FTCE Mathematics 6-12 key terms

Any number that can be written as a fraction of two integers with a nonzero denominator. Its decimal form either terminates or repeats.

A real number that cannot be written as a ratio of two integers. Its decimal expansion never terminates and never repeats, as with the square root of two or pi.

The set of all rational and irrational numbers, closed under addition, subtraction, multiplication, and division by nonzero values, and ordered on a number line.

The distance of a number from zero on the number line, always nonnegative. Written with vertical bars, it equals the number itself when nonnegative and its opposite when negative.

The agreed sequence for evaluating expressions: grouping symbols first, then exponents, then multiplication and division left to right, then addition and subtraction left to right.

A way of writing numbers as a product of a number between one and ten and a power of ten, useful for very large or very small quantities.

Laws governing powers, including adding exponents when multiplying like bases, subtracting when dividing, and multiplying exponents for a power of a power.

An expression using a root symbol. The square root of a nonnegative number is the value whose square equals that number; cube roots apply to all reals.

A number of the form a + bi where a and b are real and i is the square root of negative one. Arithmetic follows the same field rules as reals.

An equation whose graph is a straight line, typically written as y equals mx plus b or in standard form, with variables raised only to the first power.

The constant rate of change of a line, computed as the change in the dependent variable divided by the change in the independent variable between two points.

Two or more linear equations considered together. A solution is an ordered pair or set of values satisfying every equation simultaneously; systems may have none, one, or infinitely many solutions.

A function of degree two, whose graph is a parabola opening upward or downward. Its standard form is a x squared plus b x plus c with a nonzero.

The quantity b squared minus four a c in a quadratic equation. Its sign indicates whether the equation has two real, one real, or two complex solutions.

An expression formed from variables and constants using only addition, subtraction, and nonnegative whole-number exponents. Its degree is the highest exponent present.

A polynomial has a factor of x minus c exactly when evaluating the polynomial at c gives zero. This links factors to zeros and x-intercepts.

A rule listing possible rational zeros of a polynomial with integer coefficients as factors of the constant term divided by factors of the leading coefficient.

The exponent to which a base must be raised to produce a given number. Logarithms are inverses of exponential functions and convert products into sums.

A rule assigning exactly one output value to each input in its domain. It passes the vertical line test when graphed.

The domain is the set of all valid inputs of a function; the range is the set of all outputs it produces. Both may be restricted by context or formula.

A function that undoes another, exchanging inputs and outputs. It exists only when the original function is one-to-one, and its graph reflects across the line y equals x.

A function of the form a times b to the x where the base b is positive and not one. It models constant percentage growth or decay.

How much one quantity changes relative to another, often the slope of a secant line over an interval. For a linear function it is constant.

The highest or lowest point on a parabola, located at x equals negative b over two a for a quadratic in standard form. It represents the extreme value.

A line that a graph approaches arbitrarily closely as the input or output grows without bound, though the graph may never touch it.

A function that is a ratio of two polynomials. Its graph often has vertical asymptotes where the denominator is zero and horizontal or slant asymptotes.

Conditions such as side-angle-side, angle-side-angle, and side-side-side that guarantee two triangles are congruent without checking every part.

Triangles with equal corresponding angles and proportional corresponding sides. Their scale factor relates all corresponding lengths.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. The converse identifies right triangles from side lengths.

The distance between two points in the coordinate plane, derived from the Pythagorean theorem as the square root of the sum of squared coordinate differences.

Measures of three-dimensional space enclosed by a solid and of the total area of its outer faces. Formulas differ by shape, such as prisms, cylinders, cones, and spheres.

For an acute angle in a right triangle, the ratio of two sides such as opposite over hypotenuse for sine. These ratios extend to all angles via the unit circle.

A circle of radius one centered at the origin. Points on it define sine and cosine of any angle as the y and x coordinates respectively.

An angle measure based on arc length, where one radian subtends an arc equal to the radius. A full revolution equals two pi radians.

A relation stating that each side of a triangle divided by the sine of its opposite angle is constant. It solves triangles when an angle and opposite side are known.

A generalization of the Pythagorean theorem relating three sides of any triangle to one angle, used to solve triangles without a known angle-side pair.

Three measures of center: the arithmetic average, the middle value of ordered data, and the most frequent value. Each behaves differently with skew and outliers.

A measure of how spread out data values are around the mean, equal to the square root of the variance. Larger values indicate greater variability.

A symmetric bell-shaped probability distribution defined by its mean and standard deviation, in which roughly 68 percent of values fall within one standard deviation of the mean.

The number of standard deviations a data value lies from the mean, computed by subtracting the mean and dividing by the standard deviation. It allows comparison across distributions.

A graph plotting paired data to reveal association, together with a coefficient between negative one and one describing the strength and direction of a linear relationship.

A regression line chosen to minimize squared vertical distances from data points, used to model trends and make predictions from paired data.

The probability that an event occurs given that another event has already occurred, computed as the joint probability divided by the probability of the condition.

Two events whose joint probability equals the product of their individual probabilities, meaning one occurring does not change the chance of the other.

Counts of arrangements where order matters versus selections where order does not. Their formulas use factorials and underlie many probability calculations.

The probability of exactly k successes in n independent trials, each with the same success probability, computed with the binomial coefficient and powers of p and its complement.

The long-run average outcome of a random variable, found by multiplying each value by its probability and summing the results.

The value a function approaches as its input approaches a specified point. Limits underpin continuity, derivatives, and integrals.

A property of a function whose graph has no breaks, jumps, or holes at a point; formally the limit there equals the function value.

The instantaneous rate of change of a function, defined as the limit of a difference quotient. Geometrically it gives the slope of the tangent line.

A limit of sums used to compute accumulated change, interpreted as signed area under a curve. The fundamental theorem links it to antiderivatives.

An ordered list of terms and the sum of those terms. Arithmetic and geometric types have explicit formulas for terms and partial sums.

A proof technique establishing a base case and an inductive step showing that if a statement holds for one value it holds for the next, proving it for all positive integers.