Bagrut Mathematics (3/4/5 Units) cheat sheet

Difference of squares: a² − b² = (a − b)(a + b); sum/difference of cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²).

Perfect-square trinomials: a² ± 2ab + b² = (a ± b)²; recognize by checking whether the middle term equals twice the product of the square roots of the end terms.

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = [−b ± √(b² − 4ac)]/(2a); the discriminant Δ = b² − 4ac determines the number of real roots (Δ > 0 two, Δ = 0 one, Δ < 0 none).

Vieta's formulas: if x₁, x₂ are roots of ax² + bx + c = 0, then x₁ + x₂ = −b/a and x₁x₂ = c/a.

Factor theorem: (x − r) is a factor of polynomial P(x) exactly when P(r) = 0; remainder theorem: the remainder of P(x) ÷ (x − r) is P(r).

Laws of exponents: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.

Laws of logarithms (base a > 0, a ≠ 1): logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐ(xⁿ) = n logₐx, logₐ1 = 0, logₐa = 1.

Rational expressions: simplify only after factoring numerator and denominator; the domain excludes values making any denominator zero.

Absolute value: |x| = x if x ≥ 0, −x if x < 0; |x| = c (c > 0) means x = c or x = −c; |x| < c means −c < x < c.

Linear equation ax + b = 0 (a ≠ 0) has the unique solution x = −b/a.

A quadratic equation with discriminant Δ = b² − 4ac has two distinct real roots when Δ > 0, one double root when Δ = 0, and no real roots when Δ < 0.

Solving by substitution: isolate one variable in one equation and substitute into the other; check for extraneous solutions introduced by squaring or multiplying by expressions.

Multiplying or dividing an inequality by a negative number reverses the inequality sign; adding or subtracting a constant preserves it.

Quadratic inequality ax² + bx + c > 0 (a > 0): find the roots r₁ < r₂; the solution is x < r₁ or x > r₂ outside the roots, and r₁ < x < r₂ between them for the < 0 case.

Rational inequality: move all terms to one side, combine over a common denominator, find zeros of numerator and denominator, and test sign intervals on a number line; exclude values that make a denominator zero.

Absolute-value inequality: |f(x)| < c (c > 0) ⇔ −c < f(x) < c; |f(x)| > c ⇔ f(x) < −c or f(x) > c.

Exponential equation aᶠ⁽ˣ⁾ = aᵍ⁽ˣ⁾ (a > 0, a ≠ 1) implies f(x) = g(x); if bases differ, take logarithms of both sides.

System of two linear equations: unique solution when the coefficient determinant a₁b₂ − a₂b₁ ≠ 0; no solution or infinitely many when it equals zero.

A function is a rule assigning exactly one output to each input in its domain; the vertical-line test identifies whether a graph represents a function.

Domain restrictions to check: denominators ≠ 0, even roots require radicand ≥ 0, logarithms require argument > 0.

Transformations of y = f(x): y = f(x) + k shifts up by k; y = f(x − h) shifts right by h; y = −f(x) reflects over the x-axis; y = f(−x) reflects over the y-axis; y = af(x) stretches vertically by factor |a|.

A function is even if f(−x) = f(x) for all x in its domain (graph symmetric about the y-axis); odd if f(−x) = −f(x) (symmetric about the origin).

Inverse function f⁻¹ exists only if f is one-to-one; its graph is the reflection of f's graph over the line y = x, and the domain of f⁻¹ equals the range of f.

Composition: (f ∘ g)(x) = f(g(x)); the domain of f ∘ g consists of x in the domain of g with g(x) in the domain of f.

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

The graph of y = (x − p)² + q is a parabola with vertex (p, q); y = a(x − p)² + q has the same vertex and vertical stretch factor |a|.

Exponential function f(x) = aˣ with a > 1 is increasing; with 0 < a < 1 it is decreasing; its domain is all reals and its range is (0, ∞).

The graph of y = aˣ passes through (0, 1) and has the x-axis as a horizontal asymptote.

Logarithm definition: logₐb = c ⇔ aᶜ = b, with a > 0, a ≠ 1, b > 0.

Change of base: logₐb = log꜀b / log꜀a; in particular logₐb = ln b / ln a.

The natural logarithm ln x has base e ≈ 2.718; ln(eˣ) = x for all x, and e^(ln x) = x for x > 0.

Solving aˣ = b: take logarithms to get x = logₐb = ln b / ln a.

Solving logₐf(x) = c: rewrite as f(x) = aᶜ, then verify f(x) > 0 because logarithms are defined only for positive arguments.

The logarithmic function y = logₐx is the inverse of y = aˣ; its domain is (0, ∞), range is all reals, and it passes through (1, 0).

Exponential growth/decay model: A(t) = A₀·bᵏᵗ, where b > 1 gives growth and 0 < b < 1 gives decay.

In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Pythagorean identity: sin²θ + cos²θ = 1; also tan θ = sin θ / cos θ (cos θ ≠ 0).

Law of sines: a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.

Law of cosines: c² = a² + b² − 2ab·cos C; use it when two sides and the included angle or all three sides are known.

Area of a triangle: S = ½ab·sin C (two sides and included angle); also S = ½·base·height.

Exact values: sin 30° = ½, cos 30° = √3/2, sin 45° = cos 45° = √2/2, sin 60° = √3/2, cos 60° = ½, tan 45° = 1.

Sine rule for supplementary angles: sin(180° − θ) = sin θ; cosine changes sign: cos(180° − θ) = −cos θ.

Trigonometric identity for double angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

General solution of sin x = k (|k| ≤ 1): x = α + 360°n or x = 180° − α + 360°n, where α is the principal solution.

Arithmetic sequence: aₙ = a₁ + (n − 1)d, where d is the common difference.

Arithmetic series sum: Sₙ = n(a₁ + aₙ)/2 = n[2a₁ + (n − 1)d]/2.

Geometric sequence: aₙ = a₁·qⁿ⁻¹, where q is the common ratio.

Geometric series sum (q ≠ 1): Sₙ = a₁(qⁿ − 1)/(q − 1) = a₁(1 − qⁿ)/(1 − q).

Infinite geometric series converges only when |q| < 1, with sum S = a₁/(1 − q); it diverges when |q| ≥ 1.

A sequence is arithmetic iff consecutive differences are constant; geometric iff consecutive ratios are constant and nonzero.

Sum of the first n natural numbers: 1 + 2 + … + n = n(n + 1)/2.

Recursive definition: aₙ₊₁ = aₙ + d (arithmetic) or aₙ₊₁ = q·aₙ (geometric), with a₁ given.

Derivative definition: f′(x) = lim_{h→0} [f(x + h) − f(x)]/h; it gives the slope of the tangent line at x.

Power rule: d/dx (xⁿ) = n·xⁿ⁻¹ for any real n.

Derivative of a constant is 0; derivative of a sum is the sum of derivatives; (cf)′ = c·f′.

Product rule: (fg)′ = f′g + fg′; quotient rule: (f/g)′ = (f′g − fg′)/g², g ≠ 0.

Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x).

Derivatives: (eˣ)′ = eˣ; (ln x)′ = 1/x for x > 0; (sin x)′ = cos x; (cos x)′ = −sin x.

Tangent line at x = a: y − f(a) = f′(a)(x − a).

Critical points occur where f′(x) = 0 or f′(x) is undefined; f′ changes from positive to negative at a local maximum and from negative to positive at a local minimum.

On a closed interval [a, b], the absolute maximum and minimum of a continuous function occur at critical points or at the endpoints.

An antiderivative F of f satisfies F′ = f; all antiderivatives differ by a constant.

Power rule for integration: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫x⁻¹ dx = ln|x| + C.

Basic integrals: ∫eˣ dx = eˣ + C; ∫cos x dx = sin x + C; ∫sin x dx = −cos x + C.

Fundamental theorem of calculus: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f.

The definite integral ∫ₐᵇ f(x) dx gives the signed area between the graph of f and the x-axis from a to b.

Area between two curves: ∫ₐᵇ |f(x) − g(x)| dx, where a and b are the intersection x-values.

Linearity: ∫[αf(x) + βg(x)] dx = α∫f(x) dx + β∫g(x) dx.

Reversing limits changes sign: ∫ₐᵇ f(x) dx = −∫ᵦᵃ f(x) dx; ∫ₐᵃ f(x) dx = 0.

Triangle angle sum: the interior angles of a triangle add to 180°; an exterior angle equals the sum of the two non-adjacent interior angles.

Pythagorean theorem: in a right triangle, a² + b² = c², where c is the hypotenuse.

Thales' theorem: an angle inscribed in a semicircle is a right angle.

Inscribed angle theorem: an angle inscribed in a circle equals half the measure of its intercepted arc; angles subtending the same arc are equal.

Tangent–radius theorem: a tangent to a circle is perpendicular to the radius at the point of tangency.

Similar triangles: corresponding angles equal and corresponding sides proportional; the ratio of areas equals the square of the similarity ratio.

Midsegment of a triangle is parallel to the third side and half its length.

In a parallelogram, opposite sides and opposite angles are equal; diagonals bisect each other.

Cyclic quadrilateral: opposite angles sum to 180°.

Distance between points (x₁, y₁) and (x₂, y₂): d = √[(x₂ − x₁)² + (y₂ − y₁)²].

Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope of a line through two points: m = (y₂ − y₁)/(x₂ − x₁), x₁ ≠ x₂.

Line equations: slope-intercept y = mx + b; point-slope y − y₁ = m(x − x₁); general Ax + By + C = 0.

Parallel lines have equal slopes (m₁ = m₂); perpendicular lines have slopes satisfying m₁·m₂ = −1.

Circle with center (a, b) and radius r: (x − a)² + (y − b)² = r².

Distance from point (x₀, y₀) to line Ax + By + C = 0: |Ax₀ + By₀ + C| / √(A² + B²).

Parabola y = ax² + bx + c has axis of symmetry x = −b/(2a) and vertex (−b/(2a), c − b²/(4a)).

Probability axioms: 0 ≤ P(A) ≤ 1; P(Ω) = 1; P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Complement rule: P(A′) = 1 − P(A).

Conditional probability: P(A|B) = P(A ∩ B)/P(B), P(B) > 0; events A and B are independent iff P(A ∩ B) = P(A)·P(B).

Total probability: P(B) = Σ P(Aᵢ)·P(B|Aᵢ) over a partition {Aᵢ} of the sample space.

Mean of a data set: x̄ = (Σxᵢ)/n; weighted mean uses x̄ = Σ(wᵢxᵢ)/Σwᵢ.

Variance: σ² = Σ(xᵢ − x̄)²/n; standard deviation σ is its square root.

Median is the middle value of ordered data (average of the two middle values when n is even); mode is the most frequent value.

Binomial probability: P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ, where C(n, k) = n!/[k!(n − k)!].

Expected value of a discrete random variable: E(X) = Σ xᵢ·P(X = xᵢ).