Algebra 2 Terms

rational function

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

Formula

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

exponential equation

an equation in which the variable appears as an exponent.

radical equation

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

rational equation

an equation involving rational expressions.

proportion

an equation stating that 2 rational expressions are equal.

logarithmic equation

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

linear equation

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

Polynomial

an expression consisting of terms separated by some combination of + signs, - signs, or both.

Trinomial

an expression containing three terms separated by + or - signs.

Binomial

an expression containing two terms separated by a + or - sign.

sum of cubes

an expression in the following form:

difference of cubes

an expression in the form of a^3-b^3

Sequence

an ordered list of numbers.

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.

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 thedirectrix, and the given point is called the focus.

circle

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

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.

quadratic formula

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

complex fraction

a fraction containing one or more additional fractions in the numerator or denominator.

composite function

a function in which the variable name has been replaced by another function.

inverse function

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

logarithmic function

a function of the form, logn(x)

Pascal's triangle

a graphical representation of binomial expansion, named after the French mathematician Blaise Pascal.

linear inequality

a linear sentence not containing an equal sign.

Inequality

a mathematical sentence using a relational symbol other than the equal sign .

compound inequality

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

factor (n.)

a number that is multiplied by another number to make a product. For example, 2 &3, or 1 & 6 for 6.

Graph

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

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 p*q different ways.

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.

matrix (pl. matrices)

a rectangular array of numerals or variables that can be used to represent systems of equations.

Function

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

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.

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.

relation

a set of ordered pairs.

synthetic division

a shortcut for dividing a polynomial by a binomial of the form x -a; only coefficients are used.

Monomial

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

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.

determinant

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

Equation

a statement that says two mathematical expressions are equal.

inconsistent system

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

completing the square

a technique for solving quadratic equations.

natural logarithm

a term that represents log base e (also loge), which is written as ln. See logarithm.

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.

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.

point-slope form (of a non-vertical line)

takes the following form, where (x -x1) =; difference in x-coordinates, and (y - y1) =; difference in y-coordinates; m is the slope.

quotient

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

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 P3.

conjugate axis

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

radical

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

radius

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

direct variation

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

inverse variation

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

first degree equation

another name for a linear equation. See linear equation.

quadratic equation

any equation in the following form:

quadratic form

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

complex number

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

polynomial function

any function of the form P(x) + a0x^n+a1x^n-1+ a2x^n-1+ . . . an-1x+an where the coefficients a0, a1, a2, . . . , an are real numbers, and n is a whole number.

exponential function

any function with a number based raised to a variable

Term

any number in a sequence or piece of a polynomial separated by a + or - sign.

pure imaginary number

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

zero of a function

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

varies inversely

as one quantity increases another decreases, and vice versa. Seeinverse variation.

varies directly

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

minor axis

axis connecting endpoints separated my smaller distance in ellipse.

common difference

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

conic section

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

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.

vertex (of hyperbola)

either of the two points of intersection of the hyperbola and the transverse axis. See hyperbola.

minor intercepts

endpoints along minor axis of ellipse

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.

common ratio

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

imaginary value

i represents sqrt(-1), which is an expression with no real value.

Divisor

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

Dividend

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

Index

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

completely factored

incapable of being further simplified by division.

Cramer's rule

method of solving systems of equations by using determinants.

general term

nth term of a sequence; a term of some order to be determined.

like radical expressions

radical expressions with identical index and radicand.

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.

dependent system

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

greatest common factor

see GCF.

Domain

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

range

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

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 8C3.

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.

square trinomial

the expression produced by squaring a binomial:

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.

descending order

the general practice of writing polynomials in more than one variable so that the exponents decrease from right to left

x-axis

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

GCF (greatest common factor)

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

transverse axis

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

Directrix

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

major axis

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

axis of symmetry (of a parabola)

the line that passes through the vertex and focus.

slope of a line

the line's rise over its run (or its change in y divided by its change inx) 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.

vertex (of parabola)

the midpoint of the perpendicular segment from the focus to the directrix.

constant of proportionality

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

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 nth root of a."

x-coordinate

the number to the left of the comma in an ordered pair.

y-coordinate

the number to the right of the comma in an ordered pair.

radicand

the number under the radical. See radical expression.

coordinates of a point

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

Origin

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

x-intercept

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

y-intercept

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

Focus

the point from which the set of points in a conic section are equidistant. In a circle, the focus is called the center.

Center

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

major intercepts

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

rational expression

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

inverse relation

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

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.

geometric series

the sum of the terms in a geometric sequence.

arithmetic series

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

y-axis

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

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.

Conjugates

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

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.

common logarithm

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

identity function

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

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.