HKDSE Maths (Core) key terms

An equation that can be written as ax² + bx + c = 0, where a is not zero. Its solutions are the values of x that make the statement true.

The quantity b² − 4ac taken from a quadratic equation ax² + bx + c = 0. Its sign tells whether the equation has two, one, or no real roots.

A method that rewrites a quadratic expression as a perfect square plus a constant, usually to find the vertex of its graph or to solve the equation.

The values of the unknown that satisfy an equation. For a quadratic, they are the x-intercepts of its graph and can be found by factorising or the quadratic formula.

The highest or lowest point on the graph of a quadratic function. Its coordinates can be read from the completed-square form of the equation.

The vertical line through the vertex of a parabola that divides the curve into two mirror-image halves. Its equation is x equals the x-coordinate of the vertex.

The U-shaped curve that is the graph of a quadratic function. It opens upward when the coefficient of x² is positive and downward when it is negative.

A rule that assigns exactly one output value to each input value in its domain. Notation such as f(x) names the output for input x.

The set of all input values for which a function is defined. For example, a function with x in a denominator excludes values that make the denominator zero.

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

A function of the form y = aˣ, where the base a is a positive constant other than 1. Its graph rises or falls steadily and never touches the x-axis.

The power to which a base must be raised to produce a given number. For example, log₂ 8 = 3 because 2³ = 8.

Rules that turn products into sums, quotients into differences, and powers into multipliers, such as log(ab) = log a + log b and log(aⁿ) = n log a.

A formula that rewrites a logarithm in one base as a quotient of logarithms in another base, allowing calculation with a calculator's built-in log keys.

A list of numbers in which each term after the first is obtained by adding a fixed constant, called the common difference.

A list of numbers in which each term after the first is obtained by multiplying the previous term by a fixed non-zero constant, called the common ratio.

The constant amount added to each term of an arithmetic sequence to produce the next term. It equals the difference between any two consecutive terms.

The constant factor by which each term of a geometric sequence is multiplied to produce the next term. It equals any term divided by the one before it.

The limiting total of a geometric series when the common ratio lies strictly between −1 and 1. It equals the first term divided by one minus the common ratio.

A relationship in any triangle stating that each side divided by the sine of its opposite angle is the same for all three sides.

A relationship in any triangle linking a side to the other two sides and the included angle, often written c² = a² + b² − 2ab cos C.

The angle formed between two named sides of a triangle at the vertex they share. It is the angle used in the cosine rule and the area formula.

For a triangle with sides a and b and included angle C, the area equals one half of a times b times the sine of C.

The angle measured upward from the horizontal to the line of sight when looking at an object above the observer's level.

The angle measured downward from the horizontal to the line of sight when looking at an object below the observer's level.

A direction given as an angle measured clockwise from due north, written with three digits. True bearings use north as the reference direction.

The study of angles, chords, tangents, and arcs in circles. Key results include the angle in a semicircle being a right angle and the tangent-radius perpendicularity.

A straight line that touches a circle at exactly one point. At that point the tangent is perpendicular to the radius drawn to it.

A four-sided figure whose vertices all lie on one circle. Its opposite angles add up to 180 degrees.

The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment subtended by that chord.

An arrangement of objects in which order matters. The number of arrangements of r objects chosen from n distinct objects is written nPr.

A selection of objects in which order does not matter. The number of ways to choose r objects from n distinct objects is written nCr.

The product of all positive integers from 1 up to a given whole number n, written n!. By convention, 0! equals 1.

A number between 0 and 1 measuring how likely an event is. For equally likely outcomes it equals the number of favourable outcomes divided by the total number of outcomes.

Two events that cannot happen at the same time. For such events the probability that either occurs is the sum of their separate probabilities.

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

The probability that one event happens given that another event is known to have happened, written P(A|B).

A measure of how spread out a set of data is around its mean. A larger value means the data values are more dispersed.

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

The arithmetic average of a set of values, found by adding all the values and dividing by how many there are.

The middle value of a data set once the values are arranged in order. If there is an even number of values, it is the average of the two middle ones.

The value that appears most often in a set of data. A data set may have no mode or more than one mode.

The difference between the upper quartile and the lower quartile of a data set. It measures the spread of the middle half of the data.

The set of all points that satisfy a given geometric condition, such as all points a fixed distance from a given point forming a circle.

The study of geometric figures using coordinates, including formulas for distance, slope, midpoint, and equations of straight lines and circles.

A measure of the steepness of a line, calculated as the change in y divided by the change in x between two points on it.

Two lines that meet at a right angle. In coordinate geometry their slopes multiply to give −1, provided neither line is vertical.