Secondary 4 Mathematics-Technical and Scientific Option Ministerial Examination - Québec cheat sheet

Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a) for ax² + bx + c = 0, a ≠ 0.

Vertex of a parabola y = a(x - h)² + k is (h, k); axis of symmetry x = h. For y = ax² + bx + c, h = -b/(2a).

Discriminant Δ = b² - 4ac: Δ > 0 two real roots, Δ = 0 one double root, Δ < 0 no real roots.

Exponential function f(x) = a·b^x: growth if b > 1, decay if 0 < b < 1; asymptote y = 0.

Logarithm laws: log(xy) = log x + log y; log(x/y) = log x - log y; log(x^n) = n log x; log_b b = 1; log_b 1 = 0.

Change of base: log_b x = log_c x / log_c b (c > 0, c ≠ 1).

Rational function f(x) = P(x)/Q(x): vertical asymptote where Q(x) = 0 (if not cancelled); horizontal asymptote determined by degrees of P and Q.

Composition (f ∘ g)(x) = f(g(x)); inverse f⁻¹ satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x on appropriate domains.

Absolute value |x| = x if x ≥ 0, -x if x < 0; |x| = k (k ≥ 0) gives x = k or x = -k.

Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). Distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²).

Translation by vector (a, b): (x, y) → (x + a, y + b).

Rotation 90° counterclockwise about origin: (x, y) → (-y, x); 180°: (x, y) → (-x, -y); 270° counterclockwise (or 90° clockwise): (x, y) → (y, -x).

Reflection across x-axis: (x, y) → (x, -y); across y-axis: (x, y) → (-x, y); across line y = x: (x, y) → (y, x).

Dilation (homothety) centered at origin with scale factor k: (x, y) → (kx, ky); k > 1 enlargement, 0 < k < 1 reduction, k < 0 reflection through center.

Similar figures: corresponding angles equal and corresponding sides proportional; ratio of areas = (ratio of sides)²; ratio of volumes = (ratio of sides)³.

Area formulas: triangle A = ½bh; parallelogram A = bh; trapezoid A = ½(b₁ + b₂)h; circle A = πr²; sector area = (θ/360°)πr².

Volume formulas: prism/cylinder V = Bh; pyramid/cone V = ⅓Bh; sphere V = (4/3)πr³.

A composition of transformations is a single transformation; isometries (translations, rotations, reflections) preserve distances and angles.

Right-triangle ratios: sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent.

Reciprocal identities: csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ.

Pythagorean identity: sin²θ + cos²θ = 1; also 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.

Law of sines: a/sin A = b/sin B = c/sin C (works for any triangle).

Law of cosines: c² = a² + b² - 2ab cos C; use when given SAS or SSS.

Area of a triangle: A = ½ab sin C (two sides and included angle).

Sine function y = a sin(b(x - h)) + k has amplitude |a|, period 2π/|b| (or 360°/|b|), phase shift h, vertical shift k.

Unit circle reference angles: sin(180° - θ) = sin θ; cos(180° - θ) = -cos θ; sin(180° + θ) = -sin θ; cos(180° + θ) = -cos θ.

Ambiguous case of law of sines (SSA): may yield 0, 1, or 2 triangles; check whether the given angle is acute and compare height h = b sin A with side a.

Mean = sum of values / number of values; median = middle value when ordered (average of two middle values if n even); mode = most frequent value.

Range = max - min; interquartile range IQR = Q3 - Q1; a value is an outlier if below Q1 - 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures spread about the mean; larger standard deviation means more dispersion.

Probability of an event P(A) = number of favourable outcomes / total number of equally likely outcomes; 0 ≤ P(A) ≤ 1.

Complement rule: P(A') = 1 - P(A).

Addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B); if mutually exclusive, P(A ∪ B) = P(A) + P(B).

Multiplication rule: P(A ∩ B) = P(A) × P(B | A); if independent, P(A ∩ B) = P(A) × P(B).

Conditional probability: P(B | A) = P(A ∩ B) / P(A), provided P(A) > 0.

Expected value E(X) = Σ x·P(x); for a fair game E(X) = 0.

Define variables, write the objective function to maximize or minimize, and list all constraints as inequalities.

Graph the feasible region (intersection of half-planes); vertices (corner points) are candidates for the optimum.

Linear programming theorem: if an optimum exists, it occurs at a vertex of the feasible region (or along an entire edge).

Evaluate the objective function at each vertex; choose the largest value for maximization, smallest for minimization.

For a quadratic model, the optimum occurs at the vertex: maximum if a < 0, minimum if a > 0.

Constraint types: physical limits (materials, time, budget), non-negativity (x ≥ 0, y ≥ 0), and proportionality between variables.

Validate the model: check that the solution satisfies all constraints and is realistic in context (e.g., integer numbers of items).

Break-even point: where revenue R(x) = cost C(x); profit P(x) = R(x) - C(x).

Sensitivity: changing a constraint coefficient can change the feasible region and the optimal vertex; re-evaluate vertices after any change.

Justify each step by citing a definition, property, theorem, or previously established result.

Counterexample: a single case where a statement is false disproves a universal statement; one example never proves a universal statement.

Direct proof: assume hypotheses and derive the conclusion by logical deduction.

Proof by contradiction: assume the negation of the conclusion and derive a contradiction.

Converse of “if P then Q” is “if Q then P”; it is not logically equivalent to the original.

Contrapositive of “if P then Q” is “if not Q then not P”; it is logically equivalent to the original.

Necessary vs sufficient: “P is sufficient for Q” means P ⇒ Q; “P is necessary for Q” means Q ⇒ P.

In a written solution, state the given information, the goal, the method, and the conclusion with units where applicable.

Use precise vocabulary: distinguish “solve” (find values), “simplify” (rewrite in equivalent simpler form), and “factor” (write as a product).