📚 Year 12 AQA Mathematics Vocabulary Quick-Memorisation Guide | 12年级AQA数学词汇术语速记指南
Welcome to this quick-memorisation guide for Year 12 AQA Mathematics. Mastering the precise vocabulary and notation is not just about learning definitions - it is about training your brain to recognise patterns, recall formulas and apply concepts under exam pressure. This article breaks down the most essential terms from pure mathematics, statistics and mechanics, pairing each with a clear definition and a memorable memory tip. Use these pages to reinforce your understanding and speed up your recall during revision.
欢迎阅读本12年级AQA数学词汇术语速记指南。掌握精确的词汇和符号不仅是为了记住定义，更是为了训练大脑识别模式、回忆公式并在考试压力下应用概念。本文拆解了纯数、统计和力学中最关键的术语，每个术语都配有一个清晰的定义和一个难忘的记忆技巧。利用这些内容巩固理解，在复习时加快回忆速度。
1. Indices and Logarithms | 指数与对数
The three core index laws are aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Memory tip: " add the powers; when raising a power to another power, multiply; when dividing, subtract." Negative indices give reciprocals: a⁻ⁿ = 1/aⁿ. Zero index: a⁰ = 1 (a ≠ 0).
指数（幂）是重复乘法的简写：aⁿ = a × a × ... × a（n 次）。三条核心指数法则是 aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ 和 aᵐ ÷ aⁿ = aᵐ⁻ⁿ。记忆技巧：“乘法指数相加，幂的幂指数相乘，除法指数相减”。负指数表示倒数：a⁻ⁿ = 1/aⁿ。零指数：a⁰ = 1（a ≠ 0）。
Surds are irrational roots that cannot be simplified to an integer, like √2 or ∛5. A key skill is rationalising the denominator, e.g. turning 1/√a into √a/a. Memory tip: "Surds live in an irrational world; make the denominator rational to compare or add them."
根式是不能化简为整数的无理根，如√2或∛5。一个关键技巧是分母有理化，例如将 1/√a 化为 √a/a。记忆技巧：“根式生活在无理世界里；让分母有理化才好比较或相加。”
ithm answers the question: to what power must the base be raised to produce a given number? If bˣ _b x - log_b y, log_b(xⁿ) = n log_b x. The natural logarithm ln x uses base e. Memory tip: "Logs turn multiplication into addition - they are the inverse of powers." Think of 'log' as 'power finder'.
对数回答这样一个问题：底数需要几次幂才能得到某个数？如果 bˣ = y，那么 log_b y = x。基本法则：log_b(xy) = log_b x + log_b y，log_b(x/y) = log_b x - log_b y，log_b(xⁿ) = n log_b x。自然对数 ln x 以 e 为底。记忆技巧：“对数把乘法变成加法——它们是指数的逆运算。”把‘log’想象成‘幂寻找器’。
2. Quadratic Functions and the Discriminant | 二次函数与判别式
f roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots (complex roots at A2). Memory tip: "Discriminant decides - if it's positive, x-crossings exist; zero touches; negative never meets x-axis."
标准二次函数是 f(x) = ax² + bx + c，其中 a ≠ 0。图像是抛物线。判别式 Δ = b² - 4ac。它判别根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 无实根（到A2会学复根）。记忆技巧：“判别式做主——正数有两交点，零相切，负数永不相交。”
Completing the square rewrites a quadratic in the form a(x + p)² + q, where the vertex is (-p, q). It is essential for finding turning points and solving equations. Memory tip: "Halve the x-coefficient, square it, add and subtract inside the bracket." The process reveals the maximum or minimum value directly.
配方法将二次式改写为 a(x + p)² + q 的形式，顶点为 (-p, q)。这是求驻点和解方程的关键。记忆技巧：“x 系数取一半，平方，括号内加减。”此过程直接揭示最大值或最小值。
The quadratic formula x = [-b ± √b² - 4ac] / (2a) solves any quadratic equation ax² + bx + c = 0. Memory tip: "Formula rhymes - sing it to remember: 'x equals negative b plus or minus square root, b squared minus four a c, all over two a'."
求根公式 x = [-b ± √b² - 4ac] / (2a) 可解任何二次方程 ax² + bx + c = 0。记忆技巧：“公式有韵律——可以唱出来：‘x等于负b加减根号下b方减4ac，整体除以2a’”。
3. Polynomials and the Factor Theorem | 多项式与因式定理
(p) = 0. Combined with polynomial division, it helps factorise cubics and higher. Memory tip: "Plug in the candidate root - if the output is zero, you have a factor." he remainder theorem extends this: when f(x) is divided by (x - p), the remainder is f(p).
多项式是形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ 的表达式。因式定理指出：(x - p) 是多项式 f(x) 的因式当且仅当 f(p) = 0。结合多项式除法，可用于分解三次及以上多项式。记忆技巧：“代入候选根——若结果为零，则找到因式。”余式定理延伸了这一点：f(x) 除以 (x - p) 时，余式为 f(p)。
factor. It works just like numerical long division but with descending powers. Memory tip: "Focus on the leading term; divide, multiply, subtract, bring down the next term - repeat." The quotient and remainder let you write f(x) = (x - p)Q(x) + R.
多项式长除法或综合除法通过线性因式拆分多项式。方法与数值长除法类似，只是处理降幂。记忆技巧：“紧盯首项；除、乘、减、移下一位——重复。”商式与余式给出 f(x) = (x - p)Q(x) + R。
points, point of inflection and behaviour as x → ±∞ depends on the sign of a. Memory tip: "Positive a: up on the right; negative a: down on the right."
三次函数图像特征：y = ax³ + bx² + cx + d 可以有 1 或 3 个实根。图像显示驻点、拐点，且当 x → ±∞ 时的走势取决于 a 的正负。记忆技巧：“a 正，右端上升；a 负，右端下降。”
4. Coordinate Geometry | 坐标几何
(x₂ - x₁). For parallel lines, m₁ = m₂; for perpendicular lines, m₁ × m₂ = -1. Memory tip: " means same steepness; perpendicular means flip and change sign - the negative reciprocal."
过两点 (x₁, y₁) 和 (x₂, y₂) 的直线梯度（斜率）为 m = (y₂ - y₁) / (x₂ - x₁)。平行线满足 m₁ = m₂；垂直线满足 m₁ × m₂ = -1。记忆技巧：“平行则坡度相同；垂直则取倒数并变号——即负倒数。”
s y - y₁ = m(x - x₁) using a known point. The general form is ax + by + c = 0. Memory tip: "y = mx + c is your go-to; m for how steep, c for where it cuts the y-axis."
直线方程常写为 y = mx + c，其中 m 是梯度，c 是 y 轴截距。另一种实用形式为已知点斜式 y - y₁ = m(x - x₁)。一般形式为 ax + by + c = 0。记忆技巧：“y = mx + c 是你的首选；m 管陡度，c 管割 y 轴的位置。”
two points is ((x₁+x₂)/2, (y₁+y₂)/2) and distance d = √[(x₂-x₁)² + (y₂-y₁)²]. Memory tip: "Midpoint: average the x's, average the y's; distance: Pythagoras in disguise."
两点中点 M 坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)，距离公式 d = √[(x₂-x₁)² + (y₂-y₁)²]。记忆技巧：“中点：对 x 和 y 分别求平均；距离：不过是伪装的勾股定理。”
5. Binomial Expansion | 二项式展开
For a positive integer n, (a + b)ⁿ expands using Pascal's triangle or the formula Σ (n choose r) aⁿ⁻ʳ bʳ, where (n choose r) = n! / [r!(n-r)!]. The coefficients are symmetric. Memory tip: "Powers of a go down, powers of b go up; row n of Pascal's triangle gives coefficients."
对于正整数 n，(a + b)ⁿ 的展开使用帕斯卡三角形或公式 Σ (n选r) aⁿ⁻ʳ bʳ，其中 (n选r) = n! / [r!(n-r)!]。系数是对称的。记忆技巧：“a 的幂次递降，b 的幂次递升；帕斯卡三角形第 n 行给出系数。”
tive integer n, but the binomial expansion can be written as (1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + ... when |x| < 1. Validity condition is vital. Memory tip: "If n is negative or fractional, the series is infinite and x must be small enough."
在 AS 阶段我们只处理正整数 n，但二项展开也可写作 (1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + ...，当 |x| < 1 时。有效性条件至关重要。记忆技巧：“若 n 为负或分数，级数无穷，且 x 必须足够小。”
Use binomial expansions to approximate values like √1.001 or (0.998)³. Substitute x accordingly and use the first few terms. Memory tip: "Rewrite the expression as (1 + small value)ⁿ, expand the first two or three terms, ignore higher powers."
利用二项展开可对 √1.001 或 (0.998)³ 等值做近似。代入相应的 x 并保留前几项。记忆技巧：“将表达式写成 (1 + 小量)ⁿ，展开前两三项，忽略高次幂。”
6. Trigonometry | 三角学
ns; radian is the angle subtended by an arc equal to the radius. π rad = 180°. Memory tip: "A full circle is 2π rad; to convert, π = 180°, so multiply by π/180 or 180/π."
角度以度或弧度度量；弧度是弧长等于半径时所对的角度。π rad = 180°。记忆技巧：“一整圈为 2π 弧度；换算时，π = 180°，故乘以 π/180 或 180/π。”
Exact trigonometric values for 30°, 45°, 60° are frequently tested. sin30° = 1/2, cos45° = √2/2, tan60° = √3, etc. Memory tip: "Draw the two triangles: right isosceles (45-45-90) with sides 1,1,√2 and half-equilateral (30-60-90) with sides 1,√3,2."
30°、45°、60° 的精确三角函数值常考。sin30° = 1/2，cos45° = √2/2，tan60° = √3 等。记忆技巧：“画两个三角形：等腰直角（45-45-90）边比 1:1:√2，半等边（30-60-90）边比 1:√3:2。”
sinθ / cosθ. They are essential for solving equations and proving statements. Memory tip: "Pythagoras in disguise for sin-cos; tan is just the ratio."
基本恒等式：sin²θ + cos²θ = 1 和 tanθ = sinθ / cosθ。这些是解方程和证明题的基础。记忆技巧：“sin-cos 恒等式不过是伪装的勾股定理；tan 仅是比值。”
le. Sine rule: a/sinA = b/sinB = c/sinC. Cosine rule: a² = b² + c² - 2bc cosA. Memory tip: " osite angle); cosine rule when you have two sides and the included angle or need an angle."
正弦和余弦规则将直角三角形三角学推广至任意三角形。正弦规则：a/sinA = b/sinB = c/sinC。余弦规则：a² = b² + c² - 2bc cosA。记忆技巧：“正弦规则用于边角配对；余弦规则当已知两边夹角或求角时使用。”
7. Differentiation | 微分
The derivative f'(x) is the gradient function: f'(x) = limh→0 h)-f(x)]/h. In AS, you differentiate using the power rule: d/dx (xⁿ) = n xⁿ⁻¹. Memory tip: "Bring down the power, reduce the power by one."
导数 f'(x) 是梯度函数：f'(x) = limh→0 [f(x+h)-f(x)]/h。在 AS 中，使用幂规则求导：d/dx (xⁿ) = n xⁿ⁻¹。记忆技巧：“把指数带下来，指数减一。”
The chain rule differentiates a function of a function: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x). In words: "Derivative of outside with inside unchanged, times derivative of inside." Memory tip: "See y = (stuff)ⁿ, do n(stuff)ⁿ⁻¹, then multiply by the derivative of stuff."
链式法则对复合函数求导：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。简言之：“外层函数求导并保持内层不变，乘以内层导数。”记忆技巧：“见到 y = (里层)ⁿ，先写 n(里层)ⁿ⁻¹，再乘以里层的导数。”
The product rule: if y = uv, then dy/dx = u'v + v'u. Quotient rule: if y = u/v, then dy/dx = (u'v - v'u) / v². Memory tip: "Product: first derivative times second plus second derivative times first. Quotient: 'low d-high minus high d-low over low squared'."
乘法法则：若 y = uv，则 dy/dx = u'v + v'u。除法法则：若 y = u/v，则 dy/dx = (u'v - v'u) / v²。记忆技巧：“乘积：第一的导数乘第二，加第二的导数乘第一。商：‘下乘上导减上乘下导，除以下方’。”
8. Integration | 积分
f differentiation. The indefinite integral of xⁿ (n ≠ -1) is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c. The '+c' constant of integration is crucial. Memory tip: "Add one to the power, divide by the new power - and never forget +c."
积分是微分的逆运算。xⁿ（n ≠ -1）的不定积分为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c。积分常数 '+c' 至关重要。记忆技巧：“指数加一，除以新指数——切勿忘记 +c。”
the curve and the x-axis from x=a to x=b. Use the fundamental theorem: F(b) - F(a) where F'(x)=f(x). Memory tip: "Find antiderivative, plug in upper limit minus lower limit - area!"
定积分 ∫ₐₚ f(x) dx 计算从 x=a 到 x=b 区间曲线与 x 轴之间的有号面积。利用基本定理：F(b) - F(a)，其中 F'(x)=f(x)。记忆技巧：“求原函数，代入上限减下限——即得面积！”
Area between a curve and a line or between two curves is found by integrating the difference of the functions. Always sketch and check which function is above. Memory tip: "Top minus bottom, then integrate over the intersection interval."
曲线与直线间或两条曲线间的面积可通过积分两函数之差求得。务必画草图并确认哪个函数在上方。记忆技巧：“上减下，再在交点区间上积分。”
9. Statistics – Data and Probability | 统计学：数据与概率
, interquartile range, variance (Σ(x-mean)²/n for a population) and standard deviation σ = √variance. Memory tip: "Mean the average, median splits ordered data, mode popular; variance and standard deviation measure how spread out data are - larger values, more spread."
集中趋势量数：均值 (Σx/n)、中位数（中间值）、众数（最频繁）。离散量数：极差、四分位距、方差（总体 Σ(x-均值)²/n）和标准差 σ = √方差。记忆技巧：“均值是平均，中位数分割有序数据，众数是人气王；方差和标准差衡量数据分布有多散——数值越大，越分散。”
tion models the number of successes in n independent trials, each with probability p. If X ~ B(n, p), then P(X = r) = (n choose r) pʳ (1-p)ⁿ⁻ʳ. Mean μ = np, variance σ² = np(1-p). Memory tip: "Fixed number of trials, two outcomes, constant probability, independent - binomial."
二项分布模拟 n 次独立试验中的成功次数，每次成功概率为 p。若 X ~ B(n, p)，则 P(X = r) = (n选r) pʳ (1-p)ⁿ⁻ʳ。均值 μ = np，方差 σ² = np(1-p)。记忆技巧：“固定试验次数，两种结果，概率
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