Slovakia Maturita Math flashcards

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

Absolute value

For a real number x, |x| equals x if x ≥ 0 and −x if x < 0; it represents the distance from x to 0 on the number line.

Logarithm definition

log_a b = c means a^c = b, where a > 0, a ≠ 1, and b > 0. The logarithm is the exponent to which the base must be raised.

Discriminant

For a quadratic ax² + bx + c = 0, the discriminant is Δ = b² − 4ac. If Δ > 0 there are two real roots, if Δ = 0 one, if Δ < 0 none.

Divisibility by 9

A positive integer is divisible by 9 exactly when the sum of its digits is divisible by 9. The same digit-sum test with 3 works for divisibility by 3.

Interval notation

Closed interval ⟨a, b⟩ includes endpoints a and b; open interval (a, b) excludes them. Mixed forms (a, b⟩ and ⟨a, b) include only one endpoint.

Scientific notation

A number written as a · 10^n with 1 ≤ |a| < 10 and n an integer. For example, 0.00042 = 4.2 · 10⁻⁴ and 53 000 = 5.3 · 10⁴.

Percent change

If a quantity changes from A to B, the percent change is ((B − A)/A) · 100 %. A positive result is an increase, a negative result a decrease.

Inequality with negative multiplier

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign: if a < b and c < 0, then ac > bc.

Function definition

A function f from set A to set B assigns to every element x of A exactly one element y of B, written y = f(x). The set A is the domain.

Linear function slope

For f(x) = kx + q, the slope k equals the change in f(x) per unit increase in x. The graph is increasing when k > 0 and decreasing when k < 0.

Quadratic vertex

The vertex of the parabola y = ax² + bx + c has x-coordinate −b/(2a). Its y-coordinate is found by substituting this x into the equation.

Even and odd functions

A function is even if f(−x) = f(x) for all x in its domain, and odd if f(−x) = −f(x). Even graphs are symmetric about the y-axis, odd about the origin.

Exponential function

A function of the form f(x) = a^x with a > 0 and a ≠ 1. It is increasing for a > 1, decreasing for 0 < a < 1, and always positive.

Inverse function

The inverse f⁻¹ of a one-to-one function f satisfies f⁻¹(f(x)) = x and f(f⁻¹(y)) = y. Its graph is the reflection of f's graph across y = x.

Domain of logarithm

The function f(x) = log_a x is defined only for x > 0, because a logarithm is the exponent needed to obtain a positive number from a positive base.

Graph translation

The graph of y = f(x − h) + k is the graph of y = f(x) shifted h units horizontally (right if h > 0) and k units vertically (up if k > 0).

Absolute value function

The function f(x) = |x| is V-shaped with vertex at the origin. It equals x for x ≥ 0 and −x for x < 0, and its range is ⟨0, ∞).

Pythagorean theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: c² = a² + b². It is used to find an unknown side.

Circle area and circumference

For a circle of radius r, the area is A = πr² and the circumference is C = 2πr. Diameter d = 2r gives A = πd²/4 and C = πd.

Volume of a sphere

The volume of a sphere of radius r is V = (4/3)πr³, and its surface area is S = 4πr². Both formulas use only the radius.

Similar triangles

Two triangles are similar if their corresponding angles are equal. Then corresponding sides are proportional, and the ratio of areas equals the square of the side ratio.

Cosine rule

In any triangle with sides a, b, c and angle C opposite side c: c² = a² + b² − 2ab·cos C. It generalises the Pythagorean theorem to all triangles.

Interior angle sum of polygon

The interior angles of a convex n-gon sum to (n − 2)·180°. For a triangle n = 3 gives 180°, for a quadrilateral n = 4 gives 360°.

Distance between two points

The distance between points A[x₁, y₁] and B[x₂, y₂] in the plane is √((x₂ − x₁)² + (y₂ − y₁)²), derived from the Pythagorean theorem.

Perpendicular lines condition

Two lines with direction vectors u and v are perpendicular exactly when their dot product u·v = 0. For slopes, the product of slopes equals −1.

Cylinder volume

A right circular cylinder with base radius r and height h has volume V = πr²h and lateral surface area S_l = 2πrh. Total surface adds two bases.

Multiplication rule

If one event can occur in m ways and a second independent event in n ways, the two together can occur in m·n ways. This underlies most counting problems.

Factorial

For a positive integer n, n! = n·(n − 1)·…·2·1, the number of orderings of n distinct objects. By definition 0! = 1 and 1! = 1.

Combinations

C(n, k) = n!/(k!(n − k)!) counts the ways to choose k objects from n without regard to order. It equals C(n, n − k) and C(n, 0) = 1.

Variations

V(n, k) = n!/(n − k)! counts ordered selections of k distinct objects from n. With repetition allowed, the count is n^k instead.

Classical probability

If all outcomes are equally likely, the probability of event A is P(A) = (number of outcomes in A)/(total number of outcomes). It always lies between 0 and 1.

Independent events

Events A and B are independent when P(A ∩ B) = P(A)·P(B), equivalently P(A | B) = P(A). Occurrence of one does not affect the other's probability.

Complement rule

For any event A, the probability of its complement is P(A′) = 1 − P(A). This is useful when the complement is easier to count than A itself.