Pre-U Edexcel Mathematics: Key Term Memorisation Guide | Pre-U Edexcel 数学：词汇术语速记指南

📚 Pre-U Edexcel Mathematics: Key Term Memorisation Guide | Pre-U Edexcel 数学：词汇术语速记指南

Mastering the precise vocabulary of Pre-U Edexcel Mathematics is half the battle. This guide pairs essential definitions with clever memory hooks, so you can recall the right term in the exam and use it accurately in your reasoning. Each entry is followed by its Chinese equivalent to support bilingual learners and deepen conceptual understanding.

掌握 Pre-U Edexcel 数学中的精确词汇是成功的一半。这份指南将核心定义与巧妙的记忆钩子配对，帮助你在考试中准确回忆并正确运用术语。每个条目后附有对应的中文解释，以支持双语学习者并加深概念理解。

1. Algebraic Terminology | 代数学术语

A polynomial is an expression built from variables, coefficients and non‑negative integer exponents – for example, 3x⁴ − 2x² + 5. Think ‘poly‑many, nomial‑term’ – many terms linked by addition or subtraction.

多项式是由变量、系数和非负整数指数构成的表达式，例如 3x⁴ − 2x² + 5。可联想 “多‑项” 来记忆，多项式即许多项通过加减连接。

The degree of a polynomial is its highest exponent. In 7x⁵ − x³ + 2, the degree is 5. A helpful mnemonic: ‘degree decides the leading shape’ – the largest power dominates the graph’s end behaviour.

次数是多项式中变量的最高指数。在 7x⁵ − x³ + 2 中，次数为 5。可记：次数决定图形的主导行为，最高次幂掌控两端走向。

A coefficient is the numerical factor in front of a variable. In 9y², 9 is the coefficient. Memorise it as the ‘co‑pilot’ – it flies alongside the variable but doesn’t change the power.

系数是变量前面的数字因数。在 9y² 中，9 是系数。可联想 “协同飞行员”，它与变量同行但不改变幂次。

Discriminant: Δ = b² − 4ac

The discriminant Δ tells you how many real roots a quadratic has. If Δ > 0, two real roots; Δ = 0, one repeated root; Δ < 0, no real roots. Visualise Δ as a detective examining the equation’s ‘root‑scene’.

判别式 Δ = b² − 4ac 决定二次方程实根的数量。Δ > 0 时有两个实根；Δ = 0 时有一个重根；Δ < 0 时无实根。可将 Δ 想象成侦探，勘察方程的 “根现场”。

The Factor Theorem states that if f(a)=0, then (x − a) is a factor of f(x). Reverse the logic: ‘zero gives the factor’. When you spot a root, you’ve found a linear factor.

因式定理：若 f(a)=0，则 (x − a) 是 f(x) 的因式。反向记忆：“零点给出因式”。一旦找到根，你就发现了一个一次因式。

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2. Functions & Graphs | 函数与图像术语

A function is a mapping that assigns exactly one output to each input. The vertical line test checks this: any vertical line should meet the graph at most once. Think ‘one x, one y’ – like a faithful partner.

函数是一种映射，每个输入恰好对应一个输出。竖直线检验可验证：任何竖直线与图像最多相交一次。可以记 “一对一输出”，就像忠诚的伙伴。

The domain is the set of all allowed inputs, while the range is the set of all possible outputs. ‘Domain starts with d‑ for door’ – you can only enter through the allowed door.

定义域是所有允许输入的集合，值域是所有可能输出的集合。可联想 “定义域” 中的 “域” 就像入口区域，只允许合法输入进入。

The inverse function f⁻¹(x) reverses the original mapping. Graphically it is a reflection in the line y = x. Say: ‘swap x and y, then solve for y’ – and beware the domain swap.

反函数 f⁻¹(x) 反转了原来的映射。图像上它是关于直线 y = x 的反射。操作口诀：“交换 x 和 y，然后解出 y”——注意定义域也会互换。

A composite function (f ∘ g)(x) means apply g first, then f. The order matters – put on socks before shoes. ‘Rightmost acts first’ can be remembered as ‘righty‑tight first’.

复合函数 (f ∘ g)(x) 表示先应用 g，再应用 f。顺序如同穿鞋袜，先穿袜子。“最右先动” 可联想为 “右先紧”。

An even function satisfies f(−x)=f(x) (symmetric about the y‑axis), an odd function satisfies f(−x)=−f(x) (rotational symmetry about the origin). Remember: ‘even like y² (exponent even)’, ‘odd like y³’.

偶函数满足 f(−x)=f(x)，关于 y 轴对称；奇函数满足 f(−x)=−f(x)，关于原点旋转对称。可借助指数：x 的偶次幂如 x² 是偶，奇次幂如 x³ 是奇。

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3. Sequences & Series | 数列与级数术语

An arithmetic sequence changes by a constant difference d. To find the nth term: uₙ = a + (n−1)d. The word ‘arithmetic’ sounds like ‘add‑metic’ – you add the difference each step.

等差数列通过常数公差 d 变化。第 n 项公式为 uₙ = a + (n−1)d。发音可联想 “算术” 即每次加上同一差值。

The sum of the first n terms of an arithmetic series: Sₙ = n/2 [2a + (n−1)d] or n/2 (a+l). Picture pairing first and last terms – each pair has the same sum.

等差数列前 n 项和：Sₙ = n/2 [2a + (n−1)d] 或 n/2 (a+l)。想象将首项与末项配对，每对和相同。

A geometric sequence multiplies by a constant ratio r. If |r| < 1, the sum to infinity S∞ = a/(1−r). ‘Geometric’ hints at growing or shrinking by the same multiplier, like zooming a geometric figure.

等比数列以常数公比 r 相乘变化。若 |r| < 1，无穷项和为 S∞ = a/(1−r)。“几何” 暗示通过相同倍数缩放，如同几何图形的相似变换。

Binomial expansion: (1+x)ⁿ = 1 + nx + [n(n−1)/2!]x² + …

When n is a positive integer, the expansion stops at xⁿ. For rational n, it becomes an infinite series, provided |x|<1. Memorise the pattern: factorial denominators and descending n factors.

二项展开式中，当 n 为正整数时展开到 xⁿ 为止；若 n 为有理数且 |x|<1，则变成无穷级数。记忆要点：分母是阶乘，分子是递降的 n 因子。

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4. Trigonometry & Circular Functions | 三角学与圆函数术语

A radian is the angle subtended by an arc equal to the radius. 180° = π radians. Think: ‘radius‑angle’ or simply π ≈ 3.14, half a circle’s circumference ratio.

弧度是弧长等于半径时所对的圆心角。180° = π 弧度。可记 “半径‑弧度” 或记住半圆周长比是 π。

The fundamental identity sin²θ + cos²θ = 1 links sine and cosine. It comes directly from the unit circle: the coordinates (cosθ, sinθ) lie on x²+y²=1. Visualise a right triangle with hypotenuse 1.

基本恒等式 sin²θ + cos²θ = 1 将正弦与余弦联系起来。源自单位圆：点 (cosθ, sinθ) 满足 x²+y²=1。可视作斜边为 1 的直角三角形。

SOH CAH TOA is the classic mnemonic for right‑angled triangles: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. In Chinese, ‘正对斜, 余邻斜, 正对邻’.

SOH CAH TOA 是经典的直角三角形助记法：正弦=对边/斜边，余弦=邻边/斜边，正切=对边/邻边。中文可记：“正对斜，余邻斜，正对邻”。

Compound angle formulas: sin(A±B) = sinAcosB ± cosAsinB (the sign stays the same). For cosine: cos(A±B) = cosAcosB ∓ sinAsinB (sign flips). A rhyme helps: ‘sine: sign keeps, cosine: cos‑sign flips’.

和差角公式：sin(A±B) = sinAcosB ± cosAsinB（符号保持不变）。cos(A±B) = cosAcosB ∓ sinAsinB（符号翻转）。口诀：“正弦符号跟，余弦符号反”。

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5. Differentiation | 微分学术语

The derivative f′(x) gives the gradient of the tangent at any point. It measures instantaneous rate of change. Think ‘derive’ as ‘drive the slope’.

导数 f′(x) 给出任意点处切线的斜率，衡量瞬时变化率。可将 “导” 理解为 “引导出斜率”。

Power rule: d/dx (xⁿ) = nxⁿ⁻¹

The exponent comes down as a multiplier, then reduce the power by 1. ‘Bring down, knock one off’ – a simple rhythm for differentiation.

幂法则：指数作为乘数下移，次幂减一。“指数下移，幂次减一” 的节奏能帮你快速求导。

The chain rule handles composite functions: dy/dx = dy/du × du/dx. Differentiate the outer function, leaving the inner untouched, then multiply by the derivative of the inner. ‘Outer‑inner‑multiply’ – like peeling an onion.

链式法则处理复合函数：dy/dx = dy/du × du/dx。先对外层函数求导，内层保持不变，再乘以内层导数。“外‑内‑乘” 如剥洋葱的顺序。

The product rule: d/dx (uv) = u′v + uv′. ‘Prime first times second plus first times prime second’ – or ‘u‑prime‑v plus u‑v‑prime’. You can chant it to a waltz beat.

乘积法则：d/dx (uv) = u′v + uv′。“第一个求导乘第二，加第一个乘第二个求导”。可用三拍节奏吟唱记忆。

A stationary point occurs where f′(x)=0. The second derivative f″(x) classifies it: positive → minimum, negative → maximum, zero → possible inflection. Imagine a car pausing – that is a stationary point.

驻点出现在 f′(x)=0 处。二阶导数 f″(x) 判定类型：正→极小，负→极大，零→可能拐点。想象汽车暂停，即为驻点。

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6. Integration | 积分类术语

Indefinite integration is the reverse of differentiation: ∫ f′(x) dx = f(x) + C. Never forget the constant C – it represents a whole family of curves shifted vertically.

不定积分是微分的逆运算：∫ f′(x) dx = f(x) + C。常数 C 绝不能丢，它代表垂直平移的整个曲线族。

The definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve, the x‑axis and the lines x=a, x=b. ‘Limit b minus limit a’ after you integrate. Remember area above x‑axis is positive, below is negative.

定积分 ∫ₐᵇ f(x) dx 给出曲线与 x 轴及垂线 x=a, x=b 之间的带符号面积。先积分再代入“上限减下限”。x 轴上方面积为正，下方为负。

Integration by substitution transforms a difficult integral by letting u = g(x), so du = g′(x)dx. Swap every x for u, including the differential. Think of it as changing the ‘language’ of the integral to make it simpler.

换元积分法令 u = g(x), 则 du = g′(x)dx，将所有 x 代换为 u，包括微分。可想象成将积分的 “语言” 更换，以简化计算。

Integration by parts comes from the product rule: ∫ u dv = uv − ∫ v du. Choose u using LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) priority. ‘Leaving Integration Always To Experts’ is a playful reminder.

分部积分法源自乘积法则：∫ u dv = uv − ∫ v du。按 LIATE 优先级选 u（对数、反三角、代数、三角、指数）。可用 “LIATE 顺序” 记忆，或 “留爱给天才”。

Area between two curves y=f(x) and y=g(x) is ∫ₐᵇ [f(x)−g(x)] dx. Always subtract the lower curve from the upper one. Check which is ‘on top’ using a quick sketch.

两曲线 y=f(x) 与 y=g(x) 之间的面积是 ∫ₐᵇ [f(x)−g(x)] dx。总是 “上减下”。快速画草图确认谁在上方，即可列式。

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7. Vectors | 向量术语

A vector has both magnitude and direction, unlike a scalar. Represented by a column matrix or as ai + bj in 2D. Think arrow: length and which way it points.

向量既有大小又有方向，与标量不同。二维中表示为列矩阵或 ai + bj。可想象成箭头：长度及指向。

The magnitude of vector v = ai + bj is |v| = √(a² + b²). This is simply the length of the arrow, found by Pythagoras.

向量 v = ai + bj 的模是 |v| = √(a² + b²)，即箭头的长度，用勾股定理求得。

A unit vector has magnitude 1. To find the unit vector in the direction of v, divide v by |v|. ‘Unit’ means size one – like a ruler of length 1 pointing that way.

单位向量的模为 1。v 方向上的单位向量等于 v / |v|。“单位” 即尺寸为 1，犹如一把指向该方向、长度为 1 的尺。

The dot product a·b = |a||b|cosθ. It helps find the angle between vectors. If a·b = 0, they are perpendicular. Think ‘dot detects the angle’ – a projector’s light.

点积 a·b = |a||b|cosθ。可用于求向量夹角。若 a·b=0，则两向量垂直。“点探测角度” 可助记。

A vector equation of a line is r = a + λb, where a is a position vector on the line and b is the direction. λ sweeps along the line. ‘Starting point plus a journey in direction b’.

直线的向量方程为 r = a + λb，a 是直线上一点的位置向量，b 是方向向量。λ 变化描绘出整条直线。“起点加沿 b 方向的旅程”。

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8. Probability & Statistics | 概率与统计术语

A random variable assigns a numerical value to each outcome of a random experiment. Discrete ones can be listed; continuous ones take any value in an interval. Think ‘variable that dances to chance’.

随机变量为随机试验的每个结果赋予数值。离散型可列举，连续型在区间内任意取值。可想象成 “随机会跳舞的变量”。

Expected value E(X) is the long‑run average, calculated as Σx·P(X=x). ‘Expectation is a weighted average – each value is pulled by its probability.’

期望值 E(X) 是长期平均值，计算公式为 Σx·P(X=x)。期望是加权平均，每个值被其概率 “拉扯”。

The binomial distribution B(n, p) counts successes in n independent trials with constant probability p. Mean = np, variance = np(1−p). Remember ‘B for binary outcomes – success/failure’.

二项分布 B(n,p) 描述在 n 次独立试验中成功次数的分布，每次成功概率为 p。均值=np，方差=np(1−p)。“二项” 暗示每次试验只有两种结果。

The normal distribution N(μ, σ²) is a bell‑shaped continuous distribution. 68% of data falls within μ±σ, 95% within μ±2σ. ‘Normal’ suggests ‘bell‑shaped normality’ – symmetric around the mean.

正态分布 N(μ, σ²) 是钟形连续分布。约 68% 的数据落在 μ±σ 内，95% 在 μ±2σ 内。“正态” 暗示对称钟形常态，以均值为中心。

In hypothesis testing, the p‑value measures the strength of evidence against the null hypothesis. A small p‑value (<0.05) suggests the result is significant. ‘P for proof against the default’.

假设检验中，p 值衡量反对原假设的证据强度。小的 p 值（通常 <0.05）表示结果显著。可记 “p 代表推翻默认假设的证明”。

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9. Mechanics Terminology | 力学词汇速记

Displacement is the vector quantity of position relative to the origin; velocity is rate of change of displacement; acceleration is rate of change of velocity. ‘Differentiate downwards from s to v to a; integrate upwards.’

位移是相对于原点的向量位置；速度是位移变化率；加速度是速度变化率。求导从 s 到 v 到 a 向下，积分

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