ACT Math key terms

A whole number that may be positive, negative, or zero, with no fractional or decimal part, such as −4, 0, or 17. Values like 3/4 or 2.5 are never integers.

Any number that can be written as a fraction of two integers with a nonzero denominator. Its decimal form either ends, like 0.25, or repeats a block forever, like 0.333….

A number that cannot be written as a fraction of two integers. Its decimal form continues forever without repeating any block, as with √2 or π.

The distance of a number from zero on a number line, which is never negative. Written with bars, |−7| equals 7 because −7 sits seven units from zero.

The amount a quantity increases or decreases, stated as a percent of its starting amount. Divide the change by the original value, then multiply by 100 to convert to a percent.

A comparison of two quantities by division, written as a : b or as the fraction a/b. It reports how much of one quantity exists for each unit of the other.

A compact way to write very large or very small numbers as a number from 1 up to 10 multiplied by a power of ten, such as 3.2 × 10⁵ or 4.1 × 10⁻³.

An equation whose graph is a straight line, commonly written y = mx + b. Each variable appears only to the first power, with no squares, products of variables, or roots.

A measure of how steep a line is: the change in y divided by the change in x between any two points on it. Horizontal lines have slope 0; vertical lines have undefined slope.

The equation of a line written as y = mx + b, where m gives the slope and b gives the y-coordinate of the point where the line crosses the y-axis.

Two or more linear equations solved together, seeking values that satisfy every equation at once. Graphed, the solution is the point where the lines intersect.

An equation in which the variable's highest power is two, usually written ax² + bx + c = 0 with a not equal to zero. Its graph is a parabola.

The expression b² − 4ac taken from ax² + bx + c = 0. Its sign reveals the number of real solutions: positive gives two, zero gives one, negative gives none.

An expression formed by adding or subtracting terms of the form (constant)·xⁿ, where n is a whole number, such as 4x³ − 2x + 7. Variables never sit in denominators or under radicals.

A statement comparing two expressions using <, >, ≤, or ≥ rather than an equals sign. Its solution is a range of values instead of a single number.

A rule that assigns exactly one output to each permitted input. A graph represents a function when no vertical line crosses it more than once.

The full set of input values for which a function is defined. Restrictions usually come from dividing by zero or from taking an even root of a negative number.

The set of all output values a function actually produces from its domain. On a graph, it is the collection of y-values the curve reaches.

A function built by feeding the output of one function into another, written f(g(x)). Evaluate from the inside out: first compute g at x, then apply f to that result.

A function that reverses another, sending each output back to the input that produced it. It exists only when the original pairs every output with exactly one input.

A pattern in which a quantity is multiplied by the same factor in every equal time period, modeled by y = a·bˣ with b greater than 1. Values of b between 0 and 1 describe decay.

The power to which a base must be raised to produce a given number. For instance, log₂ 8 = 3 because 2³ = 8. Logarithms reverse exponential relationships.

A shift, stretch, shrink, or reflection applied to a parent graph. Adding to or subtracting from x moves the graph sideways, while changes outside the function move or stretch it vertically.

In any right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². It allows you to find an unknown side length.

Triangles whose corresponding angles are equal and whose corresponding sides are proportional. Their side lengths differ by a constant scale factor, which also scales perimeters.

An angle with its vertex at the center of a circle and its sides drawn as radii. Measured in degrees, it equals the measure of the arc it cuts off.

The pie-slice piece of a circle enclosed by two radii and the arc joining their endpoints. Its area is the same fraction of the whole circle as the central angle is of 360°.

The amount of three-dimensional space a solid occupies, measured in cubic units. Familiar formulas include prism V = Bh, cylinder V = πr²h, and sphere V = (4/3)πr³.

The length of the segment joining (x₁, y₁) and (x₂, y₂) in the coordinate plane: √[(x₂ − x₁)² + (y₂ − y₁)²]. It follows directly from the Pythagorean theorem.

The point exactly halfway between two given points, found by averaging their coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2). It gives the center of the segment joining them.

A ratio of two sides of a right triangle that links an acute angle to side lengths: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.

A right triangle whose side ratios are fixed. In a 45-45-90 triangle sides follow 1 : 1 : √2; in a 30-60-90 triangle they follow 1 : √3 : 2.

The arithmetic average, computed by adding all values and dividing by how many values there are. Extreme values pull the mean toward them.

The middle value when the data are arranged in order; with an even count, it is the average of the two middle values. It is not dragged far by outliers.

A measure of how far data values typically sit from their mean. A small value means the values cluster tightly; a large value means they are widely scattered.

A number from 0 to 1 giving the chance that an event occurs, found as favorable outcomes divided by total equally likely outcomes. Zero means impossible and 1 means certain.

Two events where the occurrence of one does not alter the chance of the other. For such events, multiply their individual probabilities to get the chance that both happen.

An ordered arrangement of items, used when order matters, such as awarding first, second, and third place. The number of ways to arrange n items taken r at a time is n!/(n − r)!.

A selection of items in which order does not matter, such as choosing three pizza toppings. The count of ways to choose r items from n is n!/[r!(n − r)!].

A straight line drawn through a scatterplot that passes as close as possible to the plotted points, used to describe a trend and make predictions. Least-squares regression produces this line.