FTCE Mathematics 6-12 flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Rational number

A number expressible as a fraction p/q of integers with q ≠ 0. Its decimal expansion either terminates or eventually repeats.

Irrational number

A real number that cannot be written as a ratio of integers, such as √2 or π. Its decimal expansion is nonterminating and nonrepeating.

Absolute value

The distance of a real number from zero on the number line, written |x|. It is always nonnegative, with |x| = x when x ≥ 0 and −x when x < 0.

Scientific notation

A number written as a × 10ⁿ where 1 ≤ |a| < 10 and n is an integer. It simplifies comparing and computing with very large or small quantities.

Greatest common factor

The largest positive integer dividing each number in a set. Found by taking the lowest power of each shared prime factor.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = [−b ± √(b² − 4ac)]/(2a). The discriminant b² − 4ac determines the nature of roots.

Discriminant

The quantity b² − 4ac for a quadratic ax² + bx + c. Positive gives two real roots, zero gives one repeated root, negative gives two complex conjugate roots.

Remainder Theorem

Dividing a polynomial p(x) by (x − r) leaves remainder p(r). Consequently, (x − r) is a factor exactly when p(r) = 0.

Rational Root Theorem

For a polynomial with integer coefficients, every rational root p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient.

Vertex of a parabola

For y = ax² + bx + c, the vertex is at x = −b/(2a). It is the minimum when a > 0 and the maximum when a < 0.

Logarithm definition

log_b x = y means b^y = x, requiring b > 0, b ≠ 1, and x > 0. Logarithms are the inverses of exponential functions.

Function

A relation assigning each element of the domain exactly one element of the range. Its graph passes the vertical line test.

Inverse function

A function f⁻¹ that reverses f, so f⁻¹(f(x)) = x. It exists only when f is one-to-one, and its graph reflects f across y = x.

Even and odd functions

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis. Odd functions satisfy f(−x) = −f(x) and are symmetric about the origin.

Horizontal asymptote

A line y = L that a rational function's values approach as x grows without bound. Determined by comparing the degrees of numerator and denominator.

Composition of functions

The function (f∘g)(x) = f(g(x)), applying g first then f. Its domain excludes inputs where g(x) is outside f's domain.

Pythagorean theorem

In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse: a² + b² = c².

Similar figures

Figures with equal corresponding angles and proportional corresponding sides. If the side ratio is k, area ratios are k² and volume ratios are k³.

Interior angle sum

For a polygon with n sides, the interior angles sum to (n − 2)·180°. Each interior angle of a regular n-gon measures (n − 2)·180°/n.

Sector area

The area of a circular sector is A = ½r²θ when the central angle θ is in radians, or A = (θ/360)πr² when θ is in degrees.

Sphere volume and area

A sphere of radius r has volume (4/3)πr³ and surface area 4πr².

Pythagorean identity

For any angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ.

Law of Sines

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite its named angle. Useful for ASA, AAS, and SSA cases.

Law of Cosines

c² = a² + b² − 2ab cos C relates the three sides of a triangle to one angle. It applies to SSS and SAS cases and reduces to the Pythagorean theorem when C = 90°.

Radian measure

An angle measure based on arc length: θ = s/r. A full circle is 2π radians, equivalent to 360°, so 180° = π radians.

Double-angle formula

sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1. These follow from the sum formulas with A = B = θ.

Standard deviation

A measure of spread equal to the square root of the average squared deviation from the mean. Larger values indicate data more dispersed about the mean.

Combination versus permutation

Combinations C(n, r) count unordered selections of r items from n. Permutations P(n, r) count ordered arrangements and equal C(n, r)·r!.

Conditional probability

The probability of event A given that B occurred: P(A|B) = P(A ∩ B)/P(B), defined when P(B) > 0.

Binomial probability

For n independent trials each with success probability p, P(X = k) = C(n, k)p^k(1 − p)^(n − k), giving the chance of exactly k successes.

Chain rule

The derivative of a composite function is d/dx f(g(x)) = f′(g(x))·g′(x). It handles nested functions such as sin(3x²).

Fundamental Theorem of Calculus

If F is an antiderivative of continuous f on [a, b], then ∫ₐᵇ f(x) dx = F(b) − F(a), linking area under a curve to antiderivatives.