📚 Year 12 CIE Mathematics: Formula & Theorem Quick Reference Handbook | Year 12 CIE 数学：公式定理速查手册
This concise handbook provides a quick reference to all the essential formulas and theorems covered in the Year 12 CIE Mathematics syllabus (AS Level). It is designed for efficient revision and helps you master the core concepts of Pure Mathematics 1 and Probability & Statistics 1.
这本简明手册为 Year 12 CIE 数学大纲 (AS Level) 中所有必备公式和定理提供快速参考。它旨在帮助你高效复习，牢固掌握纯数学 1 和概率与统计 1 的核心概念。
1. Quadratics & Polynomials | 二次函数与多项式
Quadratic equations and polynomials form the foundation of algebraic manipulation. The quadratic formula provides the roots of any quadratic equation, while the discriminant reveals their nature.
二次方程和多项式是代数运算的基础。二次公式可给出任意二次方程的根，而判别式揭示了根的性质。
The quadratic formula solves ax² + bx + c = 0:
x = [-b ± √(b² – 4ac)] / (2a)
二次公式求解 ax² + bx + c = 0 的根。
The discriminant Δ = b² – 4ac determines the type of roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 means no real roots.
判别式 Δ = b² – 4ac 决定了根的类型：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
Sum and product of roots for ax² + bx + c = 0 are given by α + β = -b/a and αβ = c/a.
对于 ax² + bx + c = 0，根的和与积分别为 α + β = -b/a 和 αβ = c/a。
Completing the square rewrites the quadratic as a(x – h)² + k where h = -b/(2a) and k = c – b²/(4a).
配方法将二次式写成 a(x – h)² + k，其中 h = -b/(2a), k = c – b²/(4a)。
To solve quadratic inequalities such as (x – p)(x – q) > 0, sketch the parabola and use the critical values to determine the interval satisfying the inequality.
求解二次不等式如 (x – p)(x – q) > 0 时，先画出抛物线，再利用临界值确定满足不等式的区间。
The Remainder Theorem states: when a polynomial f(x) is divided by (x – a), the remainder is f(a). The Factor Theorem states: (x – a) is a factor of f(x) if and only if f(a) = 0.
余式定理指出：多项式 f(x) 除以 (x – a) 时，余数为 f(a)；因式定理指出：(x – a) 是 f(x) 的因式当且仅当 f(a) = 0。
2. Functions, Graphs & Transformations | 函数、图像与变换
A function maps each input from its domain to exactly one output in its range. Composite functions combine two functions: fg(x) = f(g(x)). The inverse function f⁻¹(x) exists only if f is one-to-one.
函数将定义域中的每个输入唯一地映射到值域中的一个输出。复合函数组合两个函数：fg(x) = f(g(x))。反函数 f⁻¹(x) 仅在原函数一一对应时才存在。
The domain and range of composite and inverse functions must be carefully determined; the domain of f⁻¹ equals the range of f.
必须谨慎确定复合函数和反函数的定义域与值域；f⁻¹ 的定义域等于 f 的值域。
Graph transformations include translations, stretches, and reflections. Replacing y = f(x) with y = f(x) + a shifts the graph vertically by a.
图像变换包括平移、拉伸和反射。将 y = f(x) 换为 y = f(x) + a 使图像垂直平移 a 个单位。
Replacing y = f(x) with y = f(x + a) shifts the graph horizontally by -a. y = af(x) stretches vertically by factor a, and y = f(ax) stretches horizontally by factor 1/a.
y = f(x + a) 将图像水平平移 -a；y = af(x) 沿竖直方向拉伸 a 倍；y = f(ax) 沿水平方向拉伸 1/a 倍。
Reflections are obtained by y = -f(x) (reflection in the x-axis) and y = f(-x) (reflection in the y-axis).
反射变换由 y = -f(x)（关于 x 轴反射）和 y = f(-x)（关于 y 轴反射）得到。
3. Exponentials & Logarithms | 指数与对数
The exponential function aˣ follows fundamental laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁰ = 1, a⁻ⁿ = 1/aⁿ.
指数函数 aˣ 遵循基本法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ / aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，以及 a⁰ = 1，a⁻ⁿ = 1/aⁿ。
The natural exponential eˣ and the natural logarithm ln x = logₑ x are inverses; thus ln(eˣ) = x and e^(ln x) = x.
自然指数 eˣ 与自然对数 ln x = logₑ x 互为反函数，因此 ln(eˣ) = x 且 e^(ln x) = x。
Key logarithm properties: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, and logₐ (xⁿ) = n logₐ x.
关键对数性质：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x – logₐ y，logₐ (xⁿ) = n logₐ x。
The change-of-base formula is logₐ b = log_c b / log_c a; it is especially useful for evaluating logarithms with an unfamiliar base.
换底公式为 logₐ b = log_c b / log_c a，在计算不熟悉底数的对数时尤其有用。
Exponential and logarithmic equations are solved by taking logs of both sides or rewriting the equation in index form.
指数方程和对数方程可通过对两边取对数或改写为指数形式来求解。
4. Coordinate Geometry | 坐标几何
The distance between two points (x₁, y₁) and (x₂, y₂) is d = √[(x₂ – x₁)² + (y₂ – y₁)²]. Their midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点 (x₁, y₁) 与 (x₂, y₂)
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