SPM Add Maths key terms

A relation that maps every element of a domain to exactly one element of a codomain, so no input value produces two different outputs.

The complete set of input values for which a function is defined; for a function given by a formula, values that cause division by zero or a negative square root are excluded.

The set of all possible output values declared for a function, which may be larger than the set of values the function actually produces.

The set of all actual output values a function produces, found by applying the function to every element of its domain.

A function formed by applying one function to the result of another, written fg(x) meaning f(g(x)); order matters and fg usually differs from gf.

The function that reverses another function, so f⁻¹(f(x)) = x; it exists only when the original function is one-to-one.

A function in which each output value comes from exactly one input value, so no two distinct inputs give the same answer; such functions have inverses.

The quantity b² − 4ac for the quadratic ax² + bx + c; its sign tells whether the roots are real and distinct, real and equal, or non-real.

The rule x = [−b ± √(b² − 4ac)] / 2a giving the roots of any quadratic equation ax² + bx + c = 0 with a ≠ 0.

Rewriting a quadratic expression in the form a(x + p)² + q, which reveals the turning point of the graph and helps solve equations.

The values of the unknown that make an equation true; for a quadratic they are the x-intercepts of the corresponding graph.

For ax² + bx + c = 0 the two roots add to −b/a, a shortcut for forming equations from given roots.

For ax² + bx + c = 0 the two roots multiply to c/a, useful when constructing a quadratic equation from known roots.

The maximum or minimum point on a quadratic graph, located at x = −b/2a or read directly from completed-square form a(x + p)² + q as (−p, q).

The vertical line through the turning point of a quadratic graph, with equation x = −b/2a, about which the parabola is mirror-symmetric.

An inequality involving a squared term, solved by finding the roots and testing the sign of the expression on each interval between them.

Solving simultaneous equations by adding or subtracting multiples of the equations so that one unknown cancels out.

Solving simultaneous equations by rearranging one equation to express an unknown in terms of the other, then substituting into the remaining equation.

Two or more equations that must be satisfied by the same values of the unknowns; one linear and one non-linear equation typically give two solution pairs.

Rules for combining powers, such as aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 and a⁻ⁿ = 1/aⁿ.

A power written as a fraction, where a^(m/n) means the n-th root of a raised to the power m, for example 8^(2/3) = 4.

The power to which a base must be raised to give a number: if aˣ = N then logₐ N = x, with a > 0 and a ≠ 1.

Rules such as log(MN) = log M + log N, log(M/N) = log M − log N and log Mⁿ = n log M, used to simplify and solve equations.

Rewriting a logarithm in a new base using logₐ b = log꜀ b ÷ log꜀ a, which allows evaluation on a calculator with only common or natural logs.

The slope of a straight line, calculated as the change in y divided by the change in x between two points on the line.

A relationship describing a line, commonly y = mx + c with gradient m and y-intercept c, or y − y₁ = m(x − x₁) through a known point.

Straight lines that never meet because they have equal gradients but different y-intercepts.

Lines that meet at a right angle; their gradients multiply to −1, so one gradient is the negative reciprocal of the other.

A unit of angle measure defined so that one full turn equals 2π radians; one radian is about 57.3 degrees.

The distance along a curved part of a circle, given by s = rθ when the angle θ is measured in radians.

The region enclosed by two radii and an arc, with area ½r²θ when the angle θ is in radians.

The instantaneous rate of change of a function, found as the limit of the gradient of a chord as the two points come together.

A differentiation rule for composite functions: differentiate the outer function, then multiply by the derivative of the inner function.

A point on a curve where the gradient is zero, found by solving dy/dx = 0; it may be a maximum, minimum or point of inflection.

The derivative of the derivative, written d²y/dx²; its sign at a stationary point shows whether the point is a maximum or a minimum.

The integral of a function between two limits, evaluated by subtracting the value of the antiderivative at the lower limit from its value at the upper limit; it gives signed area.

The general antiderivative of a function, written with an arbitrary constant of integration because many functions share the same derivative.

The region between a curve and the x-axis over an interval, found by definite integration; parts below the axis count as negative unless handled separately.

A quantity with both magnitude and direction, often written in component form such as xi + yj or as a directed line segment.

The length of a vector; for xi + yj it equals √(x² + y²), and it is always non-negative.

A vector of length one, obtained by dividing a vector by its own magnitude; it gives direction only.

Vectors pointing in the same or opposite directions; one is a scalar multiple of the other, so their components are proportional.

The relationship a/sin A = b/sin B = c/sin C in any triangle, used when a side and its opposite angle are known.

The relationship a² = b² + c² − 2bc cos A, used to find a side from two sides and the included angle, or an angle from three sides.

Using trigonometry, the area equals ½ab sin C where a and b are two sides and C is the angle between them.

The situation when the sine rule gives two possible triangles for a given side-side-angle set of data, because the sine of an angle equals the sine of its supplement.

A measure expressing the change in a quantity relative to a base value, calculated as (current value ÷ base value) × 100.

A single index summarising several items, found as a weighted average of their individual index numbers using the given weightages.

The relative importance assigned to an item in a composite index, often given as a ratio, percentage or number of shares.

The position of a particle relative to a fixed point on a straight line, usually denoted s and given as a function of time t.

The rate of change of displacement with respect to time, v = ds/dt; its sign shows the direction of motion along the line.

The rate of change of velocity with respect to time, a = dv/dt = d²s/dt²; negative acceleration means velocity is decreasing.

The velocity of a particle at one particular moment, obtained by differentiating the displacement function and substituting the time.

The greatest speed reached in a motion, found by setting acceleration dv/dt equal to zero and checking the resulting value.

The whole path length covered, found by integrating speed over time or adding the magnitudes of displacements between points where the particle reverses direction.