The distance a number is from 0.
The study of mathematical symbols and the rules for manipulating those symbols.
A grid used to illustrate values of algebraic expressions.
A sequence of numbers in which each number can be computed by adding the same amount to the previous number.
Associative law of addition
For any three numbers a, b, and c, it is always true that (a+b)+c=a+(b+c).
Associative law of multiplication
For any three numbers a, b, and c, it is always true that (a(b))(c)=a(b(c)).
Axis of symmetry
A line that you can flip (or reflect) a graph across that results in the same graph.
A number that is raised to a power.
Best fit line
When the points on a grid are not all on a straight line, but seem to have a somewhat linear pattern, you can find a line that is the “best fit” (closest) to the points.
A box with “whiskers” showing the median, quartiles, and extremes (least and greatest values) of a collection of data values.
Have a profit of zero (that is, make exactly as much money as you spend).
ceiling(x) is the closest integer which is greater than or equal to x.
Multiplying both sides of an equation by some nonzero number that turns all the fractions in the equation into integers.
A constant that a variable or expression is multiplied by.
Combining like terms
Using the distributive law to add any two multiples of an expression such as x. For example, you can simplify 4x+5x into 9x.
In an arithmetic sequence, the amount that can be added to each number to get the next one.
In a geometric sequence, the amount that each number can be multiplied by to get the next number.
Commutative law of addition
For any two numbers a and b, a+b=b+a.
Commutative law of multiplication
For two numbers a and b, a(b)=b(a).
Completing the square
Rewriting the equation x2+2mx=n as (x+m)2=n+m2 so that it can be more easily solved.
The composition of two functions f and g is the function f∘g that transforms x into f(g(x)).
Conditional relative frequency
A joint frequency divided by the total of its row or column in a two-way frequency table.
A single fixed number (unlike a variable, whose value can vary).
A constant term, thought of as a coefficient of 1.
A term that is a constant. For a polynomial in x, it’s the term without an x.
A point on a 2-dimensional plane is described by a pair (x,y). The coordinate x is given by the labels below the grid, and the coordinate y is given by the labels to the left of the grid.
A 2-dimensional flat surface used for plotting points, lines, curves, and regions. It contains an x and a y axis which intersect at the origin.
A grid of lines on a coordinate plane that makes it easy to see (x,y) coordinates of locations in that plane.
In economics, how much money a company spends to produce a product.
The cube root of a, written 3√a, is the number whose cube is a. That is, (3√a)3=a.
A collection of related measurements.
A fractional quantity written with a decimal point (like 0.5).
The bottom number or expression in a fraction.
The distance between two quantities, or the answer to a subtraction problem.
The discriminant of the equation ax²+bx+c=0 is the quantity b²−4ac. A quadratic equation has two solutions if its discriminant is positive, one solution if its discriminant is zero, and no real solutions if its discriminant is negative.
Distributive law of multiplication over addition
For any three numbers a, b, and c, a(b+c)=a(b)+a(c), and (b+c)(a)=b(a)+c(a).
Distributive law of multiplication over subtraction
For any three numbers a, b, and c, a(b−c)=a(b)−a(c), and (b−c)(a)=b(a)−c(a).
The set of inputs (x-coordinates) of a relation or function.
A diagram showing data values as dots above a number line.
A mathematical sentence with an equals sign (like 3x+5=11).
Two fractions are equivalent if they have the same numerical value. Two equations or inequalities are equivalent if they have the same solution set.
A function f with f(x)=f(−x) for all x. f(x)=xn is an even function if n is an even integer. A function is even if and only if its graph has the y-axis as an axis of symmetry.
Expanding an expression
Using the distributive law to turn expressions which need parentheses (like 3(x+2)) into expressions which do not (like 3x+6).
In a power, the number of times the base is multiplied by itself.
Using exponents, especially using variables in exponents.
Decreasing toward 0 due to a variable in an exponent, such as in y=2−x.
Increasing rapidly due to a variable in an exponent, such as in y=2x.
A combination of variables and numbers using arithmetic (like 6−x).
An expression that is multiplied by another expression, or that can be multiplied by another expression to produce a specified result.
Rewriting an expression as a product.
The sequence 1, 1, 2, 3, 5, … with remaining terms F(n) given by F(n)=F(n−1)+F(n−2) for n>2.
The median of the data
An expression that is used to compute a value.
In statistics, the number of times something occurs, or is observed.
A relation in which no x-coordinate appears in more than one (x,y) ordered pair. This means you can think of a function as a transformation that takes each x-coordinate to its single corresponding y-coordinate.
Fundamental theorem of arithmetic
Any integer greater than 1 can be written as a product of prime numbers, ordered smallest to largest, in exactly one way.
A sequence of numbers in which each number can be computed by multiplying the previous number by the same amount.
An image formed by plotting the solutions to an equation or inequality, or some other set of pairs of numbers, on a coordinate plane. To graph an expression containing the variable x, set y equal to that expression.
Rectangles of equal width above a number line, where each rectangles’s height shows the number of data values in that portion of the number line.
Going from side to side, like the horizon.
A mathematical sentence that uses one of the symbols <, >, ≤, or ≥.
A number that can be “put into” a relation to produce one or more “outputs.” If a relation is given by a two column table of rows (x,y), you “look up” the input x value in the first column, and the output(s) are given by the y values in those matching row(s).
A whole number or the negative of a whole number. For instance, 37 and 0 and −5 are integers.
The third quartile minus the first quartile.
Functions f and g such that g∘f(x)=x for every x in the domain of f, and f∘g(y)=y for every y in the domain of g.
A number that cannot be written as a fraction m/n where m and n are integers
Make a variable appear alone on one side of an equation or inequality, and not occur in the other side of the equation or inequality.
The number of events that satisfy both of two specified criteria.
Joint relative frequency
A joint frequency divided by the total number of events.
A straight line, or an equation or expression whose graph is a straight line. If m and b are constants, then mx+b is a linear expression, and a function f defined by f(x)=mx+b is a linear function.
For a polynomial in x, the number that x (without an exponent) is multiplied by.
An estimate for a variable using a linear expression in another variable.
Maximizing or minimizing a linear goal or cost expression, while remaining within some constraints given by linear inequalities.
The total of a row or column in a two-way frequency table.
Marginal relative frequency
A marginal frequency divided by the total number of events.
The average of a collection of data values. This can be computed by adding all the values and then dividing by the number of values.
Mean absolute deviation
The mean distance of data values from some central value, such as the mean, median, or mode of the collection. In other words, the mean of |x−m| where x is each data value and m is the mean, median, or mode of all the data values.
Mean squared deviation
The mean squared distance of data values from some central value, such as the mean, median, or mode of the collection. In other words, the mean of (x−m)2 where x is each data value and m is the mean, median, or mode of all the data values.
The middle number in an ordered list of data values. If there are an even number of values, the median is halfway between the two middle numbers in the list.
The most common value in a collection, or “modes” if more than one are tied.
A polynomial whose leading (first) coefficient is 1.
A product of variables and numbers, like 3x or 5×2. A monomial is also sometimes called a term.
Take the opposite of a number, by multiplying it by −1.
A value less than zero (like −3).
The top number or expression in a fraction.
A function f with −f(x)=f(−x) for all x. f(x)=xn is an odd function if n is an odd integer. A function is odd if and only if its graph has the point (0,0) as a point of symmetry.
A function f for which f(x) has a different value for every distinct (different) value of x.
The point on a coordinate plane where the x-axis and y-axis intersect. It is represented by the coordinates (0,0).
A value that “lies outside” (is much smaller or larger than) most of the other values in a collection.
A number produced by applying a relation or function to an input.
The shape of the graph of y=x2.
Two lines are parallel if they always have the same distance between them, so they never intersect. If two lines are parallel, they have the same slope.
A number that is the square of a rational number. For example, 1, 4, 25,16, and 0 are perfect squares. An integer is a perfect square only if it is the square of an integer, which can be proven using the fundamental theorem of arithmetic.
For a periodic function, the amount of time before it repeats. That is, if f is a periodic function, its period is the smallest possible positive h where f(x+h)=f(x) for every x.
A function that repeats after a certain period h with h>0, so that f(x+h)=f(x) for every x.
Two lines are perpendicular if they create a 90-degree angle. If two lines are perpendicular and the slope of one of them is m, then the slope of the other line is −1m
A function that is defined by different formulas at different inputs.
A location in the coordinate plane. A point has coordinates (x,y), where x is given by the labels below a coordinate grid, and y is given by the labels to the left of a coordinate grid.
Point of symmetry
A point that you can rotate a graph around by 180° that results in the same graph.
If a line contains the point (x1,y1) and has slope m, then its equation can be written as y−y1=m(x−x1). An equation in the form y−y1=m(x−x1) is said to be in point-slope form.
A sum of monomials. Usually terms with higher powers are written first.
A value greater than zero (like 3).
An expression of the form aⁿ. a is called the base, n is called the exponent, and aⁿ is called “the nth power of a”. If n is a positive integer, aⁿ means a multiplied by itself n times.
An integer greater than 1 that can only be written as a product of two whole numbers in one way: as itself multiplied by 1.
The answer to a multiplication problem.
Revenue minus cost.
Each of the four sections of a coordinate plane made by the intersecting x- and y-axes. The four quadrants are labeled I, II, III, and IV, counterclockwise from the top right.
An expression or equation in which the highest power of a variable has exponent 2.
For a polynomial in x, the number that x2 is multiplied by.
Quartiles: The first quartile, median, and third quartile are values which divide a data collection into four roughly equal parts.
The answer to a division problem.
The set of outputs (y-coordinates) of a relation or function.
Rate of change
The speed at which a variable changes over a period of time. This is given by the change in the variable divided by the change in (amount of) time.
A number that can be written as a fraction m/n where m and n are integers.
Rigid motion across a fixed line AB in a plane, like a mirror image.
A set of ordered pairs (x,y).
A frequency divided by the total number of events, often expressed as a percentage.
An observed value minus its estimated value.
A function g is a restriction of the function f if g(x)=f(x) for every x in the domain of g, but that domain may be smaller than the domain of f.
How much money a company receives in sales.
The values of x where a polynomial is zero. These are the x-coordinates of the x-intercepts of the polynomial’s graph.
Rigid motion around a fixed center A, with turning but no reflection.
Dots in the coordinate plane representing pairs of linked measurements, such as heights and weights for a group of people.
A list of numbers that may be generated by some rule.
An unordered collection of numbers or other mathematical objects, without repetitions.
Two geometric figures are similar if they have the same shape but possibly different sizes, with corresponding lengths differing by a single common scale factor.
To rewrite an expression in a way that means the same thing but is simpler (or shorter). You can simplify 3x−x+6 into 2x+6.
A number that measures how steep a line is. It shows the amount of change in the height of the line as you go 1 unit to the right. The slope of the line y=mx+b is m.
The form y=mx+b for a linear equation, where m and b are constants. The numbers m and b give the slope and y-intercept of the line that is the graph of that equation.
In an equation or inequality, a number that can be substituted for the variable to make that equation or inequality true. If the equation or inequality has more than one variable, a solution is a list of numbers that when substituted for the list of variables makes the equation or inequality true. For a system of more than one equation or inequality, a solution must make all of the equations or inequalities true. In chemistry, a solution is a liquid mixture.
All solutions to an equation, inequality, or system.
Find the solutions to an equation, inequality, or system.
A square root of a is a number b whose square is a. That is, b²=a. If b is a square root of a, then so is −b. If a≥0, “the” square root of a, written √a, is the square root of a that is positive or zero.
The square root of the variance.
For a linear equation, the form Ax+By=C where A, B, and C are constants. For a quadratic equation, either the form y=ax2+bx+c or ax2+bx+c=0, where a, b, and c are constants.
A number used to describe or summarize data.
The study of data, and the methods used to describe or summarize data.
A piecewise-defined function where each piece’s formula is a constant (doesn’t change with x). A step function’s graph looks like stair steps.
In an expression or equation, eliminating a variable by replacing it with another expression that it is equal to.
The answer to an addition problem.
Repeating pattern or shape.
For equations or inequalities, two or more equations or inequalities that are all required to be true.
In mathematics, a rectangular arrangement of rows and columns.
Element in a sum, difference, or sequence.
The quadratic formula: The formula x = (-b±√b²-4ac)/2a
which gives the solutions to any equation in the form ax2+bx+c=0 with a≠0. The equation has two solutions when b2−4ac>0; it has one solution when b2−4ac=0; and it has no real solutions when b2−4ac<0.
For n data values, the median of the n/2 largest values if n is even, and the (n-1)/2 largest values if n is odd.
Two-way frequency table
For events that can be divided into categories two different ways, a table of joint frequencies, using rows of the table to group the events one way, and columns of the table to group the events the other way.
A standard measurement, such as a meter or an hour.
A number that a variable or expression can equal.
A letter (like x) that we can use to mean different numbers at different times.
The mean squared distance of data values from their mean m. This can be computed by adding (x−m)2 for each data value x, and then dividing by the number of data values n. When measuring a sample from a population, for instance heights of people, the variance of the sample is usually different than the variance of the entire population. To estimate the population’s variance, it is usually better to divide by n−1 instead of n.
The point where a parabola crosses its axis of symmetry, or an end of a side of a polygon, or the corner point of an angle.
A quadratic equation in the form y=a(x−h)²+k.
Going up and down.
One of the numbers 0, 1, 2, 3, … .
The horizontal line running through the origin on a coordinate plane.
The horizontal value in a coordinate pair. It tells how far to the left or right the point is. The x-coordinate is always written first in the coordinate pair.
A point where a curve meets the horizontal axis (the x-axis).
The vertical line running through the origin on a coordinate plane.
The vertical value in a coordinate pair. It tells how far up or down the point is. The y-coordinate is always written last in the coordinate pair.
A point where a line or curve meets the vertical axis (the y-axis). The y-intercept of the line y=mx+b is the point (0,b).
The values of x where an expression is zero. These are the x-coordinates of the x-intercepts of the expression’s graph. For a polynomial expression, these are usually called roots.