📚 AS Maths: Mind Map Quick Memorisation | AS 数学：思维导图速记
Mind maps turn the sprawling AS Mathematics syllabus into memorable visual structures. By organising topics around central nodes, branching out to key subtopics and linking related ideas, you can move from passive reading to active recall. This article walks you through mind‑map‑based shortcuts for every core Pure and Statistics topic, helping you fix formulas, theorems and problem‑solving flows in long‑term memory.
思维导图能把零散的 AS 数学知识梳理成易于记忆的可视化结构。围绕核心主题展开分支、串联关联概念，你的复习就能从被动翻书升级为主动回想。本文逐一讲解纯数与统计核心章节的思维导图速记法，助你把公式、定理和解题流程牢牢印在脑海中。
1. Mind Map Basics | 思维导图基础
Centre bubble: ‘AS Maths’ → main branches: Pure 1, Pure 2, Statistics. Each branch splits into chapters. Use colour, symbols, and abbreviations. A good mind map turns a chapter into one page of keywords and key equations, placed spatially to mirror the logical flow of the topic.
中心气泡：『AS 数学』→ 主分支：纯数1、纯数2、统计。每个分支再拆成章节。用颜色、符号和缩写。一份好的思维导图能把一章浓缩成一页关键词和关键方程，并按逻辑结构布局。
Always start with a known structure, then add details in layers. For instance, under ‘Differentiation’ write the first‑principles definition, then rules, then applications. Draw thin connecting lines to show relationships (e.g. differentiation ↔ integration).
始终从已知框架入手，再层层补充细节。比如在『微分』下先写第一原理定义，再写求导法则，最后写应用。用细连线标出关联（如微分 ↔ 积分）。
2. Algebra & Functions | 代数与函数
The central node ‘Functions’ branches into domain/range, composite functions, inverse functions, and transformations. A quick‑check box lists all function notation: f(x), fg(x), f⁻¹(x).
中心节点『函数』分出定义域/值域、复合函数、反函数与图像变换。速查框内记下所有函数符号：f(x), fg(x), f⁻¹(x)。
Domain & range: Domain is the set of possible inputs; range is the set of possible outputs. Write them in set notation or inequalities. Always check for restrictions like square roots, denominators, or logs.
定义域与值域：定义域是所有可能输入的集合；值域是所有可能输出的集合。用集合符号或不等式表示。永远先检查限制：平方根、分母、对数等。
Transformations: y = f(x) + a → vertical shift; y = f(x + a) → horizontal shift; y = a f(x) → vertical stretch; y = f(ax) → horizontal stretch. Direction matters: f(x+2) moves 2 left. Place these inside a little table on the map.
图像变换：y = f(x) + a → 竖直平移；y = f(x + a) → 水平平移；y = a f(x) → 竖直拉伸；y = f(ax) → 水平拉伸。方向很关键：f(x+2) 向左移2。把这些放进思维导图上的一个小表格里。
Inverse functions: Swap x and y, then solve for y. Graphically, the inverse is a reflection in the line y = x. Only one‑to‑one functions have inverses unless the domain is restricted.
反函数：交换 x 和 y，再解出 y。图像上，反函数关于直线 y = x 对称。只有一一映射的函数才有反函数，除非限制定义域。
3. Quadratic Functions | 二次函数
Quadratic functions centre on the standard form ax² + bx + c. Three key forms branch out: completed square a(x-h)² + k, factorised form a(x-p)(x-q), and the discriminant Δ = b² – 4ac.
二次函数以标准形式 ax² + bx + c 为中心。三个关键形式作为分支：配方式 a(x-h)² + k、因式形式 a(x-p)(x-q) 和判别式 Δ = b² – 4ac。
Discriminant Δ: Δ > 0 → two distinct real roots; Δ = 0 → one repeated root; Δ < 0 → no real roots. Also used to find intersections: set equations equal, form a new quadratic, check Δ ≥ 0 for intersection.
判别式 Δ：Δ > 0 → 两个相异实根；Δ = 0 → 一个重根；Δ < 0 → 无实根。也用于求交点：令方程相等，构造新二次方程，检查 Δ ≥ 0 表示有交点。
Vertex & symmetry: From a(x-h)² + k, vertex is (h, k). Axis of symmetry x = h. If in standard form, h = -b/(2a), k = f(h). Sketch quickly using vertex, y‑intercept, and roots if any.
顶点与对称性：由 a(x-h)² + k 得顶点 (h, k)，对称轴 x = h。若为标准式，h = -b/(2a)，k = f(h)。画图时先用顶点、y截距和实根（若有）快速勾勒。
4. Equations & Inequalities | 方程与不等式
Start with linear equations, then quadratic equations, simultaneous equations, and inequalities. A small box lists the solving toolkit: factorising, quadratic formula, completing the square, elimination, substitution.
从一次方程出发，再到二次方程、联立方程和不等式。一个小方框列明解题工具箱：因式分解、求根公式、配方法、消元法、代入法。
Quadratic formula: x = [-b ± √(b² – 4ac)] / (2a). Memorise using a rhythm or visual placement near the Δ box. Use it when factorising fails.
二次公式：x = [-b ± √(b² – 4ac)] / (2a)。用节奏或靠近 Δ 方框的位置记忆。因式分解做不出来时就用它。
Inequality rules: Multiplying or dividing by a negative flips the sign. Quadratic inequalities: solve the corresponding equation, then use a sign diagram or sketch. Always express answer in interval notation or set notation as required.
不等式法则：乘或除以负数时要反转不等号。二次不等式：解相应的方程，再用符号表或草图判断。答案按要求用区间或集合符号表示。
5. Coordinate Geometry | 坐标几何
The mind map hub ‘Straight lines’ splits into gradient, distance, and midpoint. Gradient between (x₁, y₁) and (x₂, y₂): m = (y₂ – y₁)/(x₂ – x₁). Distance: √[(x₂ – x₁)² + (y₂ – y₁)²]. Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2).
思维导图中心『直线』分出斜率、距离和中点。两点 (x₁, y₁) 和 (x₂, y₂) 间斜率 m = (y₂ – y₁)/(x₂ – x₁)。距离：√[(x₂ – x₁)² + (y₂ – y₁)²]。中点：((x₁+x₂)/2, (y₁+y₂)/2)。
Equation forms: y = mx + c (slope‑intercept), y – y₁ = m(x – x₁) (point‑slope), and ax + by + c = 0 (general form). Connect these to parallel (m₁ = m₂) and perpendicular (m₁ m₂ = -1) lines.
方程形式：y = mx + c（斜截式）、y – y₁ = m(x – x₁)（点斜式）、ax + by + c = 0（一般式）。将它们与平行（m₁ = m₂）和垂直（m₁ m₂ = -1）条件相连。
Circles: Centre (a, b), radius r → (x – a)² + (y – b)² = r². Expanded form x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius √(g² + f² – c). Include a mini sketch linking complete‑square steps.
圆：圆心 (a, b)，半径 r → (x – a)² + (y – b)² = r²。展开式 x² + y² + 2gx + 2fy + c = 0 圆心为 (-g, -f)，半径 √(g² + f² – c)。附上迷你草图与配方法步骤。
6. Sequences & Series | 数列与级数
Branch into arithmetic and geometric sequences. Arithmetic: nth term uₙ = a + (n-1)d, sum Sₙ = n/2 [2a + (n-1)d] or Sₙ = n/2 (a + l), where l = last term.
分为等差数列和等比数列。等差：第 n 项 uₙ = a + (n-1)d，求和 Sₙ = n/2 [2a + (n-1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
Geometric sequences: uₙ = arⁿ⁻¹. Sum of first n terms: Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. Infinite sum exists only when |r| < 1: S∞ = a/(1 - r). Add a tiny note: 'r is common ratio'.
等比数列：uₙ = arⁿ⁻¹。前 n 项和：Sₙ = a(1 – rⁿ)/(1 – r)，r ≠ 1。无穷和仅当 |r| < 1 时存在：S∞ = a/(1 - r)。加上小注释：『r 为公比』。
Sigma notation: ∑ means sum. Upper and lower limits tell you where to start and end. Connect to arithmetic and geometric series by rewriting the general term.
求和符号：∑ 表示求和。上下标给出起止项。可由通项重写为等差或等比级数后求和。
7. Trigonometry | 三角学
The central bubble ‘Trig’ has three main spokes: exact values, identities, and equations. A mini unit circle shows sine, cosine and tangent for 0°, 30°, 45°, 60°, 90°. Use the hand trick or the √/2 table to recall exact values.
中心气泡『三角』分出三条主线：精确值、恒等式和方程。迷你单位圆标注 0°、30°、45°、60°、90° 的正弦、余弦和正切。用手指法或 √/2 表格记精确值。
Key identities: tan θ = sin θ / cos θ; sin²θ + cos²θ ≡ 1. These are tools for solving equations. Write them inside a highlighted box on your map.
核心恒等式：tan θ = sin θ / cos θ；sin²θ + cos²θ ≡ 1。它们是解三角方程的工具。在思维导图上用高亮框标出。
Solving trig equations: Use quadrant diagram (CAST) to find all solutions within a given interval. For sin, cos, tan, check related angles: 180° – θ, 180° + θ, 360° – θ. Always confirm the interval and degree/radian mode.
解三角方程：用象限图（CAST）求给定区间内的所有解。正弦、余弦、正切检查相关角：180° – θ、180° + θ、360° – θ。始终确认区间和角度/弧度模式。
8. Differentiation | 微分
Start with the first‑principles definition, then the power rule. For y = xⁿ, dy/dx = n xⁿ⁻¹. Branch to sums, constant multiples, and standard derivatives (sin, cos, eˣ, ln x).
从第一原理定义出发，再到幂法则。对 y = xⁿ，dy/dx = n xⁿ⁻¹。分支加上和、常数倍及标准导数（sin、cos、eˣ、ln x）。
Tangents & normals: gradient of tangent = dy/dx at a point. Normal gradient = -1/(dy/dx). Equation found using point‑slope form. Add this as a connected node under ‘Applications’.
切线与法线：切线斜率 = 函数在某点的 dy/dx。法线斜率 = -1/(dy/dx)。用点斜式写出方程。在『应用』节点下建立连接。
Stationary points: Set dy/dx = 0. Determine nature via second derivative: d²y/dx² > 0 → minimum; d²y/dx² < 0 → maximum; = 0 → check sign change. Draw a quick decision tree on the map.
驻点：令 dy/dx = 0。用二阶导数判断性质：d²y/dx² > 0 → 极小值；d²y/dx² < 0 → 极大值；= 0 → 检查梯度符号变化。在导图上画快速判断树。
9. Integration | 积分
Integration is the reverse of differentiation. For xⁿ (n ≠ -1), ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c. Remember the constant of integration. Connect this to the differentiation branch with a double‑headed arrow.
积分是微分的逆运算。对 xⁿ (n ≠ -1)，∫ xⁿ dx = xⁿ⁺¹/(n+1) + c。不要忘记积分常数。用双箭头把积分与微分分支连起来。
Definite integrals: ∫ₐᵇ f(x) dx gives the area under the curve between x = a and x = b. Compute as F(b) – F(a). Area below the x‑axis is negative; adjust with absolute value if total area is required.
定积分：∫ₐᵇ f(x) dx 给出曲线下 x = a 到 x = b 之间的面积。计算为 F(b) – F(a)。x 轴下方的面积为负；求总面积时取绝对值调整。
Area between curves: Top minus bottom, integrated over the intersection interval. Find intersection points by solving f(x) = g(x). This forms a practical workflow: find limits → set up integral → integrate → substitute.
曲线间面积：上方函数减下方函数，在交点区间上积分。由 f(x) = g(x) 求交点。这形成一条实用工作流：找区间 → 写出积分式 → 积分 → 代值。
10. Statistics Overview | 统计概览
The Statistics mind map begins with ‘Data Representation’ (histograms, box plots, cumulative frequency) and splits into ‘Measures of centre’ (mean, median, mode) and ‘Spread’ (range, interquartile range, variance, standard deviation).
统计思维导图以『数据表示』（直方图、箱线图、累积频率）为起点，分成『中心度量』（平均数、中位数、众数）和『离散程度』（极差、四分位距、方差、标准差）。
Variance formula: σ² = Σ(x – μ)² / n for a population; sample variance s² = Σ(x – x̄)² / (n-1). Use a calculator reminder node: enter data in stats mode, read off σₙ or s.
方差公式：总体方差 σ² = Σ(x – μ)² / n；样本方差 s² = Σ(x – x̄)² / (n-1)。建立一个计算器提醒节点：在统计模式下输入数据，读取 σₙ 或 s。
Probability: Branch to Venn diagrams, tree diagrams, and conditional probability. P(A|B) = P(A ∩ B) / P(B). Mutually exclusive: P(A ∪ B) = P(A) + P(B); independent: P(A ∩ B) = P(A) P(B).
概率：分支包含文氏图、树形图和条件概率。P(A|B) = P(A ∩ B) / P(B)。互斥事件：P(A ∪ B) = P(A) + P(B)；独立事件：P(A ∩ B) = P(A) P(B)。
Binomial distribution: X ~ B(n, p). P(X = k) = (n choose k) pᵏ (1-p)ⁿ⁻ᵏ. Mean μ = np, variance σ² = np(1-p). Use table or formula. Draw a mini decision flow: fixed trials → two outcomes → constant p → independent trials.
二项分布：X ~ B(n, p)。P(X = k) = (n选k) pᵏ (1-p)ⁿ⁻ᵏ。均值 μ = np，方差 σ² = np(1-p)。用表或公式。画一个小决策流：试验次数固定 → 两种结果 → p 恒定 → 各次独立。
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