Unit 1 Real Numbers
completing the square
The process used in solving quadratic equations that do not have factorable expressions.
conjugate
Used to rationalize denominators.
discriminant
The radicand b 2 - 4ac of the quadratic formula; used to determine the nature of the roots of a quadratic equation without completing the solution.
double root
Every quadratic equation has two roots; however, in certain cases the two roots are the same number.
extraneous root
In the algebraic process used for solving an equation, a number obtained that is not a root of the original equation.
fractional exponent
An exponent that is a fraction; the numerator represents the power of the number and the denominator represents the index of the root.
general form of quadratic equation
ax 2 + bx + c = 0, where a, b, and c are real numbers and where the leading coefficient, a, is greater than or equal to one.
imaginary number
A number that involves taking the square root of a negative number.
index
In 3√8, the index is 3. If no index is indicated, the index is understood to be 2.
irrational number
A real number that is not a rational number; can be a radical, a non-repeating, non-terminating decimal, or a designated constant.
principal square root
Every positive real number has two square roots, one positive and one negative. The principal square root is the positive square root and is denoted by the radical sign.
quadratic equation
An equation of the second degree. (See "general form.")
quadratic formula
A formula used to compute the roots of a quadratic equation in general form.
radical
An expression consisting of a phrase and a radical sign over it.
radical equation
An equation in which the variable appears under a radical sign or with a fractional exponent.
radical sign
A symbol indication that an expression is a radical.
radicand
The quantity under a radical sign.
rationalize
To remove the radical in the denominator of an expression without changing the value.
rational number
A real number that can be expressed in the form a/b, where a and b are integers and b does not equal 0.
square root
A number, when raised to the second power, produces the given number.
standard form of quadratic equation
ax 2 + bx + c = 0, where a ≠ 0.
Unit 2 Quadratic Relations and Systems
asymptotes of a hyperbola
Two straight lines such that as the distance from the center of the hyperbola increases, the distance between the hyperbola and the line decreases.
axis of symmetry
A line or line segment that divides a figure into two sets of points such that each point of one set is the same distance from the axis as some point on the other, measured along the same perpendicular to the axis.
center
The center point of a circle or ellipse.
circle
A set of points in a plane such that all are the same distance from a fixed point in the plane (center).
conic sections
Circles, ellipses, parabolas, and hyperbolas.
constant of proportionality
The constant ratio (or product) of two variables that vary directly (or inversely).
direct variation
A pair of variables are said to vary directly if, for all permissible values, their ratio is constant: y = kx.
directrix of a parabola
A fixed line used for reference in the definition of a parabola.
ellipse
A set of all points in a plane such that the sum of the distances of any two fixed points (foci) is a constant.
foci of an ellipse
Fixed points of reference used in the definition of an ellipse.
foci of a hyperbola
Two fixed points of reference used in the definition of a hyperbola.
focus of a parabola
A fixed point of reference used in the definition of a parabola.
hyperbola
A set of points in a plane such that the absolute value of the difference of the distance of each point from two fixed points (foci) is a constant; formed by the intersection with both ends of a right circular cone with a plane parallel to its axis.
inverse variation
Two variables are said to vary inversely if, for all permissible values, their product is constant: xy = k.
major axis of an ellipse
Longest axis of symmetry for an ellipse.
minor axis of an ellipse
Shortest axis of symmetry for the ellipse.
parabola
A set of points in a plane equally distant from a fixed line (directrix) and a fixed point (focus) not on the line; formed from the intersection of a right circular cone and a plane parallel to one and only one element of the surface.
principal square root
The non-negative real square root.
quadratic relations
Examples are circles, ellipses, parabolas, and hyperbolas.
radius of a circle
The distance between the center and any point on the circle.
rectangular hyperbola
A hyperbola with x- and y-axes as asymptotes.
second-degree inequalities
Mathematical sentences with one or more variables with terms of second degree and no terms of higher degree.
solution set of a system of equations
The set of all replacements (numbers or ordered pairs of numbers) for variables which, when substituted, give true sentences.
x-intercept point
The point at which the graph crosses the x-axis.
y-intercept point
The point at which the graph crosses the y-axis.
antilogarithm
A number that corresponds to a given logarithm.
Unit 3 Exponential Functions
asymptote of a curve
A straight line such that as distance from the origin increases without limit, the distance between the curve and the line approaches zero.
characteristic
A part of a common logarithm; an integer.
common logarithm
A base ten logarithm.
exponential equation
An equation in which variables appear as exponents or parts of exponents.
exponential function
A function defined by the equation y = b x where b is a constant.
function
A set of ordered pairs such that no two ordered pairs have the same first element.
irrational number
A real number that is not a rational number.
logarithmic function
The inverse of an exponential function.
logarithm of a number
y = log ax means y is the exponent to which you raise the base a to get x.
mantissa
The logarithm of a number N such that 1 > N ≠ 10. A mantissa is a number x such that 0 ≠ x > 1.
matrix
A rectangular array of numbers. (plural: matrices)
natural number
e = 2.718281828. Called the natural number because it occurs frequently in nature.
normal curve
The arrangement of results for many experiments and surveys.
radical
A symbol to indicate the root of a number.
rational exponent
An exponent expressed as a fraction that can be interpreted in terms of radicals.
rational number
A number that can be expressed as the quotient of two integers; the denominator cannot be zero.
scientific notation
A notation for a number showing it as the product of two factors (b)1(10 k), where 1 ≠ b < 10 and k is an integer.
Unit 4 Counting Principles
alternating sequence
A sequence in which each successive term changes sign.
arithmetic series
A series with common difference between terms.
circular permutation
An ordering of elements in a circle or closed curve.
coefficient
A factor in an expression, usually referring to the numerical factor in an algebraic expression.
combination
A subset of a given set of elements, without regard for order.
compound events
Events involving two or more activities.
conditional probability
The probability of one event given that another event has already occurred.
correspondence
An association between members of one set and those of another set.
event
A combination of one or more of the possible outcomes of an activity.
factorial
The product of all the natural numbers from a given integer down to 1.
finite
Possible to reach or exceed by counting
general term
A formula that yields the value of a term in a sequence when the term's position in the sequence is substituted into the formula.
geometric series
A series with a common ratio between terms.
independent events
Events whose outcomes do not affect the probability of each other.
infinite
Endless or without bounds.
natural numbers
The positive integers.
outcome
One of the mutually exclusive results of an activity.
Pascal's Triangle
The coefficients of powers of a binomial, arranged to form a triangle.
permutation
An arrangement of the elements of a set in a specific order.
probability
A number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences.
random
By chance; with no plan.
ratio
The quotient of two numbers.
rotation
The process of moving around a center.
sequence
A group of numbers arranged in a definite order, with a specific first term.
series
The summation of the terms of a sequence.
subset
A set of elements contained in a given set.
summation
Addition.
symmetry
Exact correspondence of form and constituent configuration on opposite sides of a dividing line or plane or about a center or an axis.
term
A number or symbol separated from other numbers or symbols by a comma (in a sequence) or an arithmetic symbol (in an equation).