Secondary 4 Mathematics-Science Option Ministerial Examination - Québec cheat sheet

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

Distributive law and factoring: a(b+c) = ab+ac; common factor first, then notable products a²−b² = (a−b)(a+b) and a²±2ab+b² = (a±b)².

Trinomial factoring: for x²+bx+c find two numbers with product c and sum b; for ax²+bx+c use the discriminant or product-sum on ac.

Rational expressions are simplified by factoring numerator and denominator and cancelling common factors; the excluded values are the zeros of the original denominator.

Solving a linear equation: isolate the variable by inverse operations; multiplying or dividing both sides by a negative number is allowed, but for inequalities it reverses the inequality sign.

Inequality solution sets are written in interval notation; a closed bracket [ ] includes the endpoint, a parenthesis ( ) excludes it, and ±∞ always takes a parenthesis.

A system of two linear equations is solved by substitution or elimination; the number of solutions is 0 (parallel), 1 (intersecting) or infinite (coincident lines).

Quadratic equations: ax²+bx+c = 0 gives x = (−b ± √(b²−4ac))/(2a); the discriminant Δ = b²−4ac gives 2, 1 or 0 real roots as Δ > 0, = 0, < 0.

A rational equation is solved by multiplying by a common denominator, then every candidate root must be checked against the excluded values (extraneous roots).

Linear function f(x) = ax+b: a is the rate of change (slope) and b the initial value; a = (y₂−y₁)/(x₂−x₁) and the zero is x = −b/a.

Quadratic function in standard form f(x) = ax²+bx+c: vertex at x = −b/(2a); in vertex form f(x) = a(x−h)²+k the vertex is (h, k) and the axis of symmetry is x = h.

The sign of a in a quadratic opens the parabola upward (a > 0, minimum) or downward (a < 0, maximum); the range is bounded by the vertex's y-value.

Exponential function f(x) = a·c^x: c > 1 gives growth, 0 < c < 1 gives decay; the asymptote is y = 0 and the y-intercept is a.

Step (greatest-integer) function: f(x) = a[bx] has steps of width 1/|b| and jump height |a|; it is defined piecewise on each interval and its graph is a staircase.

Rational function f(x) = (ax+b)/(cx+d): vertical asymptote where the denominator is zero, horizontal asymptote y = a/c when degrees are equal, and a hole where a common factor cancels.

Periodic function: f(x) = a sin b(x−h) + k has amplitude |a|, period 2π/|b| (360°/|b| in degrees), phase shift h and midline y = k.

Inverse of a function is found by swapping x and y and solving for y; the graph of the inverse is the reflection across y = x, and it exists as a function only if the original is one-to-one.

Transformations: g(x) = a·f(b(x−h)) + k stretches vertically by |a|, horizontally by 1/|b|, shifts right by h and up by k; a negative a or b reflects the graph.

Perimeter and area: rectangle A = bh; triangle A = bh/2; parallelogram A = bh; trapezoid A = (b₁+b₂)h/2; circle C = 2πr and A = πr².

Pythagorean theorem: in a right triangle a²+b² = c², where c is the hypotenuse; its converse tests whether a triangle is right-angled.

Volume: prism/cylinder V = Bh; pyramid/cone V = Bh/3; sphere V = 4πr³/3; total area of a solid is the sum of all its faces (lateral area plus bases).

Similar figures: corresponding angles are equal and corresponding sides are proportional with ratio k; areas scale by k² and volumes by k³.

Congruence criteria for triangles: SSS, SAS, ASA (with AAS) and, for right triangles, hypotenuse-leg; congruence means equal corresponding sides and angles.

Thales / similar-triangle proportionality: a line parallel to one side of a triangle divides the other two sides proportionally, so a/a′ = b/b′.

Metric relations in a right triangle with altitude h to the hypotenuse dividing it into segments m and n: h² = mn, and each leg squared equals the hypotenuse times its adjacent segment.

Angle facts: angles in a triangle sum to 180°; the exterior angle equals the sum of the two remote interior angles; angles on a straight line sum to 180° and around a point to 360°.

Arc and sector: arc length = (θ/360°)·2πr and sector area = (θ/360°)·πr² for a central angle θ in degrees.

Measures of central tendency: mean = sum of values / number of values; median is the middle value of ordered data; mode is the most frequent value (a data set can have several or none).

Measures of dispersion: range = max − min; interquartile range IQR = Q₃ − Q₁; standard deviation measures average distance from the mean and is ≥ 0.

In a five-number summary (min, Q₁, median, Q₃, max), a value is an outlier when it lies below Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR.

Probability of an event P(A) = number of favourable outcomes / number of equally likely outcomes, always with 0 ≤ P(A) ≤ 1; P(not A) = 1 − P(A).

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the last term is zero.

Multiplication rule: P(A and B) = P(A)·P(B|A); for independent events P(A and B) = P(A)·P(B).

Conditional probability P(B|A) = P(A and B)/P(A), read as the probability of B given that A has occurred.

Counting: if a task has m ways followed by n ways, the multiplication principle gives m·n outcomes; permutations order n items as n!, and combinations count unordered selections C(n,r) = n!/(r!(n−r)!).

Correlation coefficient r lies between −1 and 1; r near ±1 indicates a strong linear association, r near 0 a weak one, and correlation does not prove causation.

Justification types: deductive proof proceeds from definitions, properties and known results; inductive reasoning generalizes from examples and does not prove a statement.

Counterexample rule: a single case where the statement is false disproves a universal claim, so one counterexample suffices for refutation.

Proof by contradiction assumes the negation of the statement and derives an impossibility; direct proof chains known properties to the conclusion.

A valid argument is one where the conclusion necessarily follows from the premises; an invalid argument can have true premises and a false conclusion.

Converse, inverse and contrapositive of 'if P then Q': the contrapositive ('if not Q then not P') is logically equivalent to the original; the converse is not.

Modelling steps: identify variables, translate constraints into equations or inequalities, solve, then validate the result against the context (units, domain, feasibility).

Optimization problems are solved by expressing the quantity to optimize as a function, finding its vertex or extreme value on the allowed domain, and interpreting it in context.

Communication standard: show the given information, state the property or formula used at each step, and give the final answer with correct units and a concluding sentence.