WACE ATAR Mathematics Specialist key terms

A rule that assigns to each element of a domain exactly one element of a codomain. Passing the vertical line test confirms a graph represents one.

The complete set of input values for which a function or relation is defined, often restricted by denominators, even roots or logarithm arguments.

The set of all output values a function actually produces as its input varies across the domain.

The function that reverses another, swapping inputs and outputs. It exists only where the original is one-to-one, and its graph reflects across the line y equals x.

A function where distinct inputs always give distinct outputs, so no horizontal line meets its graph twice and an inverse can be defined.

A function formed by applying one function to the output of another, written f of g of x, and defined only where the inner output lies in the outer domain.

A line that a curve approaches ever more closely as it extends, without necessarily meeting it, such as vertical or horizontal lines in rational graphs.

The value a function's output approaches as the input approaches a specified number or grows without bound, forming the foundation of derivative and integral definitions.

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

The instantaneous rate of change of a function, defined as the limit of a difference quotient and interpreted geometrically as the gradient of the tangent.

The property that a function has a well-defined derivative at a point; it requires continuity there, though continuity alone does not guarantee it.

A differentiation rule for composite functions: the derivative of f of g of x equals f prime of g of x multiplied by g prime of x.

A differentiation rule stating 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 dividing one function 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.

The straight line touching a curve at a point with gradient equal to the derivative there, used to approximate the function near that point.

A point on a curve where the derivative equals zero, classified as a local maximum, local minimum or point of inflection using the second derivative or sign testing.

The derivative of the derivative, measuring how the gradient itself changes and used to determine concavity and the nature of stationary points.

The way a curve bends: concave up where the second derivative is positive, concave down where it is negative, changing at a point of inflection.

A function whose derivative is the given function; antiderivatives differ only by an added constant, and finding them reverses differentiation.

The limit of sums of areas of thin rectangles across an interval, giving a signed area under a curve and evaluated using an antiderivative.

The result linking differentiation and integration: the definite integral of a derivative over an interval equals the change in the original function across that interval.

A quantity possessing both magnitude and direction, represented by a directed line segment or an ordered list of components, and added componentwise.

A quantity described by magnitude alone, with no direction, such as mass, speed or temperature, in contrast to a vector.

A vector of length one, often used to specify direction; any nonzero vector can be divided by its magnitude to produce one.

The length of a vector, computed as the square root of the sum of the squares of its components in coordinate form.

A scalar obtained by multiplying corresponding components and summing, equal to the product of magnitudes times the cosine of the angle between the vectors.

Vectors meeting at right angles, characterised by a dot product of zero, so neither has any component along the other's direction.

The signed length of one vector's shadow along another's direction, found by dividing the dot product by the magnitude of the vector projected onto.

A description of a line as a fixed position vector plus a scalar multiple of a direction vector, allowing every point on the line to be generated.

A number of the form a plus b i, where a and b are real and i squared equals negative one, combining real and imaginary parts.

The number obtained by reversing the sign of the imaginary part; multiplying a complex number by its conjugate gives a real value equal to the square of the modulus.

The distance of a complex number from the origin in the complex plane, written as the square root of a squared plus b squared.

The angle a complex number's position vector makes with the positive real axis, measured anticlockwise and usually given in the interval from negative pi to pi.

Writing a complex number as its modulus multiplied by the sum of cosine of the argument and i times sine of the argument, convenient for multiplication and powers.

The result that raising a complex number in polar form to a power raises its modulus to that power and multiplies its argument by that power.

A rectangular array of numbers arranged in rows and columns, used to represent linear transformations and to solve systems of simultaneous equations.

A single number calculated from a square matrix that indicates whether the matrix is invertible and, in two dimensions, the scale factor of area change.

The matrix that multiplies a given square matrix to give the identity, existing only when the determinant is nonzero and found using the adjugate divided by the determinant.

The square matrix with ones on the main diagonal and zeros elsewhere, acting like the number one in matrix multiplication.

A square matrix whose determinant is zero, so it has no inverse and its associated linear transformation collapses area or volume.

A scalar for which a nonzero vector is mapped to a scalar multiple of itself by a matrix, found by solving the characteristic equation.

A nonzero vector whose direction is unchanged when a given matrix is applied, only its length being scaled by the corresponding eigenvalue.

A numerical value assigned to each outcome of a random experiment, either discrete with countable values or continuous with values across an interval.

A non-negative function whose total area under its curve equals one, with area over an interval giving the probability a continuous random variable lies there.

The long-run mean of a random variable, calculated by summing value-times-probability for discrete cases or integrating over the density for continuous cases.

The expected value of the squared deviation from the mean, measuring the spread of a distribution; its square root is the standard deviation.

The square root of the variance, expressing spread in the same units as the data and widely used to standardise values.

A symmetric bell-shaped continuous distribution defined by its mean and standard deviation, with probabilities found by standardising values and using tables or technology.

The distribution of the number of successes in a fixed number of independent trials, each with the same probability of success.

The average of observations in a sample, a statistic used to estimate the population mean and whose distribution tends toward normality for large samples.