Year 12 Cambridge Mathematics: Formula and Theorem Quick Reference Handbook | Year 12 Cambridge 数学：公式定理速查手册

📚 Year 12 Cambridge Mathematics: Formula and Theorem Quick Reference Handbook | Year 12 Cambridge 数学：公式定理速查手册

This quick reference handbook compiles the essential formulas and theorems required for Year 12 Cambridge AS Level Mathematics, focusing on Pure Mathematics 1. It is designed to help you rapidly review core concepts, streamline your revision, and strengthen problem-solving confidence before assessments.

本速查手册汇编了 Year 12 Cambridge AS Level 数学（纯数 1）必备的核心公式与定理，旨在帮助你在考前高效回顾关键知识，强化解题信心。

1. Algebraic Laws and Indices | 代数法则与指数

roduct of powers: aᵐ × aⁿ = aᵐ⁺ⁿ, where a is a non-zero real number and m, n are integers.

同底数幂乘法：aᵐ × aⁿ = aᵐ⁺ⁿ，其中 a 为非零实数，m、n 为整数。

Quotient of powers: aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a ≠ 0.

同底数幂除法：aᵐ ÷ aⁿ = aᵐ⁻ⁿ，a ≠ 0。

Power of a power: (aᵐ)ⁿ = aᵐⁿ.

幂的乘方：(aᵐ)ⁿ = aᵐⁿ。

Zero exponent: a⁰ = 1, for any a ≠ 0.

零指数：任何非零实数 a，a⁰ = 1。

Negative exponent: a⁻ⁿ = 1 / aⁿ.

负指数：a⁻ⁿ = 1 / aⁿ。

Fractional exponents: a^1/n = ⁿ√a, and a^m/n = (ⁿ√a)ᵐ = ⁿ√aᵐ.

分数指数：a^1/n = ⁿ√a，a^m/n = (ⁿ√a)ᵐ = ⁿ√aᵐ。

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2. Quadratic Equations and Functions | 二次方程与二次函数

Standard form: f(x) = ax² + bx + c, where a ≠ 0.

标准式：f(x) = ax² + bx + c，其中 a ≠ 0。

es two distinct real roots, Δ = 0 gives one repeated real root, Δ < 0 gives no real roots.

判别式：Δ = b² - 4ac，用于判断根的情况：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。

Quadratic formula: x = [-b ± √b² - 4ac] / (2a).

求根公式：x = [-b ± √b² - 4ac] / (2a)。

Sum and product of roots (Vieta's formulas): for roots α and β, α + β = -b/a, αβ = c/a.

根与系数的关系（韦达定理）：若根为 α, β，则 α + β = -b/a，αβ = c/a。

Completed square form: a[(x + b/(2a))² - (Δ)/(4a²)], vertex at (-b/(2a), -Δ/(4a)).

顶点式：a[(x + b/(2a))² - (Δ)/(4a²)]，顶点坐标为 (-b/(2a), -Δ/(4a))。

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3. Equations and Inequalities | 方程与不等式

Linear inequality solving: perform operations on both sides; reverse the inequality sign when multiplying or dividing by a negative number.

线性不等式求解：可在不等号两边进行运算；乘以或除以负数时，不等号方向改变。

) > 0 or < 0, then sketch a sign diagram. The solution is often an interval expressed with ∪ and ∞.

二次不等式：因式分解为 (x - α)(x - β) > 0 或 < 0，作符号草图求解，解通常用区间及 ∪ 和 ∞ 表出。

: equations of the form a(p(x))² + b(p(x)) + c = 0 can be solved by substituting u = p(x).

二次换元方程：形如 a(p(x))² + b(p(x)) + c = 0 可令 u = p(x) 转化为二次方程求解。

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4. Binomial Expansion | 二项式展开

The binomial theorem for positive integer n: (a + b)ⁿ = Σ (r=0 to n) nCr aⁿ⁻ʳ bʳ, where nCr = n! / (r! (n - r)!).

正整数指数二项式定理：(a + b)ⁿ = Σ (r=0 至 n) nCr aⁿ⁻ʳ bʳ，其中 nCr = n! / (r! (n - r)!)。

Expanding (1 + x)ⁿ for |x| < 1, n rational: (1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + ... , valid for |x| < 1.

对有理数 n 且 |x| < 1 的展开：(1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + ... ，此展开在 |x| < 1 时成立。

Pascal's triangle relationship: nCr = (n-1)C(r-1) + (n-1)Cr.

组合数递推关系：nCr = (n-1)C(r-1) + (n-1)Cr。

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5. Functions and Transformations | 函数与图像变换

Domain and range: the domain is the set of all possible input values (x), and the range is the set of all possible output values (f(x)).

定义域与值域：定义域是所有可能输入值 x 的集合，值域是所有可能输出值 f(x) 的集合。

(x): exists if f is one-to-one. Its graph is the reflection of y = f(x) in the line y = x.

反函数 f⁻¹(x)：若 f 为一一映射则存在，其图像是 y = f(x) 关于直线 y = x 的反射。

Composite function: fg(x) = f(g(x)). The range of g must be a subset of the domain of f.

复合函数：fg(x) = f(g(x))，要求 g 的值域包含于 f 的定义域。

Graph transformations: y = f(x)

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