Shared Flashcard Set

Details

Math Review
required knowledge of mathematics for MS math teachers
181
Mathematics
6th Grade
11/19/2016

Additional Mathematics Flashcards

 


 

Cards

Term
what the converse of Pythagorean Theorem says
Definition
If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.
Term
In a 30-60-90 triangle, (relative length of sides)
Definition
the length of the hypotenuse is twice the length of the shorter leg and the
length of the longer leg is sqrt(3) times the shorter leg.
Term
Pythagorean theorem
Definition
For any right triangle with legs a and b and hypotenuse c, the following relationship is
satisfied: a^2 + b^2 = c^2
Term
In a 45-45-90 triangle,
Definition
the length of the hypotenuse is sqrt(2)times the shorter leg.
Term
The area of a triangle
Definition
one-half the product of its base and its height: A = bh/2
Term
Side-Side-Side (SSS) postulate
Definition
If three sides of one triangle are congruent to three sides of another triangle, then the
two triangles are congruent.
Term
Side-Angle-Side (SAS) postulate
Definition
If two sides and the included angle of one triangle are congruent to two sides and the
included angle of another triangle, then the two triangles are congruent.
Term
A right triangle
Definition
a triangle that includes only one right angle. the side opposite the right angle is the hypotenuse, and the
other two sides are the legs.
Term
HL Theorem
Definition
If the hypotenuse and one leg of a right triangle are congruent to corresponding sides of another right triangle, then the triangles are congruent
Term
Two sides of a triangle are congruent if and only if
Definition
... the angles opposite those sides are congruent
Term
An altitude of a triangle
Definition
An altitude of a triangle is the segment from any vertex that is perpendicular to the
line containing the opposite side
Term
A triangle is called an acute, right, or obtuse triangle according to whether it includes
Definition
three acute angles, one right angle, or one obtuse angle, respectively
Term
Two lines that are not coplanar are called this
Definition
skew lines
Term
A trapezoid
Definition
a quadrilateral in which exactly one pair of sides is parallel
Term
A line segment with endpoints that lie on two nonconsecutive vertices of the polygon
Definition
a diagonal
Term
Two triangles are congruent if...
Definition
... there is a correspondence between them such that every pair of corresponding sides is congruent and every pair of angles is congruent
Term
The Triangle Angle Sum Theorem
Definition
this states that the sum of the interior angles of a triangle is 180°
Term
coincident
Definition
This means that 2 points are the same line
Term
A trapezoid
Definition
a quadrilateral with exactly one pair of parallel sides
Term
The parallel postulate states:
Definition
Given a point not on a line, there is exactly one line parallel to the given line containing that point.
Term
quadrilaterals
Definition
they are polygons with four sides
Term
to show that two triangles are similar
Definition
One can use the AA similarity postulate, SAS similarity postulate, and SSS similarity postulate
Term
Two convex polygons are similar if
Definition
there is a correspondence between their
vertices such that the corresponding angles are congruent and such that the ratios of the lengths of their corresponding sides are equal
Term
The perpendicular bisector of a line segment
Definition
the line perpendicular to the segment at its midpoint
Term
The area of a trapezoid is equal to...
Definition
one-half the product of its height and the sum of the lengths of its
bases, b1 and b2:
Term
A chord of a circle
Definition
a line segment with endpoints on the circle
Term
If the measure of θ is 180°, then A and B are actually endpoints of a diameter and the two arcs are
semicircles
Definition
The measure of minor arc is the measure of its associated central angle, the measure of a semicircle is 180°, and the measure of a major arc is 360° minus the measure of the corresponding minor arc.
Term
counting numbers or natural numbers
whole numbers
integers
Real numbers
Definition
The ______________ are the numbers 1,2,3,4,....
The _____________ are the counting numbers plus zero.
The ________ are the whole numbers and all their opposites.
___________ consist of rational and irrational numbers
Term
Rational numbers
Irrational numbers
Definition
________________ include whole numbers, fractions, finite or repeating decimals, and percents.
_______________ are numbers that cannot be written as fractions since they are non-repeating and non-terminating decimals.
Term
complex numbers
Definition
these have the form "a + bi"
Term
prime number
composite number
Definition
A __________ is a number divisible only by 1 and itself.
A ________________ is a number with factors other than 1 and itself
Term
Real numbers
Definition
_______________ are the field of all rational and irrational numbers
Term
arc addition theorem
Definition
If B is a point of the arc AC, then m(arcAC) = m(arcAB) + m(arcBC).
Term
inscribed
Definition
An angle is _________ in an arc if the sides of the angles contain the endpoints of the arc and if the vertex is a point on the circle touching the arc
Term
inscribed angle
Definition
The measure of an _______________ is equal to half the measure of its intercepted arc
Term
congruent
Definition
Two circles are _________ if they have congruent radii.
Two arcs are ____________ if they lie on the same circle (or on congruent circles) and have the same measure.
Term
congruent
Definition
If two inscribed angles intercept the same arc or _________ arcs, then the angles are
_________.
Two arcs in the same circle or _________ circles are congruent if and only if their
corresponding chords are _________
Term
secant
Definition
A ______ of a circle is any line intersecting the circle at two points
Term
tangent
point of tangency
Definition
A _______ of a circle is any line that intersects the circle at exactly one point called the ________________
Term
line
circle
Definition
A ____ is tangent to a _______ if and only if the radius drawn to the point of tangency is perpendicular to the line
Term
secants
arcs
Definition
The measure of an angle formed by two ______ that intersect in the interior of a circle is one half the sum of the ____ intercepted by the angle and by its opposite angle.
Term
secant
tangent
Definition
The measure of an angle formed by an intersecting ______ and _______ is one half the difference of the intercepted arcs
Term
Arc length formula
Definition
s = ar, where a is the arc measure and r is the radius of the circle
Term
circumference
Definition
2(pi)r , where r is the radius of the circle
(convert from degrees to radians first)
Term
(x – h)^2 + (y – k)^2 = r^2
Definition
The equation of a circle in the plane with center (h, k) and with radius r
Term
The standard form of the equation of a circle is...
Definition
x^2 + y^2 + Ax + By + C = 0
Term
chord
Definition
A _____ of a circle is a line segment with endpoints on the circle
Term
diameter
Definition
The __________ of a circle is length of a chord containing the center of the circle and is denoted by d. The ________ of a circle is twice its radius
Term
arc
Definition
An ___ is any connected part of a circle
Term
minor
major
Definition
Any two distinct points A and B of the circle divide it into two arcs called the _____
arc AB and the _____ arc AXB
Term
(x - h)^2 + (y - k)^2 = r^2
Definition
The equation of a circle in the plane with center (h, k) and with radius r
Term
Volume of a solid with trapezoidal bases and rectangular sides
Definition
(1/2)(b1 + b2)ah,where is the area of the trapezoid, and h is the height of the solid
Term
The area of a rectangle (and parallelogram)
Definition
A = bh
Term
The area of a triangle
Definition
(1/2)bh
Term
The area of a trapezoid
Definition
(1/2)(b1-b2)h
Term
The area of a circle
Definition
A = (pi)r^2
Term
The perimeter of a circle
Definition
2(pi)r
Term
area of a cylinder
Definition
(pi)r^2h
Term
Volume of a square pyramid
Definition
(1/3)s^2h, where s is the length of a side on the square base, s2 is the area of the square base, and h is the height of the pyramid
Term
Volume of a triangular pyramid
Definition
(1/3)(1/2ba)h, where (1/2ba) is the area of the triangular base, and h is the height of the pyramid
Term
Volume of a cone
Definition
(1/3)(pi^2)h, where pir^2 is the area of the circular base, and h is the height of the cone
Term
When a two-dimensional figure is changed in size by a factor of n
Definition
the area is changed by a factor of n^2
Term
volume
Definition
lwh- for solids, this is measured in cubic units
• The volume of a non-pointed solid is found by multiplying the area of the base by the height of the solid.
• The volume of a pointed solid is found by multiplying the area of the base by the height of the solid and dividing by three
When the dimensions of a figure are increased by a factor of n, this is increased by a factor of n^3
Term
surface area
Definition
• For a solid, this is found by finding the sum of the areas of all of the surfaces of the solid.
• When the dimensions of a figure are increased by a factor of n, this is
increased by a factor of n^2
Term
A sector
Definition
a portion of a circle. The area of this can be found by setting up a proportion using the angle measure of the _______, the central angle of the circle, and the area of the entire circle.
The entire area of the circle would be 2(pi)r^2
Term
arc length
Definition
This is a part of the circumference of a circle, so its length is proportion-ate in a way similar to the way a sector’s area is proportionate to its angle
Term
The volume of a non-pointed solid is found by...
Definition
... multiplying the area of the base by the height of the solid
Term
The volume of a pointed solid is found by...
Definition
... multiplying the area of the base by the height of the solid and dividing by 3
Term
The surface area of a solid is found by...
Definition
... finding the sum of the areas of all of the surfaces of the solid
Term
When the dimensions of a figure are increased by a factor of n,
Definition
... any area is increased by a factor of n^2, and any volume is increased by a factor of n^3
Term
How do you find the area of a sector of a circle?
Definition
proportions (ie- A/ (pi)r^2 = (central angle/ 360)
Term
How do you find arc lengths?
Definition
proportions (L/ 2(pi)r = #/ 360)
Term
scale drawing
Definition
Maps of states or drawings of buildings are drawn in this type
Term
A matrix
Definition
This is a rectangular array of numbers (in rows and columns)
Term
elements
Definition
the numbers in a matrix are referred to as the ________ of a matrix
Term
The dimensions of the matrix are said to be “m by n”
Definition
which is written "m x n." An matrix has m rows and n columns. This notation
does not mean that you multiply m by n.
Term
1 x n
m x 1
Definition
A matrix is sometimes called a row vector.
A matrix is sometimes called a column vector.
Term
adding & subtracting matices
Definition
There are no surprises in addition and subtraction of matrices, however there is one
requirement. The matrices must have the same dimensions. Once you establish that the
matrices you are working with have the same dimensions, you can add or subtract each entry in
the first matrix with the corresponding entry in the second matrix.
Term
scalar multiplication
Definition
this changes the size of every entry by the same factor, called a scalar. A matrix of any dimensions may be multiplied by any scalar.
Term
matrix
elements
Definition
a rectangular array of #'s
the #'s in this are the ___________ of this
Term
rules for +'ing and -'ing matrices
Definition
no surprises but the dimensions must be equal
Term
2 types of matrix multiplication
Definition
scalar multiplication
multiplication of matrices
Term
scalar multiplication
Definition
this type changes the size of every entry by the same factor in a matrix
(ex: r = 3)
Term
an (m x n) matrix can only be multiplied by an (n x p) matrix...
Definition
this means the # of columns of the 1st matrix must = the # of rows of the 2nd matrix
Term
matrix multiplication
Definition
if the 1st matrix has dimensions (m x n), and is multiplied by a 2nd matrix of dimensions (n x p), then the dimensions of the product matrix will be (m x p)
Term
how you multiply matrices
Definition
if A = [a b] and B = [e f]
[c d] and [g h], then AB =

[a b] [e f] [ab + bg af + bh]
[c d] x [g h] = [ce + dg cf + dh]
Term
when multiplying, an identity matrix is...
Definition
... a matrix whose entries are all 0's, except for the entries that lie on the main diagonal (top left to btm rgt). This for Multiplication is denoted as l
Term
division of matrices
Definition
matrix algebra focuses on using the multiplicative inverse of a matrix. The inverse of A, denoted as A^(-1), is: AA^(-1) = l = A^(-1)A

To have an inverse, a matrix must be a square. Not all square matrices have an inverse
Term
inverse of a square matrix
Definition
A^(-1) = 1/(ad - bc)[d -b]
[-c a] ad - bc =/ 0

(ad - bc) is called the determinant
Term
review of matrices
Definition
a matrix is a rectangular array of #'s
Matrices can be +'d, -'d, and x'd
The associative property exists for addition and multiplication of matrices
The commutative property exists only in matrix addition
There are Identity Matrices in both addition and multiplication
Every matrix has an additive inverse
Term
if T, then T^(-1) = ?
Definition
T^(-1) = [-2 3][a b] = [1 0]
[ 1 -2][c d] [0 1]

-2a + 3c = 1
-2b + 3d = 0
a - 2c = 0
b - 2d = 1,

a = -2, b= -3, c = -1, d = -2

--> T^(-1) = [-2 -3]
[-1 -2]
Term
row echelon form (of a matrix)
Definition
* any form w/ all 0's at h btm o h matrix
* any row th has an entry othr tn 0 z h 1st non-zero entry
* any row th has 1 z h 1st non-zero entry has th entry frthr t h right tn h 1st non-0 entry o h row above
Term
The reduced row-echelon form, also known as the row reduced echelon form, of a matrix equation
Definition
requires back-substitution in order to develop the solution to the equation.
Term
The Gauss-Jordan elimination method
Definition
while more complex and time-consuming
than reduced row-echelon method, this develops a form in which the solution is easy to identify and does not require back-substitution
Term
2 ways to solve systems of linear equations
Definition
reduced row-ecelon form
Gauss-Jordan elimination method
Term
inconsistent
Definition
If a system of equations has no solution, it is called this
(ie: if the lines are parallel)
Term
redundant
Definition
If a system of equations has an infinite number of solutions, it is said to be this
Term
219
Definition
Term
vector
Definition
this is a matrix that has either a single column or a single row.
Term
Law of Cosines
says that for any triangle ABC)
Definition
a^2 = b^2 + c^2 - 2bccosA
b^2 = a^2 + c^2 - 2accosB
c^2 = a^2 + b^2 - 2abcosC
Term
the length of a vector,
u = [u1 u2 u3 ... uz], is denoted |u|...
Definition
|u| = sqrt( u1^2 + u2^2 + u3^2 + ... un^2)
Term
a Dot Product of 2 vectors, u = [u1 u2 u3 ... in] and v = [v1 v2 v3 ... vn]...
Definition
... is denoted uv, and is defined uv = u1v1 + u2v2 + u3v3 + ... unvn
Term
(in other words) the dot product is..
Definition
the product of the lengths of the two vectors and the cosine of the angle between the vectors

cos? = (uv)/(|u||v|)
Term
a Dot Product of 2 vectors, u = [u1 u2 u3 ... in] and v = [v1 v2 v3 ... vn]...
Definition
... is denoted uv, and is defined uv = u1v1 + u2v2 + u3v3 + ... unvn
Term
cross product u x v
Definition
[ i j k]
the determinant [u1 u2 u3]
[v1 v2 u3]

This is a vector. It is perpendicular to the plane on which the original two vectors lie.
Term
the length of the cross product u x v
Definition
the area of a parallelogram that has sides of u and v
Term
the length of a cross product of
u x v x w
Definition
the volume of the parallelepiped created by u, v, and w
Term
the number of solutions of a system of linear equations
Definition
this can be determined by graphing the system

this is also evident by the symbolic manipulations of the equations
Term
a three-variable linear system...
Definition
... with 1 solution can be graphed as 3 planes meeting at exactly 1 point

... with no solution can be graphed as 3 planes that never coincide at the same time

... with an infinite # of solutions can be graphed as 2 or 3 planes that meet at a line
Term
a two-variable linear system
Definition
... with one solution can be graphed as 2 lines that meet at exactly one point
Term
a two-variable linear system
Definition
with no solution can be graphed as parallel lines
Term
a two-variable linear system...
Definition
... w/ an infinite # of solutions is graphed as 1 line b/c both equations describe the same line
Term
Determinants can...
Definition
these can be used in a # of geometric applications
Term
a dot product
Definition
between 2 vectors this produces a scalar
Term
a dot product
Definition
between 2 vectors this produces a scalar
Term
a cross product
Definition
between 2 vectors this produces a vector
Term
Trigonometry
Definition
this is sometimes referred to as pre-calculus b/c many of its concepts serve as precursors to derivatives, integrals, and rates of change (Calculus)
Term
Trigonometric ratios (3 basic)
Definition
SOH CAH TOA
Term
Unit Circle
Definition
a circle with a radius of 1 (radius = 1)

legs = x (adjacent to angle @) and
y (opposite)
hypotenuse = 1
Term
cos 0 = ?
Definition
1

@ = 0, coordinates (1, 0)
cos @ is x-coordinate (which = 1)
Term
sin 90⁰ = ?
Definition
1

@ = 90⁰, coordinates (0, 1)
sin @ is y-coordinate (which = 1)
Term
sin (pi/2)= ?
Definition
(pi/2) = 180⁰/2 = 90⁰
Term
sin 270⁰ = ?
Definition
@ = 270⁰, x- and y-coordinates (0, -1)
sin @ is y-coordinate
Term
Where is 405⁰ on a unit circle?
Definition
405⁰ – 360⁰ = 45⁰
a unit circle is an easy way to show degrees and radians)
Term
Sec @
Definition
1/ cos @
Term
csc @
Definition
1/ sin @
Term
Reciprocal identities
Definition
cotangent, secant, cosecant
Term
Ratio Identities
Definition
tan @ = sin @/ cos @
cot @ = cos @/ sin @
Term
Trigonometric Ratios
Definition
cot @ = adjacent/ opposite
sec @ = hypotenuse/ adjacent
csc @ = hypotenuse/ opposite
Term
Pythagorean trigonometric identity
Definition
* cos^2x + sin^2x = 1
* 1 + tan^2x = sec^2x
* cot^2x + 1= csc^2x
Term
Amplitude
Definition
the height of each peak, or highest point, in a wave pattern measured from the middle of the wave. Equivalently, it is half the vertical height from lowest point, or trough to the peak.

A in the formulas f(t) = A sin (Bt + C) and f(t) = A cos (Bt+ C)
Term
range of a Cosine function
Definition
-1 to 1
Term
frequency
Definition
the number of wave patterns within a distance from 0 to 2(pi)
(the period and this are reciprocals)
In y = Asin(Bt + C), B is this
Term
period
Definition
the horizontal distance on the x-axis between corresponding portions of the wave– from peak to peak, or trough to trough.
In other words, this shows the distance from where a wave pattern starts to where the wave pattern begins to repeat itself
Term
phase shift
Definition
a horizontal shift of a line on a graph (L or R)
(the amplitude, period, and frequency are unaffected)

- C (y= A sin (Bt + C)
B
Term
odd function
Definition
a function that always holds true for
f(-x) = -f(x)
Term
even function
Definition
a function that always holds true for
f(-x) = f(x)
(this is symmetric around the y-axis)
Term
on a unit circle, 90⁰ =
Definition
(pi)/2 radians
Term
on a unit circle, 270⁰ =
Definition
3(pi)/2 radians
Term
the tangent function is an odd function
Definition
The tangent graph is not symmetrical over the y-axis, tan(-@) = -tan@
(A line that is tangent to a function with respect to the x-axis can also be referred to as the slope)
Term
slope
Definition
rise over run
Term
the graph of the secant function...
Definition
like those of the tangent and cotangent functions, has undefined points at 90⁰ and -90⁰ (pi/2 & -pi/2)

symmetric about the y-axis
Term
the graph of the cosecant function...
Definition
undefined at 0, pi, and -pi

not symmetric about the y-axis, so it is an odd function
Term
inverse trigonometric functions
Definition
sin-1@, cos-1@, tan-1@, or
arcsin@, arccos@, arctan@

(NOT the reciprocals)
Term
the sine and cosine addition formulas
Definition
sin (a + b) = sin(a)cos(b) + cos(a) + sin(b)
cos (a + b) = cos(a)cos(b) - sin(a) + sin(b)
sin (a - b) = sin(a)cos(b) - sin(b) + cos(a)
cos (a - b) = cos(a)cos(b) + sin(a) + sin(b)
Term
the addition formula for tangent
Definition
tan(a + b) = (tan(a) + tan(b))/
(1 - tan(a)tan(b))
Term
the subtraction formula for tangent
Definition
tan(a - b) = (tan(a) - tan(b))/
(1 + tan(a)tan(b))
Term
double-angle formulas
Definition
(useful to find uncommon pts w/o a calculator)

sin(2a) = 2sin(a)cos(a)
cos(2a) = cos2a – sin2a – 2cos2a – 1= 1 – 2sin2a
tan(2a) = (2tan(a))/(1 – tan2a)
Term
What is tan(120⁰)?
Definition
-sqrt(3)

tan(120⁰) = tan(2 x 60⁰)

tan(2a) = (2tan(a))/(1 – tan2a)

2 x sqrt(3)/(1 – (sqrt(3)^2)

2sqrt(3)/-2
Term
half-angle formulas
Definition
sin(x/2) = +/- sqrt((1 - cosx)/2)
cos(x/2) = +- sqrt((1 + cosx)/2)
tan(x/2) = +-sqrt((1 – cos x)/(1 + cosx))
Term
sin(15⁰) = ?
Definition
(sqrt(-(sqrt(3)-2)/2

= +/- sqrt((1 – cos(30⁰))/2)
30⁰ = sqrt(3)/2 →
sqrt(1 – (sqrt(3)/2)/2)
sqrt(-(sqrt(3)-2)/2
Term
Schwarz inequality
Definition
-1 <= (u * v)/ ||u||||v|| <= 1
Term
Rectangular coodinates
Definition
coordinates in the form (x, y), such as (3, 3)
Term
Polar coordinates
Definition
coordinates in the form (r, @), where r is the distance from the origin to the point, & @ is the angle btwn the +'ve x-axis and the ray from the origin to the point
Term
(rcos@, rsin@) = (x, y)
Definition
equation for converting polar coordinates to rectangular coordinates
Term
polar coordinates (4, (pi)/3) into rectangular coordinates?
Definition
(2, 2sqrt(3))

rcos@ = 4cos(pi/3) = 4 x ½ = 2

rsin@ = 4sin(pi/3) = 4 x sqrt(3)/2 =

2sqrt(3)
Term
(sqrt(x^2 + y^2), tan^-1(y/x)) = (r, @)
Definition
equation for converting regular coordinates into polar coordinates
Term
changing rectangular coordinates
(2, 2) into polar coordinates
Definition
(sqrt(8), pi/4)

(sqrt(x^2 + y^2), tan^-1(y/x)) = (r, @)
sqrt(22 + 22) = sqrt(8)
tan^-1(y/x)) = tan^-1(1)) = pi/ 4
Term
Converting equation y = 3x + 2 into polar form?
Definition
r = 2/ (sin@ - 3cos@)

Since y = rsin@ & x = rcos@,
rsin@ = 3(rcos@) + 2
rsin@ = 3rcos@ + 2
rsin@ - 3rcos@ = 2
r(sin@ - 3cos@) = 2
r = 2/ (sin@ - 3cos@)
Term
complex numbers
Definition
numbers which contain both a real part and an imaginary part. (ex- 3 + 7i)

(z is commonly used to denote these)

z = x + yi
Term
imaginary #'s
Definition
scalar multiples of i (sqrt(-1))
Term
equation for complex #'s in polar form
Definition
r(cos@ + isin@)
Term
z = 5 – 5i in polar form?
Definition
z = 5sqrt(2)(cos(7pi/4) + isin(7pi/4))

(z = 5 – 5i)

This point will be in the 4th quadrant of the complex plane, and since the x- and y-coordinates are both equidistant from the origin, the angle must be (7pi/4). We also know that r = sqrt(x^2 + y^2), so r = sqrt(50) = 5sqrt(2). So the polar form of the complex number is therefore z = (cos + i sin ).
Term
What is y = (3i + 2)(6i – 2) in polar form?
Definition
r = (6i – 22)/ sin@ -->
y = -18 – 6i + 12i – 4 → y = 6i – 22
(x = rcos@ and y = rsin@)

rsin@ = 6i – 22
Term
DeMoivre's Theorem
Definition
if z = (rcos@) + (rsin@)i, then z^N = r^z(cos(n@) + isin(n@)

(useful when trying to find the powers of complex numbers and to simplify complex numbers.)
Term
if z = (rcos@) + (rsin@)i, where r = 4, and @ = 60°, what is z^2?
Definition
8isqrt(3) - 8

By DeMoivre’s Theorem, we know that
z^2 = r^2(cos(2@) + isin(2@). And since r = 4, and @ = 60°, we can use substitution to get z^2 = 4^2(cos(2 60°) + isin(2 60°)= 16(cos(cos(2 60°) + isin(2 60°). Using the double angle formula cos(2@) = 1 - 2sin2@, we know that cos(2 60°) = 1 - 2sin2@60 = 1 – 2(sqrt(3)/2)2 = 1 – 2(3/4) = - ½
And using the double angle formula
sin(2@) = 2sin@cos@, we find that sin(2 60°) = 2sin(60°)cos(60°)=
2sqrt(3)/2 x ½ = sqrt(3)/2.

So using substitution, we get
16(-1/2 + i(sqrt(3)/2)=

8isqrt(3) – 8
Term
sample space
Definition
the set of all possible outcomes
Term
The formula for permutations
Definition
nPr= n!/(n-r)!

This formula describes the number of ways to arrange r elements out of n elements.
Term
There are 36 people in the 6th grade class. How many different ways can the teacher line up 14 students?
Definition
36 x 35 x 34 ... 23

(36) P (14) = 36!/ (36 - 14) = 36!/22!=
36 x 35 ... 23 x 22!/ 22! = 36 x 35 ... 23
Term
factorial
Definition
! operation (ex- 3! = 3 x 2 x 1= 6)
n! = n * (n-1)(n-2)(n-3)...
(only works for nonnegative #'s)
note: 0! = 1
Term
Suppose that a license plate consists of two letters followed by four digits. The plates use only capital letters, and only the numbers 0 hrtough 9 (these letters and numbers can repeat; for example, a license plate might feature more than one 3 or more than one A). What is the total number of unique license plates possible?
Definition
6,760,000
__ __ __ __ __ __
L L # # # #

26 x 26 x 10 x 10 x 10 x 10
Term
fundamental counting principle
Definition
the total number of possible outcomes following from a series of events is determined by multiplying all the ways that each individual event can occur
Term
When choosing one card at random from a deck of cards, what is the probability of choosing a face card (i.e., a jack, queen, or king)? There are 52 cards in a deck and 12 face cards total
Definition
3/13

P(E) = 12/52 = 3/13
Term
f 2 coins are flipped simultaneously, what is the probability that both will land tails-up?
Definition
E = both coins land with their tail sides facing up. To use the formula, determine the ratio of favorable outcomes to total outcomes (H: heads,
T: tails)- 4 total outcomes: HH, HT, TH, TT. Clearly, only one of these four outcomes provides the favorable outcome of two tails (TT), and thus P(E) = ¼.
Term
sample space
Definition
total # of possible outcomes
Term
P(E)
Definition
(# of favorable occurrences)/ (total # of outcomes)
Term
Probability
Definition
the study of how likely something is
to happen; more specifically, probability helps us decide how likely it is that a certain outcome will follow from an event.

If probability is 0, it is impossible
If probability is 1, it is guaranteed
Term
Combinations
Definition
(used to count elements in a sample space in which the order does not matter

(n)C(r) = n!/ (n - r)r!
Term
Two restaurants in town offer vegetarian plates and a variety of vegetables from which to choose.
Restaurant A offers 6 different vegetables and customers can choose 2 per plate; restaurant B offers 5
vegetables and lets customers choose 3 per plate. Which statement below is correct?

A 5 more vegetarian plates are possible at restaurant B than are possible at restaurant A.

B 3 more vegetarian plates are possible at restaurant A than are possible at restaurant B.

C 3 more vegetarian plates are possible at restaurant B than are possible at restaurant A.

D 5 more vegetarian plates are possible at restaurant A than are possible at restaurant B.
Definition
D

Since the order doesn't matter, use the combination formula:

(6)C(2) = 6!/ 4!2!= 3 x 5 = 15

(5)C(2) = 5!/ 3!2! = 5 x 2 = 10
Term
Point of Inflection
Definition
any point along a curve in which the concavity changes from down to up or up to down (In other words, the point x = c is a point of inflection if f''(x) < 0 when x < 0 and f''(x) > 0 if x > c)
Term
Newton's method for approximating the roots of a function
Definition
x(n + 1) = x(n) - (x(n))/(x'(n))
Supporting users have an ad free experience!