Algebra 2
RELATED GLOSSARY TERM DEFINITIONS (80)
Absolute value:
A number's distance form zero on a number line. Distance is
expressed as a positive value.
Approximate:
A number or measurement that is close to or near its exact value.
Area:
The number of square units needed to cover a surface.
Axes:
The horizontal and vertical number lines used in a coordinate
plane system.
Coefficient:
The number that multiplies the variable(s) in an algebraic expression
(e.g., 4xy). If no number is specified, the coefficient is 1.
Complex number:
A number that can be written in the form a + bi, where a and b
are real numbers and i is the square root of -1.
Complex fraction:
A fraction with one or more fractions embedded in the numerator
and/or denominator (e.g., (3/8)/(2/7) ).
Composition of
functions:
Combining two functions by taking the output of one and using it
as the input of another. If the output of g is used as the input of f,
then the composition is referred to as "f of g of x" and is denoted
f(g(x)) or f○g(x).
Conjugate root
theorem:
If P is a polynomial in one variable with real coefficients, and a +
bi is a zero of P with a and b real numbers, then its complex
conjugate a - bi is also a zero of P.
Constant:
Any value that does not change.
Difference:
A number that is the result of subtraction
Dimension:
The number of coordinates used to express a position.
Discriminant:
An algebraic expression related to the coefficients of a quadratic
equation that can be used to determine the number and type of
solutions to the equation. If ax^2+bx+c=0, the discriminant is
D=b^2-4ac.
Domain:
The set of values of the independent variable(s) for which a
function or relation is defined.
e:
e=2.7182818284...., is an irrational number and the base of the
natural logarithm. e is sometimes known as Napier’s constant
although the symbol e honors Euler.
End behavior:
A function’s value for extreme values of its independent variable.
Equal:
Having the same value (=).
Equation:
A mathematical sentence stating that the two expressions have
the same value. Also read the definition of equality.
Expression:
A mathematical phrase that contains variables, functions,
numbers, and/or operations. An expression does not contain
equal or inequality signs.
Factor:
A number or expression that is multiplied by one or more other
numbers or expressions to yield a product.
Formula:
A rule that shows the relationship between two or more
quantities; involving numbers and/or variables.
Imaginary part:
The coefficient of i in a complex number.
Independent
variable:
The factor that is changed in an experiment in order to study
changes in the dependent variable.
Intercept:
The points where a curve or line drawn on a rectangularcoordinate-system graph intersect the vertical and horizontal
axes.
Joint variation:
A quantity varies directly with two or more quantities. For
example, z varies jointly with x and y means that z=kxy, where k
is a constant.
Length:
A one-dimensional measure that is the measurable property of
line segments.
Line:
A collection of an infinite number of points in a straight pathway
with unlimited length and having no width.
Linear equation:
An algebraic equation in which the variable quantity or quantities
are raised to the zero or first power.
Literal equations:
An equation that contains more than one variable; an implicit
equation; often mathematical formula.
Net:
A two-dimensional diagram that can be folded or made into a
three-dimensional figure.
Number line:
A line of infinite extent whose points correspond to the real
numbers according to their distance in a positive or negative
direction from a point arbitrarily taken as zero.
Operation:
Any mathematical process, such as addition, subtraction,
multiplication, division, raising to a power, or finding the square
root.
Origin:
The point of intersection of the x- and y-axes in a rectangular
coordinate system, where the x-coordinate and y-coordinate are
both zero. On a number line, the origin is the 0 point. In three
dimensions, the origin is the point (0, 0, 0).
Perpendicular:
Two lines, two line segments, or two planes are said to be
perpendicular when they intersect at a right angle.
Point:
A specific location in space that has no discernable length or
width.
Product:
The result of multiplying numbers together.
Proportional:
Having the same or a constant ratio. Two quantities that have the
same ratio are considered directly proportional. Two quantities
whose products are always the same are considered inversely
proportional.
Real number:
The set of all rational and irrational numbers.
Real-world
problem:
A problem that is an application of a mathematical concept in a
real-life situation.
Remainder:
In a whole-number division problem, the final undivided part that
is less than the divisor and “left over” after dividing.
Remainder
Theorem:
If a polynomial P(x) is divided by (x-r), then the remainder is a
constant given by P(r).
Root:
A root of a polynomial is a number x such that P(x)=0. A
polynomial of degree n has n complex roots.
Rule:
A general statement written in numbers, symbols, or words that
describes how to determine any term in a pattern or relationship.
Rules or generalizations may include both recursive and explicit
notation. In the recursive form of pattern generalization, the rule
focuses on the rate of change from one element to the next.
Example: Next = Now + 2; Next = Now x 4. In the explicit form of
pattern generalization, the formula or rule is related to the order
of the terms in the sequence and focuses on the relationship
between the independent variable and the dependent variable.
For example: y=5t - 3 Words may also be used to write a rule in
recursive or explicit notation. Example: to find the total fee,
multiply the total time with 3; take the previous number and add
two to get the next number.
Sequence:
A list of numbers set apart by commas, such as -1, 1, -1, 1, -1, …
Series:
An indicated sum of successive terms of a sequence.
Set:
A set is a finite or infinite collection of distinct objects in which
order has no significance.
Side:
The edge of a polygon (e.g., a triangle has three sides), the face
of a polyhedron, or one of the rays that make up an angle.
Similarity:
A term describing figures that are the same shape but are not
necessarily the same size or in the same position.
Simplify:
The process of converting a fraction or mixed number, to an
equivalent fraction, or mixed number, in which the greatest
common factor of the numerator and the denominator of the
fraction is one. Simplify also refers to using the rules of arithmetic
and algebra to rewrite an expression as simply as possible.
Square:
A rectangle with four congruent sides; also, a rhombus with four
right angles.
Sum:
The result of adding numbers or expressions together.
Symmetry:
An intrinsic property of a mathematical object which causes it to
remain invariant under certain classes of transformations (such
as rotation, reflection, or translation).
Synthetic division:
A shortcut method for dividing a polynomial by another
polynomial of the first degree. It can be used in place of the
standard long division algorithm. This method reduces the
polynomials factor into a set of numeric values. After these values
are processed, the resulting set of numeric outputs is used to
construct the polynomial quotient and the polynomial remainder.
System of
equations:
A group of two or more equations that are related to the same
situation and share variables. The solution to a system of
equations is an ordered number set that makes all of the
equations true.
Theorem:
A statement or conjecture that can be proven to be true based on
postulates, definitions, or other proven theorems. The process of
showing a theorem to be correct is called a proof.
Transformation:
An operation on a figure by which another image is created.
Common transformations include reflections (flips), translations
(slides), rotations (turns) and dilations.
Variable:
Any symbol, usually a letter, which could represent a number. A
variable might vary as in f(x)=2x+1, or a variable might be fixed
as in 2x+1=5.
Change of Base
Formula:
A formula that allows you to rewrite a logarithm in terms of logs
written with another base. Assume that x, a, and b are all
positive. Also assume that a ≠ 1, b ≠1. Change of base formula:
Circle:
A closed plane figure with all points of the figure the same
distance from the center. The equation for a circle with center (h,
k) and radius r is: (x - h)2 + (y - k)2 = r2
Degree:
The unit of measure for angles (°), equal to 1/360 of a complete
revolution. There are 360 degrees in a circle.
Exponent
The number of times the base occurs as a factor, for example 2 3
(exponential form): is the exponential form of 2 x 2 x 2. The number two (2) is called
the base, and the number three (3) is called the exponent.
Exponential
Function:
A function of the form y = abcx+b + e, where a,b,c,d,e,x are real
numbers, a, b, c are nonzero, b≠1, and b>0.
Fraction:
A rational number expressed in the form a/b, where a is called the
numerator and b is called the denominator. A fraction may mean
part of a whole, ratio of two quantities, or may imply division.
Function:
A relation in which each value of x is paired with a unique value
of y. More formally, a function from A to B is a relation f such that
every a A is uniquely associated with an object F(a) B.
Fundamental
Theorem of
Algebra:
Every polynomial equation with degree greater than zero has at
least one root in the set of complex numbers. Corollary: Every
polynomial P(x) of degree n (n > 0) can be written as the product
of a constant k (k ≠ 0) and n linear factors P(x) = k (x – r1) (x – r2 )
(x – r3 )…(x – rn) Thus a polynomial equation of degree n has
exactly n complex roots, namely r1, r2, r3,…, rn.
Geometric
Sequence:
A sequence in which consecutive terms have a common ratio. A
geometric sequences can be written as an=a1rn-1 (n=1, 2, 3, …)
where an is the nth term of the sequence, a1 is the first term, r is
the common ratio.
Inequality:
A sentence that states one expression is greater than (>), greater
than or equal to (≥), less than (<), less than or equal to (≤),
another expression.
Limit:
A number to which the terms of a sequence get closer so that
beyond a certain term all terms are as close as desired to that
number. A function f(z) is said to have a limit
if, for all
e>0, there exists a d>0 such that
whenever
.
Logarithm:
“The logarithm of x to the base b” is the power to which b must be
raised to be equal to x. f(x) = logbx is the inverse function of h(x) =
bx.
Monomial:
A polynomial with one term such as 5, -2xyz, or xy4
Parabola:
A locus of points whose perpendicular distances to a line, called
the directrix, and to a fixed point, called the focus, are equal. The
graph of any quadratic function is a parabola and a parabola
always has a quadratic equation. The equation for a vertical
parabola is y = a(x - h)2 + k, where (h,k) is the vertex of the
parabola.
Polynomial:
The sum or difference of terms which have variables raised to
positive integer powers and which have coefficients that may be
real or complex. Examples: 5x3 – 2x2 + x – 13, x2y3 + xy, and (1 +
i)a2 + ib2. Standard form for a polynomial in one variable: anxn +
an–1xn–1 + ... + a2x2 + a1x + a0
Even though the prefix poly- means many, the word polynomial
refers to polynomials with 1 term (monomials), 2 terms
(binomials), 3 terms, (trinomials), etc.
Polynomial
Function:
A function that can be written as f(x) = anxn + an-1xn-1 + ... + a1x1 +
a0 , where might be real or complex.
Power:
The rate at which work is done, expressed as the amount of work
per unit time and commonly measured in units such as the watt
and horsepower.
Quadratic
Equation:
A second-order polynomial equation in a single variable x with
a≠0: ax2 + bx + c = 0. Because it is a second-order polynomial
equation, the fundamental theorem of algebra guarantees that it
has two solutions that may be both real or both complex.
Radical:
The symbol
used to indicate a root. The expression
is
therefore read “x radical n” or "the nth root of x." A radical without
an index number is understood to be a square root.
Rational Root
Theorem:
If the coefficients of the polynomial dnxn + dn-1xn-1 + ... + d0 are
specified to be integers, then rational roots must have a
numerator which is a factor of d0 and a denominator which is a
factor of (with either sign possible).
Slope:
The ratio of change in the vertical axis (y-axis) to each unit
change in the horizontal axis (x-axis) in the form rise/run or ?y/?x.
Also the constant, m, in the linear equation for the slope-intercept
form y =mx + b, where
Vertex:
The point common to the two rays that form an angle; the point
common to any two sides of a polygon; the point common to
three or more edges of a polyhedron.
Volume:
A measure of the amount of space an object takes up; also the
loudness of a sound or signal.