Cambridge Year 12 Maths: Formula & Theorem Quick Reference Handbook | 剑桥12年级数学：公式定理速查手册

📚 Cambridge Year 12 Maths: Formula & Theorem Quick Reference Handbook | 剑桥12年级数学：公式定理速查手册

This quick reference handbook compiles all essential formulas and theorems needed for Year 12 Cambridge International AS Level Mathematics (9709). It focuses on Pure Mathematics 1, covering algebra, functions, coordinate geometry, sequences, trigonometry, exponentials, logarithms and calculus. Use it to quickly recall key results before the exam.

本速查手册汇编了剑桥国际 AS Level 数学 (9709) 12 年级所需的所有核心公式和定理。内容聚焦纯数学 1，涵盖代数、函数、解析几何、数列、三角学、指数对数以及微积分。将它作为考前快速回顾的关键资料。

1. Algebra & Functions | 代数与函数

A function f maps each input value x to exactly one output f(x). The set of allowable inputs is the domain; the set of possible outputs is the range.

函数 f 将每个输入值 x 映射到唯一输出 f(x)。允许的输入值集合称为定义域；可能的输出值集合称为值域。

Composite functions combine two functions: (f∘g)(x) = f(g(x)). First apply g, then f. The domain of the composite is restricted by the domain of the inner function.

复合函数组合了两个函数：(f∘g)(x) = f(g(x))。先施加 g，再施加 f。复合函数的定义域受内层函数定义域的限制。

An inverse function f⁻¹(x) reverses the action of f, satisfying f⁻¹(f(x)) = x = f(f⁻¹(x)). It exists only if f is one-to-one on its domain.

反函数 f⁻¹(x) 逆转 f 的作用，满足 f⁻¹(f(x)) = x = f(f⁻¹(x))。只有当 f 在其定义域上是一一映射时才存在反函数。

The graph of the inverse is the reflection of the original graph in the line y = x. To find an inverse algebraically, swap x and y and solve for y.

反函数的图像是原函数图像关于直线 y = x 的反射。代数上求反函数的方法是交换 x 和 y，然后解出 y。

2. Quadratics & Inequalities | 二次方程与不等式

The standard quadratic equation is ax² + bx + c = 0 with a ≠ 0. Its solutions are given by the quadratic formula:

标准二次方程为 ax² + bx + c = 0 (a ≠ 0)。其解由求根公式给出：

x = [-b ± √(b² – 4ac)] / (2a)

The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 ⇒ two distinct real roots; Δ = 0 ⇒ one repeated real root; Δ < 0 ⇒ no real roots.

判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 ⇒ 有两个不等实根；Δ = 0 ⇒ 有一个重复实根；Δ < 0 ⇒ 没有实根。

To solve quadratic inequalities, find the critical values where the quadratic equals zero, then test intervals. Write the solution set using interval notation or inequalities.

求解二次不等式时，先找出二次式等于零的临界值，然后检验区间。用区间记号或不等式写出解集。

Completing the square rewrites ax² + bx + c in the form a(x – h)² + k, which reveals the vertex (h, k) and is useful for solving and sketching.

配方法将 ax² + bx + c 改写为 a(x – h)² + k 的形式，可显示出顶点 (h, k)，对求解和绘图都有用。

3. Coordinate Geometry | 解析几何

The gradient (slope) of a line through (x₁, y₁) and (x₂, y₂) is m = (y₂ – y₁) / (x₂ – x₁). The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).

经过 (x₁, y₁) 与 (x₂, y₂) 的直线斜率为 m = (y₂ – y₁) / (x₂ – x₁)。中点为 ((x₁+x₂)/2, (y₁+y₂)/2)。

The distance between the two points is d = √[(x₂ – x₁)² + (y₂ – y₁)²].

两点间的距离为 d = √[(x₂ – x₁)² + (y₂ – y₁)²]。

Common forms of the equation of a straight line:

直线方程的常见形式：

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