MTTC Mathematics (Secondary) (022) key terms

A specific case that satisfies the hypotheses of a statement but makes its conclusion false. One counterexample is enough to disprove a universal claim.

A method proving a statement for all positive integers: establish a base case, assume it holds for an arbitrary integer, then show it must hold for the next.

A number expressible as a ratio of two integers with a nonzero denominator. Its decimal expansion either terminates or eventually repeats.

A real number that cannot be written as a ratio of two integers. Its decimal expansion neither terminates nor repeats, such as pi or the square root of two.

The unique expression of a positive integer greater than one as a product of primes, apart from the order of the factors.

The largest positive integer that divides each number in a given set without a remainder. It helps simplify fractions and solve divisibility problems.

The smallest positive integer that is a multiple of each number in a given set. It is useful for adding fractions with unlike denominators.

A number of the form a + bi, where a and b are real numbers and i squared equals negative one. Complex numbers extend the real line to a plane.

A rule assigning exactly one output value to each input value in its domain. Graphs of functions pass the vertical line test.

The domain is the set of all input values for which a function is defined; the range is the set of all output values actually produced. Together they describe a function's scope.

A function that reverses another function's action, so composing them returns the original input. It exists only when the original function is one-to-one.

The operation forming f(g(x)) by applying g first and then f. The domain of the composition excludes inputs where g is undefined or f cannot accept g's output.

A function whose graph is a straight line, writable as f(x) = mx + b. The constant m is the slope, and b is the y-intercept.

A function of the form f(x) = ax^2 + bx + c with a nonzero. Its graph is a parabola, and its vertex gives the maximum or minimum value.

A function of the form f(x) = a times b^x, where the base b is positive and not one. It models constant percentage growth or decay.

The exponent to which a base must be raised to produce a given number. Logarithms are inverses of exponential functions and turn multiplication into addition.

A sequence in which each term after the first is obtained by multiplying by a fixed nonzero constant, called the common ratio.

A relationship in which two figures have 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 not necessarily the same size. Corresponding angles are equal and corresponding side lengths are proportional.

A transformation that preserves distance and angle measure, such as a translation, rotation, or reflection. It maps a figure onto a congruent figure.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. It supports distance and right-triangle calculations.

An angle whose vertex lies on a circle and whose sides contain chords of the circle. Its measure is half the measure of the intercepted arc.

An angle whose vertex is the center of a circle and whose sides are radii. Its measure equals the measure of the intercepted arc.

The amount of three-dimensional space enclosed by a solid. Common formulas multiply a base area by a height, with adjustments for cones and spheres.

The arithmetic average, found by adding all data values and dividing by the number of values. It is sensitive to extreme values.

The middle value when data are ordered from least to greatest, or the average of the two middle values for an even count. It resists distortion by outliers.

A measure of how far data values typically lie from their mean, computed from squared deviations. A larger value indicates greater spread.

A symmetric, bell-shaped probability distribution defined by its mean and standard deviation. About 68 percent of values lie within one standard deviation of the mean.

The probability that an event occurs given that another event has already occurred, written P(A|B). It equals P(A and B) divided by P(B).

Two events are independent when the occurrence of one does not change the probability of the other. Then P(A and B) equals P(A) times P(B).

An arrangement of a set of items in a particular order. The number of permutations of n distinct items taken r at a time is n factorial divided by (n minus r) factorial.

A selection of items from a group in which order does not matter. The number of combinations of n items taken r at a time is n factorial divided by r factorial times (n minus r) factorial.

A circle of radius one centered at the origin. It defines sine and cosine for all angles as the y- and x-coordinates of points on the circle.

An angle measure defined by arc length on a unit circle. One full revolution equals two pi radians, so pi radians equals 180 degrees.

An equation involving trigonometric functions that holds for every value where both sides are defined, such as the Pythagorean identity sine squared plus cosine squared equals one.

A generalization of the Pythagorean theorem relating side lengths of any triangle to the cosine of one angle: c squared equals a squared plus b squared minus two a b cosine C.

The value a function's output approaches as the input approaches a specified point or grows without bound. Limits underlie continuity, derivatives, and integrals.

The instantaneous rate of change of a function with respect to its input, defined as a limit of difference quotients. Geometrically it gives the slope of the tangent line.

The accumulated change of a function over an interval, often interpreted as signed area under a curve and defined as a limit of sums.

The theorem connecting differentiation and integration: the definite integral of a continuous function can be evaluated using any antiderivative, and differentiation undoes integration.