📚 Cambridge IGCSE Additional Mathematics 0606: Key Topics Explained | 剑桥IGCSE附加数学0606知识点精讲
Cambridge IGCSE Additional Mathematics (0606) extends knowledge far beyond the Mathematics 0580 curriculum, introducing students to advanced algebra, trigonometry, calculus and other pure mathematics topics. It is designed for able students who intend to continue with A Level Mathematics or Further Mathematics, and is assessed by two written papers in which a calculator is permitted. This guide provides a concise yet comprehensive overview of the essential topics, complete with key formulas, standard results and concepts to aid revision pointers for the 0606 examination.
剑桥 IGCSE 附加数学(0606)在数学 0580 课程的基础上大幅拓展，引入高等代数、三角学、微积分等纯数学主题，面向计划继续学习 A Level 数学或进阶数学的优秀学生；该科目由两份允许使用计算器的笔试考查。本指南对核心知识点作简明而全面的概述，包含关键公式、标准结论以及针对 0606 考试的复习要点。
1. Functions | 函数
A function f maps each element x of its domain to exactly one element f(x) in its range. The domain is the set of all permitted inputs, while the range is the resulting set of all resulting outputs. A mapping qualifies as a function only if every input produces exactly one output, so a one-to-many mapping is not a function.
函数 f 把定义域中的每个元素 x 映射到值域中唯一确定的元素 f(x)。定义域是所有允许输入值的集合，值域是所有输出结果构成的集合。只有当每个输入都对应唯一输出时，该映射才是函数，因此"一对多"的映射不是函数。
Two functions f and g can be combined into a composite function fg(x) = f(g(x)), read as 'f of g of x' f g for which g(x) lies in the domain of f, and in general fg(x) is not the same as gf(x).
两个函数 f 与 g 可以组合成复合函数 fg(x) = f(g(x))，读作"先 g 后 f"，即 g 的输出成为 f 的输入。复合函数的定义域由 g 的定义域中那些使得 g(x) 落在 f 定义域内的 x 组成；一般地，fg(x) 与 gf(x) 并不相同。
fg(x) = f(g(x)) gf(x) = g(f(x)) fg(x) ≠ gf(x)
ing. To find it, write y = f(x), swap x and y, then solve for y. The domain of f⁻¹ equals the range of f, and the graph of f⁻¹ is the reflection of the graph of f in the line y = x.
反函数 f⁻¹ 逆转 f 的作用，对 f 定义域内的一切 x 均有 f⁻¹(f(x)) = x。只有当 f 是一一映射时反函数才存在。求法为：设 y = f(x)，交换 x 与 y，再解出 y。f⁻¹ 的定义域等于 f 的值域，f⁻¹ 的图像是 f 的图像关于直线 y = x 的对称图形。
f⁻¹(f(x)) = x f(f⁻¹(x)) = x
The modulus function |x| gives the distance of x from zero, so |x| = x when x ≥ 0 and |x| = -x when x < 0. The graph of y = |f(x)| d by reflecting any part of y = f(x) that lies below the x-axis into the upper half-plane.
绝对值函数 |x| 表示 x 到原点的距离，故当 x ≥ 0 时 |x| = x，当 x < 0 时 |x| = -x。y = |f(x)| 的图像可由 y = f(x) 位于 x 轴下方的部分关于 x 轴反射到上方得到。
2. Quadratic Functions | 二次函数
0. The axis of symmetry is the vertical line x = -b/(2a), and the turning point (vertex) lies on this line.
二次函数的一般形式为 f(x) = ax² + bx + c，其中 a ≠ 0。其图像是抛物线：a > 0 时开口向上，a < 0 时开口向下。对称轴为竖直直线 x = -b/(2a)，顶点（转折点）位于该直线上。
x = -b ÷ (2a) x = (-b ± √b² - 4ac) ÷ (2a)
splays the vertex (h, k) directly. The vertex is represents the minimum point when a > 0 and the maximum point when a < 0, and the value k is the optimum (greatest or least) value of the function.
配方法把函数改写为 f(x) = a(x - h)² + k，可直接读出顶点 (h, k)。当 a > 0 时顶点为最低点，当 a < 0 时顶点为最高点，k 即函数的最值（最大值或最小值）。
The discriminant Δ = b² - 4ac determines how many real roots the
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