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A–F

absolute value
Unit 4 Lesson 7, Unit 8 Lesson 3

A number’s distance from zero on the number line.

The symbol means the absolute value of .

Recall that distance is always positive.

The diagram shows that and .

number line explaining absolute value x–2–2–2–1–1–1111222000
absolute value function
Unit 8 Lesson 3

A function that contains an algebraic expression within absolute value symbols. The absolute value parent function, written as:

an absolute value function on a graphx–3–3–3–2–2–2–1–1–1111222333y111222333000
angle
Unit 7 Lesson 1

Two rays that share a common endpoint called the vertex of the angle.

lines creating angles
arithmetic mean
Unit 1 Lesson 8

The arithmetic mean is also known as the average. The arithmetic mean between two numbers will be the number that is the same distance from each of the numbers. It is found by adding the two numbers and dividing by .

Untitled555666777888999101010111111121212

The arithmetic mean of several numbers is found by adding all of the numbers together and dividing by the number of items in the set:

Example: Find the arithmetic mean of

arithmetic sequence
Unit 1 Lesson 2

The list of numbers represents an arithmetic sequence because, beginning with the first term, , the number has been added to get the next term. The next term in the sequence will be () or .

an arithmetic sequence beginning at 3 with 7 being added each time

The number being added each time is called the constant difference ().

The sequence can be represented by a recursive equation.

In words:

Name the .

Using function notation:

function notation the firsttermthe nthoutputthe outputone before f(n)constantdifference

An arithmetic sequence can also be represented with an explicit equation, often in the form where is the constant difference and is the value of the first term.

The graph of the terms in an arithmetic sequence are arranged in a line.

associative property of addition or multiplication
Unit 4 Lesson 3

See properties of operations for numbers in the rational, real, or complex number systems.

augmented matrix
Unit 5 Lesson 11

An augmented matrix for a system of equations is a matrix of numbers in which each row represents the constants from one equation (both the coefficients and the constant on the other side of the equal sign) and each column represents all the coefficients for a single variable.

Given the system:

Here is the augmented matrix for this system:

an augmented matrix
average rate of change
Unit 2 Lesson 5

See rate of change.

bimodal distribution
Unit 9 Lesson 6

A bimodal distribution has two main peaks.

The data has two modes.

See also: modes.

a bimodal histogram2224446662020204040406060608080800002 modesbimodal distribution
binomial
Unit 7 Lesson 3, Unit 7 Lesson 6, Unit 8 Lesson 9

A polynomial with two terms.

a binomial of (ax b)termtermaddition or subtraction
binomial expansion
Unit 8 Lesson 9

When a binomial with an exponent is multiplied out into expanded form.

Example:

Pascal’s triangle (shown) can be used to find the coefficients in a binomial expansion. Each row gives the coefficients to , starting with . To find the binomial coefficients for , use the row and always start with the beginning variable raised to the power of . The exponents in each term will always add up to . The binomial coefficients for are , , , , , and — in that order or

The first 6 rows of Pascal's triangle
bivariate data
Unit 9 Lesson 1

Deals with two variables that can change and are compared to find relationships. If one variable is influencing another variable, then you will have bivariate data that has an independent and a dependent variable (ordered pairs). This is because one variable depends on the other for change.

box and whisker plot (box plot)
Unit 9 Lesson 6

A one-dimensional graph of numerical data based on the five-number summary, which includes the minimum value, the percentile , the median, the percentile , and the maximum value. These five descriptive statistics divide the data into four parts; each part contains of the data.

a box and whisker plot 222333444555666777888999minimumvaluemedianmaximumvaluenumber line that includes the numbers in the 5-number summaryand uses appropriate units of equal distance

Boxplots can be vertical or horizontal.

categorical data or categorical variables
Unit 9 Lesson 6

Data that can be organized into groups or categories based on certain characteristics, behavior, or outcomes. Also known as qualitative data.

categories yes and no25 YES15 NO“Yes” and “No”are categories.
a chart giving categorical data categoriescategoriesGirlsBoystotalsoccerdance144054466526046106total
causation
Unit 9 Lesson 1

Tells you that a change in the value of the variable will cause a change in the value of the variable.

center (statistics)
Unit 9 Lesson 6

A value that attempts to describe a set of data by identifying the central position of the data set (as representative of a “typical” value in the set). Measure of center refers to a measure of central tendency (mean, median, or mode).

change factor (pattern of growth)
Unit 1 Lesson 3

A change factor is a multiplier that makes each dependent variable grow (or sometimes decrease) as the independent variable increases. Sometimes called the growth factor.

In a geometric sequence it is the common ratio.

In an exponential function it is the base of the exponent.

closure
Unit 8 Lesson 10

A set is closed (under an operation) if and only if the operation on any two elements of the set produces another element of the same set.

A diagram showing that 5 2=7 is closed under addition and 2-5=-3 is not closed under subtraction5 and 2 and 7 are natural numbersThe natural numbers areclosed under addition2 and 5 are natural numbersThe natural numbers are NOTclosed under subtraction-3 isNOT anaturalnumber.
common ratio (r) (constant ratio)
Unit 1 Lesson 3

The change factor or pattern of growth () in a geometric sequence. To find it divide any output by the previous output.

Example: is a geometric sequence.

Output

Input

The common ratio is

commutative property of addition or multiplication
Unit 4 Lesson 3

See properties of operations for numbers in the rational, real, or complex number systems.

completing the square
Unit 7 Lesson 3

Completing the Square changes the form of a quadratic function from standard form to vertex form. It can be used for solving a quadratic equation and is one method for deriving the quadratic formula.

compound inequality
Unit 4 Lesson 7

An inequality that combines two simple inequalities.

compound inequality in one variable
Unit 4 Lesson 5

A compound inequality contains at least two inequalities that are separated by either “and” or “or.”

  • an inequality that combines two inequalities either so that a solution must meet both conditions (and ) or that a solution must meet either condition (or ).

Examples:

can be written as

(The value of in this set must meet both conditions, and , which is the same as )

inequality on a number line –4–4–4–3–3–3–2–2–2–1–1–1111222333444555000

can be written as which is the same as

(The value of in this set may meet either condition )

inequality on a number line –4–4–4–3–3–3–2–2–2–1–1–1111222333444555000
compound inequality in two variables
Unit 5 Lesson 6

The graph of a compound inequality in two variables with an “and ” represents the intersection of the graph of the inequalities. A number is a solution to a compound inequality combined with the word “and” if the number is a solution to both inequalities (where the regions overlap). In a system of inequalities, the word “and” is implied because the solution set must work in each equation.

If the inequalities were combined using “or” the solution would be all of the shaded area.

See feasible region and solution set for a system.

compound inequality in two variables on a graph–4–4–4–3–3–3–2–2–2–1–1–1111222333444000
conditional frequency
Unit 9 Lesson 9

See two-way relative frequency table.

constant difference (d) (common difference)
Unit 1 Lesson 2

Difference implies subtraction: common difference, constant difference, equal difference refer to the same thing. In an arithmetic sequence it is the constant amount of change. To find the difference select any output and subtract the previous output.

Example: is an arithmetic sequence.

Output 

Input 

The constant difference is:

or

constraint
Unit 5 Lesson 2

A restriction or limitation

continuous function/discontinuous function
Unit 2 Lesson 1, Unit 3 Lesson 1

A function is considered continuous if its graph does not have any breaks or holes.

continuous functionx–2–2–2–1–1–1111222333y–3–3–3–2–2–2–1–1–1111222333000

A function can be continuous on an interval.

continuous functionx–4–4–4–3–3–3–2–2–2–1–1–1y–1–1–1111222333444000

A discontinuous function is a function that is not a continuous curve. When you put your pencil down to draw a discontinuous function, you must lift the pencil from the page to continue drawing the graph at least once before it is complete. The image shows a function that is discontinuous, even though the domain is continuous on the interval that is shown.

discontinuous functionx–2–2–2–1–1–1111222y111000
correlation
Unit 9 Lesson 1

The extent to which two numerical variables have a linear relationship. A correlation gives you a number , (the correlation coefficient) which can range from to . Zero correlation means there is no relation between two variables. A correlation of (either + or -) means perfect correlation.

scatter plot with positive linear correlation x–4–4–4–3–3–3–2–2–2–1–1–1111222333444y–4–4–4–3–3–3–2–2–2–1–1–1111222333444000strong positive correlation
scatter plot with weak positive linear correlation x–4–4–4–3–3–3–2–2–2–1–1–1111222333444y–4–4–4–3–3–3–2–2–2–1–1–1111222333444000moderate positive correlation
scatter plot with negative linear correlation x–4–4–4–3–3–3–2–2–2–1–1–1111222333444y–4–4–4–3–3–3–2–2–2–1–1–1111222333444000strong negative correlation
scatter plot with weak negative linear correlation x–4–4–4–3–3–3–2–2–2–1–1–1111222333444y–4–4–4–3–3–3–2–2–2–1–1–1111222333444000MMMweak negative correlation
correlation coefficient
Unit 9 Lesson 1

See correlation.

difference of two squares
Unit 7 Lesson 6

A special product obtained after multiplying two binomials with the same numbers but one is joined by an addition symbol and the other by a subtraction sign.

difference of two squares
discrete function
Unit 2 Lesson 1, Unit 3 Lesson 1

A function is discrete if it is defined only for a set of numbers that can be listed, such as the set of whole numbers or the set of integers.

The function is an example of a discrete function if it is only defined for the set of integers .

The graph would look like dots along the line.

discrete functionx–10–10–10–5–5–5555101010y–10–10–10–5–5–5555101010000
distribution of a variable (statistics)
Unit 9 Lesson 6

a description of the number of times each possible outcome will occur in a number of trials, usually displayed as a data plot.

See: center, spread, normal distribution, modes , skewed.

distributive property of multiplication over addition
Unit 4 Lesson 1, Unit 4 Lesson 3

The distributive property of multiplication over addition says it’s okay to add within the parentheses first, and then multiply.

Or it’s okay to multiply each term first and then add. The answer works out to be the same.

a diagram labeling the operation in 5(3 9)=(5 times 3) (5 times 9)=14 45=60

The distributive property makes it possible to simplify expressions that include variables. It also makes it possible to factor expressions.

See also properties of operations.

domain
Unit 2 Lesson 1, Unit 3 Lesson 1

The set of all possible -values which will make the function work and will output real -values. A continuous domain means that all real values of included in an interval can be used in the function.

Choosing a smaller domain for a function is called restricting the domain. The domain may be restricted to make the function invertible.

Sometimes the context will restrict a domain.

Other terms that refer to the domain are input values and independent variable.

dot plot
Unit 9 Lesson 6

A graphical display of data using dots. It’s used in statistics when the data set is relatively small and the categories are discrete. To draw a dot plot, count the number of data points falling in each category and draw a stack of dots that number high for each category.

dot plot
elementary row operations
Unit 5 Lesson 12

  • Replace a row in a matrix with a constant multiple of that row

  • Replace a row in a matrix with the sum or difference of that row and another row of the matrix

  • Replace a row in a matrix with the sum of that row and a constant multiple of another row of the matrix

  • Switch two rows

equation
Unit 1 Lesson 1

A mathematical statement that two things are equal. It consists of two expressions, one on each side of an equals sign .

Example 1:

Example 2:

equivalent equations
Unit 1 Lesson 1

Algebraic equations that have identical solutions.

equivalent expressions
Unit 1 Lesson 1

Expressions that have the same value, even though they may look a little different. If you substitute in the same variable value into equivalent expressions, they will each give you the same value when you change forms.

equivalent expressions
explicit equation
Unit 1 Lesson 2

Relates an input to an output.

Example: ; is the input and is the output. 

The explicit equation is also called a function rule, an explicit formula, or explicit rule.

exponent
Unit 1 Lesson 3

An exponent refers to the number of times a number (called the base) is multiplied by itself.

Also see rational exponent.

exponentexponent5 is the base
exponential form and expanded form
Unit 1 Lesson 3
exponential form and expanded formexponentialformexpandedform
exponential function
Unit 2 Lesson 1

A function in which the independent variable, or -value, is the exponent, while the base is a constant.

For example, would be an exponential function.

exponential functionoutput or thedependent variableinitial valueor start valuethe exponent is theinput or independentvariableb is the changefactor (or constant,also the base)
expression
Unit 1 Lesson 3

A mathematical phrase such as   or   .

An expression does not have an equal sign.

An equation has an equal sign. It is a mathematical sentence.

no equal sign
factor
Unit 7 Lesson 6, Unit 8 Lesson 9

Factor (verb): To factor a number means to break it up into numbers that can be multiplied together to get the original number.

Example: Factor : , or , or

Factor (noun): a whole number that divides exactly into another number. In the example above , , , and are all factors of

In algebra factoring can get more complicated. Instead of factoring a number like , you may be asked to factor an expression like .

The numbers and and the variables and are all factors. The variable is a factor that occurs twice.

factored form
Unit 7 Lesson 9

The form of a polynomial function, where . The values are the zeros of the function, and is the vertical stretch of .

factoring a quadratic
Unit 7 Lesson 6

Change a quadratic expression or equation of the form into an equivalent expression made up of two binomials. The two binomials are the dimensions of the rectangle whose area is .

The diagram depicts a rectangle with area and dimensions and .

factoring a quadratic diagram
feasible region
Unit 5 Lesson 6

The region of the graph containing all the points that make all of the inequalities in a system true at the same time.

The feasible region for the solutions to the system of inequalities

is the location where the blue and the green regions overlap.

The feasible region does not include the dotted line because of but does include the solid line because .

feasible regionx–4–4–4–2–2–2222y–2–2–2222444000
formula
Unit 4 Lesson 2

A literal equation that describes a relationship between multiple quantities. Example: is a formula that describes the relationship between the length of the base and height and the area of the triangle.

function
Unit 3 Lesson 1
functionEvery input has one output. 2345623718-9Inputs can't have two outputs. -4-3-267891011NOT a functioncontinuousdiscontinuousdiscretefunction
function notation
Unit 1 Lesson 2
function notationfunction notationfunction ruleinputoutputinputoutput
function rule
Unit 1 Lesson 2

The explicit equation is also called the function rule.

G–L

geometric sequence
Unit 1 Lesson 3

The list of numbers

Untitled

represents a geometric sequence because, beginning with the first term, , each term is being multiplied by to get the next term in the sequence.

geometric sequence where each number is multiplied by 5

The next term in the sequence will be () or .

The number being multiplied each time is called the common ratio (r).

The sequence can be represented by a recursive equation.

In words:

Name the .

Using function notation:

function notationthe firsttermthe nthoutputthe outputone beforef(n)commonratio

A geometric sequence can also be represented with an explicit equation in the form , where is the first term, is the common ratio () and is the input value.

The explicit equation for a geometric sequence is an exponential function.

The graph of the terms in a geometric sequence is arranged in a curve.

half-plane
Unit 5 Lesson 3

The part of the plane on one side of a straight line of infinite length in the plane.

The points in a half-plane are solutions to an inequality.

half plane–4–4–4–3–3–3–2–2–2–1–1–1111222333–3–3–3–2–2–2–1–1–1111222333000
histogram
Unit 9 Lesson 6

A graphical display of univariate data. The data is grouped into equal ranges and then plotted as bars. The height of each bar shows how many are in each range.

The graph shows the heights of students in a math class.

histogram595959606060616161626262636363646464656565666666676767686868696969707070717171727272frequency of height000111222333444555666777888999101010111111height in inches
horizontal shift
Unit 7 Lesson 1

See transformations on a function.

identity: additive, multiplicative
Unit 4 Lesson 3
Two faces thinking about the additive and multiplicative identity properties of 5 0=5 and 5 times 1 =5What number can Iadd to a number toget the same numberfor the answer?What can I multiply anumber by to get thesame number for theanswer?Zero is theadditive identity.One is themultiplicative identity.

See also Properties of Operations.

independent variable / dependent variable
Unit 1 Lesson 4

In a function, the independent variable is the input to the function rule and the dependent variable is the output after the function rule has been applied. Also called ordered pairs, coordinate pairs, input-output pairs. The domain describes the independent variables and the range describes the dependent variables.

diagram showing showing the independent and dependent variables in the function f(x)=5(x)-7
inequality
Unit 4 Lesson 4

A mathematical sentence that says two values are not equal.

does not equal .

The inequality symbols tell us in what way the two values are not equal.

is less than .

is less than or equal to .

is greater than .

is greater than or equal to .

input-output pair
Unit 1 Lesson 2

Input and output pairs are related by a function rule. Also called ordered pairs, coordinate pairs, independent and dependent variables. If is an input-output pair for the function , then is the input, is the output and .

A diagram representing an input/output pair for f(x)=5x-7; x=3
intercepts
Unit 1 Lesson 2

See -intercept and -intercept.

interquartile range - IQR
Unit 9 Lesson 6

Most commonly used with a box plot, the interquartile range is a measure of where the middle is in a data set. An interquartile range is a measure of where the bulk of the values lie. The shaded box shows the IQR. It starts at and ends at .

a box and whisker plot 222333444555666777888999minimumvaluemedianmaximumvaluenumber line that includes the numbers in the 5-number summaryand uses appropriate units of equal distance
interval notation
Unit 3 Lesson 2

Notation used to describe an interval is interval notation.

interval notation
interval of increase or decrease
Unit 3 Lesson 1

In an interval of increase, the -values are increasing. In an interval of decrease, the -values are decreasing. When describing an interval of increase or decrease, the -values that correspond to the increasing or decreasing -values are named.

a diagram showing increasing, decreasing and constant intervalsx–2–2–2222444666y–8–8–8–6–6–6–4–4–4–2–2–2222444000
inverse operation
Unit 4 Lesson 1, Unit 4 Lesson 2

Inverse operations undo each other.

Some inverse operations include addition/subtraction, multiplication/division, squaring/square rooting (for positive numbers).

inverse operationInverse operations undo each other.
inverse: additive, multiplicative
Unit 4 Lesson 3

The number you add to a number to get zero is the additive inverse of that number. Every nonzero real number has a unique additive inverse. Zero is its own additive inverse. . For every there exists so that

The reciprocal of a nonzero number is the multiplicative inverse of that number. The reciprocal of is because . The product of a real number and its multiplicative inverse is . Every real number has a unique multiplicative inverse.

joint events
Unit 9 Lesson 9

Events that can occur at the same time.

Two-way tables show joints. See two-way tables.

line of best fit or linear regression
Unit 9 Lesson 2

The line (written in form) that best models the data by minimizing the distance between the actual points and the predicted values on the line.

The line will have a positive slope when the correlation coefficient is positive and a negative slope when is negative.

line of best fitYear000555101010151515180001800018000200002000020000220002200022000240002400024000
line of symmetry
Unit 7 Lesson 1

The vertical line that divides the graph into two congruent halves, sometimes called axis of symmetry.

The equation for the line of symmetry in a coordinate plane is always:

line of symmetryx–6–6–6–4–4–4–2–2–2y–2–2–2222444000
linear combination
Unit 5 Lesson 1

a sum of linear terms

linear function
Unit 2 Lesson 1, Unit 5 Lesson 1, Unit 5 Lesson 3, Unit 5 Lesson 4
several diagrams modeling linear functions, including tables and graphs. Equations for linear functions are defined as y=mx b, y=m(x-x1) y1, and Ax By=Clinear functionslope-intercept formm = slopeb = y-interceptpoint-slope formyou need slope and a point:standard formdomain: all real numbersrange: all real numbersunless restricted. unless restricted.graph is a linerate of change (slope) is constant1st difference is constantThe function can increase,decrease, or remain constant. 2122
linear regression
Unit 9 Lesson 2

See regression line.

literal equation
Unit 4 Lesson 2

A literal equation is one that has several letters or variables. Solving a literal equation means given an equation with lots of letters, solve for one letter in particular.

Example: or

Solve for .

M–R

marginal frequency
Unit 9 Lesson 9

See two-way tables.

matrix (matrices)
Unit 4 Lesson 8

A matrix is a rectangular array of data. Each piece of data in a matrix represents two characteristics, one by virtue of the row it is located in and one by virtue of the column it is in.

matrix
matrix (properties of operations)
Unit 4 Lesson 8

Associative Property of Addition

Examples with Real Numbers

Examples with Matrices

Associative Property of Multiplication

Examples with Real Numbers

Examples with Matrices

Commutative Property of Addition

Examples with Real Numbers

Examples with Matrices

Commutative Property of Multiplication

Examples with Real Numbers

Examples with Matrices

Distributive Property of Multiplication Over Addition

Examples with Real Numbers

Examples with Matrices

matrix multiplication
Unit 4 Lesson 9

To multiply two matrices the dimensions must fit the diagram.

Multiply corresponding numbers in a row times the corresponding numbers in the columns. Add each product to get one number for that position. e.g. Multiply row 1 times column 1. Add each product to obtain the one number that goes in row 1 column 1 of the product matrix.

matrix multiplication
matrix multiplication
maximum / minimum
Unit 3 Lesson 1

Maximum is the point at which a function’s value is greatest.

Minimum is the point at which a function’s value is the least.

A cubic function with points showing the maximum and minimum.–2–2–2–1–1–1111222–1–1–1111000maximumminimum
mean
Unit 9 Lesson 6

See measures of central tendency.

mean absolute deviation - M.A.D
Unit 9 Lesson 6

M.A.D of a data set is the average distance between each data value and the mean. Mean absolute deviation is a way to describe variation in a data set. It tells us how far, on average, all values are from the middle. There are 3 steps for finding the M.A.D.

  1. Find the mean of all the values.

  2. Find the distance of each value from the mean. (Recall distance is +.)

  3. Find the mean of those distances.

measures of central tendency
Unit 9 Lesson 6

A single value that describes the way in which a group of data cluster around a central value. The most common measures of central tendency are the arithmetic mean, the median and the mode.

The mean (average) is found by adding all of the numbers together and dividing by the number of items in the set:

Example: .

The median is found by ordering the set from lowest to highest and finding the exact middle. The median is just the middle number: 20.

To calculate the mode, put the numbers in order. Then count how many of each number. The number that appears most is the mode. There may be no mode if no value appears more than any other. There may also be two modes (bimodal), three modes (trimodal), or four or more modes (multimodal).

median
Unit 9 Lesson 6

See measures of central tendency.

mode(s)
Unit 9 Lesson 6

The measure of central tendency for a one-variable data set that is the value(s) that occurs most often.

Types of modes include: uniform (evenly spread- no obvious mode), unimodal (one main peak), bimodal (two main peaks), or multimodal (multiple locations where the data is relatively higher than others).

a histogram with a uniform distribution111222333444555202020404040606060808080000uniform distribution
a histogram with a unimodal distribution222444666888202020404040606060808080100100100000unimodal distributionone mode
a histogram with a bimodal distribution111222333444555202020404040606060808080000bimodal distribution2 modes
a histogram with a multimodal distribution222444666888202020404040606060808080100100100000multimodal distributionmany modes

See measures of central tendency.

multi-step equation
Unit 4 Lesson 1

An equation for which multiple inverse operations will have to be applied, in the correct order, to solve the equation for its variable.

observed value
Unit 9 Lesson 1

The value that is actually observed (what actually happened).

outliers
Unit 9 Lesson 6

Values that stand away from the body of the distribution. For a box-and-whisker plot, points are considered outliers if they are more than 1.5 times the interquartile range (length of box) beyond quartiles 1 and 3. A point is also considered an outlier if it is more than two standard deviations from the center of a normal distribution.

outlier111222333444555666777888999101010111111000outlier
parabola
Unit 6 Lesson 2, Unit 7 Lesson 1

The graph of every equation that can be written in the form , where is in the shape of a parabola. It looks a bit like a U but it has a very specific shape. Moving from the vertex, it is the exact same shape on the left as it is on the right. (It is symmetric.)

The graph of the parent function or

follows the pattern:

  • move right 1 step, move up or

  • move right 2 steps, move up or

  • move right 3 steps, move up or

parabolax–3–3–3–2–2–2–1–1–1111222333y111222333444555666777888999000vertex
Pascal’s triangle
Unit 8 Lesson 9

An array of numbers forming a triangle named after a famous mathematician Blaise Pascal. The top number of the triangle is , as well as all the numbers on the outer sides. To get any term in the triangle, you find the sum of the two numbers above it. The top number is considered row of the triangle.

The first 6 rows of Pascal's triangle
piece-wise defined function
Unit 8 Lesson 1

A function which is defined by two or more equations, each valid on its own interval. A piecewise function can be continuous or not.

Each equation in a piece-wise defined function is called a sub-function.

continuous piecewise function–4–4–4–2–2–2222000continuous3 equationsintervals
discontinuous piecewise function–4–4–4–2–2–2222000discontinuous3 equationsintervals
point-slope form of a line
Unit 2 Lesson 6

You need the slope and a point. Let and use point

The traditional way:

If we use a property of equality and add to both sides of the equation, we get an equation that is more useful:

polynomial function
Unit 8 Lesson 8, Unit 8 Lesson 10

A function of the form:

where all of the exponents are positive integers and all of the coefficients are constants.

prime number
Unit 1 Lesson 3

A prime number is a positive integer that has exactly two positive integer factors, and itself. That means is not a prime number, because it only has one factor, itself. Here is a list of all the prime numbers that are less than .

product
Unit 8 Lesson 9

The result of multiplication is a product.

a diagram showing (x a)(x b)=x^2 ax bx b^2
profit
Unit 4 Lesson 2

Profit, typically called net profit, is the amount of income that remains after paying all expenses, debts, and operating costs.

properties of equality
Unit 4 Lesson 1

The properties of equality describe operations that can be performed on each side of the equal sign ( ) and still ensure that the expressions remain equivalent.

In the table below, , , and stand for arbitrary numbers in the rational, real, or complex number systems. The properties of equality are true in these number systems.

Reflexive property of equality

Symmetric property of equality

If , then

Transitive property of equality

If and , then

Addition property of equality

If , then

Subtraction property of equality

If , then

Multiplication property of equality

If , then

Division property of equality

If and , then

Substitution property of equality

If , then may be substituted for in any expression containing

properties of inequality
Unit 4 Lesson 4

In the table a, b, and c stand for arbitrary numbers in the rational, or real number systems. The properties of inequality are true in these number systems.

Exactly one of the following is true: , ,

If and then

If , then

If , then

If and , then

If and , then

If and , then

If and , then

properties of operations for numbers in the rational, real, or complex number systems
Unit 4 Lesson 3

The letters , , and stand for arbitrary numbers in the rational, real, or complex number systems. The properties of equality are true in these number systems.

Associative property of addition

Commutative Property of addition

Additive identity property of

Existence of additive inverses

For every , there exists , so that .

Associative property of multiplication

Commutative Property of multiplication

Multiplicative identity property of

Existence of multiplicative inverses

For every , there exists , so that .

Distributive property of multiplication over addition

quadratic equations
Unit 6 Lesson 1

An equation that can be written in the form

Standard form:

Example:

Factored form:

Vertex form:

Recursive form:

(Note: Recursive forms are only used when the function is discrete.)

quadratic formula
Unit 7 Lesson 12

The quadratic formula allows us to solve any quadratic equation that’s in the form . The letters , , and in the formula represent the coefficients of the terms.

quadratic formula
quadratic function
Unit 6 Lesson 1
Several diagrams representing a quadratic function, including an area model, growing steps, a 2nd difference table, and parabolic graphs.quadratic functionvertex formfactored formstandard formgraph is a parabola
quadratic inequality
Unit 7 Lesson 14

A function whose degree is and where the is not always exactly equal to the function. These types of functions use symbols called inequality symbols that include the symbols we know as less than , greater than , less than or equal to , and greater than or equal to .

Example:

quantitative variable
Unit 9 Lesson 6

Also called numerical data, these data have meaning as a measurement or a count. Can be discrete data representing items that can be listed out or continuous data whose possible values cannot be counted and can only be described using intervals.

quantity
Unit 4 Lesson 2

A quantity is an amount, number, or measurement. It answers the question “How much?”

range (statistics)
Unit 9 Lesson 6

The difference between the highest and lowest values. It’s one number.

Example: The highest number is and the lowest is . The is .

range of a function
Unit 3 Lesson 1

All the resulting -values obtained after substituting all the possible -values into a function. All of the possible outputs of a function. The values in the range are also called dependent variables.

rate of change
Unit 3 Lesson 1

A rate that describes how one quantity changes in relation to another quantity. In a linear function the rate of change is the slope. In an exponential function the rate of change is called the change factor or growth factor. Quadratic functions have a linear rate of change (the change is changing in a linear way.)

reciprocal
Unit 4 Lesson 2
reciprocal numberAlso called the multiplicative inverse.fractionreciprocalEvery number has a reciprocalexcept 0.The reciprocal of zeroNever divide by 0!reciprocalexample
rectangular coordinate system
Unit 1 Lesson 2

Also called the Cartesian coordinate system, it’s the two-dimensional plane that allows us to see the shape of a function by graphing.

Each point in the plane is defined by an ordered pair. Order matters! The first number is always the -coordinate; the second is the -coordinate.

The coordinate plane with all quadrants labeled and the points A(2,3), B(-2,2), D(2,-1), and E (-2,-3)x–12–12–12–11–11–11–10–10–10–9–9–9–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444y–4–4–4–3–3–3–2–2–2–1–1–1111222333444000AAABBBCCCDDDQ IIQ IQ IVQIII
recursive equation
Unit 1 Lesson 2

Also called recursive formula or recursive rule. See examples under arithmetic sequence and geometric sequence, and quadratic equations.

recursive thinking
Unit 1 Lesson 2

Noticing the relationship of one output to the next output.

recursive thinking Day 1Day 2Day 3I wonder how many dots are in the next figure.
reduced row echelon form
Unit 5 Lesson 12
reduced row echelon form
reflection
Unit 7 Lesson 1

A reflection is a rigid transformation (isometry). In a reflection, the pre-image and image points are the same distance from the line of reflection; the segment connecting corresponding points is perpendicular to the line of reflection.

The orientation of the image is reversed.

a reflection of a polygon over a line
regression line (statistics)
Unit 9 Lesson 2

Also called the line of best fit. The line is a model around which the data lie if a strong linear pattern exists. It shows the general direction that a group of points seem to follow. The formula for the regression line is the same as the one used in algebra .

relative frequency table (statistics)
Unit 9 Lesson 9

When the data in a two-way table is written as percentages

See two-way frequency table.

representations
Unit 1 Lesson 2

Mathematical representations are tools for thinking about and organizing information in a situation. Representations include tables, graphs, different types of equations, stories or context, diagrams, etc.

residuals, residual plot
Unit 9 Lesson 4

The difference between the observed value (the data) and the predicted value (the -value on the regression line). Positive values for the residual (on the -axis) mean the prediction was too low, and negative values mean the prediction was too high; means the guess was exactly correct.

Create a scatter plot and graph the regression line. Draw a line from each point to the regression line, like the segments drawn in blue.

A residual plot highlights:

  1. How far the data is from the predicted value.

  2. Possible outliers

  3. Patterns in the data that suggest a different type of model

  4. If a linear model fits the data.

residual plotx111222333444555666777y111222333444000scatter plot and regression linethe residuals are the(signed) lengths of thesegments
revenue
Unit 4 Lesson 2

Revenue is the total amount of income generated by the sale of goods or services related to the company’s primary operations.

row reductions of matrices
Unit 5 Lesson 11

To solve a system using row reduction of matrices:

  • Perform elementary row operations to get a 1 in one of the columns.

  • Get zeros in all of the other rows for that column by adding a constant multiple of the row to each other row.

  • Perform elementary row operations to get a 1 in another column.

  • Create zeros in all of the other rows for that column by adding a constant multiple of the row to each other row.

  • Continue this process until each column contains a 1 and there are 0’s everywhere else, except in the augmented column that will contain the solutions to the system.

S–X

satisfies an equation
Unit 5 Lesson 1

See solution to an equation.

scalar multiplication of a matrix
Unit 4 Lesson 8

Multiplying all of the data elements in a matrix by a scale factor.

scalar multiplication of a matrix135246391561218=3scalar multiplicationMultiply each element in the matrix by 3.
scatter plot
Unit 9 Lesson 1

A display of bivariate data (ordered pairs) organized into a graph. A scatter plot has two dimensions, a horizontal dimension (-axis) and a vertical dimension (the -axis). Both axes contain a number line.

scatter plotx–4–4–4–2–2–2222444666y–6–6–6–4–4–4–2–2–2222444666000(-2,-5)(-2,-5)(4,-2)(-2,-5)Each point is definedby an ordered pair
secant line
Unit 2 Lesson 5

Also simply called a secant, is a line passing through two points of a curve. The slope of the secant is the average rate of change over the interval between the two points of intersection with the curve.

secant linex222444666888101010y222444666888101010000
sequence
Unit 1 Lesson 2

A list of numbers in some sort of pattern: patterns could be arithmetic, geometric, or other.

set
Unit 3 Lesson 2

A set is a collection of things. In mathematics it’s usually a collection of numbers. When writing sets in mathematics, the numbers are listed inside of brackets { }. This is the set of the first five counting numbers.

set builder notation
Unit 2 Lesson 2

A notation for describing a set by listing its elements or stating the properties that its members must satisfy.

The set is read aloud as “the set of all such that is greater than .”

set builder notation bracketsx is greater than 5such thatall xthe set of
skewed distribution
Unit 9 Lesson 6

When most data is to one side leaving the other with a ‘tail’. Data is skewed to side of tail. (if tail is on right side of data, then it is skewed right).

a histogram with a distribution that is skewed rightxskewed rightmodemedianmean40608020123456780
a histogram with a distribution that is skewed left111222333444555666777888202020404040606060808080000skewed leftmodemedianmean
slope
Unit 1 Lesson 2

A linear function has a constant slope or rate of change. You can count the slope of a line on a graph by counting how much it changes vertically each time you move one unit horizontally. A move down is negative and a move to the left is negative.

If you know two points on the graph, you can use the slope formula. Given two different points and

is the symbol for slope.

a straight line going through the point (0,-1) that has labels along the line of up 2 and right 1. –3–3–3–2–2–2–1–1–1111222333–3–3–3–2–2–2–1–1–1111222333000
slope-intercept form of a line
Unit 5 Lesson 4

An explicit equation for a line that uses the and the .

slope intercept form of a line x–5–5–5555y555000right 2up 5
solution set for the system of inequalities
Unit 5 Lesson 6

The set of points that satisfy all of the inequalities in a system simultaneously.

Example: The solution for a system is and .

Each inequality in the solutions is graphed. The solution set is the triangle where the blue and green overlap. This is the region where each ordered pair within the region makes each inequality true.

See also: Compound inequality in two variables.

If the system represents the constraints in a modeling context, then the feasible region is the set of viable options within the solution set that satisfy all of the constraints simultaneously.

solution set for the system of inequalitiesx–4–4–4–2–2–2222y–2–2–2222444000
solution to an equation (satisfies an equation)
Unit 5 Lesson 6

The value of the variable that makes the equation true.

solution to an equation (satisfies an equation)
solve a system by elimination or substitution
Unit 5 Lesson 9

See system of equations.

special products of binomials
Unit 7 Lesson 7

Some products occur often enough in Algebra that it is advantageous to recognize them by sight. Knowing these products is especially useful when factoring. When you see the products on the right, think of the factors on the left.

special products of binomialsDifference of two squaresSquaring a binomial
spread of a distribution (statistics)
Unit 9 Lesson 6

Measures of spread describe how similar or varied the set of observed values are for a particular variable (data item). Measures of spread include the range, quartiles and the interquartile range, variance and standard deviation.

standard deviation
Unit 9 Lesson 6, Unit 9 Lesson 7

A number used to tell how measurements for a group are spread out from the average (mean), or expected value. A low standard deviation means that most of the numbers are close to the average. A high standard deviation means that the numbers are more spread out. Symbol for standard deviation . (sigma)

standard form of a quadratic function
Unit 7 Lesson 5

standard form of line
Unit 5 Lesson 4

where , , and are integers and .

sub-function
Unit 8 Lesson 1

See piece-wise defined function.

subtraction of polynomials
Unit 8 Lesson 8

Subtraction and addition are opposite operations. This is true with polynomials. The diagram shows how the parts of an addition problem and a subtraction problem are related.

a diagram showing the addition and subtraction of polynomialsaddendssumminuendsubtrahenddifference
subtrahend
Unit 8 Lesson 8

See subtraction of polynomials.

symmetric
Unit 9 Lesson 6

If a figure can be folded or divided in half so that the two halves match exactly, then such a figure is called a symmetric figure. The fold line is the line of symmetry.

symmetricline of symmetry
symmetric distribution
Unit 9 Lesson 6

Where most of the observations cluster around the central peak. The mean, median, and mode are all equal.

symmetric distribution111222333444555666777101010202020303030404040505050606060000
symmetry
Unit 7 Lesson 1

A line that reflects a figure onto itself is called a line of symmetry.

A figure that can be carried onto itself by a rotation is said to have rotational symmetry.

symmetryThe rotation of 72° will makethis figure look the same.72°72°line of symmetry
system of equations
Unit 5 Lesson 1

A set of two or more equations with the same set of unknowns (or variables), meaning that the solutions for the variables are the same in each of the equations in the set.

Example: The equations for this system are and . This can be solved in three ways:

First, observe that both equations are equal to . This means that the first equation can be substituted for in the second equation, giving the equation . Solving this shows that , and by substituting for this means that , so the solution for this system of equations is . This method is called substitution.

Second, these equations can be manipulated so that one variable can be eliminated. In this system, the second equation can be multiplied by . turns into . This is then added to the first equation:

Since the and subtract to be , the remaining equation has one variable which can be solved to show that . Once again, plugging this value into either of the other equations will give . This method is called elimination.

Finally, both equations can be graphed. The point at which they intersect, as shown below, is the solution to the system of equations.

system of equations224624-2-4-2system of equationsWhat values of x and y make both equations true?Solve by subsitution:Replace this y with this y:y:ySolve forx.You already know Put 2 in for x to find yis the solution.butxySolve by elimination:Make something match in the two equations.Multiply the 2nd equation by 2:If you add the two equations now,Substituteto findx.sois the solution.Solve by graphing
system of inequalities
Unit 5 Lesson 2

A set of two or more inequalities with the same variables. The solution to an inequality includes a range of values. The solution to a system of inequalities is the intersection of all of the solutions. See feasible region.

system of inequalities246246-2-2-4Which regions contain (x,y) pairs that make both inequalities true?System of inequalitiesThe points in the region where the blue and green overlap willmake both inequalities true.
systems: inconsistent / independent
Unit 5 Lesson 10

When a system of equations has no solution, it is called inconsistent. If a system of equations is inconsistent, then when we try to solve it, we will end up with a statement that isn’t true such as .The graphs of the equations never intersect.

systems: inconsistent / independentxy

A system of equations is considered independent if the graphs of the equations create different lines. Independent systems of equations have one solution that can be found graphically or algebraically.

system of equations independentxy
tangent to a curve
Unit 2 Lesson 5

A line that touches a curve in exactly one point.

As the two points that form a secant line are brought together (or the interval between the two points is shortened), the secant line tends to a tangent line.

tangent to a curvex111222333444y111222333444555
transformations on a function (non-rigid)
Unit 7 Lesson 1

A dilation is a nonrigid transformation because the shape changes in size. It will make the function change faster or slower depending on the value of . If , it will grow faster and look like it has been stretched. If , the function will change more slowly and will appear to be fatter. A dilation is also called a vertical stretch.

Untitledx–4–4–3–3–2–2–1–11122334455y–2–2–1–1112233445500
transformations on a function (rigid)
Unit 7 Lesson 1

A shift up, down, left, or right, or a vertical or horizontal reflection on the graph of a function is called a rigid transformation.

Vertical shift

Up when

Down when

The vertical shift of a parabolax–4–4–4–3–3–3–2–2–2–1–1–1111y–1–1–1111222333000

Horizontal shift

Left when

Right when

the horizontal shift of a parabolax–3–3–3–2–2–2–1–1–1111222333444y–2–2–2–1–1–1111222333000

Reflection

reflection over the -axis

The reflection of a parabola over the x axisx–1–1–1111y–1–1–1111000

reflection over the -axis

the reflection of a cubic function over the y axisx–1–1–1111y–1–1–1111000

A dilation is a nonrigid transformation. It will make the function changes faster or slower depending on the value of . If , it will grow faster and look like it has been stretched. If , the function will grow more slowly and will appear to be fatter.

trinomial
Unit 7 Lesson 3, Unit 7 Lesson 6, Unit 8 Lesson 9

A polynomial with three terms.

a diagram showing ax^2 bx c has 3 terms3 terms
two-way frequency and two-way relative frequency table
Unit 9 Lesson 9

A two-way frequency chart simply lists the number of each occurrence.

Average is more than 100 texts sent per day

Average is less than 100 texts sent per day

Total

# of Teenagers

20

4

24

# of Adults

2

22

24

Totals

22

26

48

In a two-way relative frequency table, each number in the cells is divided by the grand total. That is because we are looking for a percentage that shows us how the data compares to the grand total.

Average is more than 100 texts sent per day

Average is less than 100 texts sent per day

Total

% of Teenagers

42%

8%

50%

% of Adults

4%

46%

50%

% of Total

46%

54%

100%

In this table, the ‘inner’ values represent a percent and are called conditional frequencies. The conditional values in a relative frequency table can be calculated as percentages of one of the following:

  • the whole table (relative frequency of table)

  • the rows (relative frequency of rows)

  • the columns (relative frequency of column)

two-way table
Unit 9 Lesson 9

A table listing two categorical variables whose values have been paired such that the possible values of one variable make up the rows and the possible values for the other variable make up the columns. The green cells on this table are where the joint frequency numbers are located. They are called joint frequency because you are joining one variable from the row and one variable from the column. The marginal frequency numbers are the numbers on the edges of a table. On this table, the marginal frequency numbers are in the purple cells.

two-way tableAverage is more than100 texts a dayAverage is less than100 texts a day% of teenagers% of adults% of totaljoint frequencynumbersjoint frequencynumbersjoint frequencynumbersjoint frequencynumbersmarginal frequencynumbersmarginal frequencynumbersTotalmarginalfrequencyfrequencygrand total
uniform distribution
Unit 9 Lesson 6

A uniform distribution is evenly spread with no obvious mode.

uniform distribution111222333444555666101010202020303030404040505050606060707070808080000uniform distribution
units
Unit 4 Lesson 2

A measurement unit is a standard quantity used to express a physical quantity. It identifies the items that are being counted. Units could be inches, feet, or miles. Units could also be oranges, bicycles, or people.

univariate data
Unit 9 Lesson 6

Describes a type of data which consists of observations on only a single characteristic or attribute. It doesn’t deal with causes or relationships (unlike regression) and it’s major purpose is to describe. A histogram displays univariate data.

variability
Unit 9 Lesson 6

Refers to how spread out a group of data is. Values that are close together have low variability; values that are spread apart have high variability.

variable (algebra)
Unit 1 Lesson 1

A symbol for a number we don’t know yet, usually a lowercase letter, often or . If a variable is used twice in the same expression, it represents the same value. A number by itself is called a constant. A coefficient is a number used to multiply a variable.

variable (algebra)5 is thecoefficientconstantsis the variablex
variable (statistics)
Unit 9 Lesson 1

A characteristic that’s being counted, measured, or categorized.

vertex
Unit 7 Lesson 1

See angle.

vertex form
Unit 7 Lesson 2, Unit 7 Lesson 5

See quadratic function.

vertex of a parabola
Unit 6 Lesson 4

Either the maximum or the minimum point of a parabola.

vertex of a parabolax–2–2–2222444y222000vertex
vertical shift
Unit 7 Lesson 1

See transformations on a function (rigid).

vertical stretch
Unit 7 Lesson 1

See transformations on a function (non-rigid).

viable, non-viable options
Unit 5 Lesson 2

Viable options are values that work in all of the equations in a system. Non-viable options don’t work in all of the equations.

x-intercept
Unit 1 Lesson 2, Unit 3 Lesson 1

The point(s) where a line or a curve cross the -axis. The -value of the point will be . A non-horizontal line will only cross the -axis once. A curve could cross the -axis several times.

a line passing through the points (-5,0) and (0,2)x–6–6–6–4–4–4–2–2–2y222000(-5, 0)(-5, 0)(-5, 0)
a parabola with a vertex at (-1,-4) passing through the points (-3,0) and (1,0)x–4–4–4–2–2–2222y–4–4–4–2–2–2222000(-3, 0)(-3, 0)(-3, 0)(1, 0)(1, 0)(1, 0)

Y–Z

y-intercept
Unit 1 Lesson 2, Unit 3 Lesson 1

The point where a line or a curve crosses the -axis. The -value of the point will be . The -intercept is often referred to as “” when writing the equation of a line or as the point .

A function will have at most one -intercept.

y-interceptx–2–2–2y222000(0, 3)(0, 3)(0, 3)
zero property of multiplication (also called the zero product property)
Unit 7 Lesson 11
zero property of multiplication