Glossary of Terms

Absolute Value Equations

Equations involving \(|f(x)|\), solved by splitting into cases \(f(x) = k\) and \(f(x) = -k\), or by squaring both sides where appropriate.

Example: \(|x - 3| = 5\) gives \(x = 8\) or \(x = -2\).

Algebraic Manipulation

The use of equivalence-preserving operations — expanding, factoring, simplifying, combining — to rewrite an expression or equation in a more useful form.

Analytic vs Graphic Method

Two approaches to solving problems: the analytic method uses algebraic reasoning and exact manipulation, while the graphic method uses graphs (often with technology) to locate solutions visually.

Example: Solving \(x^3 = x + 1\) analytically requires a formula; graphically, one reads the intersection of \(y = x^3\) and \(y = x + 1\).

Asymptotic Behavior

The behavior of a function near its asymptotes or as \(x \to \pm\infty\). Describes how output values grow, decay, or level off in extreme regions of the domain.

Axis of Symmetry

A line about which a graph is a mirror image of itself. For the parabola \(y = ax^2 + bx + c\), the axis of symmetry is \(x = -\tfrac{b}{2a}\).

Example: For \(y = x^2 - 6x + 5\), the axis of symmetry is \(x = 3\).

Base of an Exponential

The positive constant \(b\) in \(f(x) = a \cdot b^x\). The base determines whether the function grows (\(b > 1\)) or decays (\(0 < b < 1\)).

Boundedness

A function is bounded on an interval if there exist constants \(m\) and \(M\) with \(m \leq f(x) \leq M\) for all \(x\) in the interval. Otherwise it is unbounded.

Example: \(\sin x\) is bounded on \(\mathbb{R}\) since \(-1 \leq \sin x \leq 1\).

Cartesian Coordinates

A system of locating points in the plane using an ordered pair \((x, y)\) of signed distances from two perpendicular axes. Named after René Descartes.

Example: The point \((-3, 4)\) lies three units left and four units above the origin.

Change of Base Rule

The identity \(\log_b x = \frac{\log_a x}{\log_a b}\), which converts a logarithm from base \(b\) to any other positive base \(a \neq 1\). Useful for calculator evaluation.

Coefficient

A numerical or symbolic factor multiplying a variable or power of a variable in an expression. In \(3x^2\), the coefficient is \(3\).

Example: In \(P(x) = 5x^3 - 2x + 7\), the coefficients are \(5\), \(0\), \(-2\), \(7\).

Collecting Like Terms

Combining terms that have identical variable parts by adding their coefficients, in order to simplify an expression.

Example: \(3x + 5 - x + 2 = 2x + 7\).

Common Factor

A factor shared by every term in an expression, which can be taken outside brackets: \(ab + ac = a(b + c)\).

Example: \(6x^2 + 9x = 3x(2x + 3)\).

Common Logarithm

The logarithm to base \(10\), denoted \(\log x\) (or \(\log_{10} x\)). Historically used for numerical computation and in scientific scales.

Example: \(\log 1000 = 3\).

Completing the Square

An algebraic procedure that rewrites \(ax^2 + bx + c\) in vertex form \(a(x - h)^2 + k\) by creating a perfect square trinomial. It reveals the vertex and enables solving.

Example: \(x^2 + 6x + 5 = (x + 3)^2 - 4\).

Composite Function

A function formed by applying one function to the result of another: \((f \circ g)(x) = f(g(x))\), read "\(f\) of \(g\) of \(x\)".

Example: If \(f(x) = x + 1\) and \(g(x) = x^2\), then \((f \circ g)(x) = x^2 + 1\).

Composite Transformation

A transformation obtained by applying two or more elementary transformations in sequence. The order of application generally affects the result.

Example: \(y = 2(x - 1)^2 + 3\) combines a horizontal translation, vertical stretch, and vertical translation.

Compound Interest Model

A model for investment growth: \(A = P\left(1 + \tfrac{r}{n}\right)^{nt}\), where \(P\) is principal, \(r\) the annual rate, \(n\) compounding periods per year, and \(t\) time in years.

Concavity

A qualitative description of how a graph curves. A curve is concave up on an interval if it bends upward (holds water) and concave down if it bends downward.

Example: \(y = x^2\) is concave up everywhere.

Constant

A value that does not change within a given problem or formula. Constants may be numerical (\(3\), \(\pi\)) or symbolic placeholders for fixed numbers.

Constant Function

A function of the form \(f(x) = c\) for some constant \(c\). Its graph is a horizontal line, and its range contains the single value \(c\).

Example: \(f(x) = 7\) gives the same output \(7\) for every input.

Continuity

A function \(f\) is continuous at \(x = a\) if \(f(a)\) is defined, the limit as \(x \to a\) exists, and the limit equals \(f(a)\). Intuitively, the graph has no breaks there.

Continuous Function

Informally, a function whose graph can be drawn without lifting the pen: small changes in \(x\) produce small changes in \(f(x)\), with no jumps or holes on its domain.

Example: \(f(x) = x^2\) is continuous on \(\mathbb{R}\).

Coordinate Plane

A two-dimensional plane formed by two perpendicular number lines, the \(x\)-axis and \(y\)-axis, intersecting at the origin. Every point is specified by an ordered pair \((x, y)\).

Cubic Function

A polynomial function of degree three: \(f(x) = ax^3 + bx^2 + cx + d\) with \(a \neq 0\). Its graph has at most two turning points and one inflection point.

Example: \(f(x) = x^3 - 3x\).

Curve Fitting

The process of selecting a family of functions and determining parameters so that the resulting curve passes near, or through, a set of given data points.

Decomposing Functions

Writing a given function \(h\) as a composition \(f \circ g\) of simpler functions. Useful for differentiation and understanding structure.

Example: \(h(x) = (2x + 1)^3\) decomposes as \(f(u) = u^3\) and \(g(x) = 2x + 1\).

Decreasing Function

A function \(f\) is decreasing on an interval if \(x_1 < x_2\) implies \(f(x_1) > f(x_2)\) for all \(x_1, x_2\) in the interval. The graph falls from left to right.

Example: \(f(x) = -x + 4\) is decreasing on \(\mathbb{R}\).

Degree of a Polynomial

The highest power of \(x\) that appears with a non-zero coefficient in a polynomial. The degree governs the shape, end behavior, and maximum number of roots.

Example: \(P(x) = x^4 - 3x + 7\) has degree \(4\).

Dependent Variable

The variable whose value is determined by the value of another variable through a functional relationship. Usually plotted on the vertical axis.

Example: In \(d = 5t\), distance \(d\) depends on time \(t\), so \(d\) is the dependent variable.

Difference of Squares

The factorization identity \(a^2 - b^2 = (a - b)(a + b)\), valid for any expressions \(a\) and \(b\).

Example: \(x^2 - 25 = (x - 5)(x + 5)\).

Direct Proportion

A relationship \(y = kx\) where \(y\) is a constant multiple of \(x\). Its graph is a line through the origin with gradient \(k\), the constant of proportionality.

Example: If \(y\) is directly proportional to \(x\) and \(y = 12\) when \(x = 3\), then \(k = 4\).

Discrete Function

A function whose domain consists of isolated values, typically integers, rather than a continuous interval. Its graph is a set of separate points.

Example: The number of students in a class as a function of the day is discrete.

Discriminant

The quantity \(\Delta = b^2 - 4ac\) associated with the quadratic \(ax^2 + bx + c = 0\). Its sign determines the nature and number of real roots.

Domain

The set of all permitted input values of a function. For \(f(x)\), it is every \(x\) for which \(f(x)\) is defined.

Example: The domain of \(f(x) = \sqrt{x}\) is \(\{x \in \mathbb{R} : x \geq 0\}\).

Domain of Composite

The set of \(x\) in the domain of \(g\) such that \(g(x)\) lies in the domain of \(f\). It may be smaller than the domain of \(g\) alone.

Example: For \(f(x) = \sqrt{x}\) and \(g(x) = x - 4\), the domain of \(f \circ g\) is \(x \geq 4\).

Domain Restriction Inverse

Limiting the domain of a non-one-to-one function to an interval on which it becomes one-to-one so that an inverse can be defined.

Example: Restricting \(f(x) = x^2\) to \(x \geq 0\) allows \(f^{-1}(x) = \sqrt{x}\).

Domain Restrictions

Explicit limits placed on the domain of a function to exclude values that make the formula undefined, such as zero denominators or negatives under even roots.

Example: The domain of \(f(x) = \tfrac{1}{x - 3}\) is restricted to \(x \neq 3\).

Effect on Coordinates

Each transformation maps a point \((x, y)\) on \(f\) to a specific image point on the transformed graph.

Example: Under \(y = 2 f(x - 1) + 3\), the point \((x, y)\) maps to \((x + 1, 2y + 3)\).

Elimination Method

A technique for solving simultaneous equations: add or subtract scaled copies of the equations to eliminate one variable, then solve for the remaining one.

Example: Adding \(x + y = 5\) and \(x - y = 1\) eliminates \(y\), giving \(2x = 6\).

End Behavior

The behavior of \(f(x)\) as \(x \to +\infty\) and \(x \to -\infty\). For polynomials, it is determined by the leading term \(a_n x^n\).

Example: For \(f(x) = -x^3\), \(f(x) \to -\infty\) as \(x \to \infty\) and \(f(x) \to \infty\) as \(x \to -\infty\).

Equal Real Roots

The case \(\Delta = b^2 - 4ac = 0\), where a quadratic equation has exactly one (repeated) real root and its graph touches the \(x\)-axis at a single point.

Example: \(x^2 - 4x + 4 = 0\) has \(\Delta = 0\) and root \(x = 2\).

Equation

A mathematical statement asserting that two expressions are equal, containing an \(=\) sign. Solving an equation means finding values of the variable(s) that make it true.

Example: \(2x + 1 = 7\) is an equation with solution \(x = 3\).

Euler's Number

The irrational constant \(e \approx 2.71828\ldots\), defined as \(\lim_{n \to \infty} \left(1 + \tfrac{1}{n}\right)^n\). It is the natural base for exponentials and logarithms.

Even Function

A function satisfying \(f(-x) = f(x)\) for all \(x\) in its domain. Its graph is symmetric about the \(y\)-axis.

Example: \(f(x) = x^2\) and \(f(x) = \cos x\) are even.

Exact vs Approximate

An exact answer retains symbols such as \(\pi\), \(\sqrt{2}\), or fractions; an approximate answer is a decimal rounded to a stated precision. IB distinguishes these carefully.

Example: Exact: \(\tfrac{\pi}{3}\); approximate: \(1.05\).

Expanding Brackets

Applying the distributive law to remove parentheses: \(a(b + c) = ab + ac\), and \((a + b)(c + d) = ac + ad + bc + bd\).

Example: \((x + 3)(x - 2) = x^2 + x - 6\).

Exponential Decay

Change modeled by \(f(x) = a \cdot b^x\) with \(0 < b < 1\), where the quantity shrinks by the same factor over equal intervals, approaching zero.

Example: A drug concentration halving every \(6\) hours.

Exponential Equations

Equations in which the unknown appears in an exponent, such as \(2^x = 16\) or \(e^{2x} = 5\). They are typically solved using logarithms.

Example: \(3^x = 81\) gives \(x = 4\).

Exponential Function

A function of the form \(f(x) = a \cdot b^x\) with \(a \neq 0\), \(b > 0\), \(b \neq 1\). The variable appears in the exponent, giving characteristic rapid growth or decay.

Example: \(f(x) = 3 \cdot 2^x\).

Exponential Growth

Change modeled by \(f(x) = a \cdot b^x\) with \(b > 1\), in which the quantity multiplies by the same factor over equal intervals. It accelerates as \(x\) increases.

Example: A bacterial population doubling each hour follows \(P(t) = P_0 \cdot 2^t\).

Expression

A combination of numbers, variables, and operations that represents a value but contains no equality or inequality sign.

Example: \(3x^2 - 5x + 2\) is an expression.

Extrapolation

Using a fitted model to estimate values of the dependent variable outside the range of the observed data. Extrapolation is less reliable than interpolation.

Example: Predicting population in year \(2050\) from data collected through \(2025\).

Factor Theorem

For a polynomial \(P(x)\), \((x - a)\) is a factor of \(P(x)\) if and only if \(P(a) = 0\). It links factorization and roots directly.

Example: Since \(P(2) = 0\) for \(P(x) = x^2 - 4\), \((x - 2)\) is a factor.

Factored Form Quadratic

The quadratic written as \(f(x) = a(x - p)(x - q)\), where \(p\) and \(q\) are the roots. This form reveals the \(x\)-intercepts directly.

Example: \(f(x) = 2(x - 1)(x + 3)\) has roots \(1\) and \(-3\).

Factoring Quadratics

Rewriting a quadratic expression as a product of linear factors, typically \(a(x - p)(x - q)\), in order to read off roots or simplify.

Example: \(x^2 - 5x + 6 = (x - 2)(x - 3)\).

Finding an Inverse

The algebraic process: write \(y = f(x)\), swap \(x\) and \(y\), then solve the new equation for \(y\) to obtain \(y = f^{-1}(x)\).

Example: For \(y = 3x - 1\), swap to get \(x = 3y - 1\), then \(y = \tfrac{x + 1}{3}\).

Function

A rule that assigns to each element \(x\) in a set called the domain exactly one element \(y\) in a set called the codomain. Written \(f: x \mapsto f(x)\) or \(y = f(x)\).

Example: \(f(x) = x^2\) assigns to each real number its square, so \(f(3) = 9\).

Function Behavior Analysis

A systematic description of a function in terms of domain, range, intercepts, symmetry, intervals of increase/decrease, turning points, asymptotes, and end behavior.

Function Composition Rule

The rule defining composition: \((f \circ g)(x) = f(g(x))\), where the output of \(g\) becomes the input of \(f\). The domain of \(f \circ g\) requires \(g(x)\) to lie in the domain of \(f\).

Function Intercepts

The points where the graph of \(f\) meets the coordinate axes: the \(y\)-intercept \((0, f(0))\) and any \(x\)-intercepts where \(f(x) = 0\).

Function Modeling

Representing a real-world situation by choosing a function whose behavior captures the key features of the phenomenon, enabling prediction and analysis.

Function Notation

The symbolic convention \(f(x)\) used to represent the output of a function \(f\) at input \(x\). It makes the function's name and argument explicit.

Example: If \(f(x) = x^2 + 1\), then \(f(4) = 17\) uses function notation to evaluate the function at \(x = 4\).

Function Representations

The different equivalent ways to describe a function: verbal description, algebraic formula, table of values, graph, or mapping diagram. Each representation highlights different features.

GDC Usage

The use of a graphic display calculator to plot functions, find intersections, roots, extrema, numerical derivatives, and integrals, as permitted on IB examinations.

General Form of a Line

The equation of a straight line written as \(ax + by + d = 0\), where \(a\), \(b\), \(d\) are constants and \(a, b\) not both zero. Useful when neither coordinate is isolated.

Example: \(2x - 3y + 6 = 0\) is in general form.

Gradient Calculation

The gradient between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1 \neq x_2\).

Example: Between \((1, 2)\) and \((4, 11)\): \(m = \tfrac{11 - 2}{4 - 1} = 3\).

Gradient-Intercept Form

The equation of a straight line written as \(y = mx + c\), where \(m\) is the gradient and \(c\) is the \(y\)-intercept. It makes both features immediately visible.

Example: \(y = -\tfrac{1}{2}x + 3\) has gradient \(-\tfrac{1}{2}\) and \(y\)-intercept \(3\).

Graph of a Function

The set of all points \((x, f(x))\) in the coordinate plane where \(x\) belongs to the domain of \(f\). It is a visual representation of the function's input-output pairs.

Example: The graph of \(f(x) = x^2\) is a parabola opening upward with vertex at the origin.

Graph of Modulus

The graph of \(y = |x|\) is a V-shape with vertex at the origin, consisting of the line \(y = x\) for \(x \geq 0\) and \(y = -x\) for \(x < 0\).

Graph Transformation Order

The conventional order of applying composite transformations: horizontal stretches/reflections, then horizontal translations, then vertical stretches/reflections, then vertical translations.

Graphical Solution Method

Solving an equation or system by plotting each side (or each equation) and reading off the coordinates of intersection points. Especially useful with graphing technology.

Graphing Exponentials

Sketching \(y = a \cdot b^x\) by identifying the \(y\)-intercept \(a\), the horizontal asymptote \(y = 0\), and whether the function grows or decays.

Graphing Logarithms

Sketching \(y = \log_b x\) by identifying the \(x\)-intercept at \((1, 0)\), the vertical asymptote \(x = 0\), and the domain \(x > 0\).

Graphing Polynomials

Sketching a polynomial by combining degree, leading coefficient, end behavior, roots with multiplicities, and \(y\)-intercept to determine overall shape.

Graphing Rational Functions

The process of sketching \(f(x) = \tfrac{P(x)}{Q(x)}\) by identifying intercepts, vertical and horizontal (or oblique) asymptotes, holes, and behavior on each interval.

Graphing Technology

Hardware or software, such as a graphic display calculator or computer algebra system, used to plot functions and investigate their features.

Greatest Integer Function

The function \(\lfloor x \rfloor\) returning the greatest integer less than or equal to \(x\), also called the floor function. Its graph is a right-continuous step function.

Example: \(\lfloor 2.7 \rfloor = 2\) and \(\lfloor -1.3 \rfloor = -2\).

Half-Life Model

An exponential decay model \(N(t) = N_0 \left(\tfrac{1}{2}\right)^{t/T}\), where \(T\) is the half-life — the time for the quantity to reduce to one half of its initial value.

Example: Carbon-14 has a half-life of about \(5730\) years.

Holes in Rational Graphs

A removable discontinuity at \(x = a\) where the factor \((x - a)\) cancels in both numerator and denominator. The graph has a missing point rather than an asymptote.

Example: \(f(x) = \tfrac{(x - 1)(x + 2)}{x - 1}\) has a hole at \(x = 1\).

Horizontal Asymptote

A horizontal line \(y = L\) that the graph of \(f\) approaches as \(x \to \pm\infty\), meaning \(f(x) \to L\) for large \(|x|\).

Example: \(f(x) = \tfrac{1}{x}\) has horizontal asymptote \(y = 0\).

Horizontal Line Test

A graphical test: \(f\) is one-to-one (and therefore has an inverse) if and only if no horizontal line meets its graph at more than one point.

Horizontal Stretch

The transformation \(y = f\left(\tfrac{x}{b}\right)\) with \(b > 0\): every \(x\)-coordinate is multiplied by \(b\). Values \(b > 1\) stretch; \(0 < b < 1\) compress.

Example: \(y = \sin\left(\tfrac{x}{2}\right)\) is a horizontal stretch of \(\sin x\) by factor \(2\).

Horizontal Translation

The transformation \(y = f(x - h)\), which shifts the graph of \(f\) by \(h\) units to the right (or \(|h|\) units left if \(h < 0\)).

Example: \(y = (x - 3)^2\) is \(y = x^2\) shifted \(3\) right.

Hyperbola

The curve that is the graph of a reciprocal or similar rational function, consisting of two disconnected branches approaching asymptotes.

Example: \(y = \tfrac{1}{x}\) has a hyperbola with branches in the first and third quadrants.

Identity Function

The function \(f(x) = x\), which sends every input to itself. Its graph is the line \(y = x\), and it acts as the neutral element for function composition.

Increasing Function

A function \(f\) is increasing on an interval if \(x_1 < x_2\) implies \(f(x_1) < f(x_2)\) for all \(x_1, x_2\) in the interval. The graph rises from left to right.

Example: \(f(x) = 2x + 1\) is increasing on \(\mathbb{R}\).

Independent Variable

The variable whose value is chosen freely and which determines the value of the dependent variable. Usually plotted on the horizontal axis.

Example: In \(d = 5t\), time \(t\) is the independent variable.

Inequality

A mathematical statement comparing two expressions with one of \(<\), \(\leq\), \(>\), \(\geq\), or \(\neq\). Its solution is typically a set rather than isolated values.

Example: \(x + 2 > 5\) has solution \(x > 3\).

Inflection Point

A point on a graph where the concavity changes from concave up to concave down or vice versa. The tangent line at such a point crosses the curve.

Example: \(y = x^3\) has an inflection point at the origin.

Input Variable

The variable representing an element chosen from the domain of a function, conventionally denoted \(x\). Its value is selected first and determines the corresponding output.

Example: In \(f(x) = 2x + 1\), the input variable is \(x\).

Integers

The set \(\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}\) of whole numbers together with their negatives. It is closed under addition, subtraction, and multiplication.

Interpolation

Using a fitted model or known values to estimate the dependent variable at an input lying within the observed data range. Usually more reliable than extrapolation.

Intersection of Curves

The points \((x, y)\) lying on two curves simultaneously, found by solving \(f(x) = g(x)\) for \(x\) and then computing \(y\).

Example: \(y = x^2\) and \(y = x + 2\) intersect where \(x^2 = x + 2\), giving \(x = 2\) or \(x = -1\).

Intersection of Lines

The point (or points) lying on two or more lines simultaneously, found by solving their equations as simultaneous equations. Two distinct lines either intersect once, are parallel, or coincide.

Interval Notation

A compact way of describing a set of real numbers using brackets: \([a, b]\) includes endpoints, \((a, b)\) excludes them, and \(\infty\) is always open.

Example: \([0, 5)\) means all real \(x\) with \(0 \leq x < 5\).

Inverse Function

For a one-to-one function \(f\), the function \(f^{-1}\) satisfying \(f^{-1}(f(x)) = x\) for all \(x\) in the domain of \(f\), and \(f(f^{-1}(y)) = y\) for all \(y\) in the range.

Example: If \(f(x) = 2x + 3\), then \(f^{-1}(x) = \tfrac{x - 3}{2}\).

Inverse Function Notation

The symbol \(f^{-1}\) for the inverse of \(f\). Note: \(f^{-1}(x) \neq \tfrac{1}{f(x)}\); the \(-1\) denotes inversion, not a reciprocal power.

Irrational Numbers

Real numbers that cannot be written as \(\tfrac{p}{q}\) with \(p, q\) integers. Their decimal expansions are non-terminating and non-repeating.

Example: \(\sqrt{2}\), \(\pi\), and \(e\) are irrational.

Key Graph Features

The qualitative landmarks of a graph used to describe its behavior: intercepts, turning points, asymptotes, symmetry, end behavior, and intervals of increase or decrease.

Leading Coefficient

The coefficient \(a_n\) of the highest-degree term in a polynomial. Its sign and magnitude control end behavior and vertical scaling.

Example: The leading coefficient of \(P(x) = -2x^3 + x\) is \(-2\).

Linear Equation Solving

The process of finding the value(s) of the unknown in an equation of degree one, using inverse operations to isolate the variable.

Example: Solving \(3x - 7 = 2\) gives \(x = 3\).

Linear Function

A function of the form \(f(x) = mx + c\), where \(m\) and \(c\) are constants. Its graph is a straight line with gradient \(m\) and \(y\)-intercept \(c\).

Example: \(f(x) = 3x - 2\) is linear with gradient \(3\).

Log-Exponential Inverse

The relationship \(\log_b\) and \(b^x\) are inverses: \(\log_b(b^x) = x\) for all real \(x\), and \(b^{\log_b x} = x\) for all \(x > 0\).

Logarithm Laws

The identities \(\log_b(xy) = \log_b x + \log_b y\), \(\log_b\left(\tfrac{x}{y}\right) = \log_b x - \log_b y\), and \(\log_b(x^n) = n\log_b x\), valid for \(x, y > 0\).

Logarithmic Function

The inverse of an exponential function: \(f(x) = \log_b x\), defined for \(x > 0\), satisfying \(\log_b x = y \iff b^y = x\).

Example: \(\log_2 8 = 3\) because \(2^3 = 8\).

Logarithmic Scale

A scale where equal distances represent equal ratios rather than equal differences. Used in decibels, pH, and the Richter scale to handle quantities spanning many orders of magnitude.

Mapping Diagram

A visual representation of a function that draws arrows from each element of the domain to its corresponding element in the range. Useful for small, finite sets.

Example: Two ovals labeled \(X\) and \(Y\) with arrows from \(1 \to 2\), \(2 \to 4\), \(3 \to 6\) shows \(f(x) = 2x\).

Mathematical Modeling Cycle

The iterative process: identify a problem, make assumptions, choose variables and a model, solve, interpret results, validate against data, then refine.

Maximum Point

A point \((a, f(a))\) where \(f(a) \geq f(x)\) for all \(x\) in an interval (local max) or in the entire domain (global max).

Example: The maximum point of \(y = -(x - 1)^2 + 5\) is \((1, 5)\).

Minimum Point

A point \((a, f(a))\) where \(f(a) \leq f(x)\) for all \(x\) in an interval (local min) or in the entire domain (global min).

Example: The minimum point of \(y = x^2 + 2x + 3\) is \((-1, 2)\).

Modulus Function

The function \(f(x) = |x|\), defined by \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\). It returns the non-negative distance from zero.

Multiplicity of Roots

The number of times a factor \((x - a)\) appears in the factorization of a polynomial. A root of even multiplicity touches the \(x\)-axis; odd multiplicity crosses it.

Example: In \((x - 2)^3(x + 1)\), root \(2\) has multiplicity \(3\).

Natural Exponential

The exponential function \(f(x) = e^x\), whose base is Euler's number \(e \approx 2.71828\). It is the unique exponential whose gradient at \(x = 0\) equals \(1\).

Natural Logarithm

The logarithm to base \(e\), denoted \(\ln x\). It is the inverse of the natural exponential \(e^x\).

Example: \(\ln e^5 = 5\).

Nature of Roots

A classification of the solutions of a quadratic equation as two distinct real, one repeated real, or no real (complex conjugate) roots, based on the sign of the discriminant.

Negative Gradient

A gradient \(m < 0\), indicating that \(y\) decreases as \(x\) increases. The line falls from left to right.

No Real Roots

The case \(\Delta = b^2 - 4ac < 0\), where a quadratic equation has no real solutions and its graph does not meet the \(x\)-axis.

Example: \(x^2 + x + 1 = 0\) has \(\Delta = -3 < 0\).

Number Systems

A hierarchy of sets of numbers: natural numbers \(\mathbb{N}\), integers \(\mathbb{Z}\), rational numbers \(\mathbb{Q}\), real numbers \(\mathbb{R}\), and complex numbers \(\mathbb{C}\), each extending the previous.

Oblique Asymptote

A slanted line \(y = mx + c\) (with \(m \neq 0\)) that the graph of a rational function approaches as \(x \to \pm\infty\). It occurs when \(\deg P = \deg Q + 1\).

Example: \(f(x) = \tfrac{x^2 + 1}{x}\) has oblique asymptote \(y = x\).

Odd Function

A function satisfying \(f(-x) = -f(x)\) for all \(x\) in its domain. Its graph has rotational symmetry of order two about the origin.

Example: \(f(x) = x^3\) and \(f(x) = \sin x\) are odd.

One-to-One Function

A function \(f\) such that different inputs always give different outputs: \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\). Only one-to-one functions have inverses.

Example: \(f(x) = 2x + 1\) is one-to-one; \(f(x) = x^2\) on \(\mathbb{R}\) is not.

Order of Composition

Function composition is generally not commutative: \((f \circ g)(x) \neq (g \circ f)(x)\) in general, so order matters.

Example: If \(f(x) = x + 1\) and \(g(x) = 2x\), then \(f(g(x)) = 2x + 1\) but \(g(f(x)) = 2x + 2\).

Ordered Pair

A pair of elements \((a, b)\) in which the order matters, used to represent a point in the coordinate plane or an element of a relation. \((a, b) \neq (b, a)\) unless \(a = b\).

Example: \((3, -2)\) locates a point three units right and two units below the origin.

Origin

The point \((0, 0)\) where the \(x\)-axis and \(y\)-axis intersect in the Cartesian coordinate plane. It serves as the reference point for all coordinates.

Output Variable

The variable representing the value produced by a function for a given input, conventionally denoted \(y\) or \(f(x)\). Its value depends on the input.

Example: In \(y = 3x - 5\), \(y\) is the output variable.

Parabola

The U-shaped graph of a quadratic function. It opens upward if \(a > 0\) and downward if \(a < 0\), and is symmetric about its axis of symmetry.

Parallel Lines

Two distinct lines in the plane that never intersect. Non-vertical parallel lines have equal gradients: \(m_1 = m_2\).

Example: \(y = 2x + 1\) and \(y = 2x - 4\) are parallel.

Parameter

A quantity in a formula that is held fixed in a particular case but can be varied to generate a family of related functions or curves.

Example: In \(y = mx + c\), \(m\) and \(c\) are parameters when studying the family of all lines.

Perfect Square Trinomial

A trinomial that factors as a squared binomial: \(a^2 + 2ab + b^2 = (a + b)^2\) or \(a^2 - 2ab + b^2 = (a - b)^2\).

Example: \(x^2 + 6x + 9 = (x + 3)^2\).

Periodic Function

A function \(f\) for which there exists a positive constant \(T\) such that \(f(x + T) = f(x)\) for all \(x\). The smallest such \(T\) is the period.

Example: \(\sin x\) has period \(2\pi\).

Perpendicular Lines

Two lines that meet at a right angle. Non-vertical perpendicular lines satisfy \(m_1 \cdot m_2 = -1\).

Example: \(y = 3x\) and \(y = -\tfrac{1}{3}x + 2\) are perpendicular.

Piecewise Function

A function defined by different formulas on different parts of its domain, typically using a brace notation listing each formula with its interval.

Example: \(f(x) = \begin{cases} x + 1, & x < 0 \\ x^2, & x \geq 0 \end{cases}\)

Point-Gradient Form

The equation of a line through point \((x_1, y_1)\) with gradient \(m\), written \(y - y_1 = m(x - x_1)\). Convenient when a point and gradient are given directly.

Example: A line through \((2, 5)\) with gradient \(3\): \(y - 5 = 3(x - 2)\).

Polynomial Division

The process of dividing one polynomial by another of lower or equal degree, producing a quotient and remainder: \(P(x) = D(x) Q(x) + R(x)\) with \(\deg R < \deg D\).

Polynomial Function

A function of the form \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where the coefficients \(a_i\) are real constants and \(n\) is a non-negative integer.

Example: \(P(x) = 2x^3 - 5x + 1\).

Population Growth Model

An exponential model \(P(t) = P_0 e^{kt}\) (or \(P_0 b^t\)), where \(P_0\) is initial population and \(k > 0\) the growth rate, describing unrestricted growth.

Positive Gradient

A gradient \(m > 0\), indicating that \(y\) increases as \(x\) increases. The line rises from left to right.

Product of Roots Formula

For a polynomial \(a_n x^n + \cdots + a_0\) with roots \(r_1, \ldots, r_n\), \(\prod r_i = (-1)^n \frac{a_0}{a_n}\).

Example: For \(x^2 - 5x + 6\), the product of roots is \(6\).

Projectile Motion Model

A quadratic model of the height of an object under gravity, \(h(t) = -\tfrac{1}{2}gt^2 + v_0 t + h_0\), where \(g\) is gravitational acceleration, \(v_0\) initial velocity, \(h_0\) initial height.

Example: \(h(t) = -5t^2 + 20t + 1\) models a ball thrown upward.

Quadratic Applications

Real-world problems modeled by quadratic functions, including projectile motion, area optimization, and profit/revenue analysis.

Quadratic Formula

The formula giving the roots of \(ax^2 + bx + c = 0\): \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It solves any quadratic equation, real or complex.

Quadratic Function

A polynomial function of degree two: \(f(x) = ax^2 + bx + c\) with \(a \neq 0\). Its graph is a parabola.

Example: \(f(x) = 2x^2 - 3x + 1\) is quadratic.

Quadratic Inequalities

An inequality such as \(ax^2 + bx + c > 0\). Its solution set is determined from the roots and the direction of opening of the parabola.

Example: \(x^2 - x - 6 > 0\) has solution \(x < -2\) or \(x > 3\).

Quartic Function

A polynomial function of degree four: \(f(x) = ax^4 + bx^3 + cx^2 + dx + e\) with \(a \neq 0\). It can have up to three turning points.

Example: \(f(x) = x^4 - 5x^2 + 4\).

Range

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

Example: The range of \(f(x) = x^2\) with domain \(\mathbb{R}\) is \(\{y \in \mathbb{R} : y \geq 0\}\).

Rate of Change

The ratio describing how one quantity changes relative to another. For a linear function, this is the gradient and is constant.

Example: If a car travels at \(60\) km/h, the rate of change of distance with respect to time is \(60\).

Rational Form Ax Plus B

A rational function of the shape \(f(x) = \tfrac{ax + b}{cx + d}\) with \(c \neq 0\). Its graph is a hyperbola with one vertical and one horizontal asymptote.

Example: \(f(x) = \tfrac{2x + 1}{x - 3}\) has asymptotes \(x = 3\) and \(y = 2\).

Rational Function

A function of the form \(f(x) = \tfrac{P(x)}{Q(x)}\), where \(P\) and \(Q\) are polynomials and \(Q(x) \neq 0\). Its domain excludes the zeros of \(Q\).

Example: \(f(x) = \tfrac{x + 1}{x^2 - 4}\) is rational with domain \(x \neq \pm 2\).

Rational Numbers

The set \(\mathbb{Q}\) of numbers expressible as \(\tfrac{p}{q}\) with \(p, q \in \mathbb{Z}\) and \(q \neq 0\). Their decimal expansions terminate or repeat.

Example: \(\tfrac{3}{4} = 0.75\) and \(\tfrac{1}{3} = 0.\overline{3}\).

Real Number Line

A straight line on which every point corresponds to exactly one real number, providing a geometric model of \(\mathbb{R}\). Positive numbers lie to the right of \(0\), negatives to the left.

Real Numbers

The set \(\mathbb{R}\) consisting of all rational and irrational numbers, corresponding to every point on the real number line.

Real-World Linear Models

Applications in which a linear function \(y = mx + c\) represents a quantity changing at a constant rate, such as cost, distance-time, or temperature conversion.

Example: Taxi fare \(C = 2.5d + 4\) models cost in terms of distance.

Reciprocal Function

The function \(f(x) = \tfrac{1}{x}\), defined for \(x \neq 0\). It is its own inverse, and its graph is a hyperbola with asymptotes \(x = 0\) and \(y = 0\).

Reflection in X-Axis

The transformation \(y = -f(x)\), which reflects the graph across the \(x\)-axis by negating every \(y\)-coordinate.

Example: \(y = -x^2\) is \(y = x^2\) reflected in the \(x\)-axis.

Reflection in Y-Axis

The transformation \(y = f(-x)\), which reflects the graph across the \(y\)-axis by negating every \(x\)-coordinate.

Example: \(y = \sqrt{-x}\) is \(y = \sqrt{x}\) reflected in the \(y\)-axis.

Reflection Over Y Equals X

A geometric transformation that maps each point \((a, b)\) to \((b, a)\). The graph of \(f^{-1}\) is the reflection of the graph of \(f\) in the line \(y = x\).

Regression Model

A function fitted to a set of data points by minimizing an error measure (commonly least squares), producing parameter estimates and a best-fit curve.

Example: Linear regression fits \(y = mx + c\) to bivariate data.

Relation

A set of ordered pairs \((x, y)\) that connects elements of one set to elements of another. Every function is a relation, but not every relation is a function.

Example: \(\{(1, 2), (1, 3), (2, 4)\}\) is a relation but not a function because \(1\) maps to both \(2\) and \(3\).

Remainder Theorem

When a polynomial \(P(x)\) is divided by \((x - a)\), the remainder equals \(P(a)\). It provides a quick evaluation method.

Example: Dividing \(P(x) = x^3 + 1\) by \((x - 2)\) gives remainder \(P(2) = 9\).

Revenue Optimization

Using a quadratic revenue function \(R(x) = x \cdot p(x)\), where \(p(x)\) is a linear price function, to find the selling quantity that maximizes revenue at the vertex.

Root Finding

The task of determining values of \(x\) satisfying \(P(x) = 0\), using factoring, the factor theorem, the quadratic/cubic formulas, or numerical and graphical methods.

Roots of a Function

The values of \(x\) for which \(f(x) = 0\). Graphically, they are the \(x\)-coordinates of the intersections of the graph with the \(x\)-axis.

Example: The roots of \(f(x) = x^2 - 9\) are \(x = \pm 3\).

Scale Factor

The constant multiplying coordinates in a stretch transformation. A vertical stretch by factor \(a\) multiplies \(y\)-values; a horizontal stretch by factor \(b\) multiplies \(x\)-values.

Self-Inverse Function

A function \(f\) that is its own inverse: \(f(f(x)) = x\) for all \(x\) in its domain. Its graph is symmetric about the line \(y = x\).

Example: \(f(x) = \tfrac{1}{x}\) and \(f(x) = -x\) are self-inverse.

Set Builder Notation

A notation for defining a set by stating a property its members satisfy, written \(\{x : P(x)\}\) or \(\{x \mid P(x)\}\).

Example: \(\{x \in \mathbb{R} : x > 2\}\) is the set of all real numbers greater than \(2\).

Sign Diagram

A number line marked with the roots of an expression, showing the sign (\(+\) or \(-\)) of the expression on each resulting interval. Used to solve inequalities.

Significant Figures

The digits in a number that contribute to its precision, counted from the first non-zero digit. IB answers are typically given to three significant figures unless otherwise specified.

Example: \(0.004560\) has \(4\) significant figures.

Signum Function

The function \(\operatorname{sgn}(x)\) equal to \(-1\) for \(x < 0\), \(0\) for \(x = 0\), and \(1\) for \(x > 0\). It returns the sign of its input.

Simultaneous Equations

A set of two or more equations in the same variables that must be solved together so that each equation is satisfied. Also called a system of equations.

Example: \(\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}\) has solution \((3, 2)\).

Sketching vs Drawing

Sketching shows key features (intercepts, asymptotes, turning points, shape) without plotting point-by-point; drawing is an accurate plot to scale. IB exams usually require sketches.

Slope

The measure of steepness of a line, equal to the ratio of vertical change to horizontal change between any two points on the line. In IB notation this is called the gradient.

Solution Set

The set of all values of the variable(s) that make an equation or inequality true. It may be empty, finite, or infinite.

Example: The solution set of \(x^2 = 9\) is \(\{-3, 3\}\).

Solving Log Equations

Finding the unknown in an equation involving logarithms, by applying log laws, converting to exponential form, and checking domain validity.

Example: \(\log_2(x + 1) = 3\) gives \(x + 1 = 8\), so \(x = 7\).

Solving Quadratic Equations

Finding the values of \(x\) that satisfy \(ax^2 + bx + c = 0\) by factoring, completing the square, the quadratic formula, or technology.

Standard Form Quadratic

The quadratic expressed as \(f(x) = ax^2 + bx + c\). This form makes the \(y\)-intercept (\(c\)) and the coefficients for the discriminant and quadratic formula explicit.

Example: \(f(x) = x^2 + 4x + 3\).

Step Function

A piecewise-constant function whose graph consists of horizontal line segments at different heights, producing jump discontinuities at the transitions.

Example: Postage cost as a function of package weight.

Substitution Method

A technique for solving simultaneous equations: solve one equation for one variable, then substitute that expression into the other equation.

Example: From \(y = x + 1\) and \(2x + y = 7\), substitute to get \(2x + (x + 1) = 7\), so \(x = 2\).

Sum and Difference Cubes

The factorization identities \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) and \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Example: \(x^3 - 8 = (x - 2)(x^2 + 2x + 4)\).

Sum of Roots Formula

For a polynomial \(a_n x^n + \cdots + a_0\) with roots \(r_1, \ldots, r_n\) (counted with multiplicity), \(\sum r_i = -\frac{a_{n-1}}{a_n}\).

Example: For \(x^2 - 5x + 6\), the sum of roots is \(5\).

Symmetry Testing

The algebraic procedure of substituting \(-x\) into \(f(x)\) to decide whether a function is even (\(f(-x) = f(x)\)), odd (\(f(-x) = -f(x)\)), or neither.

Synthetic Division

A shorthand algorithm for dividing a polynomial by a linear factor \((x - a)\), using only the coefficients. Faster than long division when applicable.

Table of Values

A finite list of ordered pairs \((x, f(x))\) used to represent or plot a function numerically, typically organized in two rows or columns.

Example: For \(f(x) = 2x\): \(x: 0, 1, 2, 3\) and \(f(x): 0, 2, 4, 6\).

Technology in Mathematics

The role of calculators, computer algebra systems, and dynamic geometry software in exploring, solving, and visualizing mathematical ideas, complementing analytic methods.

Transformation Notation

The symbolic description of how \(y = f(x)\) changes, such as \(y = a f(b(x - h)) + k\), capturing stretches, reflections, and translations in one formula.

Transformation of Points

Applying a given transformation to a specific point on the original graph to find its image on the transformed graph, without redrawing the whole curve.

Translation

A transformation that shifts a graph horizontally, vertically, or both without changing its shape or orientation. Described by a vector \(\begin{pmatrix} h \\ k \end{pmatrix}\).

Turning Points

Points on a graph where the function changes from increasing to decreasing or vice versa. They are local maxima or local minima.

Example: The parabola \(y = x^2 - 4\) has one turning point at \((0, -4)\).

Turning Points Count

A polynomial of degree \(n\) has at most \(n - 1\) turning points. The actual count depends on the specific function.

Example: A cubic has at most \(2\) turning points; a quartic at most \(3\).

Two Real Roots

The case \(\Delta = b^2 - 4ac > 0\), where a quadratic equation has two distinct real solutions and its graph crosses the \(x\)-axis at two points.

Example: \(x^2 - 5x + 6 = 0\) has \(\Delta = 1 > 0\) and roots \(2, 3\).

Undefined Gradient

The gradient of a vertical line \(x = k\), where the change in \(x\) is zero and the ratio \(\tfrac{\Delta y}{\Delta x}\) is not defined.

Units and Dimensions

The physical quantities (e.g., meters, seconds, kg) attached to variables and constants in an applied model. Dimensional consistency is essential for valid results.

Example: In \(d = vt\), if \(v\) is in m/s and \(t\) in s, then \(d\) is in m.

Variable

A symbol representing a number that may change or take different values within a problem, as opposed to a fixed constant.

Example: In \(A = \pi r^2\), \(r\) is a variable while \(\pi\) is a constant.

Vertex Form Quadratic

The quadratic written as \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex. This form reveals the turning point and axis of symmetry directly.

Example: \(f(x) = (x - 2)^2 - 5\) has vertex \((2, -5)\).

Vertex of a Parabola

The turning point of a parabola, where the axis of symmetry meets the curve. For \(f(x) = ax^2 + bx + c\), it lies at \(x = -\tfrac{b}{2a}\).

Example: \(y = x^2 - 4x + 1\) has vertex \((2, -3)\).

Vertical Asymptote

A vertical line \(x = a\) that the graph of \(f\) approaches but never meets, typically where the function's output grows without bound as \(x \to a\).

Example: \(f(x) = \tfrac{1}{x - 2}\) has a vertical asymptote at \(x = 2\).

Vertical Line Test

A graphical test: a curve in the Cartesian plane represents a function of \(x\) if and only if no vertical line intersects the curve at more than one point.

Example: A circle fails the vertical line test, so \(x^2 + y^2 = 1\) is not a function of \(x\).

Vertical Stretch

The transformation \(y = a f(x)\) with \(a > 0\): every \(y\)-coordinate is multiplied by \(a\). Values \(a > 1\) stretch; \(0 < a < 1\) compress.

Example: \(y = 3x^2\) is a vertical stretch of \(y = x^2\) by factor \(3\).

Vertical Translation

The transformation \(y = f(x) + k\), which shifts the graph of \(f\) by \(k\) units upward (or \(|k|\) units downward if \(k < 0\)).

Example: \(y = x^2 + 4\) is \(y = x^2\) shifted \(4\) up.

Window Settings

The ranges of \(x\) and \(y\) displayed on a graphing tool, usually written \([x_{\min}, x_{\max}] \times [y_{\min}, y_{\max}]\). Choosing appropriate settings is essential to reveal key features.

X-Axis

The horizontal number line in the Cartesian coordinate plane. Points on the \(x\)-axis have \(y\)-coordinate equal to \(0\).

X-Intercept

The \(x\)-coordinate of a point where a graph crosses the \(x\)-axis; a value of \(x\) for which \(f(x) = 0\). Also called a root or zero of the function.

Example: The \(x\)-intercepts of \(y = x^2 - 4\) are \(x = \pm 2\).

Y Equals F of Mod X

The transformation \(y = f(|x|)\): the portion of \(f\) for \(x \geq 0\) is kept, and its reflection in the \(y\)-axis replaces the portion for \(x < 0\). The result is always an even function.

Y Equals F of X Squared

The transformation \(y = [f(x)]^2\): values of \(f\) are squared, so negatives become positive and the graph sits on or above the \(x\)-axis, with roots of \(f\) preserved.

Y Equals Mod F of X

The transformation \(y = |f(x)|\): any part of the graph of \(f\) lying below the \(x\)-axis is reflected above it, while the rest is unchanged.

Example: \(y = |x^2 - 4|\) reflects the dip between \(x = -2\) and \(x = 2\) upward.

Y-Axis

The vertical number line in the Cartesian coordinate plane. Points on the \(y\)-axis have \(x\)-coordinate equal to \(0\).

Y-Intercept

The \(y\)-coordinate of the point where a graph crosses the \(y\)-axis; the value of \(f(0)\). For a line \(y = mx + c\), the \(y\)-intercept is \(c\).

Example: The \(y\)-intercept of \(y = 2x + 5\) is \(5\).

Zero Gradient

A gradient \(m = 0\), indicating that \(y\) is constant as \(x\) changes. The line is horizontal with equation \(y = c\).

Zeros of a Function

Synonym for roots: the inputs \(x\) satisfying \(f(x) = 0\). The terms zero, root, and \(x\)-intercept refer to the same concept.