📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This quick reference handbook brings together the key formulas and theorems required for the CAIE IGCSE Additional Mathematics syllabus. It is designed to help Year 10 students find the tools they need quickly and revise efficiently before assessments.
这本速查手册汇集了 CAIE IGCSE 进阶数学大纲要求的核心公式与定理，旨在帮助 10 年级学生快速定位所需知识，高效备考。
1. Functions and Inverses | 函数与反函数
A function f maps each input x to exactly one output f(x). The domain is the set of all permissible inputs, and the range is the set of all possible outputs.
函数 f 将每一个输入 x 对应到唯一输出 f(x)。定义域是所有允许输入的集合，值域是所有可能输出的集合。
f: x → 2x + 3 or f(x) = 2x + 3
Composite function (f ∘ g)(x) means f(g(x)). Apply g first, then f.
复合函数 (f ∘ g)(x) 表示 f(g(x))，先作用 g 再作用 f。
The inverse function f⁻¹(x) exists only if f is one-to-one. To find it, write y = f(x), swap x and y, then solve for y. The graphs of f and f⁻¹ are symmetric about the line y = x.
反函数 f⁻¹(x) 仅在 f 是一一映射时存在。求反函数时将 y = f(x) 中的 x 与 y 交换，再解出 y。f 与 f⁻¹ 的图像关于直线 y = x 对称。
2. Quadratic Functions & Discriminant | 二次函数与判别式
The general quadratic is f(x) = ax² + bx + c (a ≠ 0). Completing the square gives the vertex form:
一般二次函数为 f(x) = ax² + bx + c (a ≠ 0)。配方可得顶点式：
f(x) = a(x – h)² + k, vertex (h, k) = (-b/(2a), (4ac – b²)/(4a))
The discriminant Δ = b² – 4ac decides the nature of the roots.
判别式 Δ = b² – 4ac 决定了根的性质。
- Δ > 0: two distinct real roots / 两个不等实根
- Δ = 0: one repeated real root / 一个重实根
- Δ < 0: no real roots / 没有实根
If the roots are α and β, then α + β = -b/a and αβ = c/a. For quadratic inequalities, sketch the parabola to identify the solution intervals.
若两根为 α 与 β，则 α + β = -b/a 且 αβ = c/a。解二次不等式时，借助抛物线草图确定解区间。
3. Indices, Surds & Logarithms | 指数、根式与对数
aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a
Surds can be simplified: √a × √b = √(ab), and rationalising the denominator uses (√a + √b)(√a – √b) = a – b.
根式可化简：√a × √b = √(ab)，分母有理化利用 (√a + √b)(√a – √b) = a – b。
The logarithm logₐ b = c means aᶜ = b. Key laws:
对数 logₐ b = c 意为 aᶜ = b。主要法则：
- logₐ (xy) = logₐ x + logₐ y / 积的对数等于对数之和
- logₐ (x/y) = logₐ x – logₐ y / 商的对数等于对数之差
- logₐ (xⁿ) = n logₐ x / 幂的对数等于指数乘对数
- Change of base: logₐ b = (log_c b) / (log_c a) / 换底公式
The natural logarithm ln x uses base e; ln e = 1 and ln 1 = 0.
自然对数 ln x 的底为 e；ln e = 1, ln 1 = 0。
4. Polynomials & Factor Theorem | 多项式与因式定理
Polynomial division works much like long division. The Remainder Theorem states: when f(x) is divided by (x – a), the remainder is f(a).
多项式除法与长除法类似。余式定理指出：f(x) 除以 (x – a) 的余式为 f(a)。
If f(a) = 0, then (x – a) is a factor of f(x).
若 f(a) = 0，则 (x – a) 是 f(x) 的因式。
To factorise a cubic, find one root by trial, use the factor theorem, then factorise the resulting quadratic. Solving polynomial equations may yield repeated roots.
分解三次多项式时，先通过试根找到一个根，利用因式定理，再对得到的二次式继续分解。解多项式方程可能出现重根。
5. Binomial Expansion | 二项式展开
For a positive integer n, (a + b)ⁿ = Σ (nCr) a^(n-r) b^r, where r runs from 0 to n.
对于正整数 n，(a + b)ⁿ = Σ (nCr) a^(n-r) b^r，r 取 0 到 n。
nCr = n! / [r! (n – r)!]
The (r+1)th term is T_(r+1) = nCr × a^(n-r) × b^r. The coefficients are symmetric because nCr = nC(n-r).
第 (r+1) 项为 T_(r+1) = nCr × a^(n-r) × b^r。二项式系数对称，因为 nCr = nC(n-r)。
Always simplify the term by combining constants inside the power when a or b contain numbers.
当 a 或 b 含有数字时，先将常数乘入幂次中进行化简。
6. Trigonometry & Identities | 三角学与恒等式
Angles can be measured in degrees or radians. π rad = 180°. To convert: multiply degrees by π/180 to get radians.
角度可用度或弧度表示。π rad = 180°。换算：度数乘以 π/180 得弧度。
sin²θ + cos²θ = 1, tanθ = sinθ / cosθ
1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ
Exact values for special angles:
特殊角的精确值：
- sin 30° = sin(π/6) = 1/2, cos 30° = √3/2, tan 30° = 1/√3
- sin 45° = sin(π/4) = √2/2, cos 45° = √2/2, tan 45° = 1
- sin 60° = sin(π/3) = √3/2, cos 60° = 1/2, tan 60° = √3
Sine rule: a / sin A = b / sin B = c / sin C = 2R. Cosine rule: a² = b² + c² – 2bc cos A. Area of a triangle: ½ ab sin C.
正弦定理：a / sin A = b / sin B = c / sin C = 2R。余弦定理：a² = b² + c² – 2bc cos A。三角形面积：½ ab sin C。
7. Vectors in 2D | 二维向量
A vector can be written as a column (x y) or in terms of i and j: xi + yj. The magnitude is |v| = √(x² + y²).
向量可以表示为列向量 (x y) 或用 i，j 表示：xi + yj。模长为 |v| = √(x² + y²)。
Addition: (x₁ y₁) + (x₂ y₂) = (x₁+x₂ y₁+y₂). Scalar multiplication: k(x y) = (kx ky).
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