BGCSE Additional Mathematics 4037 key terms

A rule that assigns to each input value in a stated set exactly one output value. The set of allowed inputs is the domain and the set of resulting outputs is the range.

A function formed by applying one function to the output of another, written fg(x) meaning f(g(x)). Its domain is restricted to inputs where g is defined and g(x) lies in the domain of f.

The function that reverses the effect of a one-to-one function, so f and its inverse undo each other. Its graph is the reflection of the original graph in the line y = x.

A function in which different inputs always produce different outputs, so no two points on its graph share the same y-value. Only such functions possess an inverse that is itself a function.

An expression of the form ax squared plus bx plus c with a not equal to zero. Its graph is a parabola, and completing the square or the discriminant reveals its key features.

Rewriting a quadratic in the form a(x + p) squared + q, which reveals the turning point and the range of the function. It also underpins the quadratic formula.

The quantity b squared minus 4ac for the quadratic ax squared + bx + c. Its sign tells whether the related equation has two, one or no real roots.

A value of the variable that makes the equation true. Graphically it is where the curve crosses the horizontal axis, so solving and finding intercepts are the same task.

A polynomial has (x - a) as a factor exactly when substituting a gives zero. It is used to find factors and to solve polynomial equations by division.

When a polynomial is divided by (x - a), the remainder equals the value of the polynomial at x = a. This gives remainders without performing long division.

An expression made from a variable raised to whole-number powers with constant coefficients, added together. Its degree is the highest power present, which limits the number of roots.

A statement comparing two expressions using less than, greater than, or the versions allowing equality. Solving it means finding all values satisfying the comparison, often shown on a number line.

An inequality in which the variable appears only to the first power. Its solution is an interval, and multiplying or dividing by a negative number reverses the inequality sign.

An inequality involving a squared term, solved by finding the roots of the related equation and testing which intervals satisfy it. A sketch of the parabola makes the answer clear.

A sequence in which each term differs from the previous one by a fixed constant. The nth term is a + (n - 1)d and the sum of n terms is n/2 times the first plus last term.

A sequence in which each term is obtained by multiplying the previous term by a fixed ratio. The nth term is a times r to the power n - 1.

The constant amount added to each term of an arithmetic progression to obtain the next. It may be negative, producing a decreasing sequence.

The constant multiplier linking consecutive terms of a geometric progression. Its size determines whether the series converges or diverges.

The limiting total of a geometric series when the common ratio has magnitude less than one, given by a divided by (1 - r). If the ratio is larger in size, no finite total exists.

A formula expanding (a + b) to the power n as a sum of terms involving binomial coefficients. Each term has powers of a and b that add to n.

The number of ways of choosing r objects from n, written as n choose r and found using factorials. It is the coefficient of the corresponding term in a binomial expansion.

An expression for the term in position r + 1 of a binomial expansion, used to find one particular coefficient without writing the whole expansion.

A measure of how steeply a line or curve rises or falls, equal to the change in y divided by the change in x. For a curve it is found from the tangent or derivative.

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

Two lines meeting at a right angle. Their gradients multiply to give minus one, so one gradient is the negative reciprocal of the other.

A unit of angle measure defined so that a full turn is two pi radians. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

The distance along a curved part of a circle. Measured in radians, it equals the radius multiplied by the angle subtended at the centre.

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

An equation involving trigonometric functions that holds for every angle where both sides are defined. Familiar examples link sine, cosine and tangent, or connect squared sine and cosine.

An equation containing a trigonometric function of an unknown angle, usually solved by finding a principal value and then using symmetry or periodicity to list further solutions in the required range.

The rate at which a function's output changes as its input changes, interpreted as the gradient of the tangent to the curve at a point.

A straight line touching a curve at a point and having the same gradient as the curve there. Its gradient equals the derivative evaluated at that point.

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

A point on a curve where the gradient is zero, so the tangent is horizontal. The second derivative or a sign test classifies it as a maximum, minimum or point of inflection.

The derivative of the derivative, describing how the gradient itself changes. Its sign at a stationary point distinguishes a maximum from a minimum.

The reverse process of differentiation, used to recover a function from its gradient. Applied to a rate it gives a total, and it is also used to find areas under curves.

The integral evaluated between two limits, giving a numerical value rather than a family of functions. It represents the signed area between the curve and the horizontal axis over that interval.

The region between a curve and the horizontal axis, found by integrating the function between the relevant limits. Parts below the axis count as negative unless handled separately.

A quantity with both magnitude and direction, often written as a column of components or in terms of unit vectors. It differs from a scalar, which has size only.

A vector describing the location of a point relative to a fixed origin. The vector from one point to another equals the difference of their position vectors.

A vector of length one, obtained by dividing any vector by its magnitude. It records direction only and is often written with a hat symbol.

A rectangular array of numbers arranged in rows and columns. Matrices can be added, subtracted and multiplied under specific size conditions, and some have inverses.

A single number calculated from a square matrix that indicates whether the matrix has an inverse. For a two-by-two matrix it is ad minus bc.

The matrix that multiplies a given square matrix to give the identity. It exists only when the determinant is non-zero, and it is used to solve simultaneous equations.

The power to which a base must be raised to produce a given number. Logarithms convert multiplication into addition and are the inverse of exponential functions.

A function in which the variable appears in the exponent, such as a to the power x. Its graph rises or falls ever more steeply and passes through the point (0, 1).

Rules allowing logarithms of products, quotients and powers to be rewritten as sums, differences and multiples. They are used to simplify expressions and to solve exponential equations.

A logarithm taken to the base e, written ln x. It is the inverse of the exponential function with base e and appears whenever continuous growth or decay is modelled.

A style of argument that assumes the opposite of what is to be proved and shows this assumption leads to an impossible conclusion, so the original statement must be true.

A single case that shows a general statement is false. One counterexample is enough to disprove a claim, however many other cases support it.

A condition that must hold for a statement to be true and also guarantees it. Necessary alone is not enough, and sufficient alone is stronger than required.

How quickly one quantity changes compared with another, usually found by differentiating. Connected rates problems link two changing quantities through the chain rule.

A method for differentiating a function of a function, multiplying the derivative of the outer function by the derivative of the inner one. It also underlies related rates of change.

A line that a curve approaches ever more closely as the variable grows large or approaches a particular value, without necessarily meeting it.