Secondary 4 Mathematics-Technical and Scientific Option Ministerial Examination - Québec flashcards

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Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (-b ± √(b² - 4ac)) / (2a). It gives all real roots when the discriminant is non-negative.

Discriminant

The quantity Δ = b² - 4ac for a quadratic ax² + bx + c = 0. If Δ > 0 there are two distinct real roots; if Δ = 0 one double root; if Δ < 0 no real roots.

Vertex of a parabola

For y = a(x - h)² + k, the vertex is (h, k). For y = ax² + bx + c, the x-coordinate is -b/(2a) and the y-coordinate is found by substitution.

Exponential growth vs decay

In f(x) = a·b^x with a > 0, the function models growth when b > 1 and decay when 0 < b < 1. The horizontal asymptote is y = 0.

Logarithm product law

For positive x and y and base b > 0, b ≠ 1: log_b(xy) = log_b x + log_b y. The logarithm of a product equals the sum of the logarithms.

Change of base formula

log_b x = log_c x / log_c b for any valid base c (c > 0, c ≠ 1). It allows evaluation of logarithms with a calculator that only has base 10 or e.

Inverse function

The inverse f⁻¹ reverses f: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x on suitable domains. The graph of f⁻¹ is the reflection of the graph of f across the line y = x.

Midpoint formula

The midpoint of the segment joining (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2). It averages the x-coordinates and the y-coordinates separately.

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) is √((x₂ - x₁)² + (y₂ - y₁)²). It follows from the Pythagorean theorem applied to horizontal and vertical differences.

Rotation 90° counterclockwise

About the origin, the transformation maps (x, y) to (-y, x). A 180° rotation maps (x, y) to (-x, -y); a 270° counterclockwise rotation maps (x, y) to (y, -x).

Reflection across y = x

The transformation maps (x, y) to (y, x), swapping coordinates. Reflection across the x-axis maps (x, y) to (x, -y); across the y-axis to (-x, y).

Dilation centered at origin

A dilation with scale factor k maps (x, y) to (kx, ky). If k > 1 it is an enlargement; if 0 < k < 1 a reduction; if k < 0 the image is also reflected through the origin.

Similar figures area ratio

If two figures are similar with side ratio k, the ratio of their areas is k² and the ratio of their volumes is k³. Corresponding angles remain equal.

Volume of a cone

V = ⅓Bh, where B is the base area and h is the perpendicular height. For a circular cone with radius r, B = πr², so V = ⅓πr²h.

Sine ratio

In a right triangle, sin θ = length of side opposite θ divided by length of hypotenuse. It depends only on the angle, not on the size of the triangle.

Law of cosines

In any triangle, c² = a² + b² - 2ab cos C, where C is the angle opposite side c. It is used with SAS or SSS data to find a side or angle.

Law of sines

In any triangle, a/sin A = b/sin B = c/sin C, where side a is opposite angle A, and so on. It is useful when a side and its opposite angle are known.

Pythagorean identity

For any angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ.

Amplitude and period

For y = a sin(b(x - h)) + k, the amplitude is |a| and the period is 2π/|b| (or 360°/|b| in degrees). The graph oscillates between k - |a| and k + |a|.

Ambiguous case SSA

When two sides and a non-included angle are given, the law of sines may produce zero, one, or two triangles. Compare the given side with the height h = b sin A.

Mean, median, mode

The mean is the sum divided by the count; the median is the middle ordered value (or average of two middle values); the mode is the most frequent value. Each measures a different aspect of centre.

Interquartile range

IQR = Q3 - Q1, the range of the middle 50% of the data. A value is flagged as an outlier if it lies below Q1 - 1.5·IQR or above Q3 + 1.5·IQR.

Complement rule

For any event A, P(A') = 1 - P(A), where A' is the complement (A does not occur). The probabilities of an event and its complement sum to 1.

Addition rule

P(A ∪ B) = P(A) + P(B) - P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

Conditional probability

P(B | A) = P(A ∩ B) / P(A), provided P(A) > 0. It gives the probability of B given that A has occurred, restricting the sample space to A.

Expected value

For a discrete random variable X, E(X) = Σ x·P(x). It is the long-run average value. In a fair game, the expected net gain is zero.

Feasible region

The set of points satisfying all constraints of a linear programming problem, usually a polygon. The optimum of a linear objective occurs at a vertex of this region.

Linear programming vertex principle

If a linear objective function has an optimum on a non-empty bounded feasible region, that optimum occurs at a vertex (corner point) of the region, or along an entire edge.

Break-even point

The production level x where revenue equals cost: R(x) = C(x). Profit is P(x) = R(x) - C(x); it is zero at break-even, negative before, positive after.

Quadratic optimization

For a quadratic model y = ax² + bx + c, the optimum is at the vertex x = -b/(2a). It is a maximum when a < 0 and a minimum when a > 0.

Counterexample

A single case in which a statement is false. One counterexample disproves a universal statement, but one example never proves a universal statement.

Contrapositive

The contrapositive of “if P then Q” is “if not Q then not P”. It is logically equivalent to the original statement, unlike the converse “if Q then P”.