📚 Maths Formula Booklet | 数学公式手册
This booklet is a concise, exam-focused reference for the formulas used in A-level and IGCSE Mathematics. Keep it beside you when you work through past papers, and use it to check definitions, identities and standard results in the final days before the exam. It is not a replacement for understanding: it is a revision aid that makes recall faster and more accurate, and it should always be read alongside your own notes and worked examples.
本文是一份简洁、紧扣考点的数学公式参考，适用于 A-level 与 IGCSE 数学。做历年真题时请把它放在手边，在考前最后几天用它核对定义、恒等式与标准结论。它不能替代理解，而是一种复习工具，让回忆更快、更准，并且应当与你的笔记和例题一起使用。
1. Algebraic Identities and Surds | 代数恒等式与根式
Begin with the three binomial identities that appear in almost every paper. They are the engine behind expansion, factorisation and completing the square, and they also explain why the difference of two squares factorises so cleanly.
先掌握三个几乎每份卷子都会出现的二项式恒等式。它们是展开、因式分解与配方法背后的发动机，也解释了平方差为何能如此干净地分解。
(a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²
Surds obey √ab = √a × √b and √a/b = √a / √b. To rationalise a denominator such as 1/√a, multiply numerator and denominator by √a to obtain √a / a; for a denominator of the form a + √b, multiply by the conjugate a − √b ⁿ = aᵐⁿ, a⁰ = 1, and a⁻ⁿ = 1/aⁿ for a ≠ 0. Fractional indices connect directly to roots: a1/n = ⁿ√a, and am/n = ⁿ√aᵐ.
根式运算遵循 √ab = √a × √b 与 √a/b = √a / √b。要把 1/√a 这样的分母有理化，可把分子分母同乘 √a，得到 √a / a；若分母形如 a + √b，则乘以其共轭式 a − √b。指数律包括 aᵐ × aⁿ = aᵐ⁺ⁿ、aᵐ ÷ aⁿ = aᵐ⁻ⁿ、(aᵐ)ⁿ = aᵐⁿ、a⁰ = 1，以及 a ≠ 0 时 a⁻ⁿ = 1/aⁿ。分数指数与根式直接相通：a1/n = ⁿ√a，am/n = ⁿ√aᵐ。
2. Quadratic Equations and Inequalities | 二次方程与不等式
iminant Δ = b² − 4ac decides how many real roots exist, and this single value answers most "show that" questions about intersections of curves and lines.
对于一般二次方程 ax² + bx + c = 0，其根由求根公式给出。判别式 Δ = b² − 4ac 决定实根的个数，这一个数值就能回答大多数关于曲线与直线交点个数的证明题。
x = (−b ± √b² − 4ac) / (2a) Δ = b² − 4ac
tes ax² + bx + c as a(x + b/2a)² + (c − b²/4a), which immediately reveals the vertex of the parabola at (−b/2a, c − b²/4a) and the line of symmetry x = −b/2a. The sum and product of the roots of a monic quadratic are useful too: the roots add to −b/a and multiply to c/a.
若 Δ > 0，有两个不同实根；若 Δ = 0，有一个重根；若 Δ < 0，没有实根。配方把 ax² + bx + c 改写成 a(x + b/2a)² + (c − b²/4a)，由此立刻读出抛物线顶点 (−b/2a, c − b²/4a) 与对称轴 x = −b/2a。首项系数为 1 时的两根之和与两根之积也很常用：两根之和为 −b/a，两根之积为 c/a。
For a quadratic inequality such as ax² + bx + c > 0, find the real roots first, sketch the parabola, then test the sign in each interval. The sign can only change at a real root, so the sketch alone usually gives the correct solution set, written with the standard inequality notation.
对于 ax² + bx + c > 0 这类二次不等式，先求出实根，画出抛物线草图，再判断每个区间上的符号。符号只可能在实根处改变，因此仅凭草图通常就能写出正确的解集，并用标准不等式记号表示。
3. Functions, Graphs and Transformations | 函数、图像与变换
unction the inverse f⁻¹ reverses the mapping, satisfying f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. The graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x.
函数把每一个输入对应到唯一输出。定义域是允许输入的集合，值域是可能输出的集合；对于一一函数，反函数 f⁻¹ 会逆转这一对应，满足 f(f⁻¹(x)) = x 与 f⁻¹(f(x)) = x。y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的镜像。
). To find an inverse, write y = f(x), swap x and y, then rearrange to make y the subject.
复合函数记作 fg(x)，意思是先作用 g 再作用 f；顺序很重要，通常 fg(x) 不等于 gf(x)。求反函数的方法是令 y = f(x)，交换 x 与 y，再整理成以 y 为主元的形式。
(x) are worth memorising as a table, because graph questions are quick marks once the rules are automatic.
y = f(x) 的标准变换值得当作表格背下来，因为一旦规则变成本能，图像题就是快速得分点。
| Transformation | Effect on the graph | 中文说明 |
| y = f(x) + a | Translation a units upwards | 向上平移 a 个单位 |
| y = f(x + a) | Translation a units to the left | 向左平移 a 个单位 |
| y = af(x) | Vertical stretch, scale factor a | 竖直方向拉伸 a 倍 |
| y = f(ax) | Horizontal compression, factor 1/a | 水平方向压缩为 1/a |
| y = −f(x) | Reflection in the x-axis | 关于 x 轴反射 |
4. Sequences and Series | 数列与级数
terms can be written in two equivalent ways, and the second form is faster when the last term is already known.
等差数列有固定的公差 d，因此第 n 项为 uₙ = a + (n − 1)d。前 n 项和有两种等价写法，当末项已知时第二种更快。
Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)
ₙ = arⁿ⁻¹. The partial sum and the sum to infinity are standard results, and the condition |r| < 1 must be stated whenever the infinite sum is used.
等比数列有固定的公比 r，因此第 n 项为 uₙ = arⁿ⁻¹。部分和与无穷和都是标准结论，只要用到无穷和，就必须写出条件 |r| < 1。
Sₙ = a(1 − rⁿ) / (1 − r), r ≠ 1 S∞ = a / (1 − r), |r| < 1
Binomial expansion completes the picture: (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ, with the general term ⁿCᵣ aⁿ⁻ʳ bʳ and ⁿCᵣ = n! / (r!(n − r)!). The binomial coefficients form Pascal's triangle, which is the quickest route for small values of n.
二项式展开补全这一块：(a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ，通项为 ⁿCᵣ aⁿ⁻ʳ bʳ，且 ⁿCᵣ = n! / (r!(n − r)!)。二项式系数构成帕斯卡三角形，在 n 较小时是最快的算法。
5. Coordinate Geometry | 坐标几何
These four results cover most coordinate geometry questions on straight lines and circles. Learn them as a connected set rather than as separate facts.
以下四条结论覆盖了直线与圆的大部分坐标几何题。请把它们当作一个相互关联的整体来记，而不是孤立的事实。
m = (y₂ − y₁) / (x₂ − x₁) d = √((x₂ − x₁)² + (y₂ − y₁)²)
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2) y − y₁ = m(x − x₁)
rpendicular rule is the one most often forgotten under time pressure, so check it explicitly when a question mentions normals, tangents or altitudes.
两条直线平行当且仅当 m₁ = m₂，垂直当且仅当 m₁ × m₂ = −1。垂直关系是考试时间紧张时最常被遗忘的一条，因此当题目提到法线、切线或高时，务必明确检查。
². To find the centre and radius from an expanded form, complete the square in both x and y; the radius is then the square root of the constant term after completing the square.
圆心为 (a, b)、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。若给出展开式，要对 x 与 y 分别配方求出圆心与半径；配平后常数项的平方根就是半径。
6. Trigonometry and Triangle Results | 三角学与解三角形
The Pythagorean identity and the definition of tangent underpin every trigonometric simplification. Learn the exact values for 0°, 30°, 45°, 60° and 90° as well, because they remove a whole class of avoidable errors.
勾股恒等式与正切的定义是一切三角化简的基础。同时要记住 0°、30°、45°、60°、90° 的精确值，它们能消除一整类本可避免的错误。
sin²θ + cos²θ = 1 tanθ = sinθ / cosθ
For any triangle labelled with sides a, b, c opposite angles A, B, C, the sine rule and cosine rule solve every non-right-angled case, while the area formula gives the area from two sides and the included angle.
对于边 a、b、c 分别对角 A、B、C 的任意三角形，正弦定理与余弦定理可以解决所有非直角三角形问题，而面积公式用两边及其夹角求面积。
a / sin A = b / sin B = c / sin C
a² = b² + c² − 2bc cos A Area = ½ ab sin C
r²θ, where θ must be in radians. Check your calculator mode before any trigonometry question; a wrong mode is one of the most expensive mistakes in the exam.
弧度制下 π rad = 180°。弧长与扇形面积分别为 l = rθ 与 A = ½ r²θ，其中 θ 必须以弧度表示。做任何三角题之前先检查计算器模式；模式选错是考场上代价最高的错误之一。
7. Exponentials and Logarithms | 指数与对数
A logarithm answers the question "to what power must the base be raised?". The laws of logarithms turn multiplication into addition and powers into multiples, which is exactly why they are used to solve equations where the unknown is an index.
对数回答的问题是"底数要升到多少次幂"。对数律把乘法变成加法、把幂变成倍数，这正是它被用来解未知量位于指数位置的方程的原因。
logₐ(xy) = logₐx + logₐy logₐ(x/y) = logₐx − logₐy
logₐ(xⁿ) = n logₐx logₐa = 1 logₐ1 = 0
s of a base your calculator supports: logₐx = log_b x / log_b a. The natural exponential eˣ and the natural logarithm ln x are inverse functions, so eln x = x for x > 0 and ln(eˣ) = x for all x. Any exponential can be written in base e using aˣ = ex ln a.
换底公式可以把任意底数改写为计算器支持的底数：logₐx = log_b x / log_b a。自然指数 eˣ 与自然对数 ln x 互为反函数，因此 x > 0 时 eln x = x，而对所有 x 有 ln(eˣ) = x。任何指数式都可以用 aˣ = ex ln a 改写成以 e 为底。
8. Differentiation | 微分
The power rule is the foundation of every derivative in the course. Combined with the product, quotient and chain rules, it differentiates essentially any function you will meet at this level.
幂法则是本课程一切导数的基础。配合乘积法则、商法则与链式法则，它几乎能对你在这一阶段遇到的所有函数求导。
d/dx (xⁿ) = n xⁿ⁻¹ d/dx (eˣ) = eˣ d/dx (ln x) = 1/x
in x) = cos x d/dx (cos x) = −sin x d/dx (tan x) = sec²x
ction of u and u is a function of x. In function notation the chain rule reads d/dx f(g(x)) = f′(g(x)) × g′(x).
乘积法则为 (uv)′ = u′v + uv′，商法则为 (u/v)′ = (u′v − uv′)/v²，链式法则为：当 y 是 u 的函数、u 是 x 的函数时，dy/dx = dy/du × du/dx。用函数记号写出即为 d/dx f(g(x)) = f′(g(x)) × g′(x)。
est is inconclusive and you should examine the sign of f′(x) on either side. The tangent at a point has gradient f′(x), and the normal has gradient −1/f′(x).
在驻点处 f′(x) = 0。二阶导数可用于判定其性质：若 f″(x) > 0 为极小值，若 f″(x) < 0 为极大值；若 f″(x) = 0 则此法失效，需要考察 f′(x) 在两侧的符号。某点处切线的斜率为 f′(x)，法线的斜率为 −1
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