Cambridge IAL Mathematics key terms

For a quadratic ax² + bx + c, the quantity b² − 4ac. Its sign tells you how many distinct real roots the equation ax² + bx + c = 0 has: two, one, or none.

The expression x = (−b ± √(b² − 4ac)) / (2a), which gives the roots of any quadratic equation ax² + bx + c = 0 directly from its coefficients, provided a is not zero.

A value that solves a polynomial equation twice over, occurring when a factor is squared. For a quadratic it happens exactly when the discriminant equals zero, so the curve touches the x-axis at one point.

A polynomial f(x) has (x − a) as a factor exactly when f(a) = 0. It converts the task of factorising into evaluating the polynomial at a suspected root.

When a polynomial f(x) is divided by (x − a), the remainder equals f(a). This lets you find remainders by substitution instead of long division.

A straight line that a curve approaches ever more closely as x or y grows without bound, without necessarily meeting it. Graphs of rational functions and exponentials commonly have them.

A function formed by dividing one polynomial by another. Its graph may have vertical asymptotes where the denominator is zero and a horizontal asymptote determined by the degrees involved.

Rewriting a single algebraic fraction as a sum of simpler fractions whose denominators are the factors of the original. Useful for integration and for binomial expansions of rational expressions.

Writing (a + b)ⁿ as a sum of terms using binomial coefficients, valid for any positive integer n and, with care about convergence, for fractional or negative n as an infinite series.

Changes to a graph produced by altering its equation, such as shifting, stretching, or reflecting. Knowing whether a change acts inside or outside the function tells you which direction the effect goes.

The function that returns the non-negative size of a number, written |x|. Its graph is V-shaped, and equations involving it usually split into separate cases to solve.

A function of the form aˣ where the variable sits in the exponent. Its rate of change is proportional to its current value, which is why it models growth and decay.

The logarithm taken to base e, written ln x. It undoes the exponential function eˣ, and its derivative is 1/x, which makes it central to calculus and to solving exponential equations.

Finding a derivative by taking the limit of the gradient of a chord as the gap shrinks to zero. It uses the definition of the derivative as a limit of a difference quotient.

A rule for differentiating a function of a function: differentiate the outer function, then multiply by the derivative of the inner one. Essential for composite expressions such as trigonometric or exponential nestings.

A rule for differentiating a product of two functions: the derivative of the first times the second, plus the first times the derivative of the second. Written compactly as (uv)′ = u′v + uv′.

A rule for differentiating one function divided by another, giving (u′v − uv′)/v². It applies whenever the expression is a fraction with variable denominator.

A point on a curve where the gradient is zero. It may be a maximum, a minimum, or a point of inflection, and the second derivative or a sign check distinguishes these cases.

The derivative of the derivative, measuring how the gradient itself changes. A negative value at a stationary point indicates a maximum, a positive value a minimum.

A technique for integrating composite expressions by introducing a new variable, replacing part of the integrand and adjusting the differential accordingly. Limits must be converted when the integral is definite.

A technique derived from the product rule that rewrites an integral of a product as a boundary term minus another integral. Choosing which factor to differentiate is the key decision.

The integral of a function between two limits, interpreted as the signed area under the curve. Its value is found by evaluating an antiderivative at the upper limit and subtracting its value at the lower limit.

The region bounded by a graph, the x-axis, and two vertical lines, computed with a definite integral. Parts below the axis contribute negative values, so geometry and sign must be handled separately.

The solid formed by rotating a region about an axis. Its volume is found by integrating πy² with respect to x, or πx² with respect to y, over the relevant interval.

An expansion of a function as an infinite sum of powers of x, built from the function's derivatives evaluated at zero. It approximates functions such as eˣ and sin x near the origin.

A quantity with both magnitude and direction, often written as a column of components or as a combination of unit vectors. Vectors add componentwise and can be multiplied by scalars.

A way of combining two vectors to produce a number, equal to the product of their magnitudes and the cosine of the angle between them. It is zero exactly when the vectors are perpendicular.

A vector describing the location of a point relative to a fixed origin. Differences between position vectors give displacement vectors along lines and between points.

The single force that has the same effect as all the forces acting on a body combined. Finding it usually means resolving forces into components and adding those components.

Splitting a force into perpendicular components, typically horizontal and vertical or parallel and perpendicular to a plane. It turns awkward angled forces into manageable separate equations.

The state of a body when the resultant force on it is zero and the resultant moment is also zero, so it stays at rest or moves with constant velocity.

A measure of the turning effect of a force about a point, equal to the force multiplied by the perpendicular distance from the point to its line of action.

The point at which the whole weight of a body may be treated as acting. For a uniform lamina it coincides with the centroid of its shape.

A contact force opposing relative sliding between surfaces. Before motion begins it can take any value up to a limiting maximum, which is proportional to the normal reaction.

The constant relating the limiting friction between two surfaces to the normal reaction pressing them together. It is dimensionless and depends on the materials in contact.

The force a surface exerts on a body perpendicular to that surface, preventing it from passing through. Its size follows from resolving forces perpendicular to the surface.

The principle that the resultant force on a body equals its mass times its acceleration. Applied along and perpendicular to motion, it produces the equations used to solve dynamics problems.

The motion of an object moving freely under gravity alone, with constant horizontal velocity and constant vertical acceleration. The two directions are analysed independently and linked by time.

A complete description of the possible values of a random variable together with their probabilities. For discrete variables it is often given as a table or formula.

The distribution of the number of successes in a fixed number of independent trials, each with the same probability of success. It is defined by the trial count and the success probability.

A continuous, symmetric bell-shaped distribution determined by its mean and variance. Probabilities are found by standardising values and using tables or a calculator for the standard normal curve.

The probability of one event given that another has already occurred, written P(A|B). It restricts attention to the part of the sample space where the condition holds.

Events whose occurrence does not affect each other's probabilities, so the probability of both equals the product of their individual probabilities. Distinct from mutually exclusive events.

Events that cannot happen at the same time, so their combined probability is the sum of their separate probabilities. They are not independent unless one has probability zero.

An arrangement of objects in which order matters, counted using factorials and products of descending integers. Contrasted with a combination, where order is irrelevant.

A selection of objects from a larger group where order does not matter, counted with binomial coefficients. Used when choosing a committee or a subset rather than arranging items.

A procedure that judges a claim about a population parameter by comparing observed data with what would be expected if a null hypothesis were true, using a stated significance level.

The threshold probability, chosen before a test, at which the null hypothesis is rejected. It fixes the acceptable risk of rejecting a true null hypothesis.

The probability of obtaining a result at least as extreme as the observed one, assuming the null hypothesis holds. A small value suggests the data are inconsistent with that hypothesis.

A number between minus one and one summarising the strength and direction of a linear relationship between two variables. It measures association, not causation.

The straight line fitted to bivariate data by least squares, used to predict one variable from another. Its equation is expressed in terms of the means and standard deviations of the data.