📚 Cambridge IGCSE Additional Mathematics (0606): Essential Topics Overview | 剑桥 IGCSE 附加数学(进阶数学)：核心知识点梳理
Cambridge IGCSE Additional Mathematics (0606) builds on the foundation of IGCSE Mathematics and introduces advanced concepts essential for A Level studies. This article provides a systematic overview of the core topics, including functions, quadratics, logarithms, trigonometry, calculus and more. Every key idea is presented with clarity to support effective revision.
剑桥 IGCSE 附加数学(0606)在 IGCSE 数学基础上引入了对 A Level 学习至关重要的高级概念。本文系统梳理了核心知识点，涵盖函数、二次曲线、对数、三角学、微积分等内容，并以清晰的方式呈现，助力高效复习。
1. Functions and Relations | 函数与关系
A function f is a rule that assigns each input x from the domain to exactly one output f(x) in the range. We write f: x ↦ f(x) or y = f(x).
函数 f 是一种规则，将定义域中的每个输入 x 唯一对应到值域中的一个输出 f(x)。记作 f: x ↦ f(x) 或 y = f(x)。
The domain Df is the set of allowed inputs; the range Rf is the set of all output values. A relation is a function only if it passes the vertical line test.
定义域 Df 是所有允许输入值的集合；值域 Rf 是所有输出值的集合。一个关系只有通过垂直线检验才是函数。
Composite functions combine two functions: (g ∘ f)(x) = g(f(x)). The order matters; f is applied first, then g.
复合函数将两个函数组合：(g ∘ f)(x) = g(f(x))。顺序很重要，先应用 f，再应用 g。
An inverse function f-1 reverses the mapping of f. For f-1 to exist, f must be one-to-one (injective). The graphs of f and f-1 are reflections in the line y = x.
反函数 f-1 逆转 f 的映射。f-1 存在的条件是 f 必须是一一映射（单射）。f 与 f-1 的图像关于直线 y = x 对称。
2. Quadratic Functions and Their Graphs | 二次函数及其图像
The standard form of a quadratic function is f(x) = ax² + bx + c, with a ≠ 0. The graph is a parabola opening upward if a > 0, downward if a < 0.
二次函数的标准形式为 f(x) = ax² + bx + c，其中 a ≠ 0。其图像是一条抛物线，当 a > 0 时开口向上，a < 0 时开口向下。
The vertex form f(x) = a(x – h)² + k reveals the vertex (h, k) directly. Completing the square converts the standard form to vertex form.
顶点式 f(x) = a(x – h)² + k 直接显示顶点 (h, k)。通过配方法可将标准形式转化为顶点式。
The discriminant Δ = b² – 4ac determines the nature of roots:
判别式 Δ = b² – 4ac 决定根的性质：
| Δ > 0 – Two distinct real roots | Δ > 0 – 两个不相等的实根 |
| Δ = 0 – One repeated real root | Δ = 0 – 一个重实根 |
| Δ < 0 – No real roots | Δ < 0 – 无实根 |
Roots: x = (-b ± √(b² – 4ac)) / (2a)
求根公式：x = (-b ± √(b² – 4ac)) / (2a)
3. Equations, Inequalities and Modulus | 方程、不等式与绝对值
Linear inequalities are solved similarly to equations, but multiplying or dividing by a negative number reverses the inequality sign.
解线性不等式与解方程类似，但当两边同乘或除以负数时，不等号方向需反转。
Quadratic inequalities are best tackled by sketching the associated quadratic graph and identifying intervals where the curve is above or below the x‑axis.
解二次不等式最好的方法是先画对应的二次函数图像，然后找出曲线在 x 轴上方或下方的区间。
The modulus function |x| gives the distance of x from zero. |x| = a means x = a or x = –a. For inequalities, |x – k| < d is equivalent to –d < x – k < d.
绝对值函数 |x| 表示 x 到零的距离。|x| = a 表示 x = a 或 x = –a。对于不等式，|x – k| < d 等价于 –d < x – k < d。
When a linear and a quadratic equation are solved simultaneously, substitution is used: replace y from the linear into the quadratic, yielding a quadratic in one variable.
当联立一个一次方程和一个二次方程时，使用代入法：将一次方程中的 y 代入二次方程，得到一个一元二次方程。
4. Indices and Surds | 指数与根式
The laws of indices are essential for simplifying expressions: am × an = am+n, (am)n = amn, a0 = 1, a-n = 1/an.
指数定律是化简表达式的基础：am × an = am+n，(am)n = amn，a0 = 1，a-n = 1/an。
Surds involve roots that cannot be simplified to rational numbers. Key rules: √a × √b = √(ab) and (√a + √b)(√a – √b) = a – b.
根式指的是无法化为有理数的方根。核心规则：√a × √b = √(ab)，(√a + √b)(√a – √b) = a – b。
Rationalising the denominator means eliminating surds from the denominator, often by multiplying numerator and denominator by the conjugate. For instance, 1/(√a + b) becomes (√a – b)/(a – b²).
分母有理化是指消去分母中的根号，通常用共轭式同乘分子分母。例如，1/(√a + b) 化为 (√a – b)/(a – b²)。
Example: √12 + √27 = 2√3 + 3√3 = 5√3
示例：√12 + √27 = 2√3 + 3√3 = 5√3
5. Polynomials and Factorisation | 多项式与因式分解
The factor theorem states that (x – a) is a factor of a polynomial f(x) if and only if f(a) = 0.
因式定理指出，(x – a) 是多项式 f(x) 的因式当且仅当 f(a) = 0。
According to the remainder theorem, when f(x) is divided by (x – a), the remainder is f(a). This can be used to find unknown coefficients.
根据余式定理，当 f(x) 除以 (x – a) 时，余数等于 f(a)。这常用于求待定系数。
Polynomials of degree 3 or higher can be factorised by combining long division (or synthetic division) with the factor theorem. Start by testing small integer values for a.
三次及更高次多项式可通过综合运用长除法（或综合除法）与因式定理进行因式分解。先从整数试探 a 开始。
If f(2) = 0, then (x – 2) is a factor. Divide f(x) by (x – 2) to find the remaining quadratic factor.
若 f(2) = 0，则 (x – 2) 是一个因式。用 f(x) 除以 (x – 2) 得到二次因式。
6. Simultaneous Equations | 联立方程组
Two linear equations in two unknowns can be solved by elimination or by substitution. Elimination involves adding/subtracting multiples to remove one variable.
两个二元一次方程可通过消元法或代入法求解。消元法通过加减倍乘公式消去一个未知数。
For one linear and one quadratic equation, express one variable from the linear equation, substitute it into the quadratic, and solve the resulting quadratic.
对于一个一次和一个二次方程，从一次方程中解出一个变量，代入二次方程，再解对应的二次方程。
Solutions represent intersection points of a straight line and a curve – there may be 0, 1 or 2 points.
方程组的解即为直线与曲线的交点，可能有 0 个、1 个或 2 个交点。
Example: Solve y = 2x + 1 and x² + y² = 13.
示例：求解 y = 2x + 1 和 x² + y² = 13。
7. Logarithmic and Exponential Functions | 对数与指数函数
The logarithm definition: ax = b ⇔ x = loga b, where a > 0, a ≠ 1. Logarithms are the inverses of exponential functions.
对数定义：ax = b ⇔ x = loga b，其中 a > 0，a ≠ 1。对数是指数函数的反函数。
Fundamental laws of logs: loga(xy) = loga x + loga y; loga(x/y) = loga x – loga y; loga xn = n loga x.
对数基本定律：loga(xy) = loga x + loga y；loga(x/y
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