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