Section 1
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Relation
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Section 1
(51 cards)
Relation
Any set of ordered pairs
formula
an equation that relates two or more quantities, usually represented by variables
Literal Equation
An equation that contains two or more variables
Associative
changes the grouping of parentheses, but the numbers do not move
Vertical-Line Test
If a vertical line crosses the graph of a relation in more than one point, the relation is not a function
X-Intercept
The x-coordinate of the point where the graph crosses the x-axis
Direct Variation
The equation y=kx describes a direct variation, where y varies directly as x, k is the constant of variation, and k≠0
Identity
Adding a 0, and multiplying by 1 - it does nothing
Compound Inequality
A pair of inequalities combined by the words and or or
Correlation Coefficient
A number represented by the variable r, where -1≤r≤1, that describes how closely points in a scatter plot cluster around the least-squares line
Inconsistent System
A system of equations or inequalities that has no solution
rational numbers
numbers such as 5/3, 7/9, that result when an integer is divided by a non zero integer
like radical terms
radical terms having the same radicand and index.
infinite set
a set with an unlimited, or infinite, number of elements.
Simplified Expression/Function
An expression/function that has had all like terms combined and all the parentheses have been removed
Independent Variable
The input of a function
Rational Numbers
Numbers that can be expressed as the quotient of two integers, where the denominator is not equal to zero
Y-Intercept
The y-coordinate of the point where the graph of a line crosses the y-axis
extraneous solution
an apparent solution that must be rejected because it does not satisfy the original equation
Range
The set of possible values for the second coordinate of a function
empty set
a set with no elements.
Domain
The set of possible values for the first coordinate of a function
opposite
additive inverse; the ______ of any number "b" is "−b"
power
an expression that represents repeated multiplication of the same factor
Dependent Variable
The output of a function
finite set
a set with a definite, or finite, number of elements.
absolute value
the __________ of a number "x", represented by the symbol "l x l", is the distance the number is from 0 on a number line
algebraic expression
an expression that consists of numbers, variables, operations, and grouping symbols; also called variable expression
System of Equations
A set of equations in the same variables
Commutative
changes order, but the statement remains the same
Distributive
2(4x-3) = 8x - 6
Consistent System
A system of equations or inequalities that has at least one solution
Slope-Intercept Form
y=mx+b
reciprocal
multiplicative inverse; the _______ of any nonzero number "b" is "1/b"
Function
A relation in which, for each first coordinate, there is exactly one corresponding second coordinate
integers
{-3,-2,-1,0,1,2,3...}
whole numbers
{0,1,2,3...}
radical symbol
the symbol used to denote a root
interval notation
a way of writing the set of all real numbers between two endpoints. The symbols [ and ] are used to include an endpoint in an interval, and the symbols ( and ) are used to exclude an endpoint from an interval.
variable
a letter that is used to represent one or more numbers
natural numbers
{1,2,3...}(counting numbers)
Rationalizing the denominator
Rationalizing the denominator is a method of rewriting a fraction by multiplying by another fraction that is equivalent to 1 in order to remove radical terms from the denominator.
radicand
a number or expression under a radical sign
Standard Form
Ax+By=C, where A and B are not both 0
identity
a statement that equates two equivalent expressions
element
a value in a set or matrix; also called an entry.
Irrational Numbers
Numbers whose decimal part does not terminate or repeat
Inverse
equals 0
principal root
the positive square root of a number, indicated by the radical sign.
like term
terms that have the same variable parts; also called constant terms
Section 2
(51 cards)
opposite/additive inverse
-a is the opposite/additive inverse of a
simplify
replace it with its most simplest symbol having the same value 1) substitution can be used 2) order of operations can be used (grouping symbols, exponents and roots, x/, +-
equivalent
equal for every value of each variable they contain
reciprocal/multiplicative inverse
the value a number can by multiplied by to get to 1 the reciprocal of A would by 1/A
Grouping symbols
( ) [ ] or ---- (division bar)
open sentence
an equation or inequality that contains a variable such as 2t - 1=5
algebraic expression
a numerical expression, a variable, sum, difference, product, quotient that contains one or more variables
evaluation an expression/finding the value of
the process of replacing each variable in an expression by a value and simplifying it
base
the repetitive factor 3 is the base of 3^4
origin
the graph of 0
equivalent equations
equations that have the same solution set
irrational numbers
numbers such as √12, √52 (non-repeating, not rational)
quotient
the result of dividing one number by the other 7 is the quoteint of 35/5= 7
product
the result of multiplying numbers, called factors 3 and 5 are the factors of the product 15 in 3 x 5 = 15
Constant Term
Term with a variable
equation
A sentence formed by placing a "=" between two expressions (called the sides of of the equation) it is true if both sides have the same value
absolute value
The distance a number is from zero on a number line. ALWAYS POSITIVE
value of a numeral expression
12 is the value of 3x4
values of a variable
what the variable could symbolize
Exponent
The number of times the base is multiplied
is greater than
>
similar terms/like terms
terms that contain the same variable factors
terms
numbers that are added together 6 and 15 are the terms of 6+15=21
formula
an equation that states a relationship between 2 or more variables h= -16t^2+vt
exponent
tells the number of times the base is a factor (the little number on the top)
domain/replacement set
the set that the variable belongs to (whole, natural rational sets...)
inequality
A sentence formed by placing an inequality symbol between the two expressions
difference
the result of subtracting one number from another 2 is the difference of 8-2
solution/root
any value of the variable that makes an open sentence true
unique field number
only that number has the properties
power
product of of equal factors first power = 3^1, second power = 3^2
Base
Number or factor that's being exponent times
solution set
the set of all solutions of an open sentence
constant
the fixed numbers in an equation or formula
Variable
A letter that is used to represent a number or numbers
sum
the result of adding numbers, called the terms 21 is the sum of 6+15
solve an equation
to change/transform it into a simple equivalent equation
identity
an equation that is satisfied by all values of the variable 6s-9=6s-9
numeral expression
a symbol or group of symbols used to represent a number
field
a set of numbers (real numbers a types of field)
numeral
a simplified version of a numeral expression (a number)
variable
a symbol used to represent a number (a, x, y)
Term
Expression with a variable
is less than
<
Power
Repeated times of the same number or factor
inequality symbol
one of the symbols less than (<), greater than (>), less than or equal to ( <= ) or greater than or equal to ( >= )
graph
a point on a number line that is paired with a real number
empty/null set
a solution set with no members
Reciprocal
Multiplicative inverse of a non zero number
coordinate
a real number paired with a point on the number line
Section 3
(52 cards)
Point-Slope Form
Y2-Y1=m(x2-x1)
Absolute Value
Distance from zero
direct variation
linear function where y-intercept is 0 -y=mx -m is a constant of variation(use k rather than m) -y varies directly as x -y/x=k
Absolute Value Function
F(x)=|x|
Distributive Property
a(b+c)=ab=ac
Independent Variable
X
Extraneous Solution
A solution that doesn't satisfy the original equation
Slope, Rate of Change
rise over run
Parallel
same slope
Coefficient
A Numerical factor of a term
Extraneous Solution
A solution of an equation derived from an original equation that is not a solution of the original solution
Slope-Intercept Form
Y=mx+b
Domain
Everything X can be
Standard Form
Ax+By=C, no fraction, no negative x value
Translation
Shift left or right, up or down on a graph
Irrational Numbers
Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are nonending and nonrepeating. Also not perfect squares.
Best Fitting Line
A line closest to all data points
Linear Function
Constant rate of change and each input has exactly 1 output
Direct Variation
Y=Kx, linear, always goes through (0,0)
Coefficient
Number in front of a variable
PEMDAS
Order of operations
Identity Property
a + 0=a a times 1 = a 0 + a = a 1 times a = a
Relation
Pairing of inputs and outputs
Positive Correlation
Positive slope on a scatter plot
Inverse Property
a + (-a) = 0 a times 1/a = 1
Dependent Variable
Y
Whole Numbers
Natural numbers ( counting numbers) and zero; 0, 1, 2, 3...
Vertex
Highest or lowest point on a graph, written as an ordered pair
Integers
Natural numbers,their opposites, and zero..-2,-1,0,1,2..Etc
Reflection
Flip over X or Y axis
Correlation Coefficient
numerical value between -1 and 1. Tells strength of relationship of two variables
Transformation
Changes graphs shape, size or position
Perpendicular
negative reciprocal slope
Range
Everything Y can be
Negative Correlation
Negative slope on a scatter plot
Compound Inequality
Two inequalities joined by and "or" or
Function
Every input has exactly 1 output
Associative Property
(a+b)+c=a+(b+c) (ab)c=a(bc)
Function Notation
f(x)=
Equivalent Expressions
Two expressions that equal each other
Formula
An equation that relates two or more quantities. Has an equal sign
Natural Numbers
The set of numbers 1, 2, 3, 4, ... Also called counting numbers.
Term
A Variable,a number or the product of those
Compound Inequality
A pair of inequalities joined by "and" or "or"
Parent Function
Most basic function shared in a family
Like Terms
Same variable, same exponent
Variable
Letter that represents a number
Commutative Property
a+b=b+a ab=ba
Rational Numbers
All numbers that can be written as quotients of integers.
inverse variation
indirect variation -power of two number is a constnat -xy=k -x1=y2(partners are diagonal) -x1/x2=y2/y1
Section 4
(52 cards)
rational numbers
can be written as quotients of integers but can be written as decimals that terminate or repeat
exponent
in a power, the number of times the base is used as a factor.
describing functions
-ordered pairs -mapping -rule(tells what to do to x in order to get y) -graph
addition property of equality
add the same number to each side
algebraic expression
expression with a variable
commutative property of multiplication
ab=ba
closure property of addition
a+b is a real number
inverse property of multiplication
aX(1/a)=1, a = 0
table of values
-chose x values -substitute x -slove for y -plot points -connect lines
domain
set of the 1st coordinate(x) -allows values of the independant clause
joint variation
involves three or more variables(direct variation)
linear equation
an equation that can be written in the form ax+b=0 where a and b are constants and a=0
coefficient
the product of a number and a variable
composite function
g(f(x)) -replace the x in g(x) with the function f to get a new function
x and y intercepts solving equations
-let y=0 -solve for x -let x=0 -solve for y -plot points -connect line
opposite
additive inverse (exp. subtraction is the opposite of addition)
identity property of addition
a+0=a, 0+a=a
verticle line test
draw a verticle line anywhere(everywhere) if it crosses the graph in one place, the graph is a function
identity property of multiplication
aX1=a, 1Xa=a
reciprocal
multiplicative inverse (exp. of any nonzero number b is 1/b)
numerical expression
consists of numbers, operations, and grouping symbols
point-slop form
y-b=m(x-a)
irrational numbers
cannot be written as quotients of integers and cannot be written as decimals that terminate or repeat
variable
letter that is used to represent one or more numbers
evaluate funtion
replace the x and calculate y -f(x) is the same as y
inverse property of addition
a+(-a)=0
terms
in an expression that can be written as a sum, the parts added together are called terms
slop intercept form
y=mx+b -b is the y-intercept
order of operations
Please Excuse My Dear Aunt Sally
rate of change
change in y divided by the change in x -slope
number line
a graph of real numbers that are graphed on a line in increasing money from left to right
power
expression formed by repeated multiplication of the same factor
equation
a statement that two expressions are equal
base
in a power, the factor
relation
set of ordered pairs -each number is a coordinate -1st coordinate is x(abscissa) -2nd coordinate is y(ordinate)
subtraction property of equality
subtract the same number from each side
distributive property
a(b+c)=ab+ac
multiplication property of equality
multiply each side by the same nonzero number
closure property of multiplication
ab is a real number
inverse of a function
any relation that swaps the dependant and independant variable -new relation may or may not be a function
associative property of multiplication
(ab)c=a(bc)
slop of a line
-change in y is the differnece in y coordinates (vertical change) -change in x is diffrence in x coordinates(horizontal change) -slop is the same of any 2 points
y-intercept
line crosses the y-axis -(0,b) -let x=0 to solve for y
associative property of addition
(a+b)+c=a+(b+c)
parrallel
lines that have same slop -slopes are negative reciprocals
function
set of ordered pairs in which no first coordinate is not repeated
x-intercept
line crosses x-axis -(x,0) -let y=0 to solve for x
commutative property of addition
a+b=b+a
disjunction
has "or" -ex. x id less than, or euql to five
range
set of 2nd coordinates(y) -resulting value of the dependant varible
Section 5
(52 cards)
Compound Inequality
has word and/or in it
associative property
(a+b)+c=a+(b+c)
verbal model
an equation written in words before numbers
union
the graph of a compound inequality containing 'or' is the union of the solution sets of the two inequalities
algebraic expression
expressions that contain at least one variable
distance formula
d=rt
Union
OR <=-=>
temperature formula
F=9/5C+32
Set-Builder Notation
{x|x>7}
area of a triangle
A=1/2bh
absolute value
a number's distance from 0 on the number line l x l
intersection
the graph of a compound inequality containing 'and' is the intersection of the solution sets of the two inequalities
Circumference of a circle
C=2(pi)r
Area of a trapezoid
A=1/2(b1+b2)h
identity property
a+0=a=0+a
division property of equality
divide each side by the same nonzero number
Formula
Something that relates quantitites to calculate things
Trichotomy Property
a=b, a>b, a<b
Area of a rectangle
A=lw
reflexive property
for any real numbera: a=a ex: -7+n=-7+n
element of a set
item in a set
Algebraic Expressions
Minimum of a variable
irrational numbers
real numbers that are not rational ex: square root of 5, pi
Empty Set
{ } No Solution
Variables
symbols
set-builder notation
the solution of an inequality can be expressed by using this ex: {x1x>9}
open sentence
a mathamatical sentence containing one or more variables
compound inequality
consists of two inequalities joined by the word 'and' or the word 'or'
commutative property
a+b=b+a
Area of a circle
A= r(pi)r
set
collection of items called elements
Order of Operations
PEMDAS
Interval Notation
(7, infinity)
rational numbers
can be expressed as a ratio m/n where m and n are integers and n is not 0
Open Sentence
Sentence using < or > sign
Solution
Replacement that makes a sentence true
subset
a set whose elements all belong to another set
distributive property
a(b+c)=ab+ac and (b+c)a=ba+ca
Absolute Value
Distance from 0 on the number line
Perimeter of a rectangle
P=2l+2w
Intersection
AND <-=->
formula
equation that relates two or more quantities, usually represented by variables
symmetric property
for all real numbers a and b, if a=b then b=a ex: if 3=5x-6, then 5x-6=3
substitution property
if a=b, then a may be replaced by b and b may be replaced by a ex: if (4+5)m=18, then 9m=18
Equation
Sentence containing an = sign
trichotomy property
aka the property of order for any two real numbers a and b, exactly one of the following is true: a<b a=b a>b
absolute value
how far away a number is from zero
transitive property
for all real numbers a, b, and c if a=b and b=c, then a=c ex: if 2x+1=5x-8, then 2x+1=5x-8
empty set
the solution for an equation with no solution, symbolized by { } or 0 with /
inverse property
a+(-a)=0=(-a)+a
Section 6
(51 cards)
zero
neither positive nor negative
parent function
simplest function defining characteristics of the family
translation
or "slide" moving each point in a figure the same distance same direction
subtraction
x-y means x+(-y)
function
relation in which the domain is never repeated
closure
real numbers are closed under addition and under multiplication (x+y is unique and real) (xy is unique and real)
function notation
equation satisying the definition of a function
scientific notation
method of writing numbers by using powers of ten
multiplicative inverse
1/x=(x*1/x=1)
additive identity
zero is, x+0=x if x is a real number
positive numbers
numbers greater than 0
symmetry
if x=y then y=x
reflection
transformation that flips a figure across a line called line of reflection
division
x-y=x*1/y
imaginary numbers
square roots of negative numbers, have no points on the number line
integers
whole numbers and their opposites
transformation
s change in posiion, size shaoe of a figure
independent variable
input x is independant
compression
pushing points towards y-axis
infinite set
unlimited, or infinite number of elements.
irrational numbers
cannot be expressed exactly as a ratio of two integers, but are real numbers
transitivity
x=y and y=z then x=z
range
output or Y
natural numbers
positive integers or counting numbers
distributivity
multiplication distributes over addition (x(y+z) = xy+xz
associativity
addition and multiplication of real numbers is (x+y)+z = x + (y+z)
relation
pairing of unput and output values
radicals
involve square root, cube root, etc... of integers
digits
numbers from which the numerals are made
finite set
has a definite, or finite, number of elements
stretch
pulling all points away from y-axis
empty set
set containing no elements
additive inverse
adding the opposite (x+(-x)=0)
real numbers
have points on the number line
complex numbers
form of a+bi, a and b are real i is equal to sqrt of -1
rational numbers
can be expressed exactly as a ratio of two integers
multiplicative identity
one is, 1*x=x for real numbers
negative numbers
numbers less than 0
radicand
The number expression under the radical symbol
principal root
the radical symbol indicates both the positive and negative square root of a number
reflexive property
if x is a real number, the x=x
roster notation
elements of a set listed between braces
non-integers
fractions, numbers between the integers
interval notation
the symbols [ and ] are used to include an endpoint in an interval, and the symbols ( and ) are used to exclude an endpoint from an interval
commutativity
addition and multiplications of real numbers are x+y=y+x xy=yx
like radical terms
sqare roots that have the same radicand
domain
input or X
set builder notation
uses the propertied of the elements in a set to define the set
trichotomy
either x<y, x>y, or x=y
dependent variable
depends on inout value of function example: f(x)
Section 7
(52 cards)
polynomial
a monomial or a sum of monomials
Quotient Property
(a^m)/(a^n)= a^m-n
Distributive Property
a(b+c)= ab+ac
absolute value
The distance a number is from zero on a number line
Associative Property
(a+b) +c= a+ (b+c)
algebraic expression
an expression with one or more variables
open sentence
an equation with one or more variables
Product Property
a^m *a^n= a^m+n
Additive Inverse Property
a+(-a)=0
power
an expression that represents repeated multiplication of the same factor
Power of a Product
(ab)^m= a^m*b^m
Power of a Quotient
(a/b)^m= (a^m)/(b^m)
no solution
The answer to a contradiction is:
empty set
set with no elements: zero or {}.
Zero Property
(ab)^0= 1
Commutative Property
a+b=b+a
Multiplicative Inverse
a*(1/a)= 1
Power Property
(a^m)^n= a^m*n
Interval Notation
Used to express sets of real numbers. An open interval is expressed as (a,b) is the set of real numbers not including a and b. a and b are endpoints of the interval. But if written [a,b], the set does include a and b.
radicant
number under the radical
coefficient
number in front of a variable
set-builder notation
The expression of a solution set of an inequality
union
The graph of two or more inequalities containing or >
compound inequailty
two inequalities joined by and or or
set
a collection of items, called elements.
Negative Property
b^-n= 1/(b^n)
subset
a set whose elements all belong to another set.
equation
a mathematical sentence that shows that two expressions are equal
Closure Property
a∈W, b∈W then a+b∈W
empty set
the solution set for and equation that doesn't have a solution
degree
the sum of the exponents of the variables of a monomial
trinomial
a polynomial with three unlike terms
constant
a term that does not contain a variable
monomial
an expression with only one term
rational numbers
numbers that can be written as fractions, including terminating and repeating decimals, and integers
like terms
Expressions that contain the same variables to the same power
irrational numbers
A real number that isn't rational that doesn't terminate or repeat
variable
symbols used to represent unknown quantities x, y
principle root
√16 = 4←
real numbers
all numbers we use in everyday life
Roster Notation
finite: {8,9,12} a definite # of elements. infinite: {1,2,3,4...} unlimited # of elements.
intersection
The graph of a compound inequality that contains and
Sets of Numbers
1. natural numbers (bold N) {1,2,3,4,5...} 2. whole (bold W) {0,1,2,3,4...} 3. Integers (bold Z) {-3,-2,-1, 0, 1...} 4. Rational (bold Q) {3/4, 1/2, 1/4, 7/8} 5. Irrational {√2, √8, ∏, -√3} 6. Real (bold R)
Additive Identity Property
a+0=a
multiplication property of -1
-1*x=-x
Multiplicative Identity Property
a*1=a
order of operations
PEMDAS
radical
√
multiplication property of 0
if xy=0 then x=0 or y=0
Set Builder Notation
ex. {x⎮3<x<8, x∈R
Section 8
(50 cards)
subtraction property of equality
if a = b, then a - c = b - c
Constant Function
f(x)=c D: (-∞, ∞) R: [c]
property of opposites of a sum
-(a+b) = -a + (-b)
definition of subtraction
a - b = a + (-b)
vertical translation
change the y-values to their opposite. move x-value up or down.
Square Root
F(x)= √x D: [0, ∞) R: [0, ∞)
Quadratic
f(x)= x^2 D: (-∞, ∞) R: [0, ∞)
Stretch
I. Vertical -change y. L(x)= af(x) |a|>1 II. Horizontal- change x. n(x)=f(1/b(x)) |b|>1...ex: horizontal stretch by 4-> n(x)= ((1/4)x)^2
multiplicative property of -1
a*-1 = -a -1*a = -a
distributive property of division
(a+b)/c = a/c + b/c (must have a monomial in denominator - does not work for the reciprocals of the above)
closure field property of real numbers
a + b and ab are unique real numbers
Function
for every input there is ONE unique output. the first coordinate is never repeated.
the absolute value of the product of of 2 or more numbers is the product of their absolute values
|ab| = |a| * |b|
Transformation
changes the position size or shape of the figure .
multiplicative property of 0
a*0 = a 0*a = 0
Linear
f(x)= x D: (-∞, ∞) R: (-∞, ∞)
horizontal translation
change the x-values to their opposite. move x-value right or left.
One-to-One Function
a function in which the x- and y-values can NOT repeat.
definition of division
a/b = a * 1/b b does not = 0 and ab E R
Restriction on Domain
1. Never have zero in the denominator. 2. Radicant ≥ 0.
Compression
I. Vertical- m(x)=af(x) 0<|a|<1 II. Horizontal- p(x) = f(1/b (x)...ex: p(x)= x^2 horizontal comp. by 1/3= p(x) = f(3x)^2
rule for addition if a is positive and b is negative and | a | < | b |
a + b = - ( | b | - | a | )
rule for addition if a and b are both negative numbers
a + b = - ( | a | + | b | )
addition property of equality
if a = b, then a + c = b + c
1 positive and 1 negative numbers' products are
negative
Parent Functions
constant, linear, cubic, quadratic, absolute value, square root.
distributive field property of real numbers
a(b+c) = ab + ac
property of reciprocals field property of real numbers
a * 1/a = 1
How do you determine if a set is closed under addition?
If you take any number in the set and add it to another number in the set and get an answer as number in the set, the set is closed under addition.
substitution field property of real numbers
if a = b, then a + c = b + c
2 positive or 2 negative numbers' products are
positive
identity field property of real numbers
a + 0 = a a * 1 = a
Relation
any pairing of an input(domain) with the output(range).
reflexive property of equality
a = a
the quotient of 2 positive or 2 negative numbers is
positive
symmetric property of equality
if a = b, then b = a
Cubic
f(x)= x^3 D: (-∞, ∞) R: (-∞, ∞)
Reflection
I. across x-axis. change y-value. j(x)= -f(x) II. Across y-axis. change x-value. j(x)= f(-x)
the quotient of 1 positive and 1 negative number is
negative
Absolute value
f(x)= |x| D: (-∞, ∞) R: [0, ∞)
property of opposite products
-ab = b(-a) -ab = (-a)b
transitive property of equality
if a = b and b = c, then a = c
Is a relation a function?
y^even is NOT a function...why? ex: y^2= x+3 (1, 2) (1, -2) |y| is NOT a function...why? ex: |y|=3x+2 (5,√17) (5, −√17)
How do you determine if a set is closed under multiplication?
If you take any number in the set and multiply it by another number in the set and get an answer as a number in the set, then the set is closed under multiplication.
to divide by a fraction,
multiply by its reciprocal
commutative field property of real numbers
a + b = b + a ab = ba
associative field property of real numbers
a + (b+c) = (a+b) + c a(bc) = (ab)c
multiplication property of equality
if a = b, then ac = bc
rule for addition if a is positive and b is negative and | a | > | b |
a + b = | a | - | b|
property of opposites field property of real numbers
a + (-a) = 0
Section 9
(52 cards)
Inverse Property of Multiplication
a ⋄ 1/a = 1, a ≠ 0 *Answer is ALWAYS 0 because you multiply by the reciprocal
Infinite set
A set having an unlimited number of members
Finite set
a set with a definite number of elements
Substitution POE
If a = b, then b may be substituted for a in an expression to obtain an equivalent expression
Intersection
Symbol: upside down U definition: overlap
element
one individual number/unit in a set
Commutative Property of Multiplication
ab = ba *Order of numbers/variables doesn't effect the answer
Inverse Property of Addition
a + (-a) = 0 *Answer is ALWAYS 0 because you add by the opposite
Distributive Property
a(b + c) = ab + ac
Variable
a quantity that can assume any of a set of values
subset
symbol: c definition: a part of a set including the set itself, all elements, and zero
Commutative Property of Addition
a + b = b + a *Order of numbers/variables doesn't effect the answer
Origin
the point of intersection of coordinate axes
irrational numbers
symbol: I definition: any number that cannot be written as a fraction example: square root -2, pi
Empty set
Set containing no elements
whole numbers
symbol: W definition: any counting number that is positive or negative excluding 0
integers
symbol: Z definition: any counting number that is positive or negative including 0
Transitive Property of Inequalities
If a ≤ b and b ≤ c, then a ≤ c
imaginary numbers
definition: any number that is not real example: square root -1
Identity Property of Addition
a + 0 = a, 0 + a = a *a value is never changed
Opposites
two numbers that have the same absolute value but have different signs
Division POE
If a = b and c ≠ 0, then a/c = b/c
Closure Property of Multiplication
ab is a real number
Addition Property of Inequalities
If a ≤ b, then a + c ≤ b + c
Set
Collection of elements
Multiplication Property of Inequalities
If a ≤ b and c > 0, then ac ≤ bc If a ≤ b and c < 0, then ac ≥ bc Watch out when c is negative
Subset
a set whose members are members of another set
natural numbers
symbol: N definition: any positive whole number
Algebraic expression
A mathematical phrase involving at least one variable and sometimes numbers and operation symbols.
Subtraction Property of Inequalities
If a ≤ b, then a - c ≤ b - c
value of a numerical expression
the number represented by the equation
What does absolute value usually represent?
a distance
Set
a group of numbers
Division Property of Inequalities
If a ≤ b and c > 0, then a/c ≤ b/c If a ≤ b and c < 0, then a/c ≥ b/c Watch out when c is negative
Subtraction POE
If a = b, then a - c = b - c
Interval notation
a way of writing the set of all real numbers between two endpoints. the symbols [and] are used to include an endpoint in an interval, and the symbols (and) are used to exclude endpoint from an interval
Closure Property of Addition
a + b is a real number
Identity Property of Multiplication
a ⋄ 1 = a, 1⋄ a = a *a value is never changed
grouping symbols
(parenthesis) {braces} [brackets] ____ fraction bar l absolute value bars l
Multiplication POE
If a = b, then ac = bc
Associative Property of Addition
(ab)c = a (bc) *Grouping doesn't effect the answer
Roster notation
the elements of a set are listed between braces
rational numbers
symbol: Q definition: any number that can be written as a fraction example: 2/3, 2.3, 23
real numbers
symbol: R definition: any number that is not imaginary
substitution
an expression that can be replaced by another expression that represents the same value
Associative Property of Addition
(a + b) + c = a + (b + c) *Grouping doesn't effect the answer
Set-builder notation
A formal method of describing a set. Often used for inequalities. For example {x| x>1}, which is read "x such that all x is greater than 1."
Union
symbol: U definition: everything together
domain
anything that the variable can be
Element
one of four substances thought in ancient and medieval cosmology to constitute the physical universe
Section 10
(51 cards)
discrete graph
a graph whose points are not connected.
Opposite of a Product
-(ab) = - a ⋄ b = a ⋄ (-b) [Only for multiplication, has to end up negative if the numbers are positive to begin with]
Reflexive POE
a = a [*Reflexive sounds like reflection]
continuous graph
a graph of a line or a smooth curve
Multiplication by Negative one
-1⋄ a = - a
Multiplication by Zero
0 ⋄ a = 0
independent variable
the input value;the values make up the domain
Opposite of a Difference
-(a - b) = b - a [or -a +b] [Only with subtraction]
Irrational Numbers
A number that cannot be expressed in the form a/b, where a and b are both integers Ex. .0101110101000..., √2, pi, √3 etc.
Linear Equations
has only addition, subtraction, and multiplication as operators of a variable and a constant. The variable has 1 as an exponent.
function
a mathematical relation such that each element of the domain is paired with exactly one element of the range.
dimensional analysis
the process of carrying units throughout a computation
vertical line test
a method to determine if a graph is a function or not
Ordered Pair
A pair of numbers that can be used to locate a point on a coordinate plane
Opposite
Also known as the additive inverse. When the sum of the opposite and the original number is added the sum is zero. Ex. The opposite of A is -A
equivalent equations
two equations that have the same solution
Natural Numbers
1,2,3... ✩Only positive numbers, and NO zero
Whole Numbers
0,1,2,3... ✩Only positive numbers and zero, NO fractions
formula
a rule for the relationship between certain quantities
mapping
Illustrates how each element of the domain is paired with an element in the range
Quadrant
any of the four areas into which a plane is divided by two orthogonal coordinate axes
Rational Numbers
Any number that can be written in the form a/b, where both a and b are integers and b ≠ 0 Ex. 5/8 , .75, 4, - 2/3 etc. -Some repeating and terminating decimals can be rational numbers
Real Numbers
All numbers
range
the set of second coordinates of the ordered pairs of a relation.
Opposite of an Opposite
-(-a) = a [If a is a positive number, then the answer will be positive as well]
Integers
...-3,-2,-1,0,1,2,3... ✩Negative and positive numbers including zero, NO fractions
multi-step equation
an equation that requires more than one step to solve
ratio
a comparison of two numbers by division
percent of change
the ratio of the change in an amount to the original amount expressed as a percent
Absolute Value
A number's distance from zero. The value is always positive
proportion
an equation stating two ratios are equal
Transitive POE
If a = b and b = c, then a = c
Symmetric POE
If a = b, then b = a
solve an equation
to find the value of the variable that makes the equation true
Reciprocal
Also known as the multiplicative inverse, the inverse of a nonzero number Ex. The reciprocal of 4 is 1/4
Addition POE
If a = b, then a + c = b + c
Whole Numbers
0,1,2,3... ✩Only positive numbers and zero, NO fractions
function notation
replacing the y with the symbol f(x)
domain
the set of first coordinates of the ordered pairs of a relation.
one-to-one function
a function where each element of the range is paired with exactly one element of the domain.
relation
A set of ordered pairs
identity
equations that are true for all values of the variables
Definition of Subtraction
a - b = a + (-b)
Opposite of a Sum
-(a + b) = - a + (- b) [Only with addition]
scale model
a three-dimentional reproduction of an item that has been reduced or increased in size proportionally
rate
the ratio of two measurements having different units of measure
Definition of Division
a ÷ b = a/b = a ⋄ 1/b, b≠0
Distributive Property for Subtraction
a(b - c) = ab - ac
Cartesian Coordinate Plane
a plane divided into four quadrants by the intersection of the x-axis and the y-axis at the origin
dependent variable
the output value; the values make up the range.
Section 11
(52 cards)
Parameter
a determining or characteristic element; a factor that shapes the total outcome; a limit, boundary
Linear Function
a function in which the graph of the solutions forms a line
Dependent Variable
a factor that can change in an experiment in response to changes in the independent variable
Vertex
The point in a function where the function reaches a min or a max
Linear Inequality
an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line
Relation
A set of ordered pairs
domain
set of input values
dependent variable
the output variable, depends on the value of the input variable
Standard Form
ax+by=c
range
set of output values
Absolute Value Function
a function written in the form y = /x/, and the graph is always in the shape of a v
x-intercept
The x-coordinate of the point where a line crosses the x-axis.
perpendicular
slope = - reciprocal
Scatter Plot
a graph with points plotted to show a possible relationship between two sets of data.
Mapping Diagram
a way to show a relation that links elements of the domain with cooresponding elements of the range
Independent Variable
variable that is changed in an experiment
output
y
Direct Variation
y=kx
weighted average
the sum of the product of the number of units and the value per unit divided by the sum of the number of units, represented by M
y-intercept
the y-coordinate of the point where the line crosses the y-axis.
function notation
using f(x) to represent the dependent variable of a function. basically y
slope
the change in y over the change in x m y1-y2 over x1-x2 rise over run
Translation
A transformation that "slides" each point of a figure the same distance in the same direction.
VLT
vertical line test
Stretch
Multiplies all y values by the same factor greater than 1
Point slope form
Y-Y1=m(x-x1)
equation in two variables
an equation that contains two variables
input
x
Function Notation
an equation in the form of 'f(x)=' to show the output value of a function, f, for an input value x
Vertical Line Test
a method to determine if a graph is a function or not
Shrink
Reduces y values by a factor between 1-0
relation
a mapping or pairing of input values with output values
parallel
slope = the same
linear function
can be written as y=mx+b where m and b are constants (x has to be to the first power)
Function
A relation which one element from the domain is paired with range
Transformation
An operation that moves or changes a geometric figure in some way to produce a new figure
unit rate
a rate that tells how many of one item is being compaired to 1 of another item
Range
The y-coordinates of the set of points on a graph. Also, the y-coordinates of a given set of ordered pairs.
Linear Equation
an equation whose graph is a straight line
function
a relation for which each input has exactly one output
Constant of Variation
the number k in equations of the form y=kx
Slope
the steepness of a line, equal to the ratio of a vertical change to the corresponding horizontal change
Trend Line
a line that approximates the relationship between the data sets of a scatter plot
Domain
The set of x-coordinates of the set of points on a graph; the set of x-coordinates of a given set of ordered pairs. The value that is the input in a function or relation.
HOY VIXN
-horizontal directional 0 slope y=k equation -vertical directional x=k equation no slope
independent variable
the input variable
Slope Intersept Form
the equation of a line in the form of y=mx+b, where m is the slope and b is the y intersept
Parent Function
The simplest function in a family; all functions in the family are transformations of it
-horizontal directional 0 slope y=k equation -vertical directional x=k equation no slope
HOY VIXN
Reflection
A transformation that "flips" a figure over a mirror or reflection line.
Section 12
(50 cards)
slope intercept form
y=mx+b m=slope b=y-intercept
Ray
a half line; the portion of a line on one side of a fixed point on the line.
y=ax
direct variation a=constant of variation y varies directly with x
Slope-intercept form
one form of writing the equation of a line - (y=mx+b)
subtract from input
translate graph right
relation
a mapping, or pairing, of input values with output values; can be represented by ordered pairs, a table, a graph, or a mapping diagram
transformation
a change to a function
slope
the ratio of vertical change (the rise) to horizontal change (the run) for a nonvertical line passing through the points (x₁, y₁) and (x₂, y₂), the ________ is m=(y₂ - y₁)/(x₂ - x₁)
change sign of input
reflect over y-axis
trend line
approximates the relationship between the variables, or data sets, of a scatter plot
Curve
the path traced by a moving point; an unbroken connection of points.
dependent variable
the output variable in an equation in two variables; in y=3x+5, the __________ is y
independent variable
the input variable in an equation in two variables; in y=3x+5, the __________ is x
≤ < less than (or equal to)
shade down
y=k
horizontal line 0 slope
Variable
a letter used to represent a number
weak
correlation when data points are loosely packed
Like Terms
terms whose literal components represent the same number regardless of the numbers used to replace the variables
dilation
changes the size of a graph by stretching or compressing
add to output
translate graph up
perpendicular lines have _____ _____ slopes
negative reciprocal
piecewise functions
a function determined by at least 2 equations, each of which applies to a different part of the functions domain restricted domain
translation
shifting or sliding a function up, down, left, right
scatter plot
a graph that relates two sets of data by plotting the data as ordered pairs
≥ > greater than (or equal to)
shade up
point slope form
(y-y1)=m(x-x1) m=slope (x1, y1)= point
parent
simplest function form in a set that forms a family
2-intercept form
x y - + - = 1 a b a= x-intercept b = y-intercept
Line Segment
part of a line that consists of two end points and all points between them
x=k
vertical line no slope
Constant Term
a quantity whose value does not change
best fit
line that gives the most accurate model of related data (two words)
strong
correlation when data points are tightly packed
axis of symmetry
vertical line; x=h; separates graph of absolute value function into two equal, but mirrored, parts
multiply by value bigger than 1
vertically stretch graph
parallel lines have _____ slopes
same
subtract from output
translate graph down
Point
A location in space with no physical size
add to input
translate graph left
rate of change
a comparison of how much one quantity changes, on average, relative to the change of another quantity
standard form
ax+by=c a, b, c are integers
Slope
the ratio of the change in the y coordinate to the change in the x coordinate from a point on a line to any other point on a line
change sign of output
reflect over x-axis
multiply by fraction between 0 and 1
vertically compress graph
correlation coefficient
r; tells the strength of the correlation
reflection
a flip; it changes the sign of the input or the output
LIne
a straight curve that has no ends, or extends indefinetly in two opposite directions.
vertex
single maximum or minimum point of an absolute value function; (h,k)
function
a relation for which each input has exactly one output; if any input of a relation has more than one output, the relation is not a function
absolute value
function that has a V-shaped graph; y=a|x-h|+k; a is the slope of the right branch; has symmetry and one vertex; can open up or down
Section 13
(52 cards)
Similar Triangles
Two triangles whose angles have equal measures
Proportion
an equation or other statement that indicates that two ratios equal
Transversal
a line that cuts or intersects one or more other lines in the same plane
Factor
one of two or more expressions that are multiplied to form a product
Substitution Axiom
equal quantities may always be substituted for one another
Rational Expression
an algebraic expression written in the form of a fraction
Scale Factor
a multiplying factor relating the corresponding lengths of a given geometric shape to its enlarged or reduced counterpart. A scale factor of 1 denotes congruence.
1st Property of a Parallelogram
The sides opposite each other have equal lengths
1st Major Rule of Algebra
the same quantity can be added to both sides of an equation without changing the solution to the equation
commutative axiom for addition
a+b=b+a
identity axiom of addition
a+0=a
Exterior Angles
the angles outside the parallel lines
Formula of Tangent
TOA: Opposite over adjacent
Pythagorean Theorem
the area of a square drawn on the hypotenuse of a right triangle equals the sum of the areas of the squares drawn on the other two sides
Corresponding Angles
angles located in matching positions when two lines are cut by a transversal.
Unit Multipliers
a ratio of named numbers whose value is 1; this ratio is used to convert unit measurements( for example to convert feet to inches or feet to miles)
associative axiom for addition
(a+b)+c=a+(b+c)
1st Property of a Rhombus
a rhombus is a parallelogram whose four sides have equal lengths
Product of Square roots rule
If m and n are nonnegative real numbers then the square root of m times the square root of n equals the square root of m time n and the square root of m times n equals the square root of m times the square of n
identity axiom of multiplication
a(1)=a
Area of a triangle
one half base times height
Congruent
identical in all characteristics
Formula of Sine
SOH: Opposite over hypotenuse
Corresponding Sides
corresponding sides are the sides opposite equal angles in similar triangles; the ratios of the lengths of corresponding sides in similar triangles are equal
3rd Property of a Rhombus
the diagonals of a rhombus are perpendicular bisectors of each other
Monomial
a polynomial of one term
Quadratic Equation
a one variable polynomial equation whose degree is 2
Binomial
a polynomial of two terms
4th Property of a Parallelogram
The diagonals bisect each other
Scientific Notation
a method of writing a number as a product of a decimal number and a power of 10
Denominator-Numerator Rule
the denominator and the numerator of a fraction may be multiplied by the same non-zero quantity without changing the value of the fraction
additive inverse axiom
a+-a=0
Radical
an expression for taking the root of a quantity indicated by the symbol
Alternate Angles
angles on opposite sides of a transversal
Formula of Cosine
CAH: Adjacent over hypotenuse
Inscribed Angle
an angle inside a circle with its vertex on the circle whose measure equals half the measure of the intercepted arc
2nd Property of a Rhombus
The diagonals of a rhombus bisect the angles of the rhombus
3rd Property of a Parallelogram
The sum of the measures of any two consecutive angles is 180 degrees
Interior Angles
the angles between parallel lines
distributive axiom for multiplication over addition
a(b+c)=ab+ac
Parallelogram
a quadrilateral that has two pairs of parallel sides
Trinomial
a polynomial of three terms
commutative axiom for multiplication
ab=ba
2nd Property of a Parallelogram
The angles opposite each other have equal measures
2nd Major Rule of Algebra
every term on both sides of an equation can be multiplied or divided by the same quantity( except zero) without changing the solution to the equation.
Product
the number or quantity obtained by multiplying two or more numbers together
3rd Major Rule of Algebra
the denominator and the numerator of an expression can be multiplied or divided by the same non zero quantity without changing the value of the expression. Only the form of the expression is changed.
associative axiom for multiplication
(ab)c=a(bc)
Reciprocal
for any non-zero real number, the number in inverted form. For example, the reciprocal of 3 is 1/3 and the reciprocal of 3/4 is 4/3
Remote Interior Angles
the two remaining interior angles of a triangle given one interior angle
Section 14
(52 cards)
Feasible Region
the intersection of the inequalities
Consistent Dependent
∞, all points on the line, the lines coincide
identity
an equation that is true for all values of the variable
Bounded
when the feasible region is a polygon
formula
a literal equation that states a rule for a relationship among quantities
Unbounded
when the feasible region is NOT a polygon
direct variation
slope k is constant y=kx
eight
How many different ways can 3 planes intersect?
Constraints
the inequalities
domain
set of all inputs, x coordinates of the ordered pairs
transitive property of equality and order
if a=b, and b=c, then a=c
proportion
a statement that 2 ratios are equal
deductive reasoning
the process of using logic to draw conclusions
addition property of order
if x>y, then x+a>y+a
solution of an equation
a value or values that make the equation true
range
set of all outputs, y coordinates of the ordered pairs
symmetric axiom of equality
if a=b, then b=a
linear function
a function whose graph is a line
multiplication property of -1
a(-1)=-a
Ax+By=C
standard form of a linear equation
multiplication property of equality
if x=y, then xa=ya
y=mx+b
slope intercept form
addition property of equality
if x=y, then x+a=y+a
linear function
y=mx+b
equivalent equations
equations that have the same solution set
Linear Programming
the process of finding the maximum and minimum value of a function for a region defined by inequalities
rate
a ration that compares 2 quantities measured in different units
literal equation
an equation that contains two or more variables
ratio
a comparison of 2 numbers or quantities
function
a relation in which each element of the domain is paired with exactly one element in the range
multiplication property of order
if x>y, then xa>ya
constant of variation
when x and y are variables k=y/x
unit rate
in which the 2nd quantity in the comparison is one unit
Substitution
What would be better in this situation (substitution or elmination)? 3w+2d=192, d=2w-13
relation
set of pairs of input and output values. can be written as an ordered pair
Trace
The line at which a plane intersects with a coordinate plane is called a ________.
Inconsistent
No solutions, parellel (don't intersect)
Elimination
What would be better in this situation (substitution or elmination)? 3w+d=14, 3w-2d=9
percent
a ratio that compares a number to 100
Consistent Independent
One solution, intersect at one point
y-y1=m(x-x1)
point slope form
multiplicative inverse axiom
a(1/a)=1
cross products
in the statement a÷b = c÷d bc and is this.
reflexive axiom of equality
a=a, x=1
y-y/x-x
slope formula
y=3x+2
linear equation, represents linear function
solution set
the set of all solutions of a particular problem
definition of division
a/b=a(1/b)
definition of subtraction
a-b=a+-b
equation
a mathematical statement that 2 expressions are equal
Section 15
(50 cards)
best-fitting line
line that lies closest to all the data point
Slope
The ratio of the vertical change to a corresponding horizontal change
scale factor
the multiplier used on each dimension to change one figure into a similar figure
correlation coefficient
denoted by r
Slope-intercept form
y=mx+b
Linear Function
A function whose graph is a line
Point-Slope form
y-y1=m(x-x1)
constant of variation
non-zero constant a
Parent Function
The simplest function in a group of functions with common characteristics
Scatter Plot
A graph that relates two different sets of data by plotting the data as ordered pairs
Linear Inequality
An inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line
y-intercept
y-coordinate where the graph intersects the y-axis
function notation
f(x)=mx+b
Vertical-line test
Used to determine whether the relation has at least one element of the domain paired with more than one element of the range
Slope
A measure of the steepness of a line.
rate of change
how much one quantity changes in relation to how much another quantity changes
Function
A relation in which each element of the domain is paired with exactly one element in the range
parallel
two lines on a plane that do not intersect
direct variation
y=ax
Vertex
The point where a function reaches a maximum or a minimum
x-intercept
x coordinate of a point where a graph crosses the x axis/ y coordinate of this point is zero
Mapping Diagram
Links the elements of the domain with corresponding elements of the range
standard form
Ax + By=C, where A, B, and C are not decimals or fractions, where A and B are not both zero, and where A is not a negative
Translation
An operation that shifts a graph horizontally, vertically, or both
y intercept
The y coordinate of the points where a graph intersects the y-axis.
Trend Line
A line that approximates the relationship between the data sets of a scatter plot
Constant of Variation
The value of K in direct variation
parent function
most basic family function
Linear Function
A function that can be written in the form of y=mx+b, where x is the independant vairable and m and b are real numbers. It's graph is a line
Absolute Value Function
A function of the form f(x)=|mx+b|+c
Indirect Measurement
A method of measurement that uses formulas, similar figures, and/or proportions
Correlation
A measure of the strength and direction of the relationship between two variables or data sets
Relation
A set of pairs of input and output values
slope-intercept form
y=, an equation written in the form y=mx+b is in slope-intercept form. The graph is a line with slope m and y-intercept b.
point-slope form
y-y1 = m(x-x1), where m is the slope and (x1,y1) is the point the line is passing through.
negative correlation
y decreases as x increases
slope
ratio of rise over run
scatter plot
graph of a set of data pairs
Absolute Value Function
A function whose rule contains absolute value expressions
Direct variation
A linear function defined by an equation of the form y=kx (k is not 0)
Range
All outputs, or y-coordinates, of the ordered pairs
Line of best fit
The line that comes closest to all of the points in a data set
positive correlation
y increases as x decreases
Domain
All inputs, or x-coordinates, of the ordered pairs
Identity
An equation that is true for all values of the variables
perpendicular
two lines intersect to form a ninety degree angle
Rate
A ratio that compares two quantities meausured in different units
Proportion
A statement that two ratios are equal; a/b =c/d
Standard form
Ax+By=C
Dependent Variable
The y-variable
Section 16
(54 cards)
flip the sign
what do you do when you multiply/divide by a negative (inequalities)
Equivalent Equations
equations that have the same solutions
Scale Model
a 3-dimensional model that is similar to a 3-dimensional object
equivalent inequalities
inequalities with the same solution set
conjunction
- 3 ≤ x and x ≤ 4
Unit Analysis
including units for each quantity in a calculation to determine the unit of the answer
Identity
an equation that is true for every value
Multiplication Property of Equality
multiplying each side of an equation by the same number produces equations
and
key word of conjunctions
Addition Property of Equality
to add the same number to each side of an equation
Isolate
using properties of equality and inverse operations to get a variable to a coefficient 1; alone on one side of the equation
Solution of an inequality
a number that produces a true statement when it is substituted for the variable in an inequality
Inverse Opertaions
operations that undo one another
Proportion
an equation that states two ratios are equal
closed circle
(graphing) greater than or equal to/less than or equal to
or
key word of disjunctions
Literal Equation
an equation involving two or more variables
open circle
(graphing) greater than/less than
Values
numbers that one substitutes for the variable
Steps to solving an equation
Get rid of parentheses. Get rid of fractions. Combine like terms. Variables on one side. Everything else on the other. Multiply or divide.
Division Property of Equality
to divide each side of the equation by the same number that produces an equivalent equation
Rate
a ratio of a and b where a and b represent quantities measured in different units
Percent Change
the ratio of the amount of change to the original amount expressed as a percent
Conversion Factor
a ratio of two equivalent measures in different units
disjunction
a sentence formed by joining two sentences with the word OR
compound inequality
conjunction or disjunction
axiom
a statement that is assumed to be true; also called postulate
Similar Figures
two figures that have the same shape but not the same size
hypothesis
a statement that is given or assumed to be true
conjunction
a sentence formed by joining two sentences with the word AND
Cross Product Property
the cross products of a proportion are equal
Scale Drawing
an enlarged or reduced drawing similar to an actual object or place
Subtraction Property of Equality
to subtract the same number from each side of an equation that produces an equivalent equation
y-intercept
The point at which the line crosses the y-axis
Function
Function a rule that establishes a relationship between two quantities called the input and output such that every input has exactly one output.
inequality
x + 4 ≤ 7
Unit Rate
a rate with a denominator of 1
Formula
an equation that states a relationship among quantities
conclusion
a statement that follows logically from the assumptions
Scale
the ratio of any length in a scale drawing to the corresponding actual length
Ratio
the comparison of two quantities by division
Percent Decrease
percent change found when the original amount decreases
Independent Variable
the x-variable
Linear equations
Represents a linear function
disjunction
x ≤ -2 or x ≥ 3
Solution of an equation
A number that when substituted for the variable in an equation, makes the equation a true statement
Cross Products
in a proportion a/b=c/d the products are ad=bc
Percent Error
the ratio of the absolute value of the difference of a measured (or estimated) value and an actual value compared to the actual value expressed as a percent
x-intercept
the point at which a line crosses the x-axis
Relative error
the ratio of the absolute value of the difference of a measured (or estimated) value and an actual value compared to the actual value
Section 17
(53 cards)
Slope-intercept form
y=mx+b y=y-coordinate m=slope x=x-coordinate b=y-intercept
transitive property
if a<b and b<c, then a<c
Greatest Integer Function
A step function written as f(x)=[[x]] where f(x) is the greatest integer less than or equal to x
reflects in the y axis
Which reflection occurs in the equation f(x) = (-x)?
line plot
a # line labeled with a scale that includes all the data and xs above each data point for each time it occurs
graph shifts up 2 units
Describe the translation in lxl +2
comparison property
exactly one of the following statements is true: a>b , a=b , a<b
Point slope form
y-y₁=m(x-x₁)
reflects in the x-axis
Which reflection occurs in the equation f(x) = -1(x)?
vertical shrink
Which transformation occurs in the graph of f(x) = 2lxl?
measures of central tendency
#s that represent the middle value in a set of data. ex: mean, medium and mode
Piecewise Function
A function that is written using two or more expressions
Constant Function
A linear function in the form of f(x)=b
at most/ no more than
< ≤
absolute value function
Which parent function is f(x) = lxl?
proof
a logical reasoning from hypothesis to conclusion
a "v"
What shape does an absolute value function make when graphed?
greater than
absolute value is an OR
Line of Fit
A line that closely approximates a set of data
less than
absolute value is an AND
stem and leaf plot
a system where you separate data into 2 numbers that are used to form a stem and leaf
horizontal shrink
Which transformation occurs in the graph of f(x) = l2xl?
multiplication property
1. if a<b and c is positive, then ac<bc 2. if a<b and c is negative, then ac>bc.
between/ between inclusive
< x < / ≤ x ≤
frequency
how often a piece of data occurs
limit
the number that the output gets closer to the specific x coordinate
theorem
a statement that can be proved
converse
the statement that results from interchanging the hypothesis and the conclusion of an if-then statement
identity function
Which parent function is f(x) = x, passing through all coordinates (a,a), and is the parent function for most linear equations?
Prediction Equation
An equation suggested by the points of a scatter plot that is used to predict other points
addition property
if a<b, then a+c< b+c
probability
the probability of a simple event is a ratio of the # of favorable outcomes for the event to the total # of possible outcomes to the event
compound inequality
two inequalities joined by an AND & OR
graph shifts to the left 2 units
Describe the translation in lx +2l
quadratic function
Which parent function is f(x) = x²?
a parabola (u shape)
What shape does a quadratic function make when graphed?
Scatter Plot
A set of data graphed as ordered pairs in a coordinate plane
at least/ no less than
> ≥
vertical stretch
which transformation occurs in the graph of f(x) = -3lxl?
simple event
a single event
counter example
a single numerical example that makes a statement false
horizontal stretch
Which tranformation occurs in the graph of f(x) = l-3xl?
shrink
What happens when the parent function is multiplied by a value greater than one?
absolute value
on a number line, ⌉a⌉ is the distance between the graph of th number and zero
stretch
What happens when a parent function is multiplied by a value less than one?
Identity Function
The function f(x)=x
constant function
Which parent function is f(x) = a?
sample space
the list of all possible outcomes
Absolute Value function
A function written as f(x)=lxl, where f(x)≥0 for all values of x
corollary
a theorem that c an be proved easily from another theorem
Section 18
(53 cards)
Relation
A set of ordered pairs
One-to-One Function
Each number is only paired once in the domain and range
slope intercept form
y=mx+b
Factorial
Factors in the coefficients of binomial expansions involve these special products.
Linear Equation
Only has addtion, subtraction, and multiplication of a variable by a constant
Partial Sum
Sn
Function
A special type of relation in which each element of the domain is paired with exactly one element from the range
Standard Form
Ax+By=C Integers, always reduce, and A cannot be negative
Funtional Notation
An equation of y in terms of x can be rewritten so that y=f(x)
graphing
the process of locating a point in the coordinate plane
Geometric Means
Missing terms between two nonsuccessive terms of a geometric sequence.
Quadrants
The four areas of the coordinate plane
Vertical Line Test
Determines if the graphed relation is a function
Dependent Variable
The variable, usually y, whose values make up the range
Rate of Change
What slope is often referred as. Measures how much a quantity changes, on average, relative to the change in another quantity, often time
Perpendicular Lines
Their slope is opposite reciprical
Common Ratio
R
graph of the equation
the line is called the ______________ in a graph
linear equation in two variables
is an equation that can be written in the form ax+by=c
regression
the process of using one variable to predict another
Geometric Series
indicated sum of the terms of a geometric sequence.
Slope
The ratio of the change in y-coordinates to the corresponding change in x-coordinates
table of solutions
constructed by choosing several values of x and finding the corresponding values of y
slope triangle
rise over run creates a ____________
Fibonacci Sequence
1,1,2,3,5,8,13
Mapping
Shows how each member of the domain is paired with each member of the range
Mathematical induction
A method of proof used to prove statements about positive integers.
Parallel Lines
Their slope is the same
mapping diagram
used to show how a function assigns to each member of the domain exactly one member of the range
average rate of change
to find this, we find the ratio of the change in the x and y
straight line appreciation
used when property or collectibles increase in value
Independent Variable
The variable, usually x, whose values make up the domain
x-intercept
The x-coordinate of the point at which a graph crosses the x-axis
Recursive Formula
Each term is formulated from one or more previous terms.
y-intercept
The y-coordinate of the point at which a graph crosses the y-axis
Coordinate Plane
Composed of the x-axis (horizontal) and the y-axis (vertical) which meet at the origin (0,0) and divide the plabe into four quadrants
Pascal's Triangle
a triangular array of numbers
point-slop form
y-y1=m(x-x1)
Iteration
process of composing a function with itself repeatedly.
Infinite Geometric Series
geometric series with an infinite amount of terms.
Inductive Hypothesis
The assumption that a statement is true for some positive integer, k
subscript notation
to distinguish between the coordinates of two general points on a line segment
Index of Summation
n in a sigma notation problem
subscript notation
to distinguish between the coordinates of the points we use ___________
Domain
The set of all first coordinates from the ordered pairs
straight line depreciation
used when aging equipment declines in value
Geometric Sequence
sequence in which each term after the first is found by multiplying the previous term by the constant r.
Range
The set of all second coordinates from the ordered pairs
Linear Function
A fuction whose ordered pairs satisfy a linear equation
function
a rule that assigns to each value one variable
Section 19
(50 cards)
Parabola
Graph of a Quadratic Function
Addition method
multipy to cancel out, add equations, cancel out variable, solve for a variable, substitute back into other equation, solve
complex fraction
rational expression whose numerator and or denominator contains a rational expression.
Arithmetic Series
indicated sum of the terms of an arithmetic sequence
Term
each number in a sequence
substitution method
solve on equation, substitute first variable, substitute that variable into the next equation, solve
Sequence
a list of numbers in a particular order.
Vertex Form
Quadratic Equation written as y=a(x-h)+k
Intercept Form
Quadratic equation written as y=a(x-q)(x-p)
Binomial
A polynomial with two terms that can be added or subtracted.
Standard Form
Quadratic equation written as y=ax²+bx+c
Axis of Symmetry
Line that goes through the parabola that splits it into two mirror images.
asymptote
line that the graph of the function approaches but never crosses.
direct variation
y=kx
positive number
a number greater than zero
dashed line
< or >
Maximum Value
The highest point on the parabola. Parabola opens down if the leading constant is negative.
rational equation
any equation the contains one or more rational equations.
point discontinuity
hole in a graph
rational inequality
inequalities containing one or more rational expressions.
Quadratic Function
An equation where the maximum exponent is 2.
transformation
an operation that maps an original geometric figure onto a new figure called an image
z axis
a vertical line through the origin of the graph
octants
the three axis determine three coordanate planes that divide space into octants
Common difference
D
solution
an ordered pair that satisfies a system of equations
reflection
when you flip a figure over a line
Monomial
A polynomial with one term that can be multiplied.
Series
indicated sum of the terms of a sequence
translation
when you slide a figure from one position to another without turning it
3D coordinate system
a graph with three systems
Minimum Value
The lowest point on the parabola. Parabola opens up if the leading constant is positive.
opposites
two numbers with the same absolute value but different signs (-4 and 4)
graph of a system
the graph of all solutios of the system
quadrant
the x-axis and the y-axis seperate the coordinate plane into four regions. (I , II, III, IV)
inverse variation
As one quanity increases, the other one decreases. xy=k
constant of variation
The constant K is used with direct or inverse variation.
Quadratic Equation
Same as a Quadratic Formula. An equation where the maximum exponent is 2.
a linear equation with thee variables
ax + by + cz = d
Trinomial
A polynomial with three terms that can be added or subtracted.
continuity
a graph of a function that can be traced with a pencil that never leaves the paper.
joint variation
occurs when one quanity varies directly as the product of two or more other quanities. y=kxz
Arithmetic Means
terms between any two nonsuccessive terms of an arithmetic sequence
ordered triple
(x,y,z)
Arithmetic Sequence
sequence in which each term after the first is rounded by adding a constant.
rational function
f(x)=x/x+3
rational expression
a ratio of two polynomial expressions.
solid line
≤ or ≥
negative number
a number less than zero
Vertex
The point where the parabola turns, or crosses the axis of symmetry. Minimum or maximum.
Section 20
(50 cards)
focus (horizontal parabola)
(h + (1/4a), k)
cot
cosθ/sinθ
tan
1/cotθ
horizontal hyperbola
(x-h)²/a² - (y-k)²/b² = 1
axis of symmetry (horizontal parabola)
y = k
vertical parabola
y = a(x-h)² + k
Geometric Sum
Sn = (a₁-a₁rⁿ) / (1-r)
vertical hyperbola
(y-k)²/a² - (x-h)² /b²= 1
Arithmetic Series
an = a₁ + (n-1)d
1
sin²θ + cos²θ =
hyperbola asymptotes (horizontal axis)
y = ± (b/a)x
ellipse with horizontal major axis
((x-h)²/a²) + ((y-k)²/b²) = 1
horizontal hyperbola with origin as center
x²/a² - y²/b² = 1
sec
1/cosθ
coordinate
the number that corresponds to the point on a number line
horizontal parabola
x = a(y-k)² + h
focus (vertical parabola)
(h, k + (1/4a))
horizontal ellipse (with origin as center)
(x²/a²) + (y²/b²) = 1
csc
1/sinθ
length of latus rectum
|(1/a)|
additive inverse
an integer and its opposite
tan
sinθ/cosθ
vertical ellipse (with origin as center)
(x²/b²) + (y²/a²) = 1
distance formula
d = √((x₂-x₁)² + (y₂-y₁)²)
Geometric Sum
Sn = a₁(1-rⁿ) / 1-r
Geometric Series
an = a₁rⁿ⁻¹
Law of Sines
(Sin A)/a = (Sin B)/b = (Sin C)/ c
integer
any number from the set {..., -3, -2, -1, 0, 1, 2, 3, ...}
cos
1/secθ
Arithmetic Sum
Sn = (n/2)(a₁ + an)
ellipse with vertical major axis
((x-h)²/b²) + ((y-k)²/a²) = 1
directrix (horizontal parabola)
x = h- (1/4a)
Law of Cosines
c²=a²+b²-2ab(cosC)
Definition of Sine
sinθ = y
absolute value
the distance the number is from zero on a number line. (-6 and 6 have the same absolute value)
inequality
compares numbers or quantities (any mathematical sentence containing < or >)
vertex of a parabola
(h,k)
cot
1/tanθ
hyperbola asymptotes (vertical axis)
y = ± (a/b)x
circle
(x-h)² + (y-k)² = r²
sin
1/cscθ
equation of a conic section
Ax² + Bxy + Cy² + Dx + Ey + F = 0 (A,B, & C cannot all be zero)
vertical hyperbola with origin as center
y²/a² - x²/b² = 1
Definition of Cosine
cosθ = x
Infinite Series Sum
S = a₁ / (1-r)
axis of symmetry (vertical parabola)
x = h
Geometric Series
an = a(n-1) - 1^r
mean
find the sum of the numbers and then divide by the number of items in a set
directrix (vertical parabola)
y = k- (1/4a)
midpoint of a line coordinates
(x₁+x₂)/2, (y₁+y₂)/2
Section 21
(50 cards)
Inconsistent
a system of equations that has no solutions
sin(x-y)
sinx cosy-cosx siny
sin (x/2)
±√(1-cosx)/2
Area of a Triangle
1/2 ab sinC
10
³√1,000
Intercept form
x/a + y/b=1
Line of fit
a line that closely approximates a set of data
identity matrix
[1/0 0/1]
inverse matrix
the products of 2 matrices is the identity matrix
equal
2 matrices that are exactly the same
Dependent
a system of equations that has infinite solutions
Regression line
a line of best fit
4
2²
parallel lines
a system with no solution
cos2x
cos²x-sin²x
sin2x
2(sinx)(cosx) =
cos2x
2cos²x-1
System of inequalities
a set of inequalities with the same variables
1
√1
8
³√512
single lines
a system with infinite solutions
scalar
number used to multiply
dimension
the size of the matrix
cos(x+y)
cosx cosy-sinxsiny
cos (x/2)
±√(1+cosx)/2
sin(x+y)
sinx cosy+cosx siny
343
7³
System of equations
two or more equations with the same variables
sec²θ
1 + tan²θ =
Prediction equation
an equation suggested by the points of a scatter plot that is used to predict other points
Substitution method
a method of solving a system of equations in which one equation is solved for one variable in terms of the other
Scatter plot
a set of data graphed as ordered pairs in a coordinate plane
Linear correlation coefficient
the number r
Median-fit line
a particular type of line of fit
Line of best fit
a line that best fits a set of data
cos2x
1-2sin²x
Consistent
a system of equations that has a least one solution
matrix
a rectangular arrangement of numbers
8
√64
determinant
a real number associated with a square matrix
7
√49
0
0²
Boundary
line or curve that separates the coordinate plane into two regions
csc²θ
1 + cot²θ =
system of equations
a collection of linear equations invoking the same set of variables
5
√25
1
1²
cos(x-y)
cosx cosy+sinx siny
Independent
a system of equations that has exactly one solution
Elimination method
eliminate on of the variables in a system of equations by adding or subtracting the equations
Section 22
(50 cards)
Slope
the ratio of the change in y-coordinates to the corresponding change in x-coordinates
Translation matrix
a matrix that represents a translated figure
Quadrants
the four areas of a Cartesian coordinate plane
Solution
each replacement that results in a true sentence
Example set-builder notation
{x I x>9}
Augmented matrix
a coefficient matrix with an extra column containing the constant terms
Arithmetic mean
the terms between any two nonconsecutive terms of an arithmetic sequence
Relation
a set of ordered pairs
Ordered Pairs
representation of data
Functional notation
an equation of y in terms of x can be rewritten so that y=f(x)
x-intercept
the x-coordinate of the point at which a graph crosses the x-axis
Arccosine
the inverse of y = cos x, written as x = arccos y
Determinant
a square array of numbers or variables enclosed between two parallel lines
Example interval notation
(-infinity, 2) or [-2, +infinity)
Axis of symmetry
a line about which a figure is symmetric
Multiplication Property of Inequality 2
For any real numbers a, b, and c, where c is negative: If a>b, then ac<bc. If a<b, then ac>bc
Third-order determinant
determinants of a (3)(3) matrix
Zero matrix
a matrix in which every element is zero
Equation
a mathematical sentence stating that two mathematical expressions are equal
Mapping
shows how each member of the domain is paired with each member of the range
Division Property of Inequality 2
For any real numbers a, b, c, where c is negative: If a>b, then a/c<b/c. If a<b, then a/c>b/c.
Asymptote
a line that a graph approaches but never crosses
Intersection
the graph of a compound inequality containing "and"
Vertices
the maximum of minimum value that a linear function has for the points in a feasible region
Union
the graph of a compound inequality containing "or"
Arctangent
the inverse of y = tan x, written as x = arctan y
Angle of elevation
the angle between a horizontal line and the line of sight from the observer to an object at a higher level
Cartesian coordinate plane
a plane divided into four quadrants by the intersection of the x-axis and the y-axis at the origin
Family of graphs
group of graphs that displays one or more similar characteristics
Division Property of Inequality
For any real numbers, a, b, c, where c is positive: If a>b, then a/c>b/c. If a<b, then a/c<b/c
Transitive Property of Equality
For all real numbers a, b, and c, if a=b and b=c, then a=c
Arithmetic sequence
a sequence in which each term after the first is found by adding a constant, the common difference d, to the previous term
Addition Property of Inequality
For any real numbers a, b, c: If a>b, then a+c>b+c. If a<b, then a+c<b+c
Multiplication Property of Inequality
For any real numbers a, b, and c, where c is positive: If a>b, then ac>bc. If a<b, then ac<bc
Translation
a figure is moved from one location to another on the coordinate plane without changing its size, shape, or orientation
Parent graph
simplest of the graphs in a family
y-intercept
the y-coordinate of the point at which a graph crosses the y-axis
Vertical line test
if no vertical line intersects a graph in more than one point, then the graph represents a function
Reflexive Property of Equality
For any real number a, a=a
Transformation
functions that map points of a pre-image onto its image
Dependent variable
the other variable in a function, usually y, whose values depend on x
Properties of Equalities
Reflexive, Symmetric, Transitive, Substitution
Angle of depression
the angle between a horizontal line and the line of sight from the observer to an object at a lower level
Arithmetic series
the indicated sum of the terms of an arithmetic sequence
Arcsine
the inverse of y = sin x, written as x = arcsin y
Symmetric Property of Equality
For all real numbers a and b, if a=b then b=a
Substitution Property of Equality
If a=b, then a may be replaced by b and b may be replaced by a
Independent variable
in a function, the variable, usually x, whose values make up the domain
One-to-one function
a function where each element of the range is paired with exactly one element of the domain
Subtraction Property of Inequality
For any real numbers a, b, c: If a>b, then a-c>b-c. If a<b, then a-c<b-c.
Section 23
(50 cards)
Feasible Region
intersection of the inequalities (shaded region)
Types of Word Problems
Integer Problems, Angle Problems, Geometry Problems, Motion Problems
function
all x's must be different, x's are used one time only,use vertical line test on the graph
relation
set of ordered pairs
Linear Equation
an equation that contains no operations other than addition, subtraction, and multiplication of a variable by a constant
Rational Numbers
numbers that can be written as a fraction, m/n, where m and n are integers and n does not equal 0; if written as a decimal, it will either terminate or it will repeat
Associative Property of Multiplication
(ab)c=a(bc)
A relation can be represented by
an equation, a list of ordered pairs, a mapping or a graph
(a^2)( a^6)
a^8
Trichotomy Property
for any two real numbers, a and b, exactly one of the following is true: a<b, a>b, or a=b
Irrational Numbers
a real number that is not rational; if written as a decimal, it does not terminate nor repeat
Slope Intercept Form
y=mx+b
Inverse Property of Addition
a+(-a)=0=(-a)+a
Counterexample
specific case that shows that a statement is false
Standard Form
Ax+By=C
Formula
mathematical sentence that expresses the relationship between certain quantities
Commutative Property of Multiplication
(a)(b)=(b)(a)
Identity Property of Multiplication
(a)(1)=a=(1)(a)
Constraints
integers
Commutative Property of Addition
a+b=b+a
Trace
The line in which a plane intersects a coordinate plane
Supplement
180-x
Linear Programming
the process of finding the maximum or minimum value of a function for a region defined by inequalities
Open Sentence
a mathematical sentence containing one or more variables
Grouping Symbols
parenthesis, brackets, braces
Function
special type of relation in which each element of the domain is paired with exactly one element of the range
Relation
set of ordered pairs
Distributive Property
a(b+c)=ab+ac and (b+c)a=ba+ca
Point Slope Form
y-y1 =m(x-x1 )
Compound Inequalities
two inequalities joined by the word "and" or the word "or"
range
y-coordinates, output
Order of Operations
1. grouping symbols; 2. exponents; 3. multiplication and division; 4. addition and subtraction
Algebraic Expression
an expression that contains at least one variable
Absolute Value
the distance an object is from zero on the number line
Absolute Value Inequalities
Get the absolute value by itself on the left side. Then check to see if it is a "greater" or "less than" statement
Five Special Functions
Identity Function, Absolute Value Function, Greatest Integer Function, Piecewise Function
Unbounded Feasible Region
when the feasible region is not closed
Domain
set of all first coordinates from the ordered pairs
Real Numbers
All of the numbers that you use in everyday life
Identity Property of Addition
a+0=a=0+a
Variable
symbol, usually letters, used to represent unknown quantities
Closure Property
The product of any two whole numbers is always a whole number. So, the set of whole numbers is said to be closed under multiplication
Range
set of all second coordinates from the ordered pairs
Scatter Plot
Data graphed in ordered pairs
domain
x-coordinates, input
Associative Property of Addition
(a+b)+c=a+(b+c)
Bounded Feasible Region
when the feasible region is a polygon
Inverse Property of Multiplication
If a does not equal 0, then (a)(1/a)=1=(1/a)(a)
Complement
90-x
To solve compound inequalities
If it is an "and" statement, graph the intersection. If it is an "or" statement, graph the union
Section 24
(50 cards)
What does the rational exponent mean?
power/root = "power over root"
Area of a triangle given coordinates for the three vertices?
Enter three vertices, (x, y) in a 3 X 3 matrix, in rows 1, 2, and 3, with 1's in the last column. Take plus or minus 1/2 of the determinant.
joint variation
y=kxz, y varies jointly with x and z
Column Matrix
Consists of one column
Standard Form for a Line
Ax + By = C
Intercept Form for a Quadratic
y = a(x - p)(x - q)
What do you know about slopes of perpendicular lines?
Perpendicular lines have slopes that are opposite reciprocals of each other!
How do you factor a sum or difference of cubes?
first one, second one, first one squared, product of the two and the last one squared, first sign's the same, second one's not, last one's always plus and here's what you've got!
log base a of x + log base a of y =
log base a of (x times y) This is the product property of logarithms.
What must be true to multiply two matrices?
The columns of the first must match the rows of the second matrix.
Write log base a of b = c in exponential form .
a ^ c = b
Completing the Square
Remember to have a coefficient of 1 for the squared term. Take 1/2 of b and square it. Add to both sides. Solve through square roots.
What must be true to add two matrices?
The dimensions must be the same. (Rows and Columns)
Compound Interest Formula
A = P(1 + r/n)^(n x t), r is the rate, n is the number of times compounded, t is time
What ways can you solve a system?
1. graph, 2. substitution, 3. linear combination or elimination, 4. use a matrix equation, if the system is linear
a^12
(a^2)^6
What type of matrix must you have to take a determinant?
A square matrix
Methods to Solve a Quadratic
1. Graphing, 2. Factoring, if factorable, 3. Completing the Square, 4. Quadratic Formula
Quadratic Formula
x = (-b plus or minus) the square root of b^2 -4ac) / 2a from y = ax^2 + bx + c
Range for a Logarithmic Function?
All real numbers
What does the value of the discriminant tell about a quadratic?
If the discriminant = 0, there is one real solution & graph "sits or bumps" x-axis. If the discriminant > than 0, there are two real solutions & graph crosses x-axis twice. If the discriminant < 0, there are no real solutions, two imaginary solutions & graph does not cross x-axis.
Standard Form for a Quadratic
f(x) = ax^2 + bx + c
Domain for an Exponential Function?
All real numbers
Standard Form Equation for a Circle with center (h, k)
(x - h)^2 + (y - k)^2 = r^2, circle with center at (h, k) and radius of r
Matrix
Rectangular array of numbers in rows and columns
Row Matrix
Consists of one row
x-intercepts for rational expressions
set numerator equal to zero and factor, remember to divide out any common factors from denominator!
Continuous Compounding
A = Pe^(rt)
What's i^2?
-1
Point-Slope Formula
y - y(sub1) = m(x - x(sub 1))
What do you know about slopes of parallel lines?
Parallel lines have the same slope!
x log base n of y =
log base n of y ^ x This is the power property of logarithms.
What is the discriminant?
b^2 minus 4ac
vertical asymptoptes for rational expressions
set denominator equal to zero and factor; remember to divide out any common factors from numerator!
horizontal asymptotes for rational expressions
(BOSTON)-if degree of Bottom is bigger, y=0; if degrees are same for numerator and denominator, use coefficients a/c; if Top has a larger degree, NO horizontal asymptote
How do you find the inverse of a function?
1. "y = " form, 2. switch x and y, 3. solve for y
Solving Absolute Value Inequalities
GOLA; Greater than - OR, Less than - AND
inverse variation
y=k/x
When solving a matrix equation, use...?
"A inverse times B" meaning A^-1 times B
What's the square root of -1?
i
Vertex Form for a Quadratic
y = a(x - h)^2 + k
Does every matrix have an inverse?
No. If the determinant equals 0, there will not be an inverse.
Slope Formula
m = (y (sub 2) - y (sub 1))/(x (sub 2) - x(sub 1)), difference of the y's over the difference of the x's
direct variation
y=kx
What is the slope for a horizontal line?
0
log base a of x - log base a of y =
log base a of (x divided by y) This is the quotient property of logarithms.
Standard Form Equation for a Circle with center at the origin
x^2 + y^2 = r^2
How do you set up a matrix equation to solve a linear system of equations?
There are three matrices in the equation. A coefficient matrix, where coefficients come from standard form equations, a variable matrix, and a constant matrix.
How do you solve an absolute value equation?
Set the expression from inside the absolute value equal to the positive constant and to the negative constant.
Slope-Intercept Form for a Line
y = mx + b
Section 25
(50 cards)
Rate
a ratio that involves two different units
FOIL
A method for multiplying binomials
How to determine if its a function?
Must pass the vertical line test. (Y only lands on 1 point)
Sn=n(a1+an/2)
Arithmatic Sum
Slope intercept form
y=mx+b
Yes, and the answer will be negative. A negative number raised to an odd power equals a negative.
Can you take a cube root of a negative number?
Vertical line has what type of slope?
Undefined slope
Composition of Functions
f ( g(x)) means that function g is the input to function f
Absolute value function
V shape Y = IxI (make y and x chart)
Constant functions
Consistent horizontal, slope = 0 (Horizontal line)
Polynomial
An algebraic expression with "many" terms
Monomial
A polynomial with one term
Discrete function
a relation in which the domain is a set of individual points that are not connected
linear equation in one variable
can be written in the form of ax=b
Quadratic
Second degree polynomials are called...
Slope-intercept form
y=mx+b
f ( g(x)) = g (f(x)) = x
How do you know that two functions are inverses of each other?
Slope Formula
y1-y2 / x1-x2
Sn=a1(1-r^n/1-r)
Geometric Sum
Trinomial
A polynomial with 3 unlike terms
An=a1+(n-1)d
Arithmatic series
y = x
What is the line of reflection for inverses?
Continuous function
When the domain of a relation has an infinite number of elements and the relation can be graphed with a line or smooth curve.
Identity function
Y= 1 x X + 0 Always: y=x (Diagonal line)
equation
mathematical statement that two expressions are equivalent
Horizontal line has what type of slope?
Zero slope
Point slope form
y-y1 = m(x-x1)
Ratio
a comparison of two numbers by division
Linear
First degree polynomials are called...
What is a determinant?
A real number associated with a square matrix.
To solve for x and y intercept
plug in zero
y = a(x - h)^2 + k
Absolute Value Equation with vertex (h, k)
Binomial
A polynomial with 2 unlike terms
similar
their corresponding angle are congruent and corresponding sides are proportional
Inequality
a statement that compares two expressions by using symbols <,>
Parallel lines
Same slope
What's the horizontal line test?
Used on an original function to determine if the inverse would be a function. Yes, you may still use the vertical line test on an inverse.
Identity
an equation that is true for all values of the variable
What is a circle?
The set of all points (x, y) that are equidistant from a fixed point called the center.
Standard form
Ax + By = C
Cubic
Third degree polynomials are called...
Vertical line test
Used to determine whether the reaction is a function
proportion
an equation stating that two ratios are equal
Quadratic function
Parabola, U shape, y = x2 (make y and x chart)
An=a1R^n-1
Geometric series
S=a1/1-r
Geometric infinite sum
Quartic
Fourth degree polynomials are called...
Two, positive and negative roots
How many roots does a square root have?
Perpendicular lines
Negative reciprocal
solution set of an equation
value or values of the variable that make an equation true
Section 26
(50 cards)
Principal
an amount of money borrowed or invested
Similar
two figures that have the same shape, but not necessarily the same size
correlation coefficient
measure of how well the data set is fit by a model
discount
an amount by which an original price is reduced
boundary line
divides the coordinate plane into two regions
x-intercept
the x-coordinate of a point where the line crosses the x-axis
Percent Change
an increase or decrease given as a percent of the original amount
absolute value
the distance from zero to x on a number line
regression
the statistical study of the relationship between two variables
disjunction
a compound statement that uses the word or
Formula
a literal equation that states a rule for a relationship among quantities
Line of best fit
a line used to make predictions that best fits the data
Scale Factor
a factor used to produce a similar object
y-intercept
the y-corrdinate of a point where the line crosses the y-axis
Unit Rate
a rate in which the second quantity in the comparison is one unit
Indirect measurement
uses known lengths, similar figures and proportions to measure objects that cannot be easily measured
Tip
an amount of money added to a bill for service
Scale Model
a three-dimensional model that uses a scale to represent an object as smaller or larger than the actual object
Scale Drawing
a drawing that uses a scale to represent an object as smaller or larger than the original object
Indirect Masurement
a method of measuring an object by that is difficult to measure
Scale
the ratio of any length in a drawing to the corresponding actual length
Linear function
functions with a constant rate of change: f(x)=mx+b
Correlation
the strength and direction of the linear relationship between two variables
Constant Ratio
A Ratio that a has constant value
Corresponding Angles
angles in the same relative position in two different polygons that have the same number of angles
Ratio
a comparison of two numbers by division
Direct Variation
The Relationship between two quantities x and y have a constant Ratio;One qantity varies directly with the other
Equation
A statement that is formed by placing an equal sign between the two expressions
slope intercept form
y=mx+b
slope
the constant rate of change for a linear function
Percent Decrease
a decrease given as a percent of the original amount
Commission
money paid to a person or company for making a sale
Solution
a value or values that make the equation or inequality true
absolute value function
a function whose rule contains an absolute value expression
Equation
a mathematical sentence that shows that two expressions are equivalent
Interest
the amount of money charged for borrowing money or the amount of money earned when saving or investing money
Percent
a ratio that compares a number to 100
point-slope form
y-y1=m(x-x1)
Corresponding Sides
sides in the same relative position in two different polygons that have the same number of sides
Rate
a ratio that compares two quantities measured in different units
Proportion
an equation that states that two ratios are equal
Sales Tax
a percent of the cost of an item that is charged by governments to raise money
Percent Increase
an increase given as a percent of the original amount
Cross Products
the product bc and the product ad in the statement a = b
Literal Equation
a decrease given as a percent of the original amount
Linear inequality
relates two variables using an inequality symbol y>2x-4
conjunction
a compound statement that uses the word and
Identity
an equation that is true for all values of the variables
Markup
the amount by which a wholesale cost is increased
Conversion Factor
the ratio of two equal quantities, each measured in different units
Section 27
(51 cards)
Sample Standard Deviation
Sx
Range
set of dependent variables (y values) assumed by a function or relation over all permitted values of the independent variable(s)
Decreasing function
a function with a graph that goes down as it is followed from left to right
Amplitude of y = a sin bx
|a|
1. Y-intercept: x=0 is not a root of the denominator, so evaluate function at x=0. y=-6 2. Horizontal Asymptote(s): oblique asymptote, divide the equation to find it. x+18 3. End behavior of graph: x3/x2 which would be a line increasing as x increasing that crosses the graph at x=2 and x=6 4. Degree of numerator/denominator: 3/2
x^3+10x^2+12x-72 ----------------------- x^2-8x+12 1. Y-intercept: 2. Horizontal Asymptote(s): 3. End behavior of graph: 4. Degree of numerator/denominator:
log a + log b
log (ab)
f(x) notation
same as y=
Period of y = a sin bx
(2π)/b or (360°)/b
unit rate
Rate per given unit
Increasing function
a function with a graph that goes up as it is followed from left to right
Continuous
a connected set of numbers such as the set of real numbers
Tax rate
The Percent used to calculate the amount of money taken out of your gross pay
Conjugate of a binomial (a+b)
Switch the sign in the middle (a-b)
Horizontal asymptote at y=a/b
Asymptote of a^n ------- b^n
Percent
A ratio whose Denominator is 100.
Gross Pay
The total amount an employee earns before any taxes or deductions are made
randomly chosen
This means that no particular rule was used to choose a person
Quadratic Function
a function that when graphed forms a parabola or curve
Net pay
The amount of money that an employee earns after all of the deductions are made from the gross pay
direct variation
two variable quantities that have the same rate or ration, regardless of the values of the variables
log a^c
c log a
corresponding sides
Two Congruent figures that are pairs of a side that are same relative position in both figures
log a - log b
log (a/b)
Exponential Function
function that shows a growth or decay rate is proportional to the current value
Population Standard Deviation
σx
Independent Variable
a variable in an equation that may have its value freely chosen without considering values of any other variable
Ratio
A way to compare two quantities in the same unit by using division
Horizontal asymptote at y=0
Asymptote of a^n ------ b^N
Sample
A group of items selected at random from a larger group for research
means
Two inside quantities of a proportion.
Representations
concrete models, tables, graphs, verbal descriptions, and algebraic generalizations of data
Inverse Function
function obtained by switching the x and y values
Proportion
An equation that states two ratios are equal.
imaginary
Nature of the roots when b2 - 4ac is negative
Oblique asymptote at (divide the equation to find it)
Asymptote of a^N ------ b^n
similiar
Figures that look the same, but have different sizes
Linear Function
function that when graphed form a line
No line asymptote
Asymptote of a^NN --------- b^n
Dependent Variable
a variable that depends on one or more other variables
Frequency of y = a sin bx
b
Relation
a set of ordered pairs (pairing a set of inputs (domain) with a set of outputs (range))
sampling methods
A method for selecting people or items from a population for a survey. Four types: random, stratified, systematic, convienence, and self selected
Domain
set of independent variables for which the function is defined
rate
A ratio in which the units of the quantities being compared are different
extremes
The greatest and the least value of a set of data
real, rational, equal
Nature of the roots when b^2 - 4ac is zero
biased
A sample that does accurately represent all of a population
Function
relations in which each element in the domain (input) is paired with exactly one element in the range (output)
Discrete
a set of elements that are disconnect such as the set of integers
Variance
(Standard Deviation)2
Section 28
(51 cards)
isosceles triangle
A triangle having two sides of equal length. The third side is called the "base". The angles opposite the equal sides are the "base angles".
input
The output is a function of (depends upon) the
area
The area of a region is the number of square units it contains.
abscissa
The first coordinate is am ordered pair pf numbers associated with a point in a coordinate plane. Also called x-coordinate.
conjunction
A sentence formed by joining two open sentences by the word and.
zeros of a quadratic function
The members of the solution set of the related quadratic equation.
whole numbers
The set consisting of zero and all the positive integers.
collinear points
Points that lie on the same line.
maximum point of a parabola
The highest poing; the point whose y-coordinate is the greatest value of the corresponding function.
variable expression
An expression that contains a variable.
degree of a polynomial
The greatest of the degrees of its terms after it has been simplified.
binomial
A polynomial of only two terms.
quartic function
can have 3 turning points
terminating decimal
When a common fraction is written as a decimal by dividing the numerator by the denominator, the result is a terminating decimal if the remainder is zero. Also called "ending decimal" or "finite decimal"
simplipying an expression
Replacing an expression containing variables by an equivalent expression with as a few terms as possible.
irreducible polynomial
A polynomial that cannot be expressed as a product of polynomials of lower degree.
vertical angles
Two angles whose sides are rays in the same line, but in opposite directions.
Polynomial Functions
When the absolute value of x is large, end/long-run behavior of the graph will tend to follow the leading term
solution set of an open sentence
The set of all solutions of the sentence.
absolute value
The positive number of any pair of opposite nonzero real numbers is the absolute value of each number. The absolute value of 0 is 0.
additive inverses
A number and it's opposite.
standard form of a polynomial equation
A form of the equation in which one side is zero and the other is a simplified polynomial arranged in the order of decreasing degree of the variable.
Vertical Line Test:
Vertical line should not intersect graph in more than one place. If it does, then it is not a function.
constant (monomial)
A monomial consisting of a numeral only; a term with no variable factor.
simplest form of an algebraic fraction
A form of the fraction in which the numerator and denominator have no common factor other than 1 and -1.
consecutive integers
Numbers obtained by counting by twos, beginning with any odd integer.
A polynomial of degree/order "n"
polynomial of degree/order "n" can have a maximum of n roots
The domain of a rational function is the set of all real numbers that are NOT roots of the denominator (the denominator≠0)
What is the domain of a rational function?
angle
A figure formed by two different rays that have the same end-point.
degree of a variable of a monomial
The number of times that the variable occurs as a factor in the monomial.
common factor
A factor of two or more integers is called a common factor of the integers.
The output variable
is dependent (it depends upon the input)
boundary
A line that separates the coordinate plane into three sets of points: the points on the line; the points below the line.
quadratic function
can have 1 turning point
Rational Zeros Theorem
Finding the rational (not irrational) zeros of a polynomial function:
The Fundamental Theorem of Algebra
Every polynomial of a degree of ≥1 with complex coefficients has at least one zero in the complex number system.
nonterminating decimal
When a common fraction is written as a decimal by dividing the numerator by the denominator, the result is a nonterminating decimal if a digit or block of digits repeats endlessly as the remainder. Also called "unending?, "infinite", "repeating", or "periodic decimal".
The input variable
is independent (may have its value freely chosen regardless of any other variable values)
cubic function
can have 2 turning points
The Linear Factorization Theorem
For every polynomial function where the degree is >0, there are complex numbers such that f(x)=a(x-c1) (x-c2)... etc... (as long as a≠0)
similar triangles
Two triangles are similar if the measures of two angles of one triangle equal the measures of two angles of the other. Angles with equal measures are called "corresponding angles". Sides opposite corresponding angles are called "corresponding sides".
average
The average of a set of numbers is the sum of the numbers divided by the number of numbers.
constant of variation
The nonzero constant k in a direct variation defined by y=kx or in an inverse variation defined by xy=k. Also called "constant of proportionality".
complex fraction
A fraction whose numerator or denominator contains one or more fractions.
perfect square trinomial
An expression with three terms that is the square of a binomial.
factored completely
A polynomial is factored completely when it is expressed as the product of a monimial and one or more prime polynomial.
random experiment
An activity whose outcome can't be predicted when the activity is repeated under essentially the same conditions.
converse
The sum of the degrees of the variables in the monomial.
simplest form of an expression having a square-root radical
The form of the expression in which no intergral radicand has a square factor other than one; no fractions are under a radical sign; no radicals are in the denominator.
hypotenuse
In a right triangle, the side opposite the right angle.
Section 29
(13 cards)