Secondary 4 Mathematics-Science Option Ministerial Examination - Québec flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Discriminant of a quadratic

The expression Δ = b² − 4ac for ax² + bx + c = 0. It determines the number of real roots: two if Δ > 0, one if Δ = 0, none if Δ < 0.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). It solves any quadratic equation, factorable or not.

Difference of squares

The identity a² − b² = (a − b)(a + b). It factors a binomial whose two terms are perfect squares separated by a subtraction sign.

Inequality sign reversal

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol. Adding or subtracting a number never reverses it.

Extraneous root

A solution produced by algebraic manipulation that does not satisfy the original equation, common in rational or radical equations. It must be rejected after checking in the original equation.

System of linear equations

Two or more linear equations solved simultaneously. The solution is the ordered pair satisfying all equations, found graphically at the intersection or algebraically by substitution or elimination.

Rational expression restrictions

Values of the variable that make any denominator zero are excluded from the domain. They are identified before simplifying, since cancelling factors does not restore them.

Slope of a line

The rate of change a in y = ax + b, computed as (y₂ − y₁)/(x₂ − x₁). It measures vertical change per unit of horizontal change between two points.

Vertex of a parabola

The highest or lowest point of a quadratic function, located at x = −b/(2a) in standard form or (h, k) in vertex form y = a(x − h)² + k.

Axis of symmetry

The vertical line x = −b/(2a) that divides a parabola into two mirror-image halves. The vertex always lies on this line.

Exponential growth vs decay

In f(x) = a·c^x, a base c greater than 1 produces growth, while a base between 0 and 1 produces decay. The y-intercept is a and the asymptote is y = 0.

Step function

A function whose graph consists of horizontal segments with jumps, such as the greatest-integer function. It is constant on intervals and changes value abruptly at each step.

Horizontal asymptote

A horizontal line y = L that the graph of a function approaches as x tends to infinity. For a rational function with equal degrees, it is y = ratio of leading coefficients.

Amplitude and period

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

Inverse function

The function obtained by interchanging x and y and solving for y, written f⁻¹. Its graph is the reflection of f across the line y = x, and it exists only if f is one-to-one.

Function transformations

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

Zero of a function

A value of x for which f(x) = 0, shown graphically as an x-intercept. A linear function ax + b has zero x = −b/a.

Pythagorean theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². Its converse identifies right triangles.

Similar figures ratio

When two figures are similar with side ratio k, corresponding angles are equal, areas are in ratio k², and volumes are in ratio k³.

Triangle congruence criteria

SSS, SAS and ASA (or AAS) guarantee two triangles are congruent. Right triangles also allow the hypotenuse-leg criterion.

Volume of a cone

The volume of a cone or pyramid is one third of the base area times the height: V = Bh/3. For a circular cone, B = πr².

Sector area

The area of a circular sector with central angle θ in degrees is (θ/360°)·πr². The matching arc length is (θ/360°)·2πr.

Altitude metric relation

In a right triangle, the altitude h to the hypotenuse creates segments m and n with h² = mn. Each leg squared equals the hypotenuse times its adjacent segment.

Thales proportionality

A line parallel to one side of a triangle intersecting the other two sides divides them into proportional segments, creating a smaller similar triangle.

Total area of a solid

The sum of the areas of all faces of a solid: lateral area plus the areas of the bases. For a cylinder it is 2πr² + 2πrh.

Mean, median, mode

The mean is the sum divided by the count, the median is the middle ordered value, and the mode is the most frequent value. The median resists extreme values better.

Interquartile range

IQR = Q₃ − Q₁, the spread of the middle half of the data. It measures dispersion and is used to flag outliers beyond 1.5 IQR from the quartiles.

Complementary probability

P(not A) = 1 − P(A). The probability that an event does not occur equals one minus the probability that it does.

Addition rule

P(A or B) = P(A) + P(B) − P(A and B). The intersection is subtracted so outcomes counted in both events are not double-counted.

Conditional probability

P(B|A) = P(A and B)/P(A), the probability of B given that A has occurred. It restricts the sample space to outcomes satisfying A.

Independent events

Two events are independent when P(A and B) = P(A)·P(B), equivalently P(B|A) = P(B). One event's occurrence does not change the other's probability.

Counterexample

A single case in which a general statement is false. One valid counterexample is sufficient to disprove a universal claim, but examples never prove one.