📚 Year 11 CAIE Additional Mathematics: Core Concepts Overview | 11年级CAIE进阶数学：核心知识点梳理
The CAIE Additional Mathematics (0606) syllabus builds a rigorous bridge from IGCSE Mathematics to A Level studies. It covers challenging topics in algebra, trigonometry, calculus and vectors. A strong grasp of these core concepts is the key to high exam performance and future mathematical confidence.
CAIE 进阶数学（0606）课程架起一座从 IGCSE 数学通向 A Level 学习的严谨桥梁。它涵盖了代数、三角学、微积分与向量等高阶内容。扎实掌握这些核心知识点是取得优异考试成绩、建立数学信心的关键。
1. Functions | 函数
A function f maps each element x from the domain to exactly one element f(x) in the range. The domain is the set of all possible inputs; the range is the set of all outputs.
函数 f 将定义域中的每个元素 x 映射到值域中恰好一个元素 f(x)。定义域是所有可能输入的集合，值域是所有输出的集合。
The vertical line test helps decide if a relation is a function: if any vertical line cuts the graph more than once, it is not a function.
垂直线检验用于判断关系是否为函数：若任意一条垂直线与图像相交超过一次，则不是函数。
A one-to-one (injective) function gives distinct outputs for distinct inputs. Only one-to-one functions have inverses. To find the inverse, swap x and y, then solve for y; the domain of f⁻¹ equals the range of f.
一一映射函数对不同的输入给出不同的输出。只有一一函数才有反函数。求反函数时交换 x 与 y 并解出 y；f⁻¹ 的定义域等于 f 的值域。
) = f(g(x)). You must consider the domain of the inner function and the overall composite.
复合函数 fg(x) 表示先作用 g 再作用 f：fg(x) = f(g(x))。必须考虑内层函数及整体复合的定义域。
g⁻¹(x) exists ↔ g is one‑to‑one; (f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹
2. Quadratic Functions | 二次函数
x + c (a ≠ 0). Completing the square gives the turning point form: a(x - h)² + k, where (h, k) is the vertex.
一般二次函数为 f(x) = ax² + bx + c (a ≠ 0)。配方法得到顶点式：a(x - h)² + k，其中 (h, k) 为顶点坐标。
s. If Δ > 0, two distinct real roots; Δ = 0, one repeated real root; Δ < 0, no real roots.
判别式 Δ = b² - 4ac 决定根的性质。若 Δ > 0，有两个不同实根；Δ = 0，有一个重根；Δ < 0，没有实根。
The graph of a quadratic is a parabola. Its shape depends on the sign of a: a > 0 gives a minimum point, a < 0 gives a maximum.
二次函数的图像是一条抛物线。其开口方向由 a 的符号决定：a > 0 时有最小值点，a < 0 时有最大值点。
Quadratic inequalities such as ax² + bx + c > 0 can be solved by sketching the graph and identifying where the curve lies above or below the x‑axis.
二次不等式如 ax² + bx + c > 0 可通过画出草图、判断曲线在 x 轴上方或下方的区间来求解。
3. Equations, Inequalities and Graphs | 方程、不等式与图像
Polynomial equations can be solved by factorisation, the factor theorem, or graphical methods. Inequalities are solved by considering sign changes across critical values.
多项式方程可通过因式分解、因式定理或图像法求解。不等式则需考虑穿越临界值时的符号变化。
For absolute value equations, e.g. |2x - 3| = 5, split into two linear equations: 2x - 3 = 5 or 2x - 3 = -5, then solve each.
对于绝对值方程，如 |2x - 3| = 5，可拆分成两个线性方程：2x - 3 = 5 或 2x - 3 = -5，分别求解。
When solving an inequality like |2x - 3| ≤ 5, rewrite as -5 ≤ 2x - 3 ≤ 5, then solve the sandwich inequality.
求解如 |2x - 3| ≤ 5 的不等式时，可改写为 -5 ≤ 2x - 3 ≤ 5，再解此类夹挤不等式。
to estimate solutions to equations like f(x) = g(x) or to interpret inequalities visually.
函数图像可用于估算方程 f(x) = g(x) 的解，或直观理解不等式的解集。
4. Indices and Surds | 指数与根式
= aᵐⁿ, a⁻ⁿ = 1/aⁿ, a¹/ⁿ = √[n]{a}. All standard index rules apply for rational exponents.
指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，a¹/ⁿ = √[n]{a}。所有标准指数规则对有理指数同样适用。
Surds can be simplified by extracting square factors: √50 = √25×2 = 5√2. Always express answers in simplest surd form.
根式可通过提取完全平方因子来化简：√50 = √25×2 = 5√2。答案应始终化为最简根式。
Rationalising the denominator involves multiplying numerator and denominator by the conjugate, e.g. 1/(2+√3) becomes (2-√3)/(4-3) = 2-√3.
分母有理化需要将分子和分母同时乘以共轭因子，例如 1/(2+√3) 化为 (2-√3)/(4-3) = 2-√3。
5. Polynomials and Simultaneous Equations | 多项式与联立方程
) if and only if f(p) = 0. This is used to factorise cubics and higher-degree polynomials.
因式定理指出，(x - p) 是 f(x) 的因式当且仅当 f(p) = 0。该定理用于对三次及更高次多项式进行因式分解。
Polynomial long division helps divide a polynomial by a linear or quadratic divisor, giving a quotient and remainder. The remainder equals f(p) when dividing by (x - p).
多项式长除法可将多项式除以一次或二次因式，得到商式和余式。当除以 (x - p) 时，余式等于 f(p)。
Solving simultaneous equations where one is linear and the other nonlinear: substitute the linear into the nonlinear to obtain a single equation in one variable.
求解一个线性、一个非线性的联立方程组：将线性方程代入非线性方程，得到关于一个未知数的单一方程。
The solutions correspond to the intersection points of a straight line and a curve; there may be 0, 1 or 2 real intersection points.
方程组的解对应于一条直线与一条曲线的交点；可能有 0 个、1 个或 2 个实交点。
6. Logarithmic and Exponential Functions | 对数函数与指数函数
The natural exponential function y = eˣ and natural logarithm y = ln x are inverses: ln(eˣ) = x, eˡⁿˣ = x.
自然指数函数 y = eˣ 与自然对数函数 y = ln x 互为反函数：ln(eˣ) = x, eˡⁿˣ = x。
ln(a/b) = ln a - ln b, ln(aⁿ) = n ln a. The change of base formula: logₐ b = ln b / ln a.
对数运算法则：ln(ab) = ln a + ln b，ln(a/b) = ln a - ln b，ln(aⁿ) = n ln a。换底公式：logₐ b = ln b / ln a。
Exponential equations like aˣ ithms: x = ln b / ln a. Logarithmic equations often require checking for extraneous roots.
指数方程如 aˣ = b 可通过取对数求解：x = ln b / ln a。对数方程常需检验是否有增根。
The graphs of eˣ and ln x are reflections in the line y = x. Asymptotes: y = 0 for eˣ (as x → -∞); x = 0 for ln x.
eˣ 与 ln x 的图像关于直线 y = x 对称。渐近线：eˣ 有水平渐近线 y = 0 (x → -∞)；ln x 有垂直渐近线 x = 0。
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