📚 IGCSE Cambridge Additional Mathematics: Core Concepts Review | IGCSE剑桥进阶数学：核心知识点梳理
The IGCSE Cambridge Additional Mathematics curriculum extends the core IGCSE Mathematics syllabus, introducing more sophisticated techniques and concepts that form a bridge to A Level Mathematics. This article systematically reviews the core topics: functions, quadratic functions, equations and inequalities, indices and surds, polynomials, logarithms, trigonometry, permutations and series, vectors, differentiation, and integration. A thorough understanding of these areas is essential for examination success and for confident progression to further study.
IGCSE剑桥进阶数学（Additional Mathematics）课程在核心IGCSE数学的基础上进行了拓展，引入了更精深的技巧与概念，为A Level数学搭建了桥梁。本文对核心知识点进行系统梳理：函数、二次函数、方程与不等式、指数与根式、多项式、对数、三角学、排列组合与数列、向量、微分以及积分。透彻掌握这些内容对于考试成功和自信地进入更高层次的学习至关重要。
1. Functions | 函数
A function is a rule that assigns each input x exactly one output y. We commonly write f(x) = 2x + 1 or use the mapping notation f : x ↦ 2x + 1. The set of allowed inputs is the domain, and the set of corresponding outputs is the range.
函数是一种规则，为每个输入的 x 分配唯一确定的输出 y。我们通常写作 f(x) = 2x + 1 或使用映射记号 f : x ↦ 2x + 1。所有允许的输入值构成定义域，所有对应的输出值构成值域。
For an inverse function f⁻¹ to exist, the original function must be one‑to‑one. The graphs of f and f⁻¹ are reflections of each other in the line y = x. To find an inverse, swap x and y and solve for y.
反函数 f⁻¹ 存在的条件是原函数必须是一对一的。f 和 f⁻¹ 的图像关于直线 y = x 互为反射。求反函数时，只需交换 x 与 y 然后解出 y。
A composite function fg(x) means apply g first, then f. Graph transformations include translations, stretches and reflections. Vertical translation: y = f(x) + a; horizontal translation: y = f(x + a). Stretches: y = a f(x) (vertical) and y = f(ax) (horizontal). Reflections: y = -f(x) across the x‑axis, y = f(-x) across the y‑axis.
复合函数 fg(x) 表示先作用 g 再作用 f。图像变换包括平移、伸缩和反射。纵向平移：y = f(x) + a；横向平移：y = f(x + a)。伸缩变换：y = a f(x)（纵向）和 y = f(ax)（横向）。反射变换：y = -f(x) 关于 x 轴反射，y = f(-x) 关于 y 轴反射。
2. Quadratic Functions | 二次函数
A quadratic function has the general form y = ax² + bx + c, where a ≠ 0. By completing the square, it can be expressed in vertex form y = a(x - h)² + k, revealing the turning point (h, k) and the axis of symmetry x = h.
二次函数的一般形式为 y = ax² + bx + c，其中 a ≠ 0。通过配方法可将其转化为顶点式 y = a(x - h)² + k，从中可直接看出顶点 (h, k) 与对称轴 x = h。
The discriminant Δ = b² - 4ac ot; if Δ < 0 there are no real roots. Quadratic inequalities are solved by sketching the graph and identifying the intervals where y > 0 or y < 0.
判别式 Δ = b² - 4ac 决定了根的性质。若 Δ > 0，有两个相异实根；若 Δ = 0，有一个重根；若 Δ < 0，则没有实根。解二次不等式时，通过草图找出 y > 0 或 y < 0 的区间即可。
Quadratic functions can be used to model maximum or minimum problems. The vertex gives the optimal value, and the line of symmetry helps to interpret the context.
二次函数常用来解决最大值或最小值问题。顶点给出了最优值，对称轴则帮助理解实际情景。
3. Equations and Inequalities | 方程与不等式
Solving equations often involves rearranging to isolate the variable. For quadratic equations, factorization, the quadratic formula, or completing the square are the standard methods. Equations involving algebraic fractions require clearing denominators, taking care to exclude values that make a denominator zero.
解方程通常需要整理变形式。对于二次方程，因式分解、求根公式或配方法都是标准方法。含有分式的方程需先去分母，
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