📚 IGCSE OCR Additional Mathematics: Core Knowledge Review | IGCSE OCR 进阶数学：核心知识点梳理
This article provides a structured review of the essential topics in the IGCSE OCR Additional Mathematics syllabus, covering functions, algebra, trigonometry, calculus, and more. Each section outlines key concepts with paired bilingual explanations to support revision and mastery.
本文系统梳理了 IGCSE OCR 进阶数学大纲的核心知识点，涵盖函数、代数、三角、微积分等模块，每部分都以中英双语对照要点进行讲解，助力复习与巩固。
1. Functions | 函数
A function f assigns every element x in its domain to exactly one value f(x) in its range. The domain is the set of all allowed inputs, and the range is the set of all possible outputs.
函数 f 将定义域内的每个元素 x 唯一对应到值域中的一个值 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。
To find the inverse f−1(x), write y = f(x), swap x and y, then solve for y. A function must be one-to-one (each output corresponds to exactly one input) for its inverse to exist.
求反函数 f−1(x) 时，写出 y = f(x)，交换 x 与 y，然后解出 y。只有一一映射（每个输出唯一对应一个输入）的函数才存在反函数。
The composite function gf(x) means applying g to the result of f: gf(x) = g(f(x)). The domain of gf consists of all x in the domain of f such that f(x) is in the domain of g.
复合函数 gf(x) 表示将 g 作用于 f(x) 的结果：gf(x) = g(f(x))。gf 的定义域由所有满足 f(x) 属于 g 的定义域的 x 组成。
2. Quadratic Functions | 二次函数
A quadratic function can be written in completed square form a(x + p)2 + q, which reveals the vertex (−p, q) and axis of symmetry x = −p.
二次函数可写成完全平方形式 a(x + p)2 + q，由此可得顶点 (−p, q) 和对称轴 x = −p。
The discriminant Δ = b2 − 4ac determines the nature of roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.
判别式 Δ = b2 − 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有两个相等实根，Δ < 0 无实根。
To solve a quadratic inequality such as ax2 + bx + c > 0, sketch the graph and identify intervals where it is above the x-axis. Alternatively, use a sign chart based on the roots.
解二次不等式 ax2 + bx + c > 0 时，可画出草图并找出图像在 x 轴上方的区间，或根据两根制作符号表。
3. Equations and Inequalities | 方程与不等式
Simultaneous equations with one linear and one quadratic are usually solved by substitution. Rearrange the linear equation for one variable and substitute it into the quadratic to form a single equation.
由一个一次和一个二次方程组成的联立方程组通常用代入法求解。将一次方程变形为某变量的表达式，代入二次方程得到一元方程。
An absolute value equation |f(x)| = k (k ≥ 0) leads to two cases: f(x) = k or f(x) = −k. Always check answers in the original equation.
绝对值方程 |f(x)| = k (k ≥ 0) 可化为两个方程：f(x) = k 或 f(x) = −k，解出后必须代回原方程检验。
Modulus inequalities like |f(x)| < c are equivalent to −c < f(x) < c, while |f(x)| > c means f(x) < −c or f(x) > c. Graphical approaches often help visualise solution sets.
绝对值不等式如 |f(x)| < c 等价于 −c < f(x) < c；|f(x)| > c 表示 f(x) < −c 或 f(x) > c。利用图像常能直观表示解集。
4. Surds and Indices | 根式与指数
Surds can be simplified by extracting perfect squares: √(a2b) = a√b. To rationalise a denominator containing a single surd, multiply numerator and denominator by that surd; for a binomial like a + √b, use the conjugate a − √b.
根式可通过提取平方因子化简，如 √(a2b) = a√b。分母含单个根式时，分子分母同乘该根式；若分母为 a + √b，则乘以其共轭式 a − √b 进行有理化。
Index laws are fundamental: am × an = am+n, (am)n = amn, a−n = 1/an, and am/n = n√am. Fractional indices represent roots, and negative indices represent reciprocals.
指数基本法则：am × an = am+n，(am)n = amn，a−n = 1/an，am/n = n√am。分数指数表示开方，负指数表示倒数。
5. Logarithms and Exponentials | 对数与指数函数
If y = ax then x = loga y. The natural logarithm ln x uses base e, so ln x = loge x. The function ax is the inverse of loga x.
若 y = ax，则 x = loga y。自然对数 ln x 以 e 为底，即 ln x = loge x。指数函数 ax 与对数函数 loga x 互为反函数。
Log laws: loga (xy) = loga x + loga y, loga (x/y) = loga x − loga y, loga (xn) = n loga x. The change of base formula loga b = logc b / logc a is used to evaluate logs with any base.
对数运算法则：loga (xy) = loga x + loga y，loga (x/y) = loga x − loga y，loga (xn) = n loga x。换底公式 loga b = logc b / logc a 用于计算任意底数的对数。
Equations like e2x = 5 are solved by taking natural logs: 2x = ln 5, so x = (ln 5)/2. Logarithmic equations may require combining log terms into a single logarithm first.
解指数方程如 e2x = 5 时，两边取自然对数得 2x = ln 5，解得 x = (ln 5)/2。对数方程常需先利用对数法则合并为单一对数。
6. Polynomials and Binomial Expansion | 多项式与二项式展开
The Remainder Theorem states that when a polynomial f(x) is divided by (x − a), the remainder is f(a). The Factor Theorem follows: (x − a) is a factor of f(x) if and only if f(a) = 0.
余式定理：多项式 f(x) 除以 (x − a) 的余式为 f(a)。因式定理随之而来：(x − a) 是 f(x) 的因式当且仅当 f(a) = 0。
Polynomial division can be carried out by long division or synthetic division, and it is used to factorise higher‑degree polynomials or simplify rational expressions.
多项式除法可以通过长除法或综合除法进行，常用于高次多项式的因式分解或化简有理分式。
For any rational n and |x| < 1, the binomial expansion is (1 + x)n = 1 + nx + n(n−1)/2! x2 + n(n−1)(n−2)/3! x3 + … This expansion is infinite when n is not a positive integer, and the condition |x| < 1 ensures convergence.
对于任意有理数 n 且 |x| < 1，二项式展开式为 (1 + x)n = 1 + nx + n(n−1)/2! x2 + n(n−1)(n−2)/3! x3 + …。当 n 不是正整数时，展开式是无穷级数，条件 |x| < 1 保证收敛。
7. Coordinate Geometry | 坐标几何
The distance between two points (x1, y1) and (x2, y2) is √[(x2 −
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