📚 Year 9 AQA Further Maths: Formula & Theorem Quick Reference Handbook | Year 9 AQA 进阶数学：公式定理速查手册
Welcome to your go-to revision guide for Year 9 AQA Further Maths. This handbook gathers all the essential formulas, identities, and theorems you need to master, presented in a clear bilingual format to support both English and Chinese learners. Use it alongside your class notes and past papers to strengthen your problem-solving skills and build confidence for assessments.
欢迎使用这本为你量身打造的 Year 9 AQA 进阶数学复习速查手册。本手册整理了你需要掌握的所有核心公式、恒等式和定理，并以清晰的双语形式呈现，同时支持英文和中文学习者的需求。请结合课堂笔记和历年试题使用，以此强化解题能力，并为测评积累信心。
1. Algebraic Identities and Expansion | 代数恒等式与展开
wo binomials: (a + b)(c + d) = ac + ad + bc + bd. Recognising special products saves time.
分配律是所有展开运算的基础。单项式乘多项式：a(b + c) = ab + ac。两个二项式相乘：(a + b)(c + d) = ac + ad + bc + bd。识别特殊乘积可以节省时间。
Difference of two squares: a² - b² = (a + b)(a - b)
平方差公式：a² - b² = (a + b)(a - b)
Perfect square: (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b²
完全平方公式：(a + b)² = a² + 2ab + b² 及 (a - b)² = a² - 2ab + b²
Expanding (x + 5)(x - 3) gives x² + 2x - 15.
展开 (x + 5)(x - 3) 得 x² + 2x - 15。
Factorising x² - 49 relies on the difference of two squares: (x + 7)(x - 7).
对 x² - 49 进行因式分解，利用平方差公式可得：(x + 7)(x - 7)。
2. Linear Equations and Inequalities | 线性方程与不等式
To solve linear equations, perform the same operation on both sides to isolate the variable. For inequalities, the inequality sign reverses when multiplying or dividing by a negative number.
解线性方程时，在等式两边执行相同的运算以分离变量。对于不等式，当两边同乘或同除以一个负数时，不等号方向需要改变。
General form: ax + b = 0 → x = -b/a (if a ≠ 0)
一般形式：ax + b = 0 → x = -b/a (a ≠ 0)
Solve 3x - 7 ≤ 5: 3x ≤ 12, so x ≤ 4.
解 3x - 7 ≤ 5：3x ≤ 12，因此 x ≤ 4。
Solve -2x > 8: dividing by -2 flips the sign, giving x < -4.
解 -2x > 8：除以 -2 后不等号转向，得到 x < -4。
ke 3 < 2x + 1 ≤ 9 are solved in one chain: subtract 1, then divide by 2, giving 1 < x ≤ 4.
像 3 < 2x + 1 ≤ 9 这样的复合不等式可一步求解：先减 1 再除以 2，得出 1 < x ≤ 4。
3. Indices and Standard Form | 指数与标准形式
Index laws simplify expressions involving powers. They are essential for manipulating algebraic terms and working with standard form (scientific notation).
指数定律能简化含有幂的表达式。掌握它们对于处理代数项以及使用标准形式（科学记数法）至关重要。
Core laws: am × an = am+n, am ÷ an = am-n, (am)n = amn, a⁰ = 1, a-n = 1/an
基本定律：am × an = am+n，am ÷ an = am-n，(am)n = amn，a⁰ = 1，a-n = 1/an
Standard form expresses numbers as a × 10n where 1 ≤ a < 10 and n is an integer. E.g., 0.00042 = 4.2 × 10⁻⁴.
标准形式将数字表示为 a × 10n，其中 1 ≤ a < 10 且 n 为整数。例如 0.00042 = 4.2 × 10⁻⁴。
Fractional indices represent roots: a1/2 = √a, a1/3 = ∛a, and am/n = (ⁿ√a)m.
分数指数代表方根：a1/2 = √a，a1/3 = ∛a，且 am/n = (ⁿ√a)m。
4. Sequences: Arithmetic and Quadratic | 数列：等差数列与二次数列
Sequences are patterns of numbers following a rule. Arithmetic (linear) sequences have a constant difference; quadratic sequences have a constant second difference.
数列是按规则排列的一串数字。等差（线性）数列具有恒定的差值；二次数列则具有恒定的二阶差值。
nth term of arithmetic sequence: un = a + (n - 1)d, where a = first term, d = common difference.
等差数列第 n 项公式：un = a + (n - 1)d，其中 a 为首项，d 为公差。
Quadratic nth term: un = an² + bn + c. Find a = half of the second difference, then solve for b and c.
二次数列第 n 项：un = an² + bn + c。先令 a = 二阶差值的一半，再解出 b 和 c。
Sequence 2, 7, 12, 17,... has d = 5, so nth term = 5n - 3.
数列 2, 7, 12, 17,... 的公差 d = 5，因此第 n 项公式为 5n - 3。
Sequence 3, 6, 11, 18,... has second difference 2, so a = 1; nth term = n² + 2.
数列 3, 6, 11, 18,... 的二阶差值为 2，故 a = 1；第 n 项公式为 n² + 2。
5. Coordinate Geometry: Straight Lines | 坐标几何：直线
describes any straight line, where m is the gradient and c is the y-intercept. Understanding parallel and perpendicular lines is key for coordinate problems.
斜截式 y = mx + c 可描述任一直线，其中 m 是斜率，c 是 y 轴截距。理解平行线与垂直线是解决坐标问题的关键。
Gradient between two points (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁)/(x₂ - x₁)
两点 (x₁, y₁) 与 (x₂, y₂) 间的斜率：m = (y₂ - y₁)/(x₂ - x₁)
Parallel lines: m₁ = m₂. Perpendicular lines: m₁ × m₂ = -1.
平行线：m₁ = m₂。垂直线：m₁ × m₂ = -1。
Midpoint of a segment: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
线段中点：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
Equation of a line given a point (x₁, y₁) and gradient m: y - y₁ = m(x - x₁).
已知一点 (x₁, y₁) 和斜率 m 的直线方程：y - y₁ = m(x - x₁)。
6. Angles, Polygons and Parallel Lines | 角、多边形与平行线
Angle facts for intersecting lines and within polygons form a major reasoning toolkit. These are used in proofs and problem-solving across geometry topics.
交叉线中的角度关系以及多边形内角规律构成了重要的推理工具，常用于证明和各类几何问题中。
| Theorem / Rule | Statement | 中文陈述 |
|---|---|---|
| Vertically opposite angles | Equal | 对顶角相等 |
| Angles on a straight line | Sum to 180° | 平角之和为 180° |
| Angles around a point | Sum to 360° | 周角之和为 360° |
| Alternate angles (parallel lines) | Equal | 内错角相等 |
| Corresponding angles (parallel lines) | Equal | 同位角相等 |
| Co-interior angles (parallel lines) | Sum to 180° | 同旁内角互补（和为 180°） |
um of interior angles of an n-sided polygon: (n - 2) × 180°. Sum of exterior angles: 360°.
n 边形内角和：(n - 2) × 180°。外角和总是 360°。
7. Perimeter, Area and Circles | 周长、面积与圆
Accurate recall of area formulas for 2D shapes is essential. Circle calculations involve π and should be left in terms of π unless a rounded answer is specified.
准确记忆二维图形面积公式至关重要。圆的相关计算含有 π，除非题目要求四舍五入，否则答案应保留 π。
| Shape | Perimeter / Circumference | Area |
|---|---|---|
| Triangle | a + b + c | ½ × base × height |
| Parallelogram | 2(a + b) | base × perpendicular height |
| Trapezium | Sum of all sides | ½(a + b) × h |
| Circle | C = 2πr or C = πd | A = πr² |
中文对照：三角形面积 = ½ × 底 × 高；平行四边形面积 = 底 × 高；梯形面积 = ½(上底 + 下底) × 高；圆的周长 C = 2πr，面积 A = πr²。
Arc length (sector) = (θ/360) × 2πr; Sector area = (θ/360) × πr².
弧长 = (θ/360) × 2πr；扇形面积 = (θ/360) × πr²。
8. Volume and Surface Area of 3D Shapes | 三维图形的体积与表面积
Volume measures the space inside a solid. Surface area is the total area of all faces. Prisms have a uniform cross-section, making their volume easy to compute.
体积衡量立体内部所占空间。表面积则是所有面的总面积。棱柱的截面均匀，因此体积易于计算。
Prism volume: V = area of cross-section × length. Cylinder: V = πr²h.
棱柱体积：V = 截面面积 × 长度。圆柱体积：V = πr²h。
Pyramid volume: V = ⅓ × base area × height. Cone: V = ⅓πr²h.
棱锥体积：V = ⅓ × 底面积 × 高。圆锥体积：V = ⅓πr²h。
Sphere: volume = ⁴⁄₃πr³, surface area = 4πr².
球体：体积 = ⁴⁄₃πr³，表面积 = 4πr²。
Surface area of a cylinder (closed) = 2πr² + 2πrh. Cone (with slant height l) = πrl + πr².
圆柱（封闭）表面积 = 2πr² + 2πrh。圆锥（母线长 l）表面积 = πrl + πr²。
9. Pythagoras' Theorem and Trigonometry | 勾股定理与三角函数
Right-angled triangle relationships form the basis of trigonometry. The three trigonometric ratios link angles to side lengths.
直角三角形的边角关系是三角学的基础。三个三角比将角度与边长联系起来。
Pythagoras' theorem: a² + b² = c², where c is the hypotenuse.
勾股定理：a² + b² = c²，其中 c 为斜边。
tios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Exact values for 30°, 45°, 60° are expected to be memorised: sin 30° = ½, cos 45° = 1/√2, tan 60° = √3, etc.
需熟记 30°、45°、60° 的精确值：sin 30° = ½，cos 45° = 1/√2，tan 60° = √3 等。
need to find the base diagonal first: d² = a² + b², then the space diagonal D² = d² + h².
在三维勾股问题中，常需先求出底面对角线：d² = a² + b²，然后再求空间对角线 D² = d² + h²。
10. Transformations and Vectors | 图形变换与向量
Transformations move or change a shape. Vectors describe translations and are written as column vectors or using i, j notation.
变换可以移动或改变图形。向量用来描述平移，通常用列向量或 i、j 符号表示。
| Transformation | Key description | 中文说明 |
|---|---|---|
| Translation | Move by a vector (ᵃ b) | 按向量 (ᵃ b) 平移 |
| Reflection | Mirror in a given line (x = k, y = k, y = x, etc.) | 以给定直线为对称轴进行反射 |
| Rotation | Turn about centre by angle (90°, 180°, 270° clockwise/anticlockwise) | 绕中心旋转指定角度（如 90°、180°、270° 顺/逆时针） |
| Enlargement | Scale factor k from centre; if k negative, also reversed | 从中心以比例因子 k 缩放；k 为负时还包括反向 |
ctor: v = (x y) represents the translation (x units across, y units up). Vector magnitude: |v| = √x² + y².
列向量 v = (x y) 表示平移 x 水平单位、y 垂直单位。向量的大小：|v| = √x² + y²。
11. Probability | 概率
Probability measures the chance of an event, ranging from 0 (impossible) to 1 (certain). The sum of probabilities of all mutually exclusive outcomes is 1.
概率用于衡量事件发生的可能性，取值范围从 0（不可能）到 1（必然）。所有互斥结果的概率之和为 1。
P(event) = (number of favourable outcomes) / (total number of outcomes).
P(事件) = 有利结果数 / 总结果数。
For independent events A and B: P(A and B) = P(A) × P(B).
对于独立事件 A 和 B：P(A 与 B) = P(A) × P(B)。
For mutually exclusive events: P(A or B) = P(A) + P(B).
对于互斥事件：P(A 或 B) = P(A) + P(B)。
Tree diagrams help with combined events: multiply along branches, add separate paths.
树状图有助于处理组合事件：沿分支相乘，不同路径相加。
Expected frequency = probability × number of trials.
期望频数 = 概率 × 试验次数。
12. Statistics and Data Representation | 统计与数据表示
Statistical measures summarise data sets. You should be able to calculate averages and construct diagrams like box plots, cumulative frequency graphs, and histograms.
统计量用于归纳数据集。你应该能够计算各种平均数，并绘制框图、累积频率图与直方图。
Averages: Mean = Σx / n. Median = middle value. Mode = most frequent. Range = max - min.
平均数：平均值 = Σx / n。中位数 = 中间值。众数 = 出现频率最高的值。极差 = 最大值 - 最小值。
(25th percentile), Q₂ = median, Q₃ = upper quartile. Interquartile range (IQR) = Q₃ - Q₁.
四分位数：Q₁ = 下四分位数（第 25 百分位数），Q₂ = 中位数，Q₃ = 上四分位数。四分位距 IQR = Q₃ - Q₁。
Cumulative frequency graph: plot upper class boundaries against running total. Find median and quartiles by reading across from the relevant cumulative frequency positions.
累积频率图：将上限与累计频数对应描点。通过从相应的累积频率位置水平读取得到中位数和四分位数。
requency density = frequency / class width. The area of each bar represents the frequency.
在直方图中，频率密度 = 频数 ÷ 组距。每个长方条的面积代表该组的频数。
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