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: Key Topics Explained | 剑桥IGCSE附加数学0606知识点精讲

The Cambridge IGCSE Additional Mathematics (0606) syllabus extends knowledge beyond the Mathematics 0580 curriculum, introducing students to advanced algebra, trigonometry, calculus and other pure mathematics topics. This guide provides a concise yet comprehensive overview of the essential topics, complete with key formulas and concepts to aid revision for the 0606 examination.

剑桥IGCSE附加数学(0606)教学大纲在数学0580课程的基础上拓展知识，引入高等代数、三角学、微积分和其他纯数学主题。本指南对核心知识点进行简明而全面的概述，包含关键公式和概念，帮助考生备战0606考试。

1. Functions | 函数

A function f maps each element x from its domain to exactly one element f(x) in its range. The domain is the set of all possible inputs, and the range is the resulting set of outputs. A function must be well-defined for every input in its domain.

函数 f 将其定义域中的每个元素 x 映射到值域中唯一一个元素 f(x)。定义域是所有可能输入值的集合，值域是产生的输出值的集合。函数必须在其定义域内对每个输入都有明确的定义。

Two functions f and g can be combined to form a composite function fg(x) = f(g(x)), meaning ‘f of g of x’. The output of g becomes the input of f. The domain of fg is restricted to those x in the domain of g for which g(x) lies in the domain of f.

两个函数 f 和 g 可以组合成复合函数 fg(x) = f(g(x))，意为“先经过 g 再经过 f”。g 的输出成为 f 的输入。复合函数的定义域限制为那些使得 g(x) 落在 f 的定义域内的 x。

The inverse function f⁻¹(x) reverses the effect of f, such that f⁻¹(f(x)) = x for all x in the domain of f. An inverse exists only if f is a one-to-one mapping. To find the inverse, set y = f(x), swap x and y, then solve for y.

反函数 f⁻¹(x) 逆转 f 的作用，满足对 f 定义域内的所有 x 均有 f⁻¹(f(x)) = x。反函数仅当 f 为一一映射时才存在。求反函数时，设 y = f(x)，交换 x 与 y，然后解出 y。

2. Quadratic Functions | 二次函数

A quadratic function takes the general form f(x) = ax² + bx + c, with a ≠ 0. Its graph is a parabola that opens upwards if a > 0 and downwards if a < 0. The axis of symmetry is a vertical line given by x = -b/(2a).

二次函数的一般形式为 f(x) = ax² + bx + c，其中 a ≠ 0。其图像是一条抛物线，a > 0 时开口向上，a < 0 时开口向下。对称轴为直线 x = -b/(2a)。

Completing the square rewrites the quadratic as f(x) = a(x – h)² + k, revealing the vertex (h, k). The vertex represents the minimum point if a > 0, or the maximum point if a < 0. The value k is the optimum value of the function.

配方法将二次函数改写为 f(x) = a(x – h)² + k，从而显示顶点 (h, k)。当 a > 0 时顶点为最小值点，a < 0 时顶点为最大值点。k 是函数的最优值。

The discriminant Δ = b² – 4ac determines the number of real roots of f(x) = 0: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots. Quadratic inequalities are solved by identifying intervals around the roots.

判别式 Δ = b

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