Georgia Unified 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 a fixed block of digits forever.

A real number that cannot be written as a fraction of two integers. Its decimal expansion never ends and never settles into a repeating block, as with the square root of two.

Any point on the number line, including every rational and irrational value. It covers positives, negatives, and zero, and is the largest number set used in high school algebra.

A whole number with no fractional part, such as negative four, zero, or seventeen. Integers are closed under addition, subtraction, and multiplication but not always under division.

A number written as a + bi, where a and b are real and i is the square root of negative one. Every quadratic equation has solutions among complex numbers.

The symbol i, defined so that i squared equals negative one. Powers of i cycle through four values, which lets you simplify expressions like i raised to a large exponent.

A way to write very large or very small numbers as a product of a number between one and ten and a power of ten. Moving the decimal point changes the exponent by the same count.

A ratio that compares a quantity to exactly one unit of another quantity, such as miles per one hour. Unit rates make it easy to compare prices or speeds written in different units.

A method of converting units by multiplying by fractions equal to one, arranged so unwanted units cancel. It catches setup errors because the final units must match the quantity requested.

A relationship between two quantities whose ratio stays constant, graphed as a line through the origin. Doubling one quantity doubles the other, and the constant ratio is the unit rate.

A mathematical phrase built from numbers, variables, and operations but containing no equals sign. It can be simplified or evaluated but not solved.

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

Expressions that produce the same output for every input value, even though they look different. Rewriting with the distributive property or combining like terms preserves equivalence.

The rule that a(b + c) equals ab + ac, letting a factor outside parentheses be multiplied into each term inside. It is the main tool for expanding and factoring expressions.

Rewriting a sum or polynomial as a product of simpler expressions. Common patterns include greatest common factor, difference of squares, and trinomials that multiply back to the original.

An equation in which each variable appears to the first power only, so its graph is a straight line. It has exactly one solution unless the variable cancels and leaves a true or false statement.

Two or more equations considered together, with solutions being values that satisfy all of them at once. Graphically, the solution is where the graphs intersect.

Solving a system by rewriting one equation for a single variable and inserting that expression into the other equation. It works well when one variable already has a coefficient of one.

Solving a system by adding or subtracting multiples of the equations so one variable cancels. It is efficient when coefficients line up or can be made opposites easily.

An equation of the form ax squared plus bx plus c equals zero with a not zero. Its solutions, called roots, may be found by factoring, completing the square, or the quadratic formula.

The expression giving the roots of a quadratic equation as negative b plus or minus the square root of b squared minus four ac, all divided by two a. It solves any quadratic equation.

The value b squared minus four ac taken from under the radical in the quadratic formula. Its sign tells whether the equation has two real roots, one repeated real root, or two complex roots.

If a product of factors equals zero, then at least one factor must equal zero. This is why setting a factored quadratic equal to zero reveals its solutions.

Shortcuts for combining powers: add exponents when multiplying like bases, subtract when dividing, and multiply when raising a power to a power. Negative and zero exponents follow from these same rules.

A rule that assigns exactly one output to each allowed input. The set of inputs is the domain, the set of outputs is the range, and a graph passes the vertical line test only if it is a function.

The domain is every input value for which a function is defined; the range is every output value it actually produces. Restrictions often come from division by zero or even roots of negatives.

A graph represents a function only if no vertical line crosses it more than once. A second crossing would mean one input paired with two different outputs.

The steepness of a line, measured as the change in output divided by the change in input between two points. It is the constant rate of change in a linear function.

How much an output quantity changes per unit change in the input, averaged over an interval. For a line it is constant and equals the slope; for other functions it varies by interval.

A function of the form a times b to the x power, where the base b is positive and not one. It grows or decays by a constant percent each step rather than a constant amount.

A function that undoes another, swapping inputs and outputs so their graphs mirror across the line y equals x. It exists only when the original function pairs each output with a single input.

Shifting, stretching, compressing, or reflecting a parent graph by changing its equation. Adding outside the function moves it vertically; adding inside moves it horizontally in the opposite direction.

The highest or lowest point of a quadratic graph, located at the axis of symmetry. In vertex form the coordinates can be read directly from the equation.

A relationship in which two figures have exactly the same size and shape, so one can be mapped onto the other by rigid motions. Corresponding sides and angles are equal.

A relationship in which two figures have the same shape but possibly different sizes, with corresponding angles equal and side lengths in a constant ratio. Dilations produce similar figures.

A transformation such as a translation, rotation, or reflection that preserves distances and angles. Rigid motions map a figure onto a congruent copy.

The multiplier that compares a length in an image to the corresponding length in the original figure. Values above one enlarge, between zero and one shrink, and negatives also flip direction.

A transformation that stretches or shrinks a figure from a fixed center by a scale factor. It preserves angle measures and shape but changes lengths unless the factor is one.

In a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. It also underlies the distance formula on the coordinate plane.

The distance between two points equals the square root of the sum of the squared differences of their coordinates. It is the Pythagorean theorem applied to a segment on the coordinate plane.

A ratio of two sides of a right triangle, such as sine, cosine, or tangent, tied to an acute angle. These ratios let you find unknown sides or angles.

The amount of space a three-dimensional solid occupies, measured in cubic units. Formulas differ by solid, and scaling every length by k multiplies volume by k cubed.

The flat shape exposed when a plane slices through a solid. Slicing a cylinder parallel to its base gives a circle, while a diagonal slice can give an ellipse.

The arithmetic average, found by adding all data values and dividing by how many there are. It is pulled strongly toward outliers, unlike the median.

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 far data values typically sit from the mean, equal to the square root of the variance. A larger value means the data are more spread out.

The difference between the third quartile and the first quartile, covering the middle half of the data. It describes spread while ignoring the most extreme quarter on each side.

A symmetric, bell-shaped curve in which most values cluster near the mean. Roughly 68 percent of values fall within one standard deviation and about 95 percent within two.

A graph plotting paired data as points on a coordinate plane to reveal a relationship between two variables. The pattern suggests whether a linear or other model fits.

A straight line drawn through a scatter plot that comes closest to the overall pattern of points. It is used to predict an output for a given input.

A number between negative one and one describing how strongly two variables move together in a linear pattern. It measures association, not proof that one variable causes the other.

The chance an event occurs given that another event has already happened, written as P of A given B. It shrinks the sample space to only the outcomes where the condition holds.

Two events are independent when the occurrence of one does not change the probability of the other. For such events, the probability both occur equals the product of their probabilities.

A number from zero to one expressing how likely an event is, computed as favorable outcomes divided by total equally likely outcomes. All possible outcomes together have probability one.