📚 Year 12 WJEC Mathematics: Glossary Quick Memorisation Guide | Year 12 WJEC 数学：词汇术语速记指南
Welcome to your comprehensive glossary guide for Year 12 WJEC Mathematics. Mastering the language of mathematics is half the battle; this resource gives you clear definitions, key formulas and memory tricks for the Pure, Mechanics and Statistics strands. Use it to build fluency and confidence before tackling exam-style problems.
欢迎查阅这份全面的 Year 12 WJEC 数学词汇指南。掌握数学的语言是成功的一半；本指南为你梳理了纯数学、力学和统计模块的核心定义、重要公式和记忆诀窍，帮助你在应对考试题之前提升熟练度与信心。
1. Algebraic Terminology | 代数术语
Polynomial: An expression composed of variables raised to non‑negative integer powers, combined using addition, subtraction and multiplication. For example, 4x³ – 2x + 7 is a cubic polynomial. Mnemonic: 'poly' means many, just like a polygon has many sides.
多项式：由变量的非负整数次幂通过加、减、乘运算构成的表达式，例如 4x³ – 2x + 7 是一个三次多项式。记忆方法：'poly' 表示多，就像多边形（polygon）有多条边。
Factor Theorem: For a polynomial p(x), if p(a) = 0, then (x – a) is a factor. This lets you break down polynomials quickly. Think: 'zero gives a factor'.
因式定理：对于多项式 p(x)，若 p(a) = 0，则 (x – a) 是其因式。这能帮助你快速分解多项式。记住：'函数值为零即得因式'。
Discriminant (∆): In the quadratic ax² + bx + c, the discriminant is b² – stinct real roots, zero → one repeated root, negative → no real roots (two complex roots).
判别式 (∆)：对于二次式 ax² + bx + c，判别式为 b² – 4ac。它揭示根的性质：正数 → 两个相异实根，零 → 一个重根，负数 → 无实根（两个共轭复根）。
Completing the Square: ratics and identify the vertex of a parabola. The standard process is x² + bx = (x + b/2)² – (b/2)².
配方法：将 ax² + bx + c 写成 a(x + p)² + q 的形式。该技巧用于求解二次方程并确定抛物线的顶点。标准步骤是 x² + bx = (x + b/2)² – (b/2)²。
Remainder Theorem: When a polynomial f(x) is divided by (x – a), the remainder is f(a). This is the companion to the Factor Theorem; use it to find remainders without full long division.
余数定理：当多项式 f(x) 除以 (x – a) 时，余数为 f(a)。这是因式定理的兄弟，让你无需进行完整长除法即可求得余数。
2. Functions and Graphs | 函数与图像
Function: domain is the set of allowed inputs, and the range is the set of possible outputs. Remember: 'one input, one output'.
函数：一种将每个输入值恰好对应一个输出值的规则，记作 f(x) = ...。定义域是允许输入值的集合，值域是可能输出值的集合。记住：'一个输入只对应一个输出'。
Composite Function: fg(x) means apply g first, then f. The order matters; fg(x) is generally not the same as gf(x). Think of putting on socks (g) then shoes (f).
复合函数：fg(x) 表示先作用 g，再作用 f。顺序至关重要；fg(x) 通常不等于 gf(x)。想象先穿袜子 (g) 再穿鞋 (f)。
Inverse Function: ) reverses the effect of f(x). Graphically, it is a reflection in the line y = x. Only one‑to‑one functions have inverses over their whole domain.
反函数：f⁻¹(x) 逆向执行 f(x) 的操作。图像上，它是关于直线 y = x 的反射。只有一一对应的函数在其全体定义域上才有反函数。
Modulus Function: |x| gives the absolute value, stripping away any negative sign. It creates V‑shaped graphs. |x| = x if x ≥ 0, and |x| = –x if x < 0.
模函数：|x| 给出绝对值，去掉负号。其图像呈 V 形。|x| = x 当 x ≥ 0，|x| = –x 当 x < 0。
Transformations of Graphs: f(x + a) translates left by a; f(x) + a translates up by a; f(ax) is a horizontal stretch by scale factor 1/a; af(x) is a vertical stretch by factor a; –f(x) reflects in the x‑axis, f(–x) reflects in the y‑axis.
图像变换：f(x + a) 向左平移 a；f(x) + a 向上平移 a；f(ax) 是水平方向拉伸，尺度因子为 1/a；af(x) 是垂直方向拉伸，因子为 a；–f(x) 关于 x 轴反射，f(–x) 关于 y 轴反射。
3. Sequences and Series | 数列与级数
Arithmetic Progression: ence where the difference between consecutive terms is constant, d. The nth term is a + (n – 1)d. Sum of first n terms: Sₙ = n/2 [2a + (n – 1)d].
等差数列：相邻两项之差为常数 d 的数列。第 n 项为 a + (n – 1)d。前 n 项和 Sₙ = n/2 [2a + (n – 1)d]。
Sₙ = n/2 [2a + (n – 1)d]
Geometric Progression: m is multiplied by a common ratio r. The nth term is arⁿ⁻¹. Sum of first n terms: Sₙ = a(1 – rⁿ) / (1 – r) for |r| < 1.
等比数列：每一项乘以公比 r 得到后一项的数列。第 n 项为 arⁿ⁻¹。前 n 项和 Sₙ = a(1 – rⁿ)/(1 – r) 适用于 |r| < 1。
Sₙ = a(1 – rⁿ) / (1 – r)
Sigma Notation: o n of aₖ. It simply means add up all the terms a₁ to aₙ. Practise expanding a few to gain confidence.
求和符号：紧凑表示求和：∑ₖ₌₁ⁿ aₖ 表示将 a₁ 到 aₙ 所有项相加。多展开几次就能熟练。
Binomial Expansion: (a + b)ⁿ expands with Pascal’s triangle or the nCr formula. The general term is ⁿCᵣ aⁿ⁻ʳ bʳ, where ⁿCᵣ = n! / [r!(n – r)!]. Valid for positive integer n at Year 12.
二项式展开：(a + b)ⁿ 借助帕斯卡三角形或 nCr 公式展开。通项为 ⁿCᵣ aⁿ⁻ʳ bʳ，其中 ⁿCᵣ = n! / [r!(n – r)!]。Year 12 中 n 为正整数。
(a + b)ⁿ = Σⁿᵣ₌₀ ⁿCᵣ aⁿ⁻ʳ bʳ
4. Exponentials and Logarithms | 指数与对数
Exponential Function: f(x) = eˣ e graph passes through (0, 1) and increases rapidly. The inverse is the natural logarithm.
指数函数：f(x) = eˣ，e ≈ 2.71828。其增长率与函数值成正比。图像过 (0, 1) 且快速递增。其反函数为自然对数。
Natural Logarithm: ln x is the power to which e must be raised to obtain x. So ln(e²) = 2, and eˡⁿ ˣ = x. Remember: ln 1 = 0, ln e = 1.
自然对数：ln x 表示以 e 为底使得 e 的该次幂等于 x 的指数。因此 ln(e²) = 2，eˡⁿ ˣ = x。记住：ln 1 = 0，ln e = 1。
Laws of Logarithms: logₐ (xy) = logₐ x + logₐ y; logₐ (x/y) = logₐ x – logₐ y; logₐ (xⁿ) = n logₐ x. These hold for any base, including ln.
对数运算法则：logₐ (xy) = logₐ x + logₐ y；logₐ (x/y) = logₐ x – logₐ y；logₐ (xⁿ) = n logₐ x。这些法则对任何底数都成立，包括 ln。
Change of Base: calculating logarithms on a calculator or simplifying expressions. Choose b = 10 or b = e.
换底公式：logₐ x = log_b x / log_b a。尤其适合在计算器上求解对数或化简表达式，常选 b = 10 或 b = e。
Exponential Growth/Decay: s growth; if k < 0, it is decay. The constant A is the initial value. The time taken for a halving/doubling can be found using logs.
指数增长与衰减：形如 y = A eᵏᵗ 的模型。k > 0 为增长，k < 0 为衰减。常数 A 是初始值。减半/倍增所需时间可用对数计算。
5. Trigonometry | 三角学
Radian Measure: = ½ r²θ (θ in radians). Quickly convert: multiply by π/180 to go from degrees to radians.
弧度制：以弧度度量角：π 弧度 = 180°。1 弧度 ≈ 57.3°。弧长 s = rθ，扇形面积 A = ½ r²θ（θ 用弧度）。快速转换：角度转弧度乘以 π/180。
Sine Rule: sin B = c / sin C. Use it when you have two angles and a side (AAS) or two sides and a non‑included angle (SSA). Be cautious of the ambiguous case.
正弦定理：a / sin A = b / sin B = c / sin C。适用于已知两角一边 (AAS) 或两边及非夹角 (SSA) 的情形。注意可能出现两解（模糊情况）。
Cosine Rule: a² = b² + c² – 2bc cos A. Rearranged: cos A = (b² + c² – a²) / (2bc). Use for SAS or SSS triangles. It is an extension of Pythagoras for non‑right triangles.
余弦定理：a² = b² + c² – 2bc cos A。变形为 cos A = (b² + c² – a²) / (2bc)。用于 SAS 或 SSS 的三角形，可视为勾股定理在非直角三角形中的推广。
Trigonometric Identities: nθ = sinθ / cosθ. These allow simplification and solving equations. Learn how to rewrite 1 – cos²θ as sin²θ instantly.
三角恒等式：sin²θ + cos²θ = 1；tanθ = sinθ / cosθ。这些恒等式用于化简和求解方程，应能立即看出 1 – cos²θ = sin²θ。
Trig Graphs: ymptotes at odd multiples of π/2. Transformations shift, stretch and reflect these shapes.
三角函数图像：y = sinθ 和 y = cosθ 的周期为 2π，振幅为 1。y = tanθ 周期为 π，且在 π/2 的奇数倍处有渐近线。图像变换可平移、拉伸和反射这些形状。
6. Differentiation | 微分
Derivative: The derivative f' = f(x) at any point. It represents the instantaneous rate of change. Notation: dy/dx or f'(x).
导数：导数 f'(x) 表示曲线 y = f(x) 在任意点处切线的斜率，即瞬时变化率。记作 dy/dx 或 f'(x)。
Power Rule: If f(x) = xⁿ, then f' the most used differentiation rule. Multiply by the old power and reduce the power by one.
幂法则：若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹。这是最常用的求导法则：乘以原指数，指数再减一。
d/dx (xⁿ) = nxⁿ⁻¹
Product Rule: If y = uv, then dy/dx = u dv/dx + v du/dx. A common mnemonic: 'first times derivative of second plus second times derivative of first'.
积的求导法则：若 y = uv，则 dy/dx = u dv/dx + v du/dx。记忆口诀：'前微后不微，后微前不微'。
Quotient Rule: If y = u/v, then dy/dx = (v du/dx – u dv/dx) / v². Use when the function is a fraction. The numerator echoes 'low d high minus high d low over low squared'.
商的求导法则：若 y = u/v，则 dy/dx = (v du/dx – u dv/dx) / v²。适用于分式函数。分子可记作 '下微上减上微下除以下方'。
Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) × g' changed, then multiply by derivative of the inner. Crucial for composite functions like e²ˣ, sin(3x).
链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。先对外层函数求导，内层保留，再乘以内层函数
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