The maximum distance from the midline (mean value) to a peak or a trough.

If lengths scale by $k$ in a 2D enlargement, areas scale by $|k|^2$.

A sequence in which the difference between consecutive terms is constant.

A bearing is an angle measured clockwise from the North direction, usually written using three digits (for example, $135^\circ$).

A graph whose vertices can be divided into two sets so that every edge connects a vertex in one set to a vertex in the other set.

A **bipartite graph** is a graph whose vertices can be divided into two sets so that every edge connects a vertex in one set to a vertex in the other set.

The line obtained by replacing an inequality sign with an equals sign. It separates the plane into regions that do or do not satisfy the inequality.

A fixed point $C$ such that each point of the image lies on the line from $C$ through the original point, and its distance from $C$ is multiplied by the scale factor.

A fixed point $C$ such that for any point $P$, the image point $P'$ lies on the line $CP$, and the distance from $C$ to $P$ is multiplied by the absolute value of the scale factor $|k|$ to get the distance from $C$ to $P'$.

For $a>0$, $a\neq 1$, $b>0$, and any convenient base $c>0$, $c\neq 1$:
$$\log_a b=\frac{\log_c b}{\log_c a}$$
Common choices are $c=10$ (using $\log$) or $c=e$ (using $\ln$).

The problem of finding a shortest closed route that traverses every edge of a weighted graph at least once.

A vector written in component form as $\binom{x}{y}$, representing a translation of $x$ units horizontally and $y$ units vertically.

A vector written as $\binom{x}{y}$ that represents a **translation** of $x$ units horizontally and $y$ units vertically.

The constant value $d$ such that $u_{n+1}-u_n=d$ for every term in an arithmetic sequence.

The constant multiplier $r$ in a geometric sequence, found by dividing a term by the previous term: $r=\frac{u_{n+1}}{u_n}$.

A graph where every pair of distinct vertices is joined by an edge.

Two inequalities written together, such as $a<x<b$, meaning both $a<x$ and $x<b$ must be true at the same time.

The conditional probability of an event $B$ given that $A$ has occurred is written $P(B \mid A)$. The conditional probability of an event $B$ given that $A$ has occurred is written $P(B \mid A)$.

A graph where every vertex can be reached from every other vertex by traveling along edges (possibly using several edges).

A number between $-1$ and $1$ that measures the strength and direction of a linear relationship between two variables.

A formula for any triangle that relates a side length to the other two side lengths and the cosine of the included angle. For triangle $ABC$: $a^{2}=b^{2}+c^{2}-2bc\cos A$ (and similarly for $b$ and $c$).

A path that starts and ends at the same vertex and does not pass through any vertex more than once.

The **degree** of a vertex is the number of edges meeting at that vertex (counting multiple edges separately).

Also called an enlargement or a reduction, is a transformation that changes the size of a figure but keeps its shape.

A collection of vertices (nodes) connected by edges (links) that have a specific direction.

For $\mathbf{u}=\binom{u_1}{u_2}$ and $\mathbf{v}=\binom{v_1}{v_2}$, the dot product is $\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2$.

A transformation from a fixed **centre** that multiplies the distance of every point from the centre by the absolute value of a constant **scale factor** $|k|$, producing a similar image.

A trail that uses every edge exactly once and ends at the starting vertex.

A trail that uses every edge exactly once but starts and ends at different vertices.

An **even function** satisfies $f(-x)=f(x)$. Its graph is symmetric about the **y-axis**.

A method that tries **every possible case** to guarantee the best solution.

The power to which a base is raised. In $a^n$, $a$ is the base and $n$ is the exponent (index).

A value found while solving an equation algebraically that does not satisfy the original equation (often because it makes a denominator zero).

The set of all points that satisfy every inequality in a system (all constraints). Graphically, it is the overlap region, often a polygon.

The number of complete cycles per unit interval. In this course, the coefficient $b$ is often referred to as the frequency, representing the number of cycles in $360^\circ$ (distinct from the physics definition of cycles per unit time).

A sequence in which the ratio between consecutive terms is constant (each term is found by multiplying the previous term by the same number).

A set of **vertices** (nodes) connected by **edges** (links). A graph may be **weighted**, meaning each edge has a numerical cost such as distance or time.

A Hamiltonian path that returns to its starting vertex.

A path that visits every vertex exactly once.

A horizontal line $y=c$ that a graph approaches as $x\to +\infty$ and/or $x\to -\infty$.

The transformation $y=f(ax)$, where $x$-values are scaled by a factor $\frac{1}{a}$. It is a dilation parallel to the $x$-axis.

A transformation that moves the graph of $y=f(x)$ left or right by changing the input to $f$.

The inverse function of $f$, written $f^{-1}$, is the function that reverses $f$: if $f(a)=b$, then $f^{-1}(b)=a$. Graphically, $f$ and $f^{-1}$ are reflections across the line $y=x$.

An inequality involving a linear expression (for example $y\le 2x-6$ or $3x+2y>4$). Its graph is a half-plane bounded by a straight line.

A method of maximizing or minimizing a linear objective function subject to linear constraints (a system of linear inequalities).

For $a>0$, $a\neq 1$, and $x>0$, $\log_a x$ is the exponent $k$ such that $a^k=x$.

For $m>0$, $a^{\log_a m}=m$.

The lower bound of a rounded or measured quantity is the smallest value the original (exact) quantity could have had, given the stated accuracy.

The magnitude of a vector A is the length of the vector A and is denoted by $|A|$.

A numerical value that describes how far data values are dispersed (spread out) within a distribution.

For a vector $\mathbf{a}$, the vector $-\mathbf{a}$ has the same magnitude but the opposite direction.

A set of objects (called nodes or vertices) that are connected together.

An inequality where the expressions are not linear, for example involving $x^2$, $\sqrt{x}$, $e^x$, $\log x$, or $\frac{1}{x}$.

An inequality involving $x$ and $y$ whose boundary is a non-linear curve (for example, a parabola, exponential curve, logarithmic curve, or rational curve). Its solution is a set of points (a shaded region) in the plane.

An inequality using $\le$ or $\ge$, meaning the boundary is included in the solution set.

Symmetric distribution, with most values close to the mean and tailing off evenly in either direction. Its frequency graph is a bell-shaped curve.

A linear function representing what you want to optimize (such as profit or cost), for example $P=10x+8y$.

An **odd function** satisfies $f(-x)=-f(x)$. Its graph has **180° rotational symmetry** about the origin.

An opposite pair is an angle and the side directly across from it (for example, $A$ and $a$).

Two vectors that point in the same or opposite direction. Equivalently, one is a scalar multiple of the other.

A standardized measure of linear correlation given by
$$r=\frac{\frac{1}{n}\sum xy-\bar{x}\bar{y}}{\sigma_x\sigma_y},$$
where $\sigma_x$ and $\sigma_y$ are the standard deviations of $x$ and $y$.

The horizontal length of one complete cycle of a periodic graph.

A function whose values repeat after a fixed input interval $T$, so $f(x+T) = f(x)$ for all $x$ in its domain. The smallest positive such $T$ is the period.

A **periodic function** is a function for which there is a positive number $T$ such that $f(x+T)=f(x)$ for all $x$ in its domain. The value $T$ is the **period**.

For a full population of size $n$ with mean $\mu$, the standard deviation is
$$\sigma = \sqrt{\frac{\sum (x-\mu)^2}{n}}$$

The vector from the origin $O$ to a point, written $\overrightarrow{OA}$.

The vector from the origin $O$ to a point $A$, written $\overrightarrow{OA}$.

$\log_a(x^n)=n\log_a x$ (for real $n$, as long as $x^n$ is defined and $x>0$).

A branching diagram that represents outcomes in sequence, labelling each branch with a probability, and allowing joint probabilities to be found by multiplying along a path.

$\log_a(xy)=\log_a x+\log_a y$.

$\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y$.

An expression of the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)\neq 0$.

An equation that includes one or more rational algebraic expressions.

An exponent of the form $\frac{m}{n}$ where $m$ and $n$ are integers and $n\neq 0$.

An exponent that can be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q\neq 0$.

Any function that can be written as a quotient $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)\neq 0$.

A transformation that maps each point $(x,y)$ to $(x,-y)$, producing the graph $y=-f(x)$.

A restriction is any value of the variable that is not allowed because it makes the denominator zero.

The single vector that has the same effect as performing several vectors in sequence, for example $\mathbf{a}+\mathbf{b}$.

Changing $>$ to $<$ (and $\ge$ to $\le$, or vice versa) when both sides of an inequality are multiplied or divided by a negative number.

A **rotation** is a transformation that turns a figure about a fixed point (the **center of rotation**) through a given angle and direction.

An estimate of a population’s standard deviation based on a sample of size $n$:
$$s_{n-1} = \sqrt{\frac{\sum (x-\bar{x})^2}{n-1}},$$ where $\bar{x}$ is the sample mean.

A quantity with magnitude only, with no direction.

A vector of the form $k\mathbf{a}$, where $k$ is a number (scalar). It is parallel to $\mathbf{a}$ and has magnitude $|k|\,|\mathbf{a}|$.

Multiplying a vector by a scalar $k$ multiplies each component by $k$, changing the vector’s magnitude (and possibly direction).

A notation meaning “sum of.” For example, $\sum x$ means add all the data values $x$ together.

Two figures are similar when (1) their corresponding angles are equal, and (2) all the corresponding side lengths are in the same ratio (they are all multiplied by the same scale factor).

Two figures are similar if corresponding angles are equal and all corresponding lengths are in the same ratio (the same scale factor).

For any triangle $ABC$, $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$

Equivalently, $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$

The set of all values of the variable(s) that make an inequality (or a system of inequalities) true.

The (population) standard deviation is the square root of the variance, that is, the square root of the mean of the squared deviations from the mean. For a population,

$$
\sigma=\sqrt{\frac{\sum(x-\mu)^2}{n}}
$$

An inequality using $<$ or $>$, meaning the boundary is not included in the solution set.

A set of two or more inequalities that must all be true at the same time. The solution set is the overlap of the individual solution regions.

A walk through a graph that does not repeat any edge.

In a triangle $ABC$, the side opposite angle $A$ is called $a$, the side opposite angle $B$ is called $b$, and the side opposite angle $C$ is called $c$.

A table that organizes counts (or probabilities) for two categorical variables, showing totals for rows and columns.

A graph where edges have no direction, so if vertex $A$ is connected to vertex $B$, then $B$ is also connected to $A$.

$\log_a a=1$.

The upper bound of a rounded or measured quantity is the largest value the original (exact) quantity could have had, given the stated accuracy. For continuous data, the true value is strictly less than this upper bound.

The mean of the squared deviations from the mean. For a population,
$$\sigma^{2}=\frac{\sum(x-\mu)^{2}}{n}$$

A quantity with both magnitude (size) and direction, often represented by a directed line segment (an arrow).

A vertical line $x=c$ that a graph approaches as the input approaches $c$. For rational functions, it often occurs where the denominator approaches 0.

The transformation $y=a f(x)$, where all $y$-values are multiplied by $a$. It is a dilation with scale factor $a$ parallel to the $y$-axis.

A change to a graph that affects the output values (the $y$-coordinates), such as adding a constant to $f(x)$ or multiplying $f(x)$ by a constant.

A transformation that adds a constant $k$ to a function so that every point moves up ($k>0$) or down ($k<0$) by the same number of units.

If lengths scale by $k$ in a 3D enlargement, volumes scale by $|k|^3$.

A graph in which every edge has an associated number (a weight), representing something like distance, time, cost, or risk.

A **weighted graph** is a graph in which every edge has an associated number (a weight), representing something like distance, time, cost, or risk.

$\log_a 1=0$.