WACE Maths Methods key terms

A rule that assigns to each input value in its domain exactly one output value. Written f(x), it pairs members of one set with members of another set.

The complete set of input values for which a function is defined. It may be restricted by context, by division by zero, or by square roots of negatives.

The set of all output values a function actually produces as the input varies across its domain. It is the set of values of f(x).

A function formed by applying one function to the output of another, written f(g(x)). Its domain excludes inputs where g(x) lies outside the domain of f.

A function that reverses another, so f and its inverse undo each other. It exists only when the original function is one-to-one, and its graph is a reflection in the line y = x.

A function where each output value comes from exactly one input value, so no two inputs share an output. Such functions pass the horizontal line test and have inverses.

A graphical check where any horizontal line meets the graph of a function at most once. If it holds, the function is one-to-one and invertible on that domain.

A line that a curve approaches ever more closely but does not normally reach. It may be vertical, horizontal, or oblique, and describes long-term or boundary behaviour.

A vertical line x = a near which a function's values grow without bound, typically where a denominator is zero. The graph rises or falls sharply alongside it.

A horizontal line y = L that the graph of a function approaches as x becomes very large positive or negative. It describes the limiting value of the function.

A change to a parent graph produced by altering its equation, such as shifting, stretching, or reflecting. Transformations move or reshape the graph without changing its basic form.

A stretching or compressing of a graph away from or towards an axis, caused by multiplying the function or its input by a factor. Factors above one stretch; between zero and one compress.

A shift of a graph without rotation or resizing, produced by adding to or subtracting from the input or output. Adding outside the function moves the graph vertically.

A circle of radius one centred at the origin. Points on it have coordinates (cos θ, sin θ), which defines trigonometric ratios for all angles and their signs.

An angle measure based on arc length, where one radian subtends an arc equal in length to the radius. A full turn equals 2π radians.

A trigonometric function giving, for an angle, the ratio of the opposite side to the hypotenuse in a right triangle, and the y-coordinate on the unit circle.

A trigonometric function giving, for an angle, the ratio of the adjacent side to the hypotenuse in a right triangle, and the x-coordinate on the unit circle.

A trigonometric function equal to sine divided by cosine. It is undefined where cosine is zero, giving vertical asymptotes in its graph.

Half the distance between the maximum and minimum values of a periodic function. For y = a sin(bx) it equals the size of the factor a.

The smallest positive input interval after which a periodic function repeats its values. For sine and cosine it is 2π before any horizontal stretching.

The horizontal displacement of a periodic graph from a standard starting position, caused by adding or subtracting a constant inside the function.

An equation involving trigonometric functions that is true for all angles where both sides are defined. Examples include relationships between sine, cosine, and tangent.

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 value a function's outputs approach as the input approaches some point. Limits underpin derivatives and describe behaviour near discontinuities or at infinity.

A property of a function whose graph has no breaks, jumps, or holes at a point. Formally, the function's limit there equals its value at that point.

The property of a function having a derivative at a point, meaning its graph has a well-defined tangent there. It requires continuity but is stronger than continuity.

A differentiation technique for composite functions, multiplying the derivative of the outer function by the derivative of the inner function. It handles nested expressions efficiently.

A differentiation technique for a function written as one expression multiplied by another. The derivative equals the first times the derivative of the second plus the reverse.

A differentiation technique for a function written as one expression divided by another. The result uses the denominator times the derivative of the numerator minus the reverse, 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, local minimum, or a point of inflection.

The derivative of the derivative of a function, describing how the gradient itself changes. Its sign indicates concavity and helps classify stationary points.

The way a curve bends, either upwards like a cup or downwards like a cap. It is determined by the sign of the second derivative over an interval.

A point where a curve changes concavity, from bending one way to bending the other. The second derivative is usually zero there, though not every such point qualifies.

A function whose derivative is the given function. It is found by reversing differentiation and includes an arbitrary constant because many functions share the same derivative.

A number representing the signed area between a curve and an axis over an interval. It is evaluated using an antiderivative and the fundamental theorem of calculus.

The region between a graph and a horizontal axis over an interval, computed with a definite integral. Regions below the axis count as negative unless absolute value is used.

A function of the form f(x) = a to the power x, where the base a is a positive constant. It grows or decays at a rate proportional to its current value.

The inverse of an exponential function, giving the exponent needed to produce a given value. The logarithm base a of y answers what power of a equals y.

A logarithm with base e, written ln x. It is the inverse of the natural exponential function and has derivative one over x for positive x.

The irrational constant e, approximately 2.718, which is the base of natural exponentials and logarithms. It arises wherever growth rate is proportional to size.

A number between zero and one expressing how likely an event is, where zero means impossible and one means certain. It may be estimated from data or calculated from equally likely outcomes.

The set of all possible outcomes of a random experiment. Each event is a subset of this set, and probabilities across all outcomes sum to one.

The event that a given event does not occur. Its probability equals one minus the probability of the original event.

Two events that cannot both happen in the same trial, so their joint probability is zero. Their combined probability is found by adding the separate probabilities.

Two events where the occurrence of one does not change the probability of the other. Their joint probability equals the product of their individual probabilities.

The probability of an event given that another event has already occurred. It is calculated by dividing the joint probability by the probability of the given event.

A numerical value assigned to each outcome of a random experiment. It may be discrete, taking separate values, or continuous, taking any value in a range.

A description of the possible values of a random variable together with their probabilities. For discrete variables it is often shown in a table or graph.

The long-run average of a random variable, found by multiplying each value by its probability and summing the results. It is also called the mean of the distribution.

A measure of how spread out values are around the mean, equal to the square root of the variance. A small value indicates values cluster near the mean.

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

A symmetric bell-shaped probability distribution defined by its mean and standard deviation. Many natural measurements approximate it, and probabilities are found from its standardised values.

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