AS Mathematics: Formula Summary Handbook | AS 数学：公式汇总手册

📚 AS Mathematics: Formula Summary Handbook | AS 数学：公式汇总手册

This article provides a comprehensive collection of essential formulas for AS Level Mathematics, covering pure mathematics and introductory statistics. It is designed as a quick reference tool for revision and exam preparation.

本文为AS数学提供核心公式汇总，涵盖纯数学和入门统计，旨在作为复习和备考的快速参考手册。

1. Algebra and Functions | 代数与函数

The quadratic formula solves ax² + bx + c = 0. The roots are given by:

二次公式解 ax² + bx + c = 0。根为：

x = (-b ± √(b² – 4ac)) / (2a)

Discriminant Δ = b² – 4ac determines the nature of 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 → 两个不等实根；Δ = 0 → 一个重实根；Δ < 0 → 无实根。

Completing the square: x² + bx = (x + b/2)² – (b/2)².

配方法：x² + bx = (x + b/2)² – (b/2)²。

Factor theorem: (x – a) is a factor of polynomial f(x) if and only if f(a) = 0.

因式定理：当且仅当 f(a)=0 时，(x – a) 是多项式 f(x) 的因式。

Remainder theorem: When f(x) is divided by (x – a), the remainder is f(a).

余数定理：f(x) 除以 (x – a) 的余数为 f(a)。

Inverse function f⁻¹(x) satisfies f(f⁻¹(x)) = x. Domain of f⁻¹ is range of f.

反函数 f⁻¹(x) 满足 f(f⁻¹(x)) = x。f⁻¹ 的定义域是 f 的值域。

Composite function: (g ∘ f)(x) = g(f(x)).

复合函数：(g ∘ f)(x) = g(f(x))。

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2. Coordinate Geometry | 坐标几何

Equation of a straight line: y = mx + c (gradient m, y-intercept c).

直线方程：y = mx + c（斜率 m，y 截距 c）。

Point-gradient form: y – y₁ = m(x – x₁).

点斜式：y – y₁ = m(x – x₁)。

Distance between two points (x₁,y₁) and (x₂,y₂):

两点距离：

d = √((x₂ – x₁)² + (y₂ – y₁)²)

Midpoint:

中点：

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Circle: (x – a)² + (y – b)² = r² with centre (a,b) and radius r.

圆：(x – a)² + (y – b)² = r²，圆心 (a,b)，半径 r。

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3. Sequences and Series | 数列与级数

Arithmetic progression: nth term uₙ = a + (n – 1)d, where a is first term, d is common difference.

等差数列：第 n 项 uₙ = a + (n – 1)d，其中 a 为首项，d 为公差。

Sum of first n terms: Sₙ = n/2 (2a + (n – 1)d) = n/2 (a + l), where l is last term.

前 n 项和：Sₙ = n/2 (2a + (n – 1)d) = n/2 (a + l)，其中 l 为末项。

Geometric progression: nth term uₙ = arⁿ⁻¹, where a is first term, r is common ratio.

等比数列：第 n 项 uₙ = arⁿ⁻¹，a 为首项，r 为公比。

Sum of first n terms: Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1.

前 n 项和：Sₙ = a(1 – rⁿ)/(1 – r)，r ≠ 1。

Sum to infinity: S∞ = a/(1 – r) valid when |r| < 1.

无穷等比级数和：S∞ = a/(1 – r)，当 |r| < 1 时成立。

Sigma notation: Σ (from k=1 to n) of aₖ.

西格玛符号：Σ（k=1 到 n）aₖ

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