NYS Math key terms

A number that can be written as a fraction of two integers, with a nonzero denominator. Its decimal form either ends or repeats in a fixed pattern forever, such as 0.75 or 0.333...

A real number that cannot be written as a fraction of two integers. Its decimal form never ends and never repeats, as with the square root of 2 or pi.

All numbers on the number line, including rational and irrational values. Every real number has a decimal representation, and the set is closed under the four basic operations except division by zero.

A whole number, its negative counterpart, or zero, with no fractional part. The set includes values like -7, 0, and 42 and is closed under addition, subtraction, and multiplication.

The distance of a number from zero on the number line, always zero or positive. Written with vertical bars, so the absolute value of -6 is 6 and of 6 is 6.

Shortcut laws for combining powers with the same base: add exponents when multiplying, subtract when dividing, and multiply when raising a power to a power. They also define negative and zero exponents.

A way to write very large or very small numbers as a number between 1 and 10 multiplied by a power of ten, such as 3.2 x 10^5, which equals 320,000.

A root of a number, shown with the square-root or nth-root symbol. For example, the square root of 49 is 7 because 7 times 7 equals 49; even roots of negative numbers are not real.

A ratio that compares a quantity to exactly one unit of another quantity, such as 55 miles per one hour. It lets you scale, compare, and convert between different measurement systems.

The difference between a new value and an original value, divided by the original value and expressed as a percent. It measures increase or decrease relative to where you started.

A relationship between two quantities whose ratio stays constant, graphed as a line through the origin. It is written y = kx, where k is the constant of proportionality.

A mathematical phrase built from numbers, variables, and operations, with no equals sign. Examples include 4x - 7 and 2(a + 3); it can be simplified or evaluated but not solved.

A statement that two expressions have equal value, joined by an equals sign. Solving it means finding every value of the variable that makes the statement true.

Two expressions that produce the same result for every value of their variables. For instance, 3(x + 2) and 3x + 6 are equivalent because distributing changes form but not value.

The rule that multiplying a sum by a number gives the same result as multiplying each addend separately and then adding. Formally, a(b + c) equals ab + ac.

Rewriting an expression as a product of simpler expressions. For example, x^2 - 9 factors as (x - 3)(x + 3), which reveals the zeros of the related quadratic function.

An equation whose graph is a straight line, with each variable raised only to the first power. Standard form is Ax + By = C, and slope-intercept form is y = mx + b.

Two or more equations involving the same variables, solved together. A solution makes every equation true at once, and it corresponds to a point where the graphs intersect.

A technique for solving systems by expressing one variable in terms of the other and replacing it in the second equation. This leaves a single equation in one variable.

A technique for solving systems by adding or subtracting the equations, sometimes after multiplying one, so that a variable cancels. The result is one equation in one variable.

A statement comparing two expressions with less than, greater than, or their or-equal-to forms. Its solution is a range of values, graphed as a shaded region or interval on a number line.

An equation of the form ax^2 + bx + c = 0 with a nonzero. It can be solved by factoring, completing the square, or the quadratic formula, and it may have zero, one, or two real solutions.

A formula giving the solutions of any quadratic equation: x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. It works even when factoring fails.

The part of the quadratic formula under the square root, b^2 - 4ac. Its sign tells how many real solutions exist: positive gives two, zero gives one, negative gives none.

A rule that assigns exactly one output to each allowed input. Passing the vertical line test confirms that a graph represents a function, since no input has two outputs.

The complete set of input values for which a function is defined. For a square-root function the radicand must be nonnegative, and for a rational function the denominator cannot be zero.

The set of all output values a function actually produces as its input varies over the domain. It is the collection of y-values visible in the graph.

A function whose graph is a straight line, written f(x) = mx + b. The slope m gives the constant rate of change, and b is the y-intercept.

The steepness of a line, calculated as the change in y divided by the change in x between two points. It describes how fast the output changes per unit of input.

How much one quantity changes relative to another, often the slope of a graph over an interval. For nonlinear functions it varies, so the average rate of change is taken between two points.

A function of the form f(x) = ax^2 + bx + c, whose graph is a parabola. The sign of a sets the direction it opens, and the vertex marks its maximum or minimum.

The highest or lowest point on a parabola, where the curve turns around. In vertex form f(x) = a(x - h)^2 + k, the vertex is the point (h, k).

The vertical line through a parabola's vertex that divides it into mirror-image halves. Its equation is x = -b/(2a) for a quadratic written in standard form.

A function of the form f(x) = a times b^x, where the base b is positive and not 1. It grows or decays by a constant percent each step rather than a constant amount.

A shift, stretch, compression, or reflection applied to a parent graph. Adding outside the function moves it up or down; adding inside moves it left or right.

The relationship between figures with the same size and shape. Rigid motions such as translations, rotations, and reflections map one congruent figure exactly onto the other.

The relationship between figures with the same shape but possibly different sizes. Corresponding angles are equal and corresponding side lengths share a constant scale factor.

A rule that moves or resizes every point of a figure, producing a new figure. Common types are translations, rotations, reflections, and dilations.

The ratio by which a figure is enlarged or reduced in a dilation or similarity. A factor greater than 1 enlarges; between 0 and 1 reduces; negative values also rotate the figure.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a^2 + b^2 = c^2. It is used to find missing side lengths and distances.

Ratios comparing side lengths in a right triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. They depend only on the acute angle.

The amount of three-dimensional space a solid occupies, measured in cubic units. Formulas differ by shape, such as length times width times height for a rectangular prism.

A rule giving the straight-line distance between two points, derived from the Pythagorean theorem. For points (x1, y1) and (x2, y2) it is the square root of the sum of squared differences.

A grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin. Each point is named by an ordered pair (x, y) showing horizontal and vertical position.

The arithmetic average of a data set, found by adding all values and dividing by how many there are. It is affected strongly by unusually large or small values.

The middle value of a data set ordered from least to greatest, or the average of the two middle values when the count is even. It resists distortion by extreme values.

A measure of how spread out data values are around the mean. A small value means values cluster near the mean; a large value means they are widely scattered.

The difference between the third quartile and the first quartile of a data set. It measures the spread of the middle half of the data and ignores extreme values.

A graph showing a data set's five-number summary: minimum, first quartile, median, third quartile, and maximum. The box spans the middle half of the data.

A symmetric, bell-shaped distribution of data centered on the mean. About 68 percent of values fall within one standard deviation of the mean and about 95 percent within two.

A number from -1 to 1 describing the strength and direction of a linear relationship between two variables. Values near 1 or -1 indicate a strong fit; near 0 indicates a weak one.

A straight line drawn through a scatter plot that comes closest to the data points overall. It models the trend and can be used to predict values.

A graph of paired data with one variable on each axis, showing each pair as a point. It reveals whether two quantities are related and how strong that relationship appears.

A number from 0 to 1 expressing how likely an event is, where 0 means impossible and 1 means certain. For equally likely outcomes it is favorable outcomes divided by total outcomes.

The chance that an event occurs given that another event is already known to have happened. It restricts attention to a smaller group of outcomes rather than the whole sample space.

The complete list or set of all possible outcomes of a probability experiment. Every event is a subset of it, and its probabilities must sum to 1.

A compact way to write a range of values using brackets and parentheses. A square bracket includes the endpoint, a parenthesis excludes it, and infinity always takes a parenthesis.