A–F
- acute angle
An angle whose measure is between
and . is an acute angle. - angle
Two rays that share a common endpoint called the vertex of the angle.
- angle of rotation
The fixed point a figure is rotated about is called the center of rotation. If one connects a point in the pre-image, the center of rotation, and the corresponding point in the image, they can see the angle of rotation. A counterclockwise rotation is a rotation in a positive direction. Clockwise is a negative rotation.
- 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
. 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 . 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:
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. - augmented matrix
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:
- auxiliary line
An extra line or line segment drawn in a figure to help with a proof.
is an auxiliary line (added to the diagram of to help prove that the sum of the angles . - 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.
- bisect (verb); bisector (noun) (midpoint)
To divide into two congruent parts.
A bisector can be a point or a line segment.
A perpendicular bisector divides a line segment into two congruent parts and is perpendicular to the segment.
- 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. 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.
- 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.
- circle
All points in a plane that are equidistant from a fixed point called the center of the circle. The circle is named after its center point. The distance from the center to the circle is the radius. A line segment from the center point to a point on the circle is also called a radius (plural radii, when referring to more than one).
Notation:
- circumscribe
To draw a circle that passes through all of the vertices of a polygon. The circle is called the circumcircle.
All of these polygons are inscribed in the circles.
- clockwise / counterclockwise
clockwise: Moving in the same direction, as the hands on a clock move.
counterclockwise: Moving in the opposite direction, as the hands on a clock move.
- coincides (superimposed or carried onto)
When working with transformations, we use words like coincide, superimposed, or carried onto to refer to two points or line segments that occupy the same position on the plane.
- 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 ) can be written as which is the same as (The value of
in this set may meet either condition ) - 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.
- concentric circles
Circles with a common center.
- conditional frequency
See two-way relative frequency table.
- congruence statement
A mathematical statement that uses the
symbol. Examples: Only figures or shapes can be congruent. Numbers are equal.
- congruent (CPCTC)
Two triangles (figures) are congruent if they are the same size and same shape. Two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other.
The symbol for congruent is
. If it’s given that two triangles (figures) are congruent, then the Corresponding Parts of the Congruent Triangles (figures) are Congruent (CPCTC).
- conjecture
A mathematical statement that has not yet been rigorously proven. Conjectures arise when one notices a pattern that holds true for many cases. However, just because a pattern holds true for many cases does not mean that the pattern will hold true for all cases. When a conjecture is proven, it becomes a theorem.
- 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
- construction
Creating a diagram of geometric figures and items such as perpendicular lines or a regular pentagon using only a compass and straightedge.
A construction yields an exactly reproducible and unambiguous result, of which all properties can be measured as expected (within the accuracy of the instruments use.)
Constructing an angle bisector:
- continuous function/discontinuous function
A function is considered continuous if its graph does not have any breaks or holes.
A function can be continuous on an interval.
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.
- 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. - correlation coefficient
See correlation.
- corresponding angles
Angles that are in the same relative position.
- corresponding parts (in a triangle)
The word corresponding refers to parts that match between two congruent figures. Corresponding angles and corresponding sides will have the same measurements in congruent figures.
- corresponding points / sides
Points, sides, and angles can all be corresponding. It means they are in the same relative position.
- counterexample
An example that disproves a statement or conjecture. One counterexample can disprove a conjecture based on many examples.
Statement: All blondes drive red cars.
Counterexample: My mom is blonde, but her car is silver.
- CPCTC
See congruent (CPCTC).
- cube root
A value that, when multiplied by itself, three times gives the number.
Example:
, so the cube root of is or . , so the cube root of is or . The mathematical symbol that indicates to find the cube root is a radical sign with a small
on the outside. . - degree
A degree is the measure of an angle of rotation that is equal to
of a complete rotation around a fixed point. A measure of degrees would be written as . - determinant of a matrix
The determinant of a matrix is a number that is specially defined only for square matrices. If the determinant is not equal to zero, then the matrix has a multiplicative inverse.
For a
matrix the determinant can be found using the following rule: (note: the vertical lines, rather than the square brackets, which are used to indicate that we are finding the determinant of the matrix) - diagonal
Any line segment that connects nonconsecutive vertices of a polygon.
- 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.
- direction of a vector
The direction of a vector is determined by the angle it makes with a horizontal line.
See vector.
- 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. - 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.
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.
- elementary row operations
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
- equality statements
A mathematical sentence that states two values are equal.
It contains an equal sign.
- 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.
- 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.
- 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.
- exponential form and expanded form
- 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. - 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.
- 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 . - 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
- function notation
- function rule
The explicit equation is also called the function rule.
G–L
- geometric sequence
The list of numbers
represents a geometric sequence because, beginning with the first term,
, each term is being multiplied by to get the next term in the sequence. 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:
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.
- hexagon
A six-sided polygon.
- 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. - 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, . As
gets larger, the graph of approaches the horizontal asymptote, . As
gets smaller, the graph of approaches the horizontal asymptote, . See also asymptote.
- identity: additive, multiplicative
See also Properties of Operations.
- image
A picture; a visual representation of a thing. See pre-image / image.
- 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.
- 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 . - inscribed in a circle
- 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 . - interval notation
Notation used to describe an interval is 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. - inverse operation
Inverse operations undo each other.
Some inverse operations include addition/subtraction, multiplication/division, squaring/square rooting (for positive numbers).
- 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.
- joint events
Events that can occur at the same time.
Two-way tables show joints. See two-way tables.
- 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 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 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 segment
A piece of a line with two endpoints.
Notation:
represents the line segment with endpoints at point and point . is an object. A line segment has length and can be measured.
The notation
(without any kind of line above it) refers to the length of segment . - linear combination
a sum of linear terms
- linear function
- 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
- magnitude of a vector
The length of a vector.
See vector.
- marginal frequency
See two-way tables.
- matrix (matrices)
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 (properties of operations)
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
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.
- 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.
- 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.
Find the mean of all the values.
Find the distance of each value from the mean. (Recall distance is +.)
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: 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).
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.
- n-gon
A polygon with
number of sides. See polygon.
- number sets (systems)
Your first experience with number sets was probably when you learned to count. This set is called the Natural numbers,
. When you added the set grew to be the Whole numbers, . The need for Integers, , arose when you subtracted a large number from a smaller number. Then you needed the Rational numbers, , when you started dividing. Other number sets (or systems) are needed in more advanced mathematics. - observed value
The value that is actually observed (what actually happened).
- obtuse angle / obtuse triangle
- octagon
An eight-sided polygon.
- opposite (or negative) reciprocal slope
Slopes of perpendicular lines are opposite reciprocals, so that the product of the slopes is
. (See perpendicular lines) - opposite angles, opposite vertices
Opposite angles in a quadrilateral do not share a side.
A vertex (plural, vertices) is part of an angle.
- opposite sides (in a parallelogram or an even-sided polygon)
If two sides in a parallelogram are parallel, they must be opposite sides.
If two sides in an even-sided polygon are parallel, they must be opposite sides.
- orientation
The orientation is determined by the order in which a figure’s vertices are labeled. In the diagram, the vertices of the green pentagon are labeled from
to to to to in a clockwise direction. In the blue pentagon, the orientation of the vertices has changed. The corresponding vertices go in a counterclockwise direction from
to to to to . - origin
The origin is a starting point. The coordinates for every other point are based on how far that point is from the origin. At the origin, both
and are equal to zero, and the -axis and the -axis intersect. - 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.
- parallel line
- parallelogram
A quadrilateral in which the opposite sides are parallel.
- parent function
The most basic form of a function. A parent function can be transformed to create a family of functions.
- pentagon
A five-sided polygon.
- perpendicular bisector
The line (line segment or ray) that divides a line segment into two equal lengths and makes a right angle with the line segment it divides.
- 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. - 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
. - 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-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: - polygon
Any 2-D shape formed with line segments that connect at their endpoints, making a closed figure. The location where any two line segments connect is called a vertex.
Triangles, quadrilaterals, pentagons, and hexagons are all examples of polygons. The name identifies how many sides the shape has. For example, a triangle has three sides, a quadrilateral has four sides, a pentagon five sides, and an octagon eight sides. A regular polygon is made up of congruent line segments.
In a regular polygon, all sides are congruent, and all angles are congruent.
- pre-image / image
The pre-image is the original figure. The image is the new figure created from the pre-image through a sequence of transformations or a dilation.
- preserves distance and angle measure
Measurements are not changed under a rigid transformation.
- 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.
- proof by contradiction
A way to justify a claim is to use a proof by contradiction method, in which one assumes the opposite of the claim is true, and shows that this leads to a contradiction of something that is known to be true.
- 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
- Pythagorean theorem
The relationship among the lengths of the sides of a right triangle that results in the sum of the squares of the lengths of the legs equaling the square of the length of the hypotenuse.
- quadrilaterals: types
A quadrilateral is a four-sided polygon. See the diagram for various types of quadrilaterals.
- 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?”
- radical
A radical is the mathematical inverse of an exponent. This is the symbol for a radical:
. It is also called a square root symbol, but that is only when it’s asking for the number that when multiplied by itself gives you the number inside the . (The is not usually written.) It can be used to indicate a cube root , a fourth root , or higher. (A root that is higher than is written in.) - radii
Plural of radius. See circle.
- 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 . 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. - rational exponent (fractional exponent)
Rational exponents (also called fractional exponents) are expressions with exponents that are rational numbers (as opposed to integers).
- ray
A part of a line that has a fixed starting point (endpoint), and then continues toward infinity.
Notation:
– ray A ray is named using its endpoint first, and then any other point on the ray.
- reciprocal
- rectangle
See quadrilaterals: types.
- 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. - 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.
- reduced row echelon form
- reflection
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.
- 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
. - regular polygon
See polygon.
- relative frequency table (statistics)
When the data in a two-way table is written as percentages
See two-way frequency table.
- 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:
How far the data is from the predicted value.
Possible outliers
Patterns in the data that suggest a different type of model
If a linear model fits the data.
- revenue
Revenue is the total amount of income generated by the sale of goods or services related to the company’s primary operations.
- rhombus
A quadrilateral in which all sides are congruent.
- right angle
An angle that measures
. The symbol for a right angle in a geometric figure is a box.
- rigid transformation
Also called an isometry. The word rigid means that the pre-image and image are congruent. The rigid transformations include translation, rotation, and reflection.
- rotation
A rotation is a rigid transformation. In a rotation, all points remain the same distance from the center of rotation, move in the same direction, and through the same central angle. The orientation of the pre-image remains the same.
- rotational symmetry
See symmetry.
- row reductions of matrices
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
See solution to an equation.
- scalar multiplication of a matrix
Multiplying all of the data elements in a matrix by a scale factor.
- scalar quantity
A scalar quantity is usually depicted by a number, numerical value, or a magnitude, but no direction.
- scalene triangle
A triangle that has three unequal sides.
- 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. - 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.
- 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 .” - 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).
- 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. - slope-intercept form of a line
An explicit equation for a line that uses the
and the . - 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 to an equation (satisfies an equation)
The value of the variable that makes the equation true.
- solve a system by elimination or substitution
See system of equations.
- 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.
- square
See quadrilaterals: types.
- square root
The square root of a number is one of the two identical factors that when multiplied together equal the number.
Example:
6, so a square root of is . Note that
too. That means is also a square root of . The mathematical symbol that indicates to find the square root is a radical sign . - 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 . - 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.
- symmetric distribution
Where most of the observations cluster around the central peak. The mean, median, and mode are all equal.
- symmetry
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.
- 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 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.
- 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. 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.
- 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.
- transformations on a function (rigid)
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
Horizontal shift
Left when
Right when
Reflection
reflection over the -axis reflection over the -axis 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. - translation
A translation is a rigid transformation.
- trapezoid
A quadrilateral with exactly one pair of parallel opposite sides.
(Note: A trapezoid can also be defined as a quadrilateral with at least one pair of opposite sides that are parallel. This definition makes it possible for parallelograms to be a special type of trapezoid.)
In an isosceles trapezoid, the two opposite sides that are not parallel are congruent and form congruent angles with the parallel sides. This feature of an isosceles trapezoid only exists if the trapezoid is not a parallelogram.
- triangle congruence criteria: ASA, SAS, AAS, SSS
Two triangles are congruent if all three sides and all three angles are congruent. But sometimes only three pieces of information are sufficient to prove two triangles congruent.
ASA stands for “angle-side-angle.”
SAS stands for “side-angle-side.”
AAS stands for “angle-angle-side.”
SSS stands for “side-side-side.”
- two-way frequency and two-way relative frequency table
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
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.
- uniform distribution
A uniform distribution is evenly spread with no obvious mode.
- 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 (statistics)
A characteristic that’s being counted, measured, or categorized.
- vector, vector quantity
A vector is a quantity that has magnitude (length) and direction.
Notation:
Unlike a geometric ray, a vector has a specific length.
The magnitude
of a vector is calculated using the Pythagorean theorem. The direction of a vector is determined by the angle it makes with a horizontal line. In the diagram, the direction will be represented by theta,
(or ). The value of the angle will be found using trigonometry. at above the horizontal. - vertical shift
See transformations on a function (rigid).
- 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.
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.