VCE Maths Methods key terms

A rule that assigns to each input value in a stated domain exactly one output value. Passing the vertical line test confirms that a graph represents a function.

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

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

The largest set of real input values for which a rule produces a real output. For example, the maximal domain of a square-root rule excludes inputs making the radicand negative.

A graphical check where a relation is a function only if no vertical line meets its graph more than once. It detects inputs paired with two or more outputs.

A graphical check where a function is one-to-one only if no horizontal line meets its graph more than once. Passing it guarantees the function has an inverse function.

A function in which distinct inputs always give distinct outputs, so no two points share the same y-value. Such functions are invertible without restricting their domain.

The function that reverses another, swapping inputs and outputs, written with a superscript minus one. Its graph is the reflection of the original graph in the line y equals x.

A function formed by applying one function to the output of another, written as f of g of x. It exists only where the range of the inner function meets the domain of the outer one.

A line that a curve approaches ever more closely as the input or output grows without bound, without necessarily meeting it. Vertical, horizontal and oblique forms commonly appear in rational and exponential graphs.

A change to a parent graph produced by dilations, reflections or translations, described by adjusting parameters in its rule. Each change shifts, stretches or flips the shape predictably.

A transformation that stretches or compresses a graph away from or towards an axis by a scale factor. A factor greater than one stretches; a factor between zero and one compresses.

A transformation that slides every point of a graph the same distance in the same direction without changing its shape or orientation. It is described by horizontal and vertical shifts.

A transformation that flips a graph across a line, commonly the x-axis or y-axis. Reflecting in the line y equals x produces the graph of the inverse relation.

Any set of ordered pairs linking values of two variables, whether or not each input has a single output. Circles and sideways parabolas are relations but not functions.

A function written as one polynomial divided by another, defined wherever the denominator is nonzero. Its graph often features vertical asymptotes where the denominator vanishes.

An expression formed from a sum of terms, each a constant multiplied by a non-negative whole-number power of a variable. Its graph is smooth and continuous with no asymptotes.

The highest power of the variable appearing with a nonzero coefficient in a polynomial. It determines the maximum number of roots and the graph's end behaviour.

A statement that a polynomial has factor x minus a exactly when substituting a gives zero. It links linear factors to the roots of the polynomial.

A result stating that dividing a polynomial by x minus a leaves a remainder equal to the polynomial evaluated at a. It gives remainders without performing long division.

A value of the variable that makes an expression or equation equal zero. On a graph, real roots appear as the x-intercepts of the curve.

The quantity b squared minus four a c in a quadratic equation, revealing how many real solutions exist. Positive means two, zero means one, and negative means none.

An expression or equation whose highest power of the variable is two, generally written in the form a x squared plus b x plus c. Its graph is a parabola.

The turning point of a parabola, where the curve reaches its maximum or minimum value. It lies on the axis of symmetry and can be found by completing the square.

The vertical line through the vertex of a parabola that divides it into two mirror-image halves. Its equation is x equals negative b over two a for a standard quadratic.

A function in which the variable appears in the exponent, such as a to the power x with a positive base other than one. It grows or decays at a rate proportional to its current value.

The exponent to which a fixed base must be raised to produce a given number. Logarithms reverse exponential rules and convert multiplication into addition.

Rules allowing logarithms of products, quotients and powers to be rewritten as sums, differences or multiples of logarithms. They are used to solve exponential equations.

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

The process of finding a derivative from a function's rule using limit definitions or standard rules. It produces a new function describing gradients or rates of change.

A differentiation rule for composite functions, multiplying the derivative of the outer function by the derivative of the inner function. It handles expressions nested inside other functions.

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

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

A straight line touching a curve at a point and matching its gradient there. Its slope equals the derivative of the function at that point.

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

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

The derivative of the derivative, measuring how the gradient itself changes. Its sign helps classify stationary points as maxima, minima or points of inflection.

A function whose derivative is the given function. Finding one reverses differentiation and is the first step in definite integration.

The signed area between a curve and the horizontal axis over an interval, computed as the difference of an antiderivative's values at the endpoints. It accumulates rates of change.

The region enclosed between a graph and the horizontal axis over an interval, found by definite integration. Areas below the axis count as negative unless handled separately.

The arithmetic average of a data set, found by adding all values and dividing by how many there are. It is sensitive to extreme values.

The middle value of an ordered data set, or the average of the two middle values when the count is even. It resists distortion by outliers.

A measure of how far values typically spread from the mean, equal to the square root of the variance. Larger values indicate more dispersed data.

The average of the squared deviations of data values from their mean. Its square root gives the standard deviation, restoring the original units.

A value dividing ordered data into four parts of roughly equal size. The lower and upper quartiles, with the median, summarise spread and support boxplots.

The difference between the upper and lower quartiles, covering the middle half of the data. It measures spread while ignoring extreme values.

A data value lying unusually far from the rest of the set, often flagged as beyond one and a half interquartile ranges from a quartile. It can strongly affect the mean.

A symmetric bell-shaped probability distribution defined by its mean and standard deviation. Roughly 68 percent of values lie within one standard deviation of the mean.

A function whose value over an interval gives the probability that a continuous variable falls in that interval, through the area beneath its graph. Total area under the curve equals one.

A function giving the probability that a variable takes a value less than or equal to a given point. It increases from zero to one and equals the accumulated area under a density curve.

The long-run average outcome of a random variable, computed by summing each value weighted by its probability. It need not equal any value the variable can actually take.

The probability of an event given that another event has already occurred, written as the probability of A given B. It restricts attention to the reduced sample space of the known event.

A relationship between two events where the occurrence of one does not change the probability of the other. Equivalently, their joint probability equals the product of their individual probabilities.

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

The set of all possible outcomes of a random experiment. Every event is a subset of this collection.