Slovakia Maturita Math cheat sheet

For real numbers a, b: |a + b| ≤ |a| + |b| (triangle inequality), and |ab| = |a||b|.

Laws of exponents: a^m · a^n = a^(m+n); a^m / a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1 for a ≠ 0.

Laws of logarithms: log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(x^k) = k log_a x; log_a b = ln b / ln a.

A number is divisible by 3 (or 9) iff the sum of its digits is divisible by 3 (or 9); by 4 iff its last two digits form a multiple of 4.

Quadratic formula: roots of ax² + bx + c = 0 (a ≠ 0) are x = (−b ± √(b² − 4ac)) / (2a); discriminant Δ = b² − 4ac determines the number of real roots.

Vieta's formulas: for ax² + bx + c = 0, x₁ + x₂ = −b/a and x₁·x₂ = c/a.

Percent change: new value = old value · (1 ± p/100); a change from A to B is ((B − A)/A)·100 %.

Interval notation: (a, b) open, ⟨a, b⟩ closed; union A ∪ B, intersection A ∩ B, difference A \ B, complement A′.

A function f: A → B assigns to each x ∈ A exactly one y ∈ B; the domain is A, the range is f(A) ⊆ B.

Linear function f(x) = kx + q has slope k and y-intercept q; it is increasing if k > 0, decreasing if k < 0, constant if k = 0.

Quadratic function f(x) = ax² + bx + c has vertex at x = −b/(2a); it opens upward if a > 0 and downward if a < 0.

Exponential function f(x) = a^x (a > 0, a ≠ 1) is increasing if a > 1 and decreasing if 0 < a < 1; it passes through (0, 1).

Logarithmic function f(x) = log_a x is the inverse of a^x; its domain is (0, ∞) and its range is ℝ.

Power function f(x) = x^n: for even n it is even (f(−x) = f(x)); for odd n it is odd (f(−x) = −f(x)).

Inverse function f⁻¹ satisfies f⁻¹(f(x)) = x and f(f⁻¹(y)) = y; the graph of f⁻¹ is the reflection of the graph of f across the line y = x.

Transformations: y = f(x − h) + k shifts the graph h units right and k units up; y = −f(x) reflects across the x-axis; y = f(−x) reflects across the y-axis.

Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c, a² + b² = c².

Area formulas: triangle A = ½ · base · height; circle A = πr², circumference C = 2πr; trapezoid A = ½(a + b)h.

Volume formulas: cylinder V = πr²h; cone V = (1/3)πr²h; sphere V = (4/3)πr³, surface area S = 4πr².

Similar triangles: corresponding sides are proportional (a/a′ = b/b′ = c/c′) and areas scale by the square of the similarity ratio.

Sine rule: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius; cosine rule: c² = a² + b² − 2ab cos C.

Sum of interior angles of an n-gon is (n − 2)·180°; each interior angle of a regular n-gon is ((n − 2)·180°)/n.

Distance between points A[x₁, y₁] and B[x₂, y₂] is √((x₂ − x₁)² + (y₂ − y₁)²); midpoint has coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2).

Line in general form ax + by + c = 0 has normal vector (a, b) and direction vector (−b, a); two lines are perpendicular iff a₁a₂ + b₁b₂ = 0.

Multiplication rule: if a choice can be made in m ways and a second in n ways, the two together can be made in m·n ways.

Permutations of n distinct objects: P(n) = n!; permutations with repetition: n!/(n₁!·n₂!·…·n_k!).

Variations (ordered selections) of k from n: V(n, k) = n!/(n − k)!; with repetition: n^k.

Combinations (unordered selections) of k from n: C(n, k) = n!/(k!(n − k)!); C(n, k) = C(n, n − k); C(n, 0) = C(n, n) = 1.

Classical probability: P(A) = (number of favourable outcomes)/(number of all equally likely outcomes), with 0 ≤ P(A) ≤ 1.

Complement rule: P(A′) = 1 − P(A); addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Independent events: P(A ∩ B) = P(A)·P(B); conditional probability: P(A | B) = P(A ∩ B)/P(B) for P(B) > 0.

Binomial coefficient identity: C(n, k) = C(n − 1, k − 1) + C(n − 1, k) (Pascal's rule).