LC Ordinary Maths key terms

The value b² − 4ac for the quadratic ax² + bx + c. Its sign tells how many real roots exist: positive gives two, zero gives one repeated root, negative gives none.

The values of the variable that make the equation true. For a quadratic, these are the x-values where its graph crosses the x-axis.

A rule giving the roots of ax² + bx + c = 0 as x = (−b ± √(b² − 4ac)) ÷ 2a, valid whenever a is not zero.

Two or more equations that must hold at the same time. Solving means finding values satisfying all of them, often by substitution or elimination.

A solving method where one variable is expressed in terms of another and inserted into a second equation, leaving a single equation in one unknown.

A statement comparing expressions with symbols such as <, ≤, > or ≥. Multiplying or dividing by a negative number reverses the direction of the symbol.

The rate at which a function's output changes as its input changes, written dy/dx or f′(x). Geometrically it gives the slope of the tangent to the curve.

Finding the derivative as the limit of the slope of a chord whose two points get closer together, using the limit of [f(x+h) − f(x)] ÷ h as h tends to zero.

A point on a curve where the slope is zero, so the tangent is horizontal. It may be a local maximum, a local minimum, or a point of inflection.

A point where a curve changes from bending one way to bending the other. The second derivative is zero or undefined there, though not every such point is an inflection.

The derivative of the derivative, written d²y/dx². Its sign at a stationary point tests whether that point is a local maximum or a local minimum.

A straight line touching a curve at one point and having the same slope as the curve there. The derivative gives the slope of this line.

The line through a point on a curve drawn perpendicular to the tangent there. Its slope is the negative reciprocal of the tangent's slope.

The limit of a sum of areas under a curve between two limits, found by integrating and subtracting values at the limits. It measures signed area.

The region between a graph and the x-axis over an interval, calculated with a definite integral; parts below the axis count as negative unless handled separately.

A function formed by applying one function to the result of another, written f(g(x)). The output of the inner function must lie within the domain of the outer one.

The function that undoes another, so f⁻¹(f(x)) = x. Its graph is the reflection of the original graph in the line y = x.

The size of a number ignoring its sign, written |x|. For a complex number a + bi it equals √(a² + b²), the distance from the origin.

A number of the form a + bi where i² = −1. The real part is a and the imaginary part is b; such numbers arise when a discriminant is negative.

The length of the straight line between two points (x₁, y₁) and (x₂, y₂), given by √[(x₂ − x₁)² + (y₂ − y₁)²].

The point halfway between two given points, found by averaging their x-coordinates and averaging their y-coordinates separately.

A measure of how steep a line is, calculated as the change in y divided by the change in x between two points on it.

Two lines meeting at a right angle. Their slopes multiply to give −1, so each slope is the negative reciprocal of the other.

Events that cannot both happen in the same trial. For these, the probability of either occurring is the sum of their separate probabilities.

Events where the occurrence of one does not change the probability of the other. For these, the probability of both is the product of their probabilities.

The chance that one event happens given that another has already happened, written P(A|B) and equal to P(A and B) divided by P(B).

A single experiment with exactly two possible outcomes, success or failure, where the probability of success stays the same each time it is repeated.

The probability pattern for the number of successes in a fixed number of independent Bernoulli trials, each with the same success probability.

A measure of how spread out data are around the mean, equal to the square root of the variance and expressed in the same units as the data.

The average of the squared deviations of data values from their mean. It measures spread but in squared units, so its square root is usually reported instead.

The arithmetic average, found by adding all values and dividing by how many there are. It is affected strongly by unusually large or small values.

In any triangle, each side divided by the sine of its opposite angle gives the same value. Useful when a matching side and angle pair is known.

A formula relating the three sides of a triangle to one angle, a² = b² + c² − 2bc·cos A. It extends Pythagoras to triangles without a right angle.

The area equals half the product of two sides multiplied by the sine of the angle between them, written ½ab·sin C.

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, so a² + b² = c².

A unit for measuring angles based on arc length, where a full turn is 2π radians. One radian is about 57.3 degrees.

A circle of radius one centred at the origin. The coordinates of a point on it give the cosine and sine of the angle measured from the positive x-axis.

The distance along a curved part of a circle's circumference, equal to the radius multiplied by the angle in radians.

The region of a circle bounded by two radii and the arc between them. Its area is half the radius squared times the angle in radians.

A list of numbers where each term after the first is found by adding a fixed amount, called the common difference, to the previous term.

A list of numbers where each term after the first is found by multiplying the previous term by a fixed value called the common ratio.

The constant amount added to each term of an arithmetic sequence to get the next term; it can be negative.

The constant multiplier linking consecutive terms of a geometric sequence. If its size is below one the terms shrink towards zero.

The total of all terms of an infinite geometric series, which exists only when the common ratio lies strictly between −1 and 1; it equals a ÷ (1 − r).

Interest calculated on both the original amount and on interest already added, so the balance grows by a fixed multiplier each period.

The amount that would need to be invested now, at a given rate, to reach a stated future sum after a stated number of periods.

A statement about geometry proved from accepted assumptions and earlier results, such as the theorem that angles in a triangle add to 180 degrees.

A result that follows directly and easily from a theorem already proved, needing little or no extra argument.

The statement formed by swapping what a theorem assumes with what it concludes. A converse is not automatically true just because the original is.

Triangles identical in shape and size, proved by side-side-side, side-angle-side, angle-side-angle, right angle-hypotenuse-side, or angle-angle-side with matching sides.

Shapes with the same angles whose corresponding sides are in a fixed ratio. Lengths scale by that ratio and areas by its square.

A four-sided figure whose vertices all lie on one circle. Its opposite angles add to 180 degrees, and each exterior angle equals the opposite interior angle.

A drawing made using only a straight edge and a pair of compasses, such as bisecting an angle or dropping a perpendicular from a point.

The set of all points satisfying a stated geometric condition, such as all points a fixed distance from a given point forming a circle.