📚 Year 12 OCR Maths: Key Terminology Quick Reference Guide | Year 12 OCR 数学：词汇术语速记指南
Mastering the precise language of mathematics is essential for success in OCR Year 12. This quick reference guide breaks down the most important terms from Pure, Statistics and Mechanics, giving you clear definitions in both English and Chinese to accelerate your learning and exam preparation.
掌握数学的精确语言对于在OCR Year 12中取得成功至关重要。这份速记指南解析了纯数、统计和力学中最重要的术语，提供中英双语清晰定义，助你加速学习与备考。
1. Core Algebra & Polynomials | 核心代数与多项式
Variable: A symbol, usually a letter such as x or y, that represents an unknown or changeable number in an expression or equation.
变量： 一个符号，通常是像 x 或 y 这样的字母，在表达式或方程中代表未知或可变的数。
Coefficient: A numerical or constant factor that multiplies a variable in a term. In 4x³, 4 is the coefficient.
系数： 项中乘以变量的数字常量因子。在 4x³ 中，4 是系数。
Polynomial: An algebraic expression consisting of terms with non‑negative integer powers of one or more variables, e.g. 2x⁴ − 7x² + x − 5.
多项式： 由一项或多项变量非负整数次幂组成的代数表达式，例如 2x⁴ − 7x² + x − 5。
Degree (of a polynomial): The highest power of the variable in a polynomial. For 3x⁵ + 2x² − 1, the degree is 5.
多项式的次数： 多项式中变量的最高次幂。如 3x⁵ + 2x² − 1 的次数为 5。
Factor: An expression that divides another expression exactly, leaving no remainder. For example, (x − 2) is a factor of x³ − 3x² + 4x − 12 if it yields zero remainder.
因式： 能够整除另一个表达式的式子，不留余式。例如，若计算后余数为零，则 (x − 2) 是 x³ − 3x² + 4x − 12 的因式。
2. Quadratics & the Discriminant | 二次函数与判别式
Quadratic expression: An algebraic expression of the form ax² + bx + c, where a ≠ 0. It is a polynomial of degree 2.
二次表达式： 形如 ax² + bx + c 的代数式，其中 a ≠ 0，是一个二次多项式。
Discriminant (Δ): For a quadratic ax² + bx + c = 0, the discriminant is Δ = b² − 4ac. It determines the nature of the roots.
判别式 (Δ)： 对于二次方程 ax² + bx + c = 0，判别式为 Δ = b² − 4ac。它决定了根的性质。
Completing the square: A method for rewriting a quadratic expression in the form a(x + p)² + q, which reveals the vertex of its graph.
配方法： 一种将二次表达式改写成 a(x + p)² + q 形式的方法，用于揭示其图像的顶点。
Roots (solutions): The values of x that satisfy the quadratic equation ax² + bx + c = 0. They are given by x = (−b ± √(b² − 4ac)) / (2a).
根（解）： 使二次方程 ax² + bx + c = 0 成立的 x 值，由公式 x = (−b ± √(b² − 4ac)) / (2a) 给出。
3. Functions & Graphs | 函数与图像
Function: A rule that assigns each input (domain element) exactly one output (range element). Written as f(x).
函数： 一种规则，为每个输入（定义域元素）分配唯一的输出（值域元素）。记作 f(x)。
Domain: The set of all possible input values (x‑values) for which a function is defined.
定义域： 函数有定义的所有可能输入值（x 值）的集合。
Range: The set of all possible output values (y‑values) that a function can produce.
值域： 函数能够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： 将一个函数作用于另一个函数的输出所构成的函数。记作 f(g(x)) 或 (f ∘ g)(x)。
Inverse function: A function f⁻¹(x) that reverses the effect of f(x), such that f⁻¹(f(x)) = x. Only one‑to‑one functions have inverses over their whole domain.
反函数： 逆转 f(x) 效果的函数 f⁻¹(x)，满足 f⁻¹(f(x)) = x。只有一一对应函数在其整个定义域上才有反函数。
Modulus function: The function |x| that gives the absolute value of x, i.e. the non‑negative distance from zero. Its graph is V‑shaped.
模函数： 给出 x 绝对值的函数 |x|，即到零的非负距离。其图像呈 V 形。
4. Differentiation | 微分
Derivative: The rate of change of a function with respect to its variable, denoted f'(x) or dy/dx. It gives the gradient of the tangent to the curve.
导数： 函数相对于其变量的变化率，记作 f'(x) 或 dy/dx。它给出了曲线切线的斜率。
Stationary point: A point on a curve where the gradient (first derivative) is zero. It can be a local maximum, local minimum, or a point of inflection.
驻点： 曲线上梯度（一阶导数）为零的点。它可以是局部极大点、局部极小点或拐点。
Chain rule: A differentiation rule used when a function is composed of two functions: if y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx).
链式法则： 当函数由两个函数复合而成时使用的微分规则：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。
Product rule: For y = u(x)v(x), the derivative is dy/dx = u'(x)v(x) + u(x)v'(x).
乘法法则： 对于 y = u(x)v(x)，导数为 dy/dx = u'(x)v(x) + u(x)v'(x)。
Second derivative: The derivative of the first derivative, denoted f”(x) or d²y/dx². It helps determine the nature of stationary points and concavity.
二阶导数： 一阶导数的导数，记作 f”(x) 或 d²y/dx²。它有助于判断驻点的性质以及凹凸性。
5. Integration | 积分
Indefinite integral: The reverse process of differentiation. The indefinite integral of f(x) is written ∫ f(x) dx and includes a constant of integration + C.
不定积分： 微分的逆运算。f(x) 的不定积分记作 ∫ f(x) dx，并包含积分常数 + C。
Constant of integration: An arbitrary constant C added when finding an indefinite integral, because the derivative of any constant is zero.
积分常数： 求不定积分时加入的任意常数 C，因为任何常数的导数为零。
Definite integral: An integral with upper and lower limits, ∫ₐᵇ f(x) dx, which represents the signed area between the curve, the x‑axis and the lines x = a and x = b.
定积分： 带有上下限的积分 ∫ₐᵇ f(x) dx，表示曲线、x 轴以及直线 x = a 与 x = b 之间的带符号面积。
Area under a curve: The area enclosed by the graph of a function and the x‑axis, found by evaluating a definite integral between the appropriate limits.
曲线下方面积： 函数图像与 x 轴围成的面积，通过在适当限之间计算定积分来求得。
Trapezium rule: A numerical method for approximating a definite integral using trapezoids. For n strips of width h, ∫ₐᵇ f(x) dx ≈ h/2 [f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)].
梯形法则： 一种用梯形近似计算定积分的数值方法。对于 n 个宽度为 h 的条形，∫ₐᵇ f(x) dx ≈ h/2 [f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)]。
6. Exponentials & Logarithms | 指数与对数
Exponential function: A function of the form f(x) = aˣ, where a > 0. The most important base is Euler’s number e ≈ 2.718, giving f(x) = eˣ.
指数函数： 形如 f(x) = aˣ 的函数，其中 a > 0。最重要的底数是欧拉数 e ≈ 2.718，得到 f(x) = eˣ。
Logarithm: The inverse of exponentiation. If aˣ = b, then logₐ b = x. For base 10 it’s common log; for base e it’s the natural logarithm ln.
对数： 指数运算的逆运算。若 aˣ = b，则 logₐ b = x。以 10 为底称为常用对数；以 e 为底称为自然对数 ln。
Natural logarithm: The logarithm to the base e, written ln x. It satisfies ln(eˣ) = x and e^(ln x) = x for x > 0.
自然对数： 以 e 为底的对数，写作 ln x。它满足 ln(eˣ) = x 且对于 x > 0，e^(ln x) = x。
Laws of logarithms: Key rules: logₐ(xy) = logₐ x + logₐ y; logₐ(x/y) = logₐ x − logₐ y; logₐ(xⁿ) = n logₐ x.
对数运算法则： 关键规则：logₐ(xy) = logₐ x + logₐ y；logₐ(x/y) = logₐ x − logₐ y；logₐ(xⁿ) = n logₐ x。
7. Trigonometry | 三角学
Radian: The angle subtended at the centre of a circle by an arc equal in length to the radius. π radians = 180°.
弧度： 圆心角所对的弧长等于半径时的角度。π 弧度 = 180°。
Sine (sin), Cosine (cos), Tangent (tan): The three primary trigonometric ratios for a right‑angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
正弦 (sin)、余弦 (cos)、正切 (tan)： 直角三角形的三个基本三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Sine rule: In any triangle, a / sin A = b / sin B = c / sin C. Used to find unknown sides or angles.
正弦定理： 在任何三角形中，a / sin A = b / sin B = c / sin C。用于求未知边或角。
Cosine rule: a² = b² + c² − 2bc cos A. Connects three sides and one angle; useful when two sides and the included angle are given.
余弦定理： a² = b² + c² − 2bc cos A。联系三条边和一个角；当已知两边及其夹角时很有用。
Trigonometric identity: An equation involving trig functions that holds for all valid angles. Fundamental: sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ.
三角恒等式： 对一切有效角度都成立的包含三角函数的等式。基本恒等式：sin²θ + cos²θ ≡ 1，tan θ ≡ sin θ / cos θ。
8. Vectors | 向量
Vector: A quantity that has both magnitude and direction, often represented as a directed line segment or a column matrix. Examples: displacement, velocity.
向量： 既有大小又有方向的量，常用有向线段或列矩阵表示。例如：位移、速度。
Scalar: A quantity that has only magnitude, no direction. Examples: speed, distance, mass.
标量： 只有大小而没有方向的量。例如：速率、距离、质量。
Magnitude of a vector: The length or size of a vector. For vector v = (x, y), magnitude |v| = √(x² + y²).
向量的模： 向量的长度或大小。对于向量 v = (x, y)，模 |v| = √(x² + y²)。
Unit vector: A vector with magnitude 1, often used to specify direction. The unit vector in the direction of v is v / |v|.
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