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arithmetic sequence

An arithmetic sequence is a number sequence in which the difference between each successive term remains constant. This difference can either be positive or negative, and dependent on the sign will result in terms of the arithmetic sequence tending towards positive or negative infinity.



a line that continually approaches a given curve but does not meet it at any finite distance.


change of base formula

Most graphing calculators have functions or keys that directly calculate the logarithms of numbers in base-1010 and base-ee. Thus, you will only see two buttons: LOG for common logarithm and LN for natural logarithm.



Radius, Diameter and Circumference ... The Radius is the distance from the center outwards. The Diameter goes straight across the circle, through the center.


combined variation

Combined proportionality (or combined variation) is just the combination of direct and inverse proportionality. If a variable has a combined proportionality with two other variables, then it has a direct proportion with one and an inverse proportion with the other.


common difference

The constant difference between consecutive terms of an arithmetic sequence is called the common difference. Example: Given the arithmetic sequence 9,7,5,3,... . To find the common difference, subtract any term from the term that follows it.


common logarithm

The common logarithm is the logarithm to base 10. The notation logx is used by physicists, engineers, and calculator keypads to denote the common logarithm.


common ratio

The constant factor between consecutive terms of a geometric sequence is called the common ratio. Example: ... To find the common ratio , find the ratio between a term and the term preceding it. r=42=2. 2 is the common ratio.


completing the square

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form ax^{2}+bx+c to the form {\displaystyle a^{2}+k} for some values of h and k


complex conjunctions

A complex number is equal to its complex conjugate if its imaginary part is zero, or equivalently, if the number is real. In other words, real numbers are the only fixed points of conjugation. The product of a complex number with its conjugate is equal to the square of the number's modulus.


complex fractions

A complex fraction can be defined as a fraction in which the denominator and numerator or both contain fractions. A complex fraction containing a variable is known as a complex rational expression. For example, 3/(1/2) is a complex fraction whereby, 3 is the numerator and 1/2 is the denominator.


complex numbers

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no "real" number satisfies this equation


composition of functions

In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. ... Intuitively, if z is a function of y, and y is a function of x, then z is a function of x.


compound inequality

A compound inequality is a sentence with two inequality statements joined either by the word “or” or by the word “and.” “And” indicates that both statements of the compound sentence are true at the same time. It is the overlap or intersection of the solution sets for the individual statements.


conic section

a figure formed by the intersection of a plane and a right circular cone. Depending on the angle of the plane with respect to the cone, a conic section may be a circle, an ellipse, a parabola, or a hyperbola.



In mathematics and particularly in algebra, a linear or nonlinear system of equations is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity.


constant of variation

The constant of variation in a direct variation is the constant (unchanged) ratio of two variable quantities. where k is the constant of variation . Example 1: If y varies directly as x and y=15 when x=24 , find x when y=25


continuous relation

A function is continuous as a relation iff it is continuous in the usual sense and a composition of continuous relations is continuous. A partial function that is continuous on its domain is continuous as a relation iff its domain is closed.


correlation coefficient

There is a way of measuring the "goodness of fit" of the best fit line (least squares line), called the correlation coefficient. It is a number between -1 and 1, inclusive, which indicates the measure of linear association between the two variables, and also shows whether the correlation is positive or negative.



The dependent variable is the one that depends on the value of some other number. If, say, y = x+3, then the value y can have depends on what the value of x is. Another way to put it is the dependent variable is the output value and the independent variable is the input value.


dimensions of a matrix

The dimensions of a matrix are the number of rows by the number of columns. If a matrix has a rows and b columns, it is an a×b matrix. For example, the first matrix shown below is a 2×2 matrix; the second one is a 1×4 matrix; and the third one is a 3×3 matrix.


direct variation

Direct variation describes a simple relationship between two variables . We say y varies directly with x (or as x , in some textbooks) if: y=kx. for some constant k , called the constant of variation or constant of proportionality .



The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.



In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same


end behavior

The end behavior of a function f describes the behavior of the graph of the function at the "ends" of the x-axis. In other words, the end behavior of a function describes the trend of the graph if we look to the right end of the x-axis (as x approaches +∞ ) and to the left end of the x-axis (as x approaches −∞ ).


exponential equation

Exponential equations are equations in which variables occur as exponents. For example, exponential equations are in the form ax=by . To solve exponential equations with same base, use the property of equality of exponential functions . ... In other words, if the bases are the same, then the exponents must be equal.


extraneous solution

In mathematics, an extraneous solution (or spurious solution) is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the problem.



Extremum, plural Extrema, in calculus, any point at which the value of a function is largest (a maximum) or smallest (a minimum). There are both absolute and relative (or local) maxima and minima.


factor theorem

In practice, the Factor Theorem is used when factoring polynomials "completely". ... Any time you divide by a number (being a potential root of the polynomial) and get a zero remainder in the synthetic division, this means that the number is indeed a root, and thus "x minus the number" is a factor.


finite sequence

A finite sequence is a list of terms in a specific order. The sequence has a first term and a last term. The order of the terms of a finite sequence follows some type of mathematical pattern or logical arrangement.



A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function.


geometric sequence

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.


greatest integer function

The greatest integer function is represented/denoted by ⌊x⌋, for any real function. The function rounds -off the real number down to the integer less than the number. This function is also known as the Floor Function.


growth factor

Growth factor is the factor by which a quantity multiplies itself over time. Growth rate is the addend by which a quantity increases (or decreases) over time.



A hyperbola is the set of all points (x,y) in a plane such that the difference of the distances between (x,y) and the foci is a positive constant. The standard form of a hyperbola can be used to locate its vertices and foci.


identity function

In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. That is, for f being identity, the equality f(x) = x holds for all x.


imaginary unit

The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication.



Inconsistent mathematics is the study of commonplace mathematical objects, like sets, numbers, and functions, where some contradictions are allowed.



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.


infinite sequence

An infinite sequence is a list or string of discrete objects, usually numbers, that can be paired off one-to-one with the set of positive integer s {1, 2, 3, ...}. Examples of infinite sequences are N = (0, 1, 2, 3, ...) and S = (1, 1/2, 1/4, 1/8, ..., 1/2 n , ...).


interval notion

Interval notation is a way to describe continuous sets of real numbers by the numbers that bound them. Intervals, when written, look somewhat like ordered pairs. However, they are not meant to denote a specific point. Rather, they are meant to be a shorthand way to write an inequality or system of inequalities.


inverse function

The inverse is usually shown by putting a little "-1" after the function name, like this: f-1(y) We say "f inverse of y" So, the inverse of f(x) = 2x+3 is written: f-1(y) = (y-3)/2.


inverse relation

An inverse relation is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function. ... this inverse function is denoted by f -1(x). Note: If the original function is a one-to-one function, the inverse will be a function.


inverse variation

An inverse variation can be represented by the equation xy=k or y=kx . ... That is, y varies inversely as x if there is some nonzero constant k such that, xy=k or y=kx where x≠0,y≠0 . Suppose y varies inversely as x such that xy=3 or y=3x . That graph of this equation shown.


joint variation

Joint variation describes a situation where one variable depends on two (or more) other variables, and varies directly as each of them when the others are held constant. We say z varies jointly as x and y if. z=kxy.


lactus rectum

"Latus rectum" is a compound of the Latin latus, meaning "side," and rectum, meaning "straight." Half the latus rectum is called the semilatus rectum. SEE ALSO: Conic Section, Conic Section Directrix, Parabola, Semilatus Rectum, Universal Parabolic Constant.


linear programing

Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.



Logarithm (log) In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.


logistic growth model

Logistic Growth Model - Background: Logistic Modeling where P0 is the population at time t = 0. In short, unconstrained natural growth is exponential growth. ... But, for the second population, as P becomes a significant fraction of K, the curves begin to diverge, and as P gets close to K, the growth rate drops to 0.



In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are commonly written in box brackets. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.


nth root

An nth root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x: Every positive real number x has a single positive nth root, called the principal nth root, which is written. . For n equal to 2 this is called the principal square root and the n is omitted.


natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x.


negative exponent

A negative exponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side. For instance, "x–2" (pronounced as "ecks to the minus two") just means "x2, but underneath, as in 1 x 2 \frac{1}{x^2} x21 ".



In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix).


parent function

A parent function is the simplest function of a family of functions. For the family of quadratic functions, y = ax2 + bx + c, the simplest function. of this form is y = x2. The "Parent" Graph: The simplest parabola is y = x2, whose graph is shown at the right.


piece wise defined function

A piecewise-defined function is one which is defined not by a single equation, but by two or more. Each equation is valid for some interval . Example 1: ... The function in this example is piecewise-linear, because each of the three parts of the graph is a line.


point slope form

Point-slope is the general form y-y₁=m(x-x₁) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept).


quadratic function

A quadratic function is a function of degree two. The graph of a quadratic function is a parabola. The general form of a quadratic function is f(x)=ax2+bx+c where a, b, and c are real numbers and a≠0. The standard form of a quadratic function is f(x)=a(x−h)2+k. The vertex (h,k) is located at h=–b2a,k=f(h)=f(−b2a).



The value inside the radical symbol. The value you want to take the root of. In √x, "x" is the radicand. See: Radical · Squares and Square Roots.


rate of change

A rate of change is a rate that describes how one quantity changes in relation to another quantity. The rate of change is 401 or 40 . This means a vehicle is traveling at a rate of 40 miles per hour.


rational exponent

A rational exponent is an exponent that is a fraction. For example, can be written as . Can't imagine raising a number to a rational exponent? They may be hard to get used to, but rational exponents can actually help simplify some problems.


rational function

A rational function is any function which can be written as the ratio of two polynomial functions, where the polynomial in the denominator is not equal to zero. The domain of f(x)=P(x)Q(x) f ( x ) = P ( x ) Q ( x ) is the set of all points x for which the denominator Q(x) is not zero.


rationalizing the denominator

To rationalize the denominator means to eliminate any radical expressions in the denominator such as square roots and cube roots. The key idea is to multiply the original fraction by an appropriate value, such that after simplification, the denominator no longer contains radicals.


recursive formula

A recursive formula is a formula that defines each term of a sequence using preceding term(s). Recursive formulas must always state the initial term, or terms, of the sequence.


regression line

A regression line is an estimate of the line that describes the true, but unknown, linear relationship between the two variables. The equation of the regression line is used to predict (or estimate) the value of the response variable from a given value of the explanatory variable.


relative maximum

A relative maximum is a point that is higher than the points directly beside it on both sides, and a relative minimum is a point that is lower than the points directly beside it on both sides.


relative minimum

A relative minimum of a function is all the points x, in the domain of the function, such that it is the smallest value for some neighborhood. These are points in which the first derivative is 0 or it does not exist.



In mathematics, a square root of a number x is a number y such that y² = x; in other words, a number y whose square is x. For example, 4 and −4 are square roots of 16, because 4² = ² = 16


scatter plot

A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data.



In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. ... This sequence differs from (A, R, M, Y).


set builder notion

In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.


step function

In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals. ... Informally speaking, a step function is a piecewise constant function having only finitely many pieces.


synthetic division

Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor -- and it only works in this case. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials.


vertex form

The vertex form of a quadratic is given by y = a(x – h)2 + k, where (h, k) is the vertex. The “a” in the vertex form is the same “a” as in y = ax2 + bx + c (that is, both a's have exactly the same value). The sign on “a” tells you whether the quadratic opens up or opens down.


vertical line test

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The vertical line test is a method that is used to determine whether a given relation is a function or not. The approach is rather simple. Draw a vertical line cutting through the graph of the relation, and then observe the points of intersection.



In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function , is a member of the domain of such that vanishes at ; that is, the function attains the value of 0 at , or equivalently, is the solution to the equation. .