DAT Quantitative Reasoning key terms

A combination of numbers, variables, and operation signs that represents a value, such as 3x + 5. Unlike an equation, it has no equals sign and cannot be solved, only simplified or evaluated.

A letter or symbol standing for an unknown or changing quantity in an expression or equation. Isolating it is usually the goal when solving for an unknown value.

An equation in which every variable appears to the first power, so its graph is a straight line. Solving typically means isolating the variable using inverse operations applied equally to both sides.

Two or more equations that must be satisfied at the same time. A solution is the set of values making all equations true simultaneously, found by substitution, elimination, or graphing.

The steepness of a line, computed as the change in y divided by the change in x between two points. A horizontal line has slope zero; a vertical line has undefined slope.

The point where a graph crosses the vertical axis, occurring where x equals zero. In slope-intercept form y = mx + b, the value b is the y-intercept.

An equation whose highest power of the variable is two, usually written ax² + bx + c = 0 with a not equal to zero. Solutions come from factoring, completing the square, or the quadratic formula.

The expression x = (−b ± √(b² − 4ac)) / 2a giving the solutions of any quadratic equation ax² + bx + c = 0. The part under the radical determines how many real solutions exist.

The quantity b² − 4ac taken from a quadratic equation. Positive means two distinct real solutions, zero means one repeated real solution, and negative means no real solutions.

Rewriting an expression as a product of simpler expressions, such as turning x² − 9 into (x − 3)(x + 3). It is a standard route to solving polynomial equations set equal to zero.

A sum of terms in which variables have only nonnegative whole-number exponents, such as 4x³ − 2x + 7. The degree equals the largest exponent present.

A rule assigning exactly one output value to each allowed input value. Passing the vertical line test on a graph confirms the relationship is a function.

The domain is the complete set of permitted input values for a function; the range is the complete set of resulting output values. Restrictions often come from division by zero or even roots.

Shortcuts for combining powers: multiplying like bases adds exponents, dividing subtracts them, a power raised to a power multiplies them, and a negative exponent means a reciprocal.

The exponent to which a base must be raised to produce a given number, so log base b of x asks what power of b equals x. Logarithms undo exponentials and turn multiplication into addition.

A comparison of two quantities by division, expressing how many times one value contains another. Ratios can be written with a colon, as a fraction, or in words.

A statement that two ratios are equal, solved by cross-multiplying. Proportions are the standard tool for scaling recipes, maps, and similar figures.

A ratio out of one hundred, meaning per hundred. Converting between decimals, fractions, and percents requires shifting the decimal point or multiplying by one hundred.

The difference between a new value and an original value, divided by the original value and multiplied by one hundred. It reports increase or decrease relative to where you started.

The sum of a set of values divided by how many values there are. Rearranging the definition lets you find a missing value when the mean and the others are known.

The middle value of an ordered data set, or the average of the two middle values when the count is even. Unlike the mean, it resists distortion by extreme outliers.

The value appearing most frequently in a data set. A set may have one mode, several modes, or none when every value occurs equally often.

A mean in which each value contributes in proportion to an assigned weight rather than equally. Multiply each value by its weight, sum the products, then divide by the total weight.

A number from zero to one expressing how likely an event is, computed as favorable outcomes divided by total equally likely outcomes. Complementary events sum to one.

An arrangement of items in which order matters, such as the finishing order of a race. The count is n factorial divided by (n − r) factorial when choosing r from n items.

A selection of items in which order does not matter, such as choosing a committee from a group. The permutation count is divided by r factorial to remove duplicate orderings.

A triangle containing one ninety-degree angle. Its sides obey the Pythagorean theorem, and its trigonometric ratios are defined relative to that right angle.

The relationship stating that in a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides, written a² + b² = c². It also underlies the distance formula.

Fixed ratios of side lengths in a right triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. They connect angles to side lengths.

The expression √((x₂ − x₁)² + (y₂ − y₁)²) giving the straight-line distance between two points on a coordinate plane. It is the Pythagorean theorem applied to horizontal and vertical gaps.

Standard relationships for common shapes: rectangle area is length times width, triangle area is one-half base times height, and cylinder volume is πr²h. Rectangular solid volume is length times width times height.

Changing a measurement from one unit to another by multiplying by a fraction equal to one, such as 12 inches over 1 foot. Units cancel diagonally to verify the setup.

A way of writing numbers as a value between one and ten multiplied by a power of ten, such as 4.2 × 10⁻⁵. It simplifies very large or very small quantities and their arithmetic.

A statement comparing two expressions with symbols such as less than, greater than, or not equal. Multiplying or dividing both sides by a negative number reverses the direction of the symbol.

The distance of a number from zero on the number line, always nonnegative. Equations containing it typically split into two separate cases to solve.

A sequence is an ordered list of numbers following a pattern; a series is the sum of such terms. Arithmetic patterns add a constant each step, while geometric patterns multiply by a constant.