A–F

acute angle

An angle whose measure is between and .

is an acute angle.

an acute angle
arithmetic mean

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

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

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

asymptote

A line that a graph approaches, but does not reach. A graph will never touch a vertical asymptote, but it might cross a horizontal or an oblique (also called slant) asymptote.

Horizontal and oblique asymptotes indicate the general behavior of the ends of a graph in both positive and negative directions. If a rational function has a horizontal asymptote, it will not have an oblique asymptote.

Oblique asymptotes only occur when the numerator of has a degree that is one higher than the degree of the denominator.

a diagram showing vertical asymptotes between curvesverticalasymptoteverticalasymptote
a diagram showing the oblique asymptote within a 1/x functionobliqueasymptote
a diagram showing the horizontal asymptote within a 1/x functionhorizontal asymptote
average rate of change

See rate of change.

bimodal distribution

A bimodal distribution has two main peaks.

The data has two modes.

See also: modes.

a bimodal histogram2224446662020204040406060608080800002 modesbimodal distribution
binomial

A polynomial with two terms.

a binomial of (ax b)termtermaddition or subtraction
bivariate data

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)

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

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

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

center (statistics)

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)

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.

common ratio (r) (constant ratio)

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

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

compound inequality in one variable

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

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
constant difference (d) (common difference)

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

A restriction or limitation

continuous function/discontinuous function

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

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

See correlation.

difference of two squares

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
directed distance

Distance is always positive. A directed distance has length and direction. Partitions occur on line segments that are referred to as directed line segments. A directed segment is a segment that has distance (length) and direction. It is important to understand that a directed segment has a starting point referred to as the initial point and a direction from which to move away from the starting point. This will clarify the location of the partition ratio on the segment.

discrete function

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)

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

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

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

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
equation

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:

equilateral, equilateral triangle

Equilateral means equal side lengths.

In an equilateral triangle, all of the sides have the same length.

equilateral, equilateral triangle
equivalent equations

Algebraic equations that have identical solutions.

equivalent expressions

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

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

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
exponential form and expanded formexponentialformexpandedform
exponential function

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

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

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 of a quadratic function

Go to quadratic function.

factoring a quadratic

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

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

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

The explicit equation is also called the function rule.

G–L

geometric sequence

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

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

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 asymptote

A line that the graph approaches but does not reach. Exponential functions have a horizontal asymptote. The horizontal asymptote is the value the function approaches as either gets infinitely larger or smaller. An asymptote is an imaginary line, but it is often shown as a dotted line on the graph.

As gets smaller, the graph of approaches the horizontal asymptote, .

the graphs of f(x)=2^x and its horizontal asymptote of y=0x–10–10–10–5–5–5555y–5–5–5555101010000

As gets larger, the graph of approaches the horizontal asymptote, .

the graphs of f(x)=2^-x and its horizontal asymptote of y=0x–5–5–5555101010y–5–5–5555101010000

As gets smaller, the graph of approaches the horizontal asymptote, .

the graphs of f(x)=2^x-3 and its horizontal asymptote of y=-3x–10–10–10–5–5–5555y–5–5–5555101010000

See also asymptote.

identity: additive, multiplicative
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

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

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

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

See -intercept and -intercept.

interquartile range - IQR

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

Notation used to describe an interval is interval notation.

interval notation
interval of increase or decrease

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

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

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.

isosceles triangle, trapezoid

The word isosceles is only used to describe a triangle or a trapezoid with two congruent sides.

isosceles triangle, trapezoid
line

A line is an undefined term because it is an abstract idea, rather than concrete like a stroke of ink. It is defined as a line of points that extends infinitely in two directions. It has one dimension, length. Points that are on the same line are called collinear points. A line is defined by two points, such as line .

Notation:

line
line of best fit or linear regression

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

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

a sum of linear terms

linear function
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

See regression line.

literal equation

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

maximum / minimum

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

See measures of central tendency.

mean absolute deviation - M.A.D

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

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

See measures of central tendency.

midpoint

A point on a line segment that divides it into two equal parts.

The formula for finding half the distance between two points (or the midpoint ) in a coordinate grid is:

midpointx–4–4–4–3–3–3–2–2–2–1–1–1111222333444y–3–3–3–2–2–2–1–1–1111222333444000(-3, -2)(-3, -2)(-3, -2)(-0.5, 1)(-0.5, 1)(-0.5, 1)(2, 4)(2, 4)(2, 4)midpoint

See also bisect.

mode(s)

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

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

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

obtuse angle / obtuse triangle
obtuse angle / obtuse triangle
opposite (or negative) reciprocal slope

Slopes of perpendicular lines are opposite reciprocals, so that the product of the slopes is . (See perpendicular lines)

outliers

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

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
parallel line
parallel lineParallel lineshave the sameslope.Two lines in a plane thatwill never intersect.The arrow headsindicate parallel.Line BC is parallel to line AD.
perpendicular lines

Two lines or line segments are perpendicular if they have opposite, reciprocal slopes, or if one is vertical and the other is horizontal. Two lines are perpendicular if their intersection forms four right () angles.

perpendicular lines4 right angles
plane

A plane is an undefined term because it is an abstract idea rather than concrete like a piece of paper. A plane has two dimensions. It can be identified by determining three noncollinear points. It is labeled according to the letters used to label the points, such as plane .

plane
point

A point is an undefined term because it is an abstract idea rather than concrete like a dot. A point in geometry is a location. It has no size, (i.e., no width, no length, and no depth). A point is labeled with a dot and a capital letter.

point
point-slope form of a line

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

A function of the form:

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

prime number

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 .

profit

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

properties of equality

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

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

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

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 function
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
quantitative variable

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

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

range (statistics)

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

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

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.)

ratio

A ratio compares the size or amount of two values.

Here is a sentence that compares apples to oranges as shown in the diagram below: “We have five apples for every three oranges.” It describes a ratio of to or . A ratio can also be written as a fraction, in this case .

Compare oranges to apples. The ratio changes to or .

The two previous ratios are called part-to-part ratios. Another way to write a ratio is to compare a part to a whole.

Compare apples to the total amount of fruit. The ratio changes to or .

ratio

Ratios can be scaled up or down. There are bags of fruit, each containing oranges and apples. The ratio to still represents the number of apples compared to the number of oranges. But the ratio to also compares the number of apples to the number of oranges.

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

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

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

recursive thinking

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.
regression line (statistics)

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 .

representations

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

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

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

right angle

An angle that measures .

The symbol for a right angle in a geometric figure is a box.

right angleright angle

S–X

satisfies an equation

See solution to an equation.

scalene triangle

A triangle that has three unequal sides.

scalene triangle
scatter plot

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

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

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

set

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

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

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

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

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

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)

The value of the variable that makes the equation true.

solution to an equation (satisfies an equation)
solve a system by elimination or substitution

See system of equations.

special products of binomials

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)

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

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 line

where , , and are integers and .

subtrahend

See subtraction of polynomials.

symmetric

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

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

symmetric distribution111222333444555666777101010202020303030404040505050606060000
system of equations

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

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

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

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
trinomial

A polynomial with three terms.

a diagram showing ax^2 bx c has 3 terms3 terms
uniform distribution

A uniform distribution is evenly spread with no obvious mode.

uniform distribution111222333444555666101010202020303030404040505050606060707070808080000uniform distribution
units

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

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

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)

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)

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

vertex

See angle.

vertex of a parabola

Either the maximum or the minimum point of a parabola.

vertex of a parabolax–2–2–2222444y222000vertex
viable, non-viable options

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

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

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)