E
Eccentricity of a Conic Section
A number that indicates how drawn out or attenuated a conic
section is. Eccentricity is represented by the letter e (no
relation to e = 2.718...). For horizontal ellipses and hyperbolas,
eccentricity . For vertical ellipses and hyperbolas, eccentricity .
Here, c = the distance from the center to a focus, a = the
horizontal distance from the center to the vertex, and b = the
vertical distance from the center to the vertex.
Interactive Feature
Conic Sections Defined by Eccentricity Java Sketchpad version
(Java enabled browser required)Geometer's Sketchpad version (GSP 4
required)
Conic Sections Defined by Eccentricity 1. Move the slider for
eccentricity and see what kind of conic section you get.2. Move the
focus to see what impact that has on the figure.3. Move a point on
the directrix to show that no matter what point is taken on the
curve, the ratio of the distance to the focus over the distance to
the directrix always equals the eccentricity.
e
e ≈ 2.7182818284.... is a transcendental number commonly
encountered when working with exponential models (growth, decay,and
logistic models, and continuously compounded interest, for example)
and exponential functions. e is also the base of the natural
logarithm.
Row-Echelon Form of a MatrixEchelon Form of a Matrix
A matrix form used when solving linear systems of equations.
Edge of a Polyhedron
One of the line segments making up the framework of a
polyhedron. The edges are where the faces intersect each other.
Element of a Matrix
One of the entries in a matrix. The address of an element is
given by listing the row number then the column number.
Ellipsoid
A sphere-like surface for which all cross-sections are
ellipses.
Ellipse
A conic section which is essentially a stretched circle.
Formally, an ellipse can be defined as follows: For two given
points, the foci, an ellipse is the locus of points such that the
sum of the distance to each focus is constant. The standard form
for the equation of an ellipse is given below.
Movie clip
Ellipse: Sum of distancesfrom the foci is constant(182K)
Element of a Set
A number, letter, point, line, or any other object contained in
a set. For example, the elements of the set {a, b, c} are the
letters a, b, and c.
End Behavior
The appearance of a graph as it is followed farther and farther
in either direction. For polynomials, the end behavior is indicated
by drawing the positions of the arms of the graph, which may be
pointed up or down. Other graphs may also have end behavior
indicated in terms of the arms, or in terms of asymptotes or
limits.
Polynomial End Behavior:1. If the degree n of a polynomial is
even, then the arms of the graph are either both up or both down.2.
If the degree n is odd, then one arm of the graph is up and one is
down.3. If the leading coefficient an is positive, the right arm of
the graph is up.4. If the leading coefficient an is negative, the
right arm of the graph is down.
Empty SetNull Set
The set with no elements. The empty set can be written or
{}.
Elliptic GeometryRiemannian Geometry
A non-Euclidean geometry in which there are no parallel lines.
This geometry is usually thought of as taking place on the surface
of a sphere. The "lines" are great circles, and the "points" are
pairs of diametrically opposed points. As a result, all "lines"
intersect.
Equation of a Line
The various common forms for the equation of a line are listed
below. In all forms, slope is represented by m, the x-intercept by
a, and the y-intercept by b.
Note: The standard form coefficients A, B, and C have no
particular graphical significance.
Forms for the Equation of a Line
Slope-intercept
y = mx + b
Used when you have the slope and the y-intercept.
Point-slope
y – y1 = m(x – x1)
(x1, y1) is a point on the line.
Standard form
Ax + By = C
If possible, A is nonnegative and A, B, and C are relatively
prime integers.
Two-intercept
Used when you have both intercepts.
Vertical
x = a
All points have x-coordinate a.
Horizontal
y = b
All points have y-coordinate b.
Movie Clips (with narration)
Horizontal Line:how to find the equation(1.43M)
Vertical Line:how to find the equation(1.46M)
Point and Slope:How to find the equation of a line(4.13M)
Two Points:How to find the equation of a line(5.5M)
Equation
A mathematical sentence built from expressions using one or more
equal signs (=).
Examples:
Properties of EqualityEquation Rules
Equivalence properties and algebra rules for manipulating
equations are listed below.
Definitions
1. a = b means a is equal to b.2. a ≠ b means a does not equal
b.
Operations
1. Addition: If a = b then a + c = b + c.2. Subtraction: If a =
b then a – c = b– c.3. Multiplication: If a = b then ac = bc. 4.
Division: If a = b and c ≠ 0 then a/c = b/c.
Reflexive Property
a = a
Symmetric Property
If a = b then b = a.
Transitive Property
If a = b and b = c then a = c.
Equiangular Triangle
A triangle with three congruent angles.
Note: In Euclidean geometry, all equiangular triangles are
equilateral and vice-versa. The angles of a Euclidean equiangular
triangle each measure 60°.
Equiangular (Equilateral) Triangle
s = length of a side
Equidistant
Equally distant. For example, any two points on a circle are
equidistant from the center.
Equilateral Triangle
A triangle with three congruent sides.
Note: An equilateral triangle is also equiangular. In Euclidean
geometry, the angles of an equilateral triangle each measure
60°.
Equilateral Triangle
s = length of a side
Equivalence Properties of Equality
The reflexive, symmetric, and transitive properties that are
satisfied by the = symbol.
Reflexive Property
a = a
Symmetric Property
If a = b then b = a.
Transitive Property
If a = b and b = c then a = c.
Equivalence Relation
Any relation that satisfies the reflexive, symmetric, and
transitive properties. For example, modular equivalence is an
equivalence relation. So is cardinality of a set.
Equivalent Systems of Equations
Systems of equations that have the same solution set.
Essential Discontinuity
Any discontinuity that is not removable. That is, a place where
a graph is not connected and cannot be made connected simply by
filling in a single point. Step discontinuities and vertical
asymptotes are two types of essential discontinuities.
Formally, an essential discontinuity is a discontinuity at which
the limit of the function does not exist.
Euclidean Geometry
The main area of study in high school geometry. This is the
geometry of axioms, theorems, and two-column proofs. It includes
the study of points, lines, triangles, quadrilaterals, other
polygons, circles, spheres, prisms, pyramids, cones, cylinders,
etc.
Note: Euclidean geometry is named for Euclid, a Greek who lived
2500 years ago and wrote Elements, a book that has survived to the
present day as the standard source book for Euclidean geometry.
Euler Line
The line segment that passes through a triangle’s orthocenter,
centroid, and circumcenter. These three points are collinear for
any triangle. In addition, the distance from the orthocenter to the
centroid is twice the distance from the circumcenter to the
centroid.
Note: Euler is pronounced "Oiler".
Euler's Formula
eiπ + 1 = 0. This remarkable equation combines e, i, π (pi), 1,
and 0, which are arguably the five fundamental numbers of
mathematics. It also includes addition, multiplication,
exponentiation, and composition, four of the fundamental operations
of mathematics.
Note: Euler is pronounced "Oiler".
Euler's Formula (Polyhedra)
The equation below:
(number of faces) + (number of vertices) – (number of edges) =
2
This formula is true for all convex polyhedra as well as many
types of concave polyhedra.
Note: Euler is pronounced "Oiler".
Evaluate
To figure out or compute. For example, "evaluate " means to
figure out that the expression simplifies to 17.
Even Function
A function with a graph that is symmetric with respect to the
y-axis. A function is even if and only if f(–x) = f(x).
Even Number
An integer that is a multiple of 2. The even numbers are { . . .
, –4, –2, 0, 2, 4, 6, . . . }.
Event
A set of possible outcomes resulting from a particular
experiment. For example, a possible event when a single six-sided
die is rolled is {5, 6}. That is, the roll could be a 5 or a 6.
In general, an event is any subset of a sample space (including
the possibility of an the empty set).
Trig Values of Special AnglesExact Values of Trig Functions
Certain angles have trig values that may be computed exactly. Of
these, the angles listed below are some of the angles most commonly
used in math classes.
Exclusive
Excluding the endpoints of an interval. For example, "the
interval from 1 to 2, exclusive" means the open interval written
either (1, 2) or ]1, 2[.
Exclusive or
A disjunction for which either statement may be true but not
both.
For example, the use of the word or in "This morning I can go to
school or I can stay home" is exclusive. Either option may be true
but not both.
Note: Mathematicians rarely use exclusive or. In math, or is
understood to be inclusive unless stated otherwise.
Expansion by Cofactors
A method for evaluating determinants. Expansion by cofactors
involves following any row or column of a determinant and
multiplying each element of the row or column by its cofactor. The
sum of these products equals the value of the determinant.
Expected ValueMean of a Random Variable
A quantity equal to the average result of an experiment after a
large number of trials. For example, if a fair 6-sided die is
rolled, the expected value of the number rolled is 3.5. This is a
correct interpretation even though it is impossible to roll a 3.5
on a 6-sided die. This sort of thing often occurs with expected
values.
Experiment
In the study of probability, the name given to any controlled,
repeatable process. For example, the following are all experiments:
tossing a coin, rolling a die, or selecting a ball from a bag.
Often experiments are composed of multiple actions. For
instance, simultaneously flipping a coin and rolling a die is an
experiment.
Explicit Formula of a Sequence
A formula that allows direct computation of any term for a
sequence a1, a2, a3, . . . , an, . . . .
Explicit Differentiation
The process of finding the derivative of an explicit function.
For example, the explicit function y = x2 – 7x + 1 has derivative
y' = 2x – 7.
Explicit Function
A function in which the dependent variable can be written
explicitly in terms of the independent variable.
For example, the following are explicit functions: y = x2 – 3, ,
and y = log2 x.
Exponent
x in the expression ax. For example, 3 is the exponent in
23.
Exponent Rules
Algebra rules and formulas for exponents are listed below.
Definitions
1. an = a·a·a···a (n times)
2. a0 = 1 (a ≠ 0)
3. (a ≠ 0)
4. (a ≥ 0, m ≥ 0, n > 0)
Combining
1. multiplication: axay = ax + y
2. division: (a ≠ 0)
3. powers: (ax)y = axy
Distributing (a ≥ 0, b ≥ 0)
1. (ab)x = axbx
2. (b ≠ 0)
Careful!!
1. (a + b)n ≠ an + bn
2. (a – b)n ≠ an – bn
Exponential Decay
A model for decay of a quantity for which the rate of decay is
directly proportional to the amount present. The equation for the
model is A = A0bt (where 0 < b < 1 ) or A = A0ekt (where k is
a negative number representing the rate of decay). In both formulas
A0 is the original amount present at time t = 0.
This model is used for phenomena such as radioactivity or
depreciation. For example, A = 50e–0.01t is a model for exponential
decay of 50 grams of a radioactive element that decays at a rate of
1% per year.
Exponential FunctionExponential Model
A function of the form y = a·bx where a > 0 and either 0 <
b < 1 or b > 1. The variables do not have to be x and y. For
example, A = 3.2·(1.02)t is an exponential function.
Note: Exponential functions are used to model exponential
growth, exponential decay, compound interest, and continuously
compounded interest.
Exponential Growth
A model for growth of a quantity for which the rate of growth is
directly proportional to the amount present. The equation for the
model is A = A0bt (where b > 1 ) or A = A0ekt (where k is a
positive number representing the rate of growth). In both formulas
A0 is the original amount present at time t = 0.
This model is used for such phenomena as inflation or population
growth. For example, A = 7000e0.05t is a model for the exponential
growth of $7000 invested at 5% per year compounded
continuously.
Exponentiation
The use of exponents.
Expression
Any mathematical calculation or formula combining numbers and/or
variables using sums, differences, products, quotients (including
fractions), exponents, roots, logarithms, trig functions,
parentheses, brackets, functions, or other mathematical operations.
Expressions may not contain the equal sign (=) or any type of
inequality.
Examples:
Exterior Angle of a Polygon
An angle between one side of a polygon and the extension of an
adjacent side.
Note: The sum of the exterior angles of any convex polygon is
360°. This assumes that only one exterior angle is taken at each
vertex.
Extreme Value TheoremMin/Max Theorem
A theorem which guarantees the existence of an absolute max and
an absolute min for any continuous function over a closed
interval.
Extraneous SolutionSpurious Solution
A solution of a simplified version of an equation that does not
satisfy the original equation. Watch out for extraneous solutions
when solving equations with a variable in the denominator of a
rational expression, with a variable in the argument of a
logarithm, or a variable as the radicand in an nth root when n is
an even number.
Extreme Values of a Polynomial
The graph of a polynomial of degree n has at most n – 1 extreme
values (minima and/or maxima). The total number of extreme values
could be n – 1 or n – 3 or n – 5 etc.
For example, a degree 9 polynomial could have 8, 6, 4, 2, or 0
extreme values. A degree 2 (quadratic) polynomial must have 1
extreme value.
Extremum
An extreme value of a function. In other words, the minima and
maxima of a function. Extrema may be either relative (local) or
absolute (global).
Note: The first derivative test and the second derivative test
are common methods used to find extrema.
Mathwords for ICP Program
