📚 Year 12 SQA Maths: Glossary & Key Terms Quick Guide | Year 12 SQA 数学：词汇术语速记指南
A strong command of mathematical vocabulary is the key to unlocking exam questions quickly and accurately. This quick guide brings together essential terminology from across the Year 12 SQA Mathematics curriculum, offering clear definitions and helpful notes to reinforce your understanding.
熟练掌握数学词汇是快速、准确解答考试题目的关键。本速记指南汇集了 SQA 12 年级数学课程中的核心术语，提供清晰的定义和有用的提示，帮助你巩固理解。
1. Algebra & Polynomials | 代数与多项式
Variable: A symbol, usually a letter such as x or y, that represents an unknown or changeable number in an expression, equation or function.
变量 (Variable): 通常用字母 x 或 y 表示的符号，代表表达式、方程或函数中未知或可变的数。
Coefficient: The numerical factor multiplying a variable in a term. For example, in 7x², the coefficient of x² is 7.
系数 (Coefficient): 一个项中与变量相乘的数字因子。例如，在 7x² 中，x² 的系数是 7。
Expression: A combination of variables, numbers and operations (such as addition or multiplication) that does not contain an equals sign.
表达式 (Expression): 由变量、数字和运算（如加法或乘法）组合而成的式子，不含等号。
Equation: A mathematical statement that shows two expressions are equal, using an equals sign. Solving an equation means finding the value(s) of the variable that make the statement true.
方程 (Equation): 用等号表明两个表达式相等的数学陈述。解方程就是求出使该陈述成立的变量的值。
Quadratic equation: An equation of the form ax² + bx + c = 0, where a ≠ 0. Its graph is a parabola.
二次方程 (Quadratic equation): 形如 ax² + bx + c = 0 且 a ≠ 0 的方程，其图像是一条抛物线。
Discriminant: real distinct roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.
判别式 (Discriminant): 取自二次公式的表达式 Δ = b² − 4ac。它决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
2. Functions & Graphs | 函数与图像
Function: A rule that assigns exactly one output to every input. Often written as f(x), g(x) or h(t).
函数 (Function): 一种为每个输入恰好分配一个输出的规则。常写作 f(x)，g(x) 或 h(t)。
Domain: The set of all possible input values (x-values) for which a function is defined.
定义域 (Domain): 函数所有可能输入值（x 值）的集合。
Range: The set of all possible output values (y-values) that a function can produce.
值域 (Range): 函数能产生的所有可能输出值（y 值）的集合。
Composite function: A function formed by applying one function to the output of another. Denoted by f(g(x)) or (f∘g)(x).
复合函数 (Composite function): 将一个函数的输出作为另一个函数的输入而构成的函数，记作 f(g(x)) 或 (f∘g)(x)。
Inverse function: iginal function. The inverse of f(x) is written as f⁻¹(x), and it satisfies f(f⁻¹(x)) = x.
反函数 (Inverse function): 逆转原函数作用的函数。f(x) 的反函数记作 f⁻¹(x)，且满足 f(f⁻¹(x)) = x。
Transformations: flections and stretches. For y = f(x), y = f(x − a) shifts the graph a units to the right.
图像变换 (Transformations): 对函数图像所做的改变，包括平移、反射和伸缩。对于 y = f(x)，y = f(x − a) 将图像向右平移 a 个单位。
3. Trigonometry | 三角学
Sine, cosine and tangent: The three primary trigonometric ratios, defined for an angle θ in a right‑ ngle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
正弦、余弦和正切 (Sine, cosine and tangent): 三种基本三角比，在直角三角形中对于角度 θ 的定义为：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Radian: ssential in calculus and when working with trigonometric functions without degree symbols.
弧度 (Radian): 一种角的度量单位，π 弧度 = 180°。弧度在微积分和处理不带度数符号的三角函数时必不可少。
Exact values: onometric values that can be written as fractions or surds, such as sin 30° = ½, cos 45° = √2/2, tan 60° = √3.
精确值 (Exact values): 常考且能以分数或根式写出的三角函数值，如 sin 30° = ½，cos 45° = √2/2，tan 60° = √3。
Double angle formulas: : Identities that express sin 2θ, cos 2θ and tan 2θ in terms of single angles. For example, sin 2θ = 2 sin θ cos θ.
倍角公式 (Double angle formulas): 用单角表示 sin 2θ，cos 2θ 和 tan 2θ 的恒等式。例如 sin 2θ = 2 sin θ cos θ。
Wave function: tten as k sin(x + α) or k cos(x ± α), which helps to find maxima, minima and phase shifts.
波函数 (Wave function): 将形如 a cos x + b sin x 或 a sin x + b cos x 的表达式改写为 k sin(x + α) 或 k cos(x ± α)，有助于求最大值、最小值和相位移动。
4. Differentiation | 微分
Derivative: The instantaneous rate of change of a function. It is denoted by f′(x) or dy/dx and gives the gradient of the tangent to the curve at any point.
导数 (Derivative): 函数的瞬时变化率，记作 f′(x) 或 dy/dx，表示曲线上任意一点处切线的斜率。
Power rule: (x) = xⁿ, then f′(x) = nxⁿ⁻¹. This rule is the foundation for differentiating polynomials.
幂法则 (Power rule): 如果 f(x) = xⁿ，那么 f′(x) = nxⁿ⁻¹。此规则是多项式微分的基础。
Stationary point: graph where f′(x) = 0. It can be a local maximum, local minimum or a point of inflection.
驻点 (Stationary point): 图像上 f′(x) = 0 的点，可能是局部极大值、局部极小值或拐点。
Chain rule: : if y = f(g(x)), then dy/dx = f′(g(x))·g′(x). In Leibniz notation, dy/dx = dy/du × du/dx.
链式法则 (Chain rule): 用于对复合函数求导：若 y = f(g(x))，则 dy/dx = f′(g(x))·g′(x)。在莱布尼茨记号中，dy/dx = dy/du × du/dx。
Product and quotient rules: ule: (u/v)′ = (u′v − uv′)/v². Both are essential when functions are multiplied or divided.
乘法法则与除法法则 (Product and quotient rules): 乘法法则：(uv)′ = u′v + uv′。除法法则：(u/v)′ = (u′v − uv′)/v²。当函数相乘或相除时，这两个法则至关重要。
5. Integration | 积分
Indefinite integral: tegral of f′(x) is f(x) + C, where C is the constant of integration. Written as ∫ f(x) dx.
不定积分 (Indefinite integral): 微分的逆过程。f′(x) 的不定积分为 f(x) + C，其中 C 是积分常数，记作 ∫ f(x) dx。
Constant of integration: An arbitrary constant added when finding an indefinite integral, representing the family of all antiderivatives.
积分常数 (Constant of integration): 求不定积分时加上的任意常数，代表整个原函数族。
Definite integral: s ∫ₐᵇ f(x) dx. It calculates the signed area under the curve y = f(x) from x = a to x = b.
定积分 (Definite integral): 在两个界限之间计算的积分，记作 ∫ₐᵇ f(x) dx。它用来求曲线 y = f(x) 从 x = a 到 x = b 所包围的有向面积。
Area under a curve: x) ≥ 0 on [a, b], the definite integral gives the exact area. When the curve crosses the x‑axis, the total area is found by summing the absolute values of the separate integrals.
曲线下方面积 (Area under a curve): 若在 [a, b] 上 f(x) ≥ 0，则定积分给出精确面积。当曲线穿过 x 轴时，总面积通过将各段积分绝对值相加而得。
Fundamental theorem of calculus: It links differentiation and integration: if F′(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a).
微积分基本定理 (Fundamental theorem of calculus): 它将微分与积分联系起来：如果 F′(x) = f(x)，那么 ∫ₐᵇ f(x) dx = F(b) − F(a)。
6. Exponentials & Logarithms | 指数与对数
Exponential function: A function of the form f(x) = aˣ where a > 0 and a ≠ 1. The most important base is e ≈ 2.718, giving y = eˣ, which is its own derivative.
指数函数 (Exponential function): 形如 f(x) = aˣ 且 a > 0, a ≠ 1 的函数。最重要的底是 e ≈ 2.718，函数为 y = eˣ，其导数等于它本身。
Natural logarithm: The logarithm to base e, written as ln x. It is the inverse of eˣ, so ln(eˣ) = x and eln x = x.
自然对数 (Natural logarithm): 以 e 为底的对数，写作 ln x。它是 eˣ 的反函数，故有 ln(eˣ) = x 和 eln x = x。
Logarithmic form: The equation logₐ y = x means aˣ = y. The number a is called the base, and y must be positive.
对数形式 (Logarithmic form): 等式 logₐ y = x 意味着 aˣ = y。数字 a 叫做底数，y 必须大于零。
Laws of logarithms: − logₐ y; logₐ(xⁿ) = n logₐ x. These laws are essential for solving exponential equations.
对数运算法则 (Laws of logarithms): logₐ(xy) = logₐ x +
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