IB Math AI SL key terms

A way of writing numbers as a value between 1 and 10 multiplied by a power of ten, such as 3.2 × 10^4. It is used to handle very large or very small quantities compactly and to judge their order of magnitude.

The meaningful digits in a measured or calculated value, counted from the first non-zero digit. Rounding to a given number of significant figures controls the precision reported in an answer, especially when a question specifies a required accuracy.

The size of the difference between an approximate or measured value and an accepted value, expressed as a percentage of the accepted value. It quantifies how inaccurate a result is relative to the true quantity.

A list of numbers in which each term is obtained by adding the same constant to the previous term. The constant is the common difference, and the nth term grows linearly with n.

The fixed amount added to each term of an arithmetic sequence to get the next term. It equals the difference between any two consecutive terms and may be positive, negative, or zero.

The sum of the terms of an arithmetic sequence. Its total can be computed quickly from the first term, the number of terms, and the common difference, without adding every term individually.

A list of numbers in which each term is obtained by multiplying the previous term by the same fixed factor. This factor is the common ratio, and the terms grow or shrink exponentially rather than linearly.

The fixed multiplier that takes each term of a geometric sequence to the next term. It is found by dividing any term by the term before it and may be positive, negative, or fractional.

The sum of the terms of a geometric sequence. When the common ratio has absolute value below one, the sum of infinitely many terms approaches a finite limiting value.

The limiting total of an infinite geometric series, which exists only when the common ratio lies strictly between negative one and one. It is calculated as the first term divided by one minus the common ratio.

A compact way of writing a sum using the Greek capital letter sigma, with an index that runs from a lower to an upper limit. It is used to express series and to manipulate sums algebraically.

Interest calculated on both the original amount and on previously earned interest, so the balance grows by a fixed percentage each period. It is modelled by an exponential relationship rather than a linear one.

The fall in value of an asset over time, often modelled as a fixed percentage decrease each period. This produces a geometric decay pattern rather than a constant annual reduction.

The yearly percentage charged or earned, together with how often interest is added within the year. More frequent compounding at the same nominal yearly rate produces a slightly larger effective growth.

A rule that assigns exactly one output value to each permitted input value. The set of allowed inputs is the domain and the set of resulting outputs is the range.

The domain is the complete set of input values for which a function is defined; the range is the set of all output values it actually produces. Both are often described with inequalities or interval notation.

A new function formed by applying one function to the output of another, written as f of g of x. The order of composition matters, and the domain of the inner function restricts the result.

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

A relationship whose graph is a straight line, described by a constant rate of change. The slope gives that rate and the intercept gives the starting value when the input is zero.

The measure of steepness of a straight line, equal to the change in the vertical direction divided by the corresponding change in the horizontal direction. It represents the constant rate of change of a linear model.

A function whose rule involves the square of the variable, producing a parabolic graph. Its key features include the vertex, the axis of symmetry, and the points where it crosses the axes.

The turning point of a parabola, where the graph reaches its maximum or minimum value. Its coordinates are read from the completed-square or standard form of the quadratic.

The vertical line through the vertex of a parabola that divides the graph into two mirror-image halves. Its equation gives the input value at which the maximum or minimum occurs.

A relationship in which the quantity is multiplied by a fixed factor for each unit increase in the input. It describes growth or decay at a constant percentage rate, such as populations or radioactive substances.

A line that a curve approaches ever more closely as the input or output grows without bound, without necessarily touching it. Exponential graphs typically have a horizontal asymptote.

The power to which a base must be raised to produce a given number. Logarithms reverse exponential relationships and allow unknown exponents in growth or decay problems to be solved.

Rules that turn products into sums, quotients into differences, and powers into multipliers of logarithms. They are used to simplify expressions and to solve equations containing unknown exponents.

A relationship stating that each side of a triangle divided by the sine of its opposite angle is constant. It is used to find unknown sides or angles when a matching pair is already known.

A generalised version of Pythagoras' theorem that relates all three sides of a triangle to one angle. It is used when two sides and the included angle, or all three sides, are known.

The enclosed region of a triangle, calculated from two sides and the angle between them using half the product of those sides and the sine of the included angle. It avoids needing the perpendicular height.

A direction measured clockwise from due north, written as a three-digit angle. Bearings are used to describe navigation and position problems involving distances and angles.

The space occupied by a solid with a constant cross-section, found by multiplying the area of that cross-section by the length or height of the solid. It applies to many everyday three-dimensional shapes.

The space inside a shape that tapers to a point, equal to one third of the base area multiplied by the perpendicular height. This factor of one third distinguishes it from a prism of the same base and height.

The total area of all the outer faces of a three-dimensional solid. It is found by summing the areas of each face, including curved surfaces where relevant.

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

The middle value of a data set once the values are arranged in order; for an even count it is the average of the two central values. It is resistant to extreme values.

A measure of how spread out values are around the mean, roughly the typical distance of a value from the mean. A small value means the data cluster tightly; a large value means they are widely scattered.

A value that divides an ordered data set into quarters, with the lower and upper quartiles marking the boundaries of the middle half. Quartiles support the five-number summary and box plots.

The difference between the upper and lower quartiles, describing the spread of the middle half of the data. It ignores the extreme quarters and is therefore resistant to outliers.

A value that lies far away from the bulk of the data, commonly flagged when it falls more than one and a half interquartile ranges beyond a quartile. Outliers can distort summaries such as the mean.

A number between negative one and one describing how strongly two variables follow a straight-line pattern, together with its direction. It measures linear association only and does not prove causation.

The straight line that best represents the trend in a scatter diagram, often found by least squares. It can be used to predict one variable from another within the observed range of data.

A graph plotting one variable against another as individual points, used to reveal the form and strength of a relationship. Patterns such as linear, curved, or no association become visible.

A number between zero and one expressing how likely an event is, where zero means impossible and one means certain. It can come from equally likely outcomes, relative frequency, or subjective judgement.

The event that a given event does not happen. Its probability equals one minus the probability of the original event, which is useful when the opposite outcome is easier to calculate.

Two events that cannot both occur in the same trial. For such events the probability of either happening is the sum of their individual probabilities, with no overlap to subtract.

Two events where the occurrence of one gives no information about the other. For such events the probability that both happen is the product of their separate probabilities.

The probability that an event happens given that another event is known to have happened. It restricts attention to a subset of outcomes and is often read from two-way tables or tree diagrams.

A variable that can take only separate, countable values, each with an associated probability. Its behaviour is summarised by an expected value and a variance computed from the probability distribution.

The long-run average of a random variable, found by multiplying each possible value by its probability and adding the results. It represents the mean outcome over many repetitions.

The probability pattern for the number of successes in a fixed number of independent trials, each with the same two possible outcomes and the same success probability. It is specified by the trial count and the success probability.

A symmetric bell-shaped probability curve defined by its mean and standard deviation. Probabilities correspond to areas under the curve, and standardising a value allows comparison across different distributions.

The instantaneous rate at which a function changes with respect to its input, equal to the gradient of the tangent to its graph. It is used to describe velocity, marginal cost, and rates of change generally.

A straight line that touches a curve at a single point and matches the curve's direction there. Its gradient at that point equals the derivative of the function.

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.

Points where a function reaches a peak or a trough within a small neighbourhood. They are located among stationary points and classified by the sign change of the derivative or the second derivative.

The derivative of the derivative, describing how the rate of change itself is changing. Its sign indicates whether a stationary point is a maximum or a minimum.

The reverse process of differentiation, producing a family of functions that differ by a constant. It is used to recover a quantity from its known rate of change.

The integral evaluated between two limits, giving a single numerical value. It represents the accumulated change of a quantity or the signed area between a curve and the horizontal axis.

The region between a graph and the horizontal axis over an interval, computed with a definite integral. Parts below the axis count as negative, so separate integrals may be needed for total area.