HSC Maths Advanced key terms

A rule that assigns to each input value in a stated domain exactly one output value. It may be given by a formula, a graph, or a table.

The complete set of input values for which a function is defined. For a formula, it is often restricted by division by zero or by square roots of negatives.

The set of all output values a function actually produces as the input runs through its domain. It can be read from the graph as the spread of y-values.

A new function formed by applying one function to the output of another, written f(g(x)). The output of the inner function must lie in the domain of the outer one.

The function that reverses another function, swapping inputs and outputs. It exists only when each output comes from exactly one input, and its graph is a reflection in y = x.

A function in which no two different inputs give the same output, so it passes both the vertical and horizontal line tests. Such functions have inverses that are also functions.

A graphical check where a horizontal line is swept across a curve. If no horizontal line meets the graph more than once, the function is one-to-one and invertible.

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

A line that a curve approaches ever more closely as it extends, without necessarily meeting it. Vertical and horizontal asymptotes help describe rational and exponential graphs.

A function written as one polynomial divided by another. Its graph often has vertical asymptotes where the denominator is zero and horizontal asymptotes determined by the degrees.

A change to a base graph produced by altering its equation, such as shifting, reflecting, stretching, or compressing. Each change has a predictable effect on the coordinates of every point.

In a right-angled triangle, the length of the side opposite a given angle divided by the hypotenuse. It also defines the sine of any angle through the unit circle.

In a right-angled triangle, the length of the side adjacent to a given angle divided by the hypotenuse. On the unit circle it gives the x-coordinate of the point at that angle.

The ratio of the opposite side to the adjacent side in a right-angled triangle, equal to sine divided by cosine. It is undefined wherever cosine is zero.

A unit of angle measure defined so that a full turn is 2 pi radians. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

The distance measured along a circle's edge between two points. For an angle measured in radians it equals the radius multiplied by the angle.

The area of the region enclosed by two radii of a circle and the arc between them. With the angle in radians it equals half the radius squared times the angle.

Half the difference between the maximum and minimum values of a periodic graph. It measures how far the wave rises above or falls below its central line.

The smallest positive input interval after which a periodic function repeats its values exactly. For sine and cosine it is 2 pi radians unless the graph is horizontally stretched.

The horizontal translation of a periodic graph from its standard position. It is read from the constant added to or subtracted from the angle inside the function.

A relationship stating that each side of a triangle divided by the sine of its opposite angle is constant. It is used to solve triangles when a matching side and angle pair is known.

A formula generalising Pythagoras to any triangle, relating one side to the other two sides and the cosine of the included angle. It solves triangles given three sides or two sides and the included angle.

An equation involving trigonometric ratios that holds for every angle in its domain. Common examples connect tangent to sine and cosine, or relate sine squared plus cosine squared to one.

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

The process of finding a derivative from a function. It uses rules such as the power, product, quotient, and chain rules rather than first principles each time.

A differentiation rule for composite functions: differentiate the outer function, then multiply by the derivative of the inner function. It is essential for expressions with brackets or powers of functions.

A rule for differentiating a product of two functions, giving the first times the derivative of the second plus the second times the derivative of the first. It applies where terms are multiplied together.

A rule for differentiating one function divided by another. The result has the denominator squared, with the numerator built from cross products of each function and the other's derivative.

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.

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

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 derivative of the derivative, describing how the gradient itself changes. Its sign at a stationary point distinguishes a local maximum from a local minimum.

A function whose derivative is the given function, also called an antiderivative. Any two primitives of the same function differ only by a constant.

The family of all primitives of a function, written with an integration sign and a constant of integration. It reverses differentiation but gives a function rather than a number.

The signed area between a curve and the horizontal axis over a stated interval, found by evaluating a primitive at the endpoints and subtracting. Areas below the axis count as negative.

A numerical method estimating a definite integral by splitting the interval into strips and treating each as a trapezium. Accuracy improves as the number of strips increases.

A numerical integration method fitting parabolas through groups of three points to estimate area. It usually needs an even number of strips and is often more accurate than the trapezoidal rule.

Interest calculated on both the original principal and previously earned interest, so the balance grows by a constant factor each period. It is modelled by an exponential formula.

The original amount of money invested or borrowed, before any interest is added. It forms the starting value in compound interest and annuity calculations.

The amount an investment or loan will be worth after interest has been applied for a number of periods. It is found by multiplying the principal by the growth factor repeatedly.

The amount that must be invested now to reach a stated future amount after interest is applied. It is found by dividing the future value by the growth factor.

A series of equal payments made at regular intervals, often used to save a lump sum or repay a loan. Its value depends on the payment size, interest rate, and number of periods.

A rule expressing each term of a sequence in terms of earlier terms, such as a balance updated by adding interest and a payment. It models repeated financial calculations step by step.

The arithmetic average of a data set, found by adding all values and dividing by how many there are. It is affected strongly by unusually large or small values.

The middle value of a data set once the values are arranged in order, or the average of the two middle values. It is resistant to extreme values.

A measure of how spread out data values are around the mean, equal to the square root of the variance. A larger value indicates greater variability.

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

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

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

A data value lying far from the rest of the set, often defined as more than 1.5 interquartile ranges beyond a quartile. It may indicate error or genuine unusual behaviour.

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

The number of standard deviations a value lies above or below the mean. It allows values from different normal distributions to be compared on a common scale.

A number between minus one and one measuring the strength and direction of a linear relationship between two variables. Values near zero suggest little linear association.

A straight line drawn through a scatterplot to summarise the trend in paired data and support predictions. It is often found by least squares regression.