the other using algebraic techniques and known identities

vertex (plural: vertices)

For an ellipse, an endpoint of the major axis; for a hyperbola, an endpoint of the transverse axis; for a parabola, the point on the curve closest to the directrix and midway between the focus and directrix.

vertical asymptote

a vertical line [latex]x = a[/latex] where the graph tends toward positive or negative infinity as the inputs approach [latex]a[/latex]

vertical compression

a transformation that compresses a function’s graph vertically by multiplying the output by a constant [latex]0 < a < 1[/latex]

vertical line test

a test to determine if a graph represents a function; if any vertical line intersects the graph more than once, it’s not a function

vertical reflection

a transformation that reflects a function’s graph across the x-axis by multiplying the output by [latex]-1[/latex]

vertical shift

a transformation that shifts a function’s graph up or down by adding a constant to the output

vertical stretch

a transformation that stretches a function’s graph vertically by multiplying the output by a constant [latex]a > 1[/latex]

zero property of logarithms

The logarithm of 1 to any base equals 0: [latex]\log_b(1) = 0[/latex].