LC Higher Maths key terms

The value b² − 4ac for the quadratic ax² + bx + c. Its sign tells how many real roots exist: positive gives two, zero gives one repeated root, negative gives none.

The values of the variable that make the equation true. For a polynomial set equal to zero, they are the values where the graph crosses or touches the horizontal axis.

A statement that (x − a) is a factor of a polynomial exactly when substituting a gives zero. It links factors, roots and remainders in one test.

A short written method for dividing a polynomial by (x − a), working only with coefficients. The final number produced is the remainder, and the others give the quotient.

The amount left over after one polynomial is divided by another. Dividing by (x − a) always leaves the value the polynomial takes at x = a.

A number written a + bi, where a and b are real and i² = −1. Such numbers allow equations with no real solutions, like x² + 1 = 0, to be solved.

For a complex number a + bi, the conjugate is a − bi. Multiplying a complex number by its conjugate always gives a real, non-negative result.

The distance of a number from zero. For a + bi it equals the square root of a² + b², and for a real number it is simply its size without sign.

A plane used to picture complex numbers, with the real part measured horizontally and the imaginary part vertically. A number a + bi appears as the point (a, b).

A straight line that a curve approaches ever more closely as it runs away to infinity, without necessarily meeting it. Graphs of rational and exponential functions often have them.

A function formed by dividing one polynomial by another. Its graph may have breaks, vertical asymptotes where the bottom is zero, and horizontal asymptotes at extreme values.

The full set of input values for which the function is defined. For example, a square root needs a non-negative input and a fraction needs a non-zero bottom.

The set of all output values the function actually produces as the input runs through its domain. It is read from the vertical spread of the graph.

The function that undoes another, sending each output back to its input. It exists only when the original is one-to-one, and its graph is the reflection in the line y = x.

A function built by feeding the output of one function into another, written as one inside the other. Order matters: reversing the order usually changes the result.

A function whose values repeat at regular intervals of the input. Sine and cosine are the standard examples, repeating every 360 degrees or 2π radians.

Half the vertical distance between the highest and lowest points of a wave-like graph. For a sine or cosine curve it is the size of the number multiplying the function.

An angle unit based on arc length: one radian is the angle subtended when the arc equals the radius. A full turn is 2π radians, equal to 360 degrees.

A relationship stating that each side of a triangle divided by the sine of the opposite angle is the same for all three sides. Useful when a matching side and angle are known.

A generalisation of Pythagoras to any triangle, giving one side from the other two sides and the angle between them. It also finds an angle when all three sides are known.

A measure of how steeply a line rises or falls, found by dividing the vertical change by the horizontal change between two points. Parallel lines share the same value.

Two lines that meet at a right angle. Their slopes multiply to give −1, so one slope is the negative reciprocal of the other.

An algebraic rule describing every point on a straight line, often written using a point and the slope. It allows any point on the line to be checked or predicted.

A straight line touching a circle at exactly one point. It is perpendicular to the radius drawn to that point of contact.

Two outcomes that cannot both happen in the same trial. For such events the probability of either one occurring is found by adding their separate probabilities.

Two events where the outcome of one does not affect the chance of the other. Their combined probability is the product of the individual probabilities.

The chance of an event given that another event is already known to have happened. It is calculated by dividing the probability of both events by that of the known one.

A single experiment with exactly two possible outcomes, usually called success and failure, where the chance of success stays the same each time it is repeated.

The pattern of probabilities for counting successes across a fixed number of repeated two-outcome trials. It needs a known number of trials and a constant success probability.

The long-run average result of a random quantity, found by multiplying each possible value by its probability and adding these products together.

A measure of how spread out data values are around their mean. A small value means the values cluster tightly; a large one means they are widely scattered.

A symmetric bell-shaped pattern of continuous data centred on the mean. About 68 percent of values lie within one standard deviation of the mean and 95 percent within two.

A procedure for judging whether sample evidence supports a claim about a population. It compares the observed result with what would be expected if the claim were true.

A range of values, built from sample data, that is likely to contain the true population value. The chosen confidence level states how often such ranges succeed.

The instantaneous rate at which a function changes at a point, equal to the slope of the tangent there. It is found as the limit of average rates over shrinking intervals.

The value a function's output settles towards as the input approaches some chosen point, whether or not the function is defined there. It underpins both derivatives and asymptotes.

A straight line that touches a curve at a point and matches its direction there. Its slope equals the derivative of the function at that point.

A number obtained by accumulating a function's values between two limits. Geometrically it gives the signed area between the curve and the horizontal axis.