UEE Mathematics key terms

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

A formula giving the solutions of ax² + bx + c = 0 as x = (−b ± √(b² − 4ac)) / 2a. It solves any quadratic whose coefficients are known.

Rewriting a quadratic expression as a perfect square plus a constant, for example x² + 6x + 1 = (x + 3)² − 8. It reveals the vertex and helps solve equations.

A list of numbers in which each term is obtained by adding a fixed constant to the previous one. The constant is called the common difference.

A list of numbers in which each term after the first is found by multiplying the previous term by a fixed nonzero constant, called the common ratio.

The distance of a number from zero on the number line, written |x|. It is never negative and equals x when x ≥ 0, and −x when x < 0.

A statement that one quantity is greater than, less than, or not equal to another, using symbols such as <, >, ≤, ≥. Solving it gives a range of values.

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

A number of the form a + bi, where a and b are real and i² = −1. The real part is a and the imaginary part is b.

For a complex number a + bi, the conjugate is a − bi. Multiplying a number by its conjugate gives the real value a² + b².

The distance of a complex number a + bi from the origin, equal to √(a² + b²). It is written |a + bi| and is always nonnegative.

A rule that assigns exactly one output value to each input in its domain. Different inputs may share an output, but one input cannot give two outputs.

The complete set of input values for which a function is defined. For example, the domain of 1/x excludes zero because division by zero is undefined.

The set of all output values a function actually produces as its input runs through the domain. It is a subset of the codomain.

A function that undoes another, so applying both returns the original input. It exists only when the original function is one-to-one on its domain.

A function formed by applying one function to the result of another, written f(g(x)). The output of the inner function becomes the input of the outer one.

A function in which distinct inputs always give distinct outputs, so no two domain values share an output. Such functions are also called injective.

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

A function of the form f(x) = a·bˣ where the base b is positive and not equal to one. It grows or decays at a rate proportional to its value.

A function written as one polynomial divided by another, such as (x + 1)/(x − 3). Its graph may have vertical asymptotes where the denominator is zero.

The highest or lowest point on the graph of a parabola. For y = a(x − h)² + k the vertex is at (h, k).

The U-shaped curve that is the graph of a quadratic function y = ax² + bx + c. It opens upward when a is positive and downward when a is negative.

A function defined by different formulas on different parts of its domain. For example, the absolute value function uses one rule for negatives and another for nonnegatives.

A function whose values repeat at regular intervals, so f(x + T) = f(x) for some fixed nonzero T. The smallest such T is the period.

Half the difference between the maximum and minimum values of a periodic function. For y = a sin x the amplitude is |a|.

The function that gives, for an angle, the ratio of the opposite side to the hypotenuse in a right triangle. Its graph is a wave with period 2π.

The function giving the ratio of the adjacent side to the hypotenuse for an angle in a right triangle. Its graph is a wave shifted from the sine curve by π/2.

The equation sin²θ + cos²θ = 1, true for every angle θ. It follows from the Pythagorean theorem applied to a unit circle.

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

The distance along a curve between two points on it. For a circle of radius r and central angle θ in radians, the arc length is rθ.

The region of a circle bounded by two radii and the arc between them. Its area is ½r²θ when the central angle θ is measured in radians.

Triangles with equal corresponding angles and proportional corresponding sides. Their matching side lengths share a single constant ratio.

Triangles that have exactly the same shape and size, so every corresponding side and angle matches. Standard tests include SSS, SAS, and ASA.

In any triangle, each side divided by the sine of its opposite angle equals the same value: a/sin A = b/sin B = c/sin C. It also equals twice the circumradius.

A relation for any triangle: c² = a² + b² − 2ab cos C. It generalises the Pythagorean theorem to triangles that are not right-angled.

The instantaneous rate at which a function changes with respect to its variable. Geometrically it is the slope of the tangent line at a point.

The value a function's output approaches as its input approaches a chosen point. It can exist even when the function is undefined at that point.

A property of a function whose graph has no breaks, jumps, or holes at a point. Formally, the limit there equals the function's value.

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

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

A quantity measuring accumulated change, often interpreted as the signed area under a curve. Definite integrals give numbers; indefinite ones give families of functions.

The result linking differentiation and integration as inverse processes. It states that the definite integral of a rate of change equals the net change in the original quantity.

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

The middle value of a data set when the values are arranged in order. With an even count it is the mean of the two central values.

A measure of how spread out values are around the mean, equal to the square root of the variance. A small value means values cluster near the mean.

The average of the squared differences between each value and the mean. It measures spread, and its square root is the standard deviation.

A value that divides ordered data into four groups of roughly equal size. The lower and upper quartiles flank the median.

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

The probability that an event occurs given that another event has already happened, written P(A|B). It equals P(A and B) divided by P(B).

Events whose outcomes do not affect each other's probabilities. For such events, the probability of both occurring is the product of their probabilities.

Events that cannot happen at the same time, so their joint probability is zero. For them, the probability of either occurring is the sum of the probabilities.

An arrangement of objects in which order matters. The number of permutations of n distinct objects taken all at once is n factorial.

A selection of objects from a group in which order does not matter. The count is n! divided by r!(n − r)! for choosing r from n.

The product of all positive integers up to a given positive integer n, written n!. For example, 4! = 4 × 3 × 2 × 1 = 24, and 0! is defined as 1.

A compact way to write a set of real numbers using brackets and parentheses. Square brackets include endpoints; round brackets exclude them.

A value of the variable that makes the equation true. Finding roots of f(x) = 0 is the same as finding where the graph crosses the horizontal axis.