SACE Stage 2 Mathematical Methods key terms

A rule that assigns exactly one output value to each input value in a stated set. Written f(x), it may be given by a formula, graph, table, or description.

The complete set of input values for which a function is defined. It may be restricted by the context, by division by zero, or by the need to take a square root of a non-negative number.

The set of all output values a function actually produces as the input runs over its domain. It is found from the graph or by analysing the rule.

A function where different inputs always give different outputs, so no horizontal line meets its graph more than once. Only such functions have inverses that are also functions.

The function that reverses another 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 formed by applying one function and then another, written f(g(x)). It exists only where the output of the inner function lies in the domain of the outer one.

The distance of a number from zero on the number line, always non-negative. Written |x|, it equals x when x is non-negative and −x when x is negative.

A function defined by different formulas on different parts of its domain. Each piece applies to a stated interval, and the graph may have jumps or corners at the boundaries.

An expression made by adding terms of the form a·xⁿ where n is a whole number and a is a constant. Its degree is the highest power of x present.

A polynomial P(x) has factor (x − a) exactly when P(a) = 0. It links factors of a polynomial to its zeros and is used to factorise cubics and higher-degree expressions.

When a polynomial P(x) is divided by (x − a), the remainder equals P(a). This gives a quick way to find remainders without long division.

A line that a curve approaches ever more closely as it extends, without necessarily meeting it. Graphs of rational, exponential, and logarithmic functions often have horizontal or vertical asymptotes.

A function that is the quotient of two polynomials. Its graph may have vertical asymptotes where the denominator is zero and horizontal asymptotes determined by the degrees involved.

A function of the form y = aˣ with a positive base not equal to one. It grows or decays by a constant factor for each equal increase in the input.

The power to which a base must be raised to produce a given number. It is the inverse of the exponential function, so log base a of aˣ equals x.

A function such as sine, cosine, or tangent that describes periodic behaviour. Its input is an angle, measured in degrees or radians, and its outputs repeat at regular intervals.

A way of measuring angles using the ratio of arc length to radius. One full turn is 2π radians, and π radians equal 180 degrees.

Half the difference between the maximum and minimum values of a periodic function. For y = A sin x it is |A|, and it measures how far the graph swings from its central value.

The smallest positive input interval after which a periodic function repeats its values. For sine and cosine of kx the period is 2π divided by k.

The value a function's output approaches as the input approaches a particular point. Limits describe behaviour near a point and underpin the definition of the derivative.

A property of a function whose graph can be drawn without lifting the pen. Formally, the function's limit at each point equals its value there.

The instantaneous rate of change of a function, found as the limit of a difference quotient. Geometrically it gives the gradient of the tangent to the curve at a point.

A property of a function that has a derivative at a point. A function must be continuous there to be differentiable, and corners or vertical tangents prevent it.

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

The line perpendicular to the tangent at a point on a curve. Its gradient is the negative reciprocal of the derivative at that point.

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

The derivative of the derivative of a function. It measures how the gradient itself changes and helps classify stationary points as maxima or minima.

A rule for differentiating a composite function. For y = f(g(x)) the derivative is f′(g(x)) multiplied by g′(x).

A rule for differentiating a product of two functions. The derivative of u times v is u′v plus uv′.

A rule for differentiating one function divided by another. The derivative of u over v is (u′v − uv′) divided by v squared.

A function whose derivative is the given function. Because constants vanish on differentiation, antiderivatives differ only by an added constant.

A number found by integrating a function between two limits. It represents the signed area between the graph and the horizontal axis over that interval.

The region enclosed between a graph and the horizontal axis over an interval. Its signed value is computed with a definite integral, with regions below the axis counting as negative.

A numerical method for estimating a definite integral by joining sample points on the curve with straight segments. It approximates the region with a series of trapeziums.

A variable that can take only separate, countable values, each with an associated probability. Its probabilities must be non-negative and sum to one.

A complete description of the possible values of a random variable together with their probabilities. For discrete variables it may be given as a table or formula.

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

A measure of how spread out the values of a random variable are, equal to the expected value of the squared deviation from the mean. Its square root is the standard deviation.

The square root of the variance, giving spread in the same units as the variable. A small value means values cluster near the mean.

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 n and p.

A symmetric bell-shaped continuous distribution determined by its mean and standard deviation. Many natural measurements follow it approximately, and probabilities are found from areas under the curve.

The normal distribution with mean zero and standard deviation one. Any normal variable can be converted to it using a z-score so that standard tables can be applied.

The number of standard deviations a value lies above or below the mean. It is calculated as the value minus the mean, divided by the standard deviation.

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

The fraction of a sample showing a particular characteristic. It estimates the population proportion and has an approximately normal distribution for large samples.

The skills and content from earlier years that students are expected to use without re-teaching, such as basic algebra, graphing, and trigonometry. External examinations rely on this foundation throughout.

The four interrelated processes of understanding, fluency, problem-solving, and reasoning that shape how mathematics is learned and assessed. Questions often test these processes alongside specific content.

Applying known mathematics to unfamiliar situations, often without a routine method being obvious. Assessment rewards clear formulation, selection of strategy, and interpretation of results.

Explaining why a mathematical result or method is valid, using logical argument and justification. It is assessed through questions requiring proof, justification, or evaluation of claims.

The formal written assessment set and marked outside the school, contributing a set proportion of the final subject grade. It samples the full syllabus and is timed under supervised conditions.

The component of assessment designed and marked by the school, following syllabus requirements. It includes tasks such as skills and applications tasks and investigations, and is moderated externally.

A school-assessed task in which students explore a mathematical situation, often using technology, and report findings. It assesses reasoning and communication as well as content knowledge.

The practical skills of managing time, showing working, checking answers, and allocating effort by mark value. Good technique prevents avoidable loss of marks in timed conditions.

Writing down the steps of a solution, not just the final answer, so that method marks can be awarded. Marks are often allocated for correct process even when an arithmetic slip occurs.

A section or question that must be answered without a calculator or other electronic aid. It tests fluency with mental and written computation and exact values.

A supplied reference containing standard formulas candidates may use during the examination. Knowing when and how to apply each formula still requires understanding of the content.