📚 Year 12 Cambridge Mathematics: Terminology & Mnemonics Guide | Year 12 剑桥数学：词汇术语速记指南
Mastering the language of mathematics is the first step to clear reasoning and exam success. This guide presents essential terms from the Cambridge Year 12 (AS Level) syllabus—Pure, Mechanics, and Statistics—with concise definitions and bilingual memory hooks. Each English entry is immediately followed by its Chinese counterpart to reinforce understanding through dual‑language exposure.
掌握数学语言是清晰推理与考试成功的第一步。本指南精选剑桥12年级（AS阶段）纯数、力学和统计中的核心术语，配以简要定义和双语记忆提示。每个英文条目紧接中文对应解释，通过双语言接触加深理解，帮助快速形成条件反射。
1. Algebra & Quadratics | 代数与二次方程
Quadratic Expression: A polynomial of degree 2, standard form ax² + bx + c, where a ≠ 0.
二次式：次数为2的多项式，标准形式为 ax² + bx + c，且 a ≠ 0。
Discriminant (Δ) wo distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.
判别式 (Δ)：来自求根公式的 b² − 4ac，用来判别根的性质。Δ > 0 时有两个不等实根，Δ = 0 时有一个重实根，Δ < 0 时无实根。
Completing the Square: Rewriting a quadratic in the form a(x + p)² + q to identify the vertex at (−p, q) and to solve equations.
配方法：将二次式写成 a(x + p)² + q 的形式，以便确定顶点 (−p, q) 并解方程。
Simultaneous Equations: A set of equations solved together to find common values for variables. Substitution and elimination are standard methods.
联立方程组：一同求解以找到变量公共值的一组方程。标准方法有代入法与消元法。
2. Functions & Transformations | 函数与变换
Function: A relation where every input (x) maps to exactly one output (y). Often written as f(x). Domain and range describe allowed inputs and possible outputs.
函数：每个输入 (x) 恰好对应一个输出 (y) 的关系，常记作 f(x)。定义域和值域分别描述允许的输入和可能的输出。
One‑to‑one Function: A function that never takes the same output for two different inputs. It passes the horizontal line test and has an inverse function.
一一函数：不同输入绝不会产生相同输出的函数。它能通过水平线检验，且存在反函数。
Inverse Function f⁻¹(x) t of f(x). Its graph is a reflection of y = f(x) in the line y = x. Domain and range swap.
反函数 f⁻¹(x)：逆转 f(x) 的作用。其图像是 y = f(x) 关于直线 y = x 的反射。定义域与值域互换。
Transformation of Graphs ) shifts horizontally, af(x) stretches vertically, f(ax) stretches horizontally by factor 1/|a|. A negative sign produces a reflection.
图像变换：y = f(x) + a 垂直平移，f(x + a) 水平平移，af(x) 垂直伸缩，f(ax) 以因子 1/|a| 水平伸缩。负号产生反射。
3. Coordinate Geometry | 坐标几何
Gradient of a Line ₁)/(x₂ − x₁). It measures steepness. Positive means upward slope, negative means downward.
直线的斜率：m = (y₂ − y₁)/(x₂ − x₁)，衡量倾斜程度。正数为上坡，负数为下坡。
Equation of a Line: y − y₁ = m(x − x₁) is the point‑gradient form; y = mx + c is the slope‑intercept form where c is the y‑intercept.
直线方程：y − y₁ = m(x − x₁) 是点斜式；y = mx + c 是斜截式，c 为 y 轴截距。
Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). The point exactly halfway between two given points.
中点：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)，两个给定点正中间的点。
Distance Between Two Points: d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Derived from Pythagoras’ theorem.
两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]，由勾股定理导出。
Perpendicular Lines: Two lines with gradients m₁ and m₂ are perpendicular if m₁m₂ = −1.
垂直线：斜率分别为 m₁ 与 m₂ 的两条直线互相垂直当且仅当 m₁m₂ = −1。
4. Sequences & Series | 数列与级数
Arithmetic Progression (AP) . The nth term is aₙ = a + (n−1)d. The sum of the first n terms is Sₙ = n/2 [2a + (n−1)d].
等差数列 (AP)：具有常数公差 d 的数列。第 n 项 aₙ = a + (n−1)d，前 n 项和 Sₙ = n/2 [2a + (n−1)d]。
Geometric Progression (GP) io r. The nth term is arⁿ⁻¹. The sum of the first n terms is Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1.
等比数列 (GP)：具有常数公比 r 的数列。第 n 项为 arⁿ⁻¹。前 n 项和 Sₙ = a(1−rⁿ)/(1−r)，r ≠ 1。
Sum to Infinity: For a GP with |r| < 1, the infinite sum converges: S∞ = a/(1−r). The series diverges if |r| ≥ 1.
无穷和：当 |r| < 1 时等比级数收敛，S∞ = a/(1−r)。若 |r| ≥ 1 则发散。
Sigma Notation (Σ): Shorthand for summing terms. Σᵢ₌₁ⁿ uᵢ means u₁ + u₂ + … + uₙ.
西格玛记法 (Σ)：求和的简写。Σᵢ₌₁ⁿ uᵢ 表示 u₁ + u₂ + … + uₙ。
5. Trigonometry & Circular Measure | 三角学与弧度
Radian equal in length to the radius. π radians = 180°. Arc length s = rθ, sector area A = ½r²θ.
弧度：弧长等于半径时所对的圆心角。π 弧度 = 180°。弧长 s = rθ，扇形面积 A = ½r²θ。
Sine Rule: In any triangle, a/sin A = b/sin B = c/sin C. Used for finding sides or angles in non‑right‑angled triangles.
正弦定理：任意三角形中 a/sin A = b/sin B = c/sin C，用于求解非直角三角形的边或角。
Cosine Rule olves triangles when two sides and the included angle are known, or three sides are given.
余弦定理：a² = b² + c² − 2bc cos A。当已知两边夹角或三边时用于解三角形。
Trigonometric Identities in θ / cos θ, sin²θ + cos²θ = 1. Helpful for simplifying equations and proving statements.
三角恒等式：核心恒等式有 tan θ = sin θ / cos θ 及 sin²θ + cos²θ = 1，用于化简方程和证明结论。
6. Exponentials & Logarithms | 指数与对数
Exponential Function: y = aˣ with a > 0, but mostly eˣ . The function grows proportionally to its current value. Its derivative is itself: d/dx(eˣ) = eˣ.
指数函数：y = aˣ (a > 0)，最常用 eˣ，其中 e ≈ 2.718。函数的增长与当前值成正比，其导数为自身：d/dx(eˣ) = eˣ。
Natural Logarithm (ln x): The inverse of eˣ. So ln(eˣ) = x and eˡⁿˣ = x for x > 0. Graph passes through (1, 0).
自然对数 (ln x)：eˣ 的反函数。因此 ln(eˣ) = x 且 eˡⁿˣ = x (x > 0)。图像过 (1, 0)。
Laws of Logarithms gₐy; logₐ(x/y) = logₐx − logₐy; logₐ(xⁿ) = n logₐx. These apply to any base, including ln.
对数运算法则：logₐ(xy) = logₐx + logₐy；logₐ(x/y) = logₐx − logₐy；logₐ(xⁿ) = n logₐx。适用于任意底数，包括 ln。
Exponential Growth & Decay: Model y = A eᵏˣ k > 0 gives growth, k < 0 gives decay. Contexts include population, interest, and cooling.
指数增长与衰减：模型 y = A eᵏˣ，k > 0 为增长，k < 0 为衰减。常用于人口、利息和冷却问题。
7. Differentiation | 微分
Derivative f'(x) or dy/dx h respect to x. Graphically it is the gradient of the tangent. For y = xⁿ, dy/dx = n xⁿ⁻¹.
导数 f'(x) 或 dy/dx：y 关于 x 的瞬时变化率。几何上是切线的斜率。对 y = xⁿ，有 dy/dx = n xⁿ⁻¹。
Chain Rule and u = g(x), then dy/dx = (dy/du) × (du/dx). Essential for handling brackets and powers.
链式法则：用于复合函数求导：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。处理括号和幂时必备。
Product Rule: For y = u(x)v(x), dy/dx = u'v + uv'. Use when multiplying two distinct functions.
乘积法则：对于 y = u(x)v(x)，有 dy/dx = u'v + uv'。两个不同函数相乘时使用。
Stationary Points: Where dy/dx = 0. Determine their nature using the second derivative: f''(x) > 0 gives a local minimum, f''(x) < 0 gives a local maximum, and f''(x) = 0 requires further testing.
驻点：导数为零的点。用二阶导数判断性质：f''(x) > 0 为局部极小值，f''(x) < 0 为局部极大值，f''(x) = 0 需进一步检验。
8. Integration | 积分
Indefinite Integral ⁺¹/(n+1) + C, for n ≠ −1. The constant C arises because derivatives of constants are zero.
不定积分：微分的逆运算。∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ −1。常数 C 因常数的导数为零而出现。
Definite Integral der y = f(x) between x = a and x = b. Evaluate using the Fundamental Theorem: F(b) − F(a).
定积分：∫ᵃᵇ f(x) dx 给出曲线下方从 x=a 到 x=b 的符号面积。用微积分基本定理计算：F(b) − F(a)。
Area Between Curves g(x)] dx, where f(x) is the upper curve on [a, b]. Be mindful of which function is on top.
曲线间面积：面积 = ∫ᵃᵇ [f(x) − g(x)] dx，其中 f(x) 是区间 [a, b] 上的上方曲线。注意确认谁在上方。
Trapezium Rule: Approximate ∫ᵃᵇ y dx using h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)], where h = (b−a)/n. Useful when exact integration is difficult.
梯形法则：用 h/2 [y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)] 近似 ∫ᵃᵇ y dx，其中 h = (b−a)/n。难以精确积分时十分有用。
9. Kinematics (Mechanics) | 运动学（力学）
Displacement (s): Vector quantity giving distance in a specific direction from a fixed origin. It can be positive or negative.
位移 (s)：矢量，表示从固定原点沿特定方向的距离，可正可负。
Velocity (v) & Acceleration (a) = d²s/dt². Speed is the magnitude of velocity. In SUVAT models, acceleration is constant.
速度 (v) 与加速度 (a)：v = ds/dt，a = dv/dt = d²s/dt²。速率是速度的大小。SUVAT 模型中加速度恒定。
SUVAT Equations ½at², s = vt − ½at², v² = u² + 2as, s = ½(u+v)t. u is initial velocity, v final velocity.
SUVAT 方程：恒定加速度下：v = u + at，s = ut + ½at²，s = vt − ½at²，v² = u² + 2as，s = ½(u+v)t，其中 u 为初速度，v 为末速度。
Distance Travelled from Velocity–Time Graph: Area between the graph and the time axis gives displacement; total area gives total distance, treating all regions as positive.
速度‑时间图求路程：图线与时间轴围成的面积表示位移；总面积（所有区域取正）给出总路程。
10. Probability & Statistics | 概率与统计
Random Variable (X): A variable whose numerical outcome depends on a chance experiment. Can be discrete (countable) or continuous (measured).
随机变量 (X)：数值结果取决于随机试验的变量，可以是离散的（可数）或连续的（测量值）。
Probability Distribution s : Lists all possible outcomes and their probabilities. For discrete X, Σ P(X=x) = 1. Expected value E(X) = Σ x·P(X=x).
概率分布：列出所有可能结果及其概率。对离散 X 有 Σ P(X=x) = 1，期望 E(X) = Σ x·P(X=x)。
Binomial Distribution: X ~ odels number of successes in n independent trials, each with probability p. P(X=r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ.
二项分布：X ~ B(n, p) 模拟 n 次独立试验中成功的次数，每次成功概率为 p。P(X=r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ。
Normal Distribution: X ~ N(μ, σ²) is a continuous symmetric bell‑shaped curve. Probabilities found using z = (x−μ)/σ and standard tables. Total area = 1.
正态分布：X ~ N(μ, σ²) 是连续对称钟形曲线。通过 z = (x−μ)/σ 和标准正态表求概率，总面积=1。
Measures of Centre & Spread nt). Variance σ² measures spread: σ² = Σ(x−μ)²/n. Standard deviation σ is its square root.
中心与离散度量：平均数（均值）、中位数（中间值）、众数（出现最频）。方差 σ² = Σ(x−μ)²/n 度量离散程度，标准差 σ 是其平方根。
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