QCE Maths Methods key terms

A rule that pairs every input value in its domain with exactly one output value. Written f(x), it may be described by a formula, a graph, or a table.

The complete set of input values for which a function is defined. Restrictions often arise from division by zero, square roots of negatives, or logarithms of non-positive numbers.

The set of all output values a function actually produces as the input varies over its domain. Found by analysing the graph or by transforming the base function's outputs.

A function where distinct inputs always give distinct outputs, so its graph passes both the vertical line test and the horizontal line test. Only such functions possess an inverse that is also a function.

The function that reverses another, undoing its operation so that f-inverse of f(x) returns x. Its graph is the reflection of the original graph in the line y = x.

A function formed by applying one function to the output of another, written f(g(x)). It is defined only when the outputs of the inner function lie within the domain of the outer one.

A line that a curve approaches ever more closely as it extends, without necessarily meeting it. Vertical, horizontal and oblique forms commonly appear in rational, exponential and logarithmic graphs.

A point on a curve where the graph changes direction between increasing and decreasing, producing a local maximum or local minimum. At such points the derivative equals zero.

A vertical line about which a parabola or other symmetric graph mirrors onto itself. For a parabola with turning point at x = h, the axis is the line x = h.

A function of the form ax squared plus bx plus c with a not zero, whose graph is a parabola. Its features include a turning point, an axis of symmetry, and up to two x-intercepts.

The quantity b squared minus 4ac from the quadratic formula, which reveals the number of real roots. Positive means two, zero means one repeated root, negative means none.

A value of the variable that makes an expression or equation equal to zero. Graphically, real roots correspond to the points where the curve crosses or touches the horizontal axis.

A statement that a polynomial has factor (x minus a) exactly when substituting a gives zero. It links algebraic division with the location of roots and graph intercepts.

A result stating that dividing a polynomial by (x minus a) leaves a remainder equal to the polynomial evaluated at a. It allows remainders to be found without long division.

An expression formed from a sum of terms, each a constant multiplied by a non-negative whole-number power of the variable. Its degree is the highest power present.

A function written as one polynomial divided by another, defined wherever the denominator is not zero. Its graph often features vertical asymptotes at the zeros of the denominator.

A function of the form a to the power x with a positive base other than one. It has a constant ratio between successive values and a horizontal asymptote along the x-axis.

The exponent to which a base must be raised to produce a given number, so log base a of n equals the power p where a to the p equals n. It is the inverse operation of exponentiation.

A logarithm taken with base e, written ln(x), where e is approximately 2.718. It is the inverse function of the natural exponential and appears widely in calculus.

The function e to the power x, whose derivative equals itself and whose graph passes through the point (0, 1). It models continuous growth and decay.

Any of the algebraic rules for combining logarithms, such as turning a product into a sum, a quotient into a difference, or a power into a multiplier. They are used to solve exponential equations.

The value a function's outputs approach as the input approaches a particular point, whether or not the function is defined there. Limits underpin the definitions of derivative and integral.

The instantaneous rate of change of a function, equal to the gradient of its tangent line at a point. It is defined as the limit of a difference quotient.

The process of finding a derivative from a function using rules such as the power, product, quotient and chain rules. The result describes how steeply the original function changes.

A differentiation rule for composite functions: differentiate the outer function, then multiply by the derivative of the inner function. For example, the derivative of f(g(x)) is f-prime(g(x)) times g-prime(x).

A differentiation rule stating that the derivative of a product of two functions equals the first times the derivative of the second plus the second times the derivative of the first.

A differentiation rule for one function divided by another, giving the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared.

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

The way a curve bends, described as concave up when it holds water or concave down when it spills. It is determined by the sign of the second derivative.

The derivative of the derivative, measuring how quickly the gradient itself changes. Its sign indicates concavity and it helps classify stationary points as maxima or minima.

A point where a curve changes concavity from up to down or the reverse. It occurs where the second derivative changes sign, though not every such point is stationary.

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

A line perpendicular to the tangent at a point on a curve. Its gradient is the negative reciprocal of the tangent's gradient.

A function whose derivative is the given function, found by reversing differentiation and adding an arbitrary constant. It is the foundation of definite and indefinite integration.

The signed area between a curve and the horizontal axis over an interval, computed as the difference of an antiderivative at the endpoints. Areas below the axis count as negative.

The result connecting differentiation and integration, stating that the definite integral of a function over an interval equals the change in any antiderivative across that interval.

The region between a graph and the horizontal axis over an interval, found by integration. Regions below the axis contribute negative values unless absolute value is applied.

A unit of angle measure defined by arc length divided by radius, so a full turn equals two pi radians. Calculus formulas for trigonometric derivatives require angles in radians.

A circle of radius one centred at the origin, used to define sine and cosine as coordinates of points on it. It explains exact values and the signs of trigonometric ratios in each quadrant.

Half the vertical distance between the maximum and minimum values of a periodic function. For a sine or cosine curve written with leading coefficient a, the amplitude is the size of a.

The smallest positive input interval after which a periodic function repeats its values. For sine and cosine of kx, the period equals two pi divided by k.

The horizontal translation of a periodic graph from its standard position, determined by the constant added inside the trigonometric expression. It indicates where a cycle begins.

An equation involving trigonometric functions that holds for every value where both sides are defined. Common examples include the Pythagorean identity and the double-angle formulas.

A numerical quantity whose value depends on the outcome of a random experiment. It may be discrete, taking separate values, or continuous, taking any value in an interval.

A complete description of the possible values of a random variable together with their probabilities. For continuous variables it is given by a density curve whose total area is one.

The long-run average of a random variable, found by summing each value multiplied by its probability or, in the continuous case, by integration. It is the mean of the distribution.

A measure of spread equal to the square root of the variance, expressing how far values typically lie from the mean. It uses the same units as the data.

The average of the squared deviations of values from their mean. It measures spread in squared units and equals the square of the standard deviation.

A symmetric bell-shaped continuous distribution fixed by its mean and standard deviation. About 68 percent of values lie within one standard deviation of the mean and 95 percent within two.

The number of standard deviations a value lies above or below the mean, calculated by subtracting the mean and dividing by the standard deviation. It allows comparison across different normal distributions.

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

The fraction of a sample showing a particular characteristic, used to estimate the corresponding population proportion. Its distribution is approximately normal for large samples.

A range of values, calculated from sample data, that is likely to contain an unknown population parameter. The confidence level states how often such intervals succeed in repeated sampling.