Year 12 CIE Mathematics Formula & Theorem Quick Reference | CIE Year 12 数学公式定理速查手册

📚 Year 12 CIE Mathematics Formula & Theorem Quick Reference | CIE Year 12 数学公式定理速查手册

This quick reference handbook summarises essential formulas and theorems for Year 12 CIE AS Mathematics (9709). It covers Pure Mathematics 1 topics along with key statistics concepts to help you revise efficiently. Keep this guide handy for quick checks and exam preparation.

本速查手册总结了 Year 12 CIE AS 数学 (9709) 的核心公式与定理，涵盖纯数 1 及部分统计内容，帮助高效复习。考试前随时查阅。

1. Quadratics & Discriminant | 二次方程与判别式

The quadratic formula for solving ax² + bx + c = 0 is x = (−b ± √(b² − 4ac)) / (2a).

二次方程 ax² + bx + c = 0 的求根公式为 x = (−b ± √(b² − 4ac)) / (2a)。

The discriminant Δ = b² − 4ac determines the nature of roots: if 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 = 0，一个重根；Δ < 0，无实根。

Completing the square transforms ax² + bx + c into a(x + p)² + q, where p = b/(2a) and q = c − b²/(4a).

配方法将 ax² + bx + c 转化为 a(x + p)² + q，其中 p = b/(2a)，q = c − b²/(4a)。

If α and β are the roots of ax² + bx + c = 0, then sum α + β = −b/a and product αβ = c/a.

若 α、β 为方程 ax² + bx + c = 0 的根，则和 α + β = −b/a，积 αβ = c/a。

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2. Equations & Inequalities | 方程与不等式

When multiplying or dividing an inequality by a negative number, reverse the inequality sign: if a < b and c < 0, then ac > bc.

不等式两边同乘或同除负数时，不等号要改变方向：若 a < b 且 c < 0，则 ac > bc。

Quadratic inequalities are solved by factorising, finding critical values, and using a sign diagram to determine intervals where the expression is > 0 or < 0.

解二次不等式需要因式分解，求出临界值，然后用符号图确定表达式大于零或小于零的区间。

For modulus equations |ax + b| = c, rewrite as ax + b = ±c. For inequalities |ax + b| ≤ c, this is equivalent to −c ≤ ax + b ≤ c.

形如 |ax + b| = c 的模方程可化为 ax + b = ±c；对于模不等式 |ax + b| ≤ c，等价于 −c ≤ ax + b ≤ c。

Always check solutions of modulus equations by substitution to avoid extraneous roots.

解模方程后务必代回原方程检验，避免增根。

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3. Functions & Graphs | 函数与图像

The domain of a function f(x) is the set of all possible input values; the range is the set of all output values.

函数 f(x) 的定义域是所有可能输入的集合，值域是所有可能输出的集合。

For composite function fg(x) = f(g(x)), first apply g, then f. The range of g must be a subset of the domain of f.

复合函数 fg(x) = f(g(x)) 先作用 g 再作用 f；g 的值域必须是 f 定义域的子集。

A function has an inverse f⁻¹ only if it is one‑one. To find f⁻¹, write y = f(x), swap x and y, then solve for y.

仅当函数是一一映射时才存在反函数 f⁻¹。求反函数：令 y = f(x)，交换 x 与 y，再解出 y。

Graph transformations: y = f(x) + a shifts vertically by a; y = f(x + a) shifts horizontally by −a; y = af(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a.

图像变换：y = f(x) + a 沿 y 轴平移 a；y = f(x + a) 沿 x 轴平移 −a；y = af(x) 纵向拉伸 a 倍；y = f(ax) 横向压缩 1/a 倍。

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4. Coordinate Geometry | 坐标几何

Distance between two points (x₁, y₁) and (x₂, y₂): d = √[(x₂ − x₁)² + (y₂ − y₁)²].

两点 (x₁, y₁) 和 (x₂, y₂) 间的距离为 d = √[(x₂ − x₁)² + (y₂ − y₁)²]。

Midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

中点 M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。

Gradient of a line through (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁)/(x₂ − x₁). Two lines are parallel if m₁ = m₂, perpendicular if m₁ × m₂ = −1.

过 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m = (y₂ − y₁)/(x₂ − x₁)。两直线平行则 m₁ = m₂，垂直则 m₁ × m₂ = −1。

Equation of a straight line: y − y₁ = m(x − x₁) or ax + by + c = 0.

直线方程：y − y₁ = m(x − x

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