MTTC Mathematics (Secondary) (022) flashcards

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

Fundamental Theorem of Arithmetic

Every integer greater than one has a unique prime factorization up to the order of the factors; this justifies prime-based divisibility and divisor-counting arguments.

Euclidean algorithm

For positive integers a and b, repeatedly replace the pair (a, b) with (b, a mod b); the last nonzero remainder is gcd(a, b).

Divisibility by 11

An integer is divisible by 11 exactly when the alternating sum of its decimal digits is divisible by 11, counting from either end with alternating signs.

Rational root theorem

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

Modular inverse condition

An integer a has a multiplicative inverse modulo m exactly when gcd(a, m) equals 1; otherwise the congruence ax is congruent to 1 modulo m has no solution.

Fermat's little theorem

If p is prime and a is not divisible by p, then a to the power p minus 1 is congruent to 1 modulo p; it is used in modular exponent reduction.

Inclusion-exclusion for two sets

The number of elements in the union of sets A and B equals the number in A plus the number in B minus the number in their intersection.

Quadratic formula

For ax squared plus bx plus c equals zero with a nonzero, the solutions are x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a.

Discriminant test

For a quadratic with real coefficients, the discriminant b squared minus 4ac is positive for two real roots, zero for one repeated real root, and negative for two complex conjugate roots.

Vertex of a parabola

For the quadratic f of x equals a x squared plus b x plus c, the vertex has x-coordinate negative b divided by 2a; its y-coordinate is f evaluated there.

Change of base formula

The logarithm of x base b equals the logarithm of x base c divided by the logarithm of b base c, for positive x and bases not equal to 1.

Vertical asymptote of rational function

A rational function has a vertical asymptote at x equals r when the denominator is zero at r but the factor x minus r does not cancel from the numerator.

Arithmetic sequence nth term

For an arithmetic sequence with first term a sub 1 and common difference d, the nth term is a sub n equals a sub 1 plus n minus 1 times d.

Inverse function test

A function has an inverse function exactly when it is one-to-one; its graph is the reflection of the original graph across the line y equals x, with domain and range swapped.

Arc length in radians

For a circle of radius r, an arc subtending a central angle theta measured in radians has length s equal to r times theta.

Sector area formula

The area of a sector with radius r and central angle theta in radians equals one half times r squared times theta; theta must be expressed in radians.

Similar solids scaling

If two similar solid figures have linear scale factor k, then their surface areas compare as k squared and their volumes compare as k cubed.

Polygon interior angle sum

A convex polygon with n sides has interior angle sum equal to n minus 2 times 180 degrees; for a regular polygon, divide this sum by n to get each interior angle.

Special right triangle ratios

A 45-45-90 triangle has side ratios 1 to 1 to square root of 2; a 30-60-90 triangle has side ratios 1 to square root of 3 to 2.

Perpendicular slope rule

In a coordinate plane, two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals; a horizontal line is perpendicular to a vertical line.

Standard normal z-score

A z-score equals x minus mu divided by sigma; it states how many standard deviations a value x lies from the mean mu in a distribution with standard deviation sigma.

Conditional probability formula

For events A and B with P of B positive, the probability of A given B equals the probability of A and B divided by the probability of B.

Combinations formula

The number of ways to choose r objects from n distinct objects without regard to order is n factorial divided by r factorial times n minus r factorial.

Binomial probability

The probability of exactly k successes in n independent trials with success probability p is C of n comma k times p to the k times 1 minus p to the n minus k.

Euler circuit criterion

A connected graph has an Euler circuit exactly when every vertex has even degree; it has an Euler path but no circuit exactly when exactly two vertices have odd degree.

Expected value discrete variable

For a discrete random variable with values x and probabilities P of x, the expected value is the sum of each x times its probability, representing the long-run mean.

Law of cosines

In a triangle, c squared equals a squared plus b squared minus 2ab cosine C, where C is the angle opposite side c; it generalizes the Pythagorean theorem.

Sine graph period

For y equals sine of b x, the period is 2 pi divided by the absolute value of b; the amplitude of a sine or cosine graph is the absolute value of its coefficient.

L'Hopital's rule

If a limit of a quotient has indeterminate form 0 over 0 or infinity over infinity, the limit equals the limit of the quotient of the derivatives, when that limit exists.

Chain rule

The derivative of f of g of x equals f prime evaluated at g of x times g prime of x; it differentiates compositions by multiplying outer and inner derivatives.

Fundamental Theorem of Calculus

If F is an antiderivative of continuous f on an interval, the definite integral of f from a to b equals F of b minus F of a.

Second derivative concavity test

On an interval where the second derivative of f is positive, the graph of f is concave up; where it is negative, the graph is concave down.