📚 IB CCEA Mathematics: End-of-Term Revision Guide | IB CCEA 数学：期末复习提纲
As final examinations approach, a well‑structured revision checklist becomes your most powerful tool. This guide brings together the essential topics from the IB and CCEA Mathematics specifications – covering algebra, functions, trigonometry, calculus, vectors, sequences, probability, statistics, and mechanics. Use it to track your progress, identify weak areas, and ensure that every key formula and technique is at your fingertips.
期末考试临近，一份条理清晰的复习提纲是你最有力的工具。本指南汇集了 IB 与 CCEA 数学大纲中的核心主题，涵盖代数、函数、三角、微积分、向量、数列、概率、统计和力学，帮助你追踪复习进度，查漏补缺，确保每一条重要公式和技巧都了然于心。
1. Algebra Fundamentals | 代数基础
Fluency in manipulating algebraic expressions is the bedrock of all advanced topics. Practise expanding brackets, factorising quadratic and cubic polynomials, and simplifying rational expressions. Be completely confident with the Remainder Theorem and Factor Theorem: for a polynomial f(x), if f(a)=0 then (x−a) is a factor.
熟练的代数运算能力是所有进阶主题的基石。请反复练习展开括号、对二次和三次多项式进行因式分解，以及化简有理式。要完全掌握余数定理与因式定理：对于多项式 f(x)，若 f(a)=0，则 (x−a) 为其因式。
Master solving linear equations, quadratic equations (by factorising, completing the square, and the quadratic formula), and simultaneous equations. For inequalities, pay careful attention to the sign when multiplying or dividing by a negative number, and sketch sign diagrams for rational inequalities.
熟练掌握解线性方程、二次方程（因式分解法、配方法、求根公式法）以及联立方程组。解不等式时，特别留意乘以或除以负数时的变号规则，对于有理不等式要绘制符号图。
x = [−b ± √(b² − 4ac)] / (2a)
| Key Skills | 核心技能 |
|---|---|
| Simplify surds and rationalise denominators | 化简根式，进行分母有理化 |
| Apply index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ | 运用指数运算法则 |
| Perform polynomial long division | 多项式长除法 |
| Solve modulus equations and inequalities | 解绝对值方程与不等式 |
2. Functions and Graphs | 函数与图像
Understand the concepts of domain, range, one‑to‑one functions, and inverse functions. For f⁻¹ to exist, the original function must be one‑to‑one. Be able to sketch and interpret graphs of polynomial, rational, exponential, logarithmic, and trigonometric functions, including key features such as intercepts, asymptotes, turning points, and behaviour at infinity.
理解定义域、值域、一一映射和反函数等概念。反函数 f⁻¹ 存在的前提是原函数为一对一映射。要能绘制并解读多项式、有理函数、指数函数、对数函数和三角函数的图像，把握截距、渐近线、驻点和无穷远处的性态等关键特征。
Transformations must be second nature: y = f(x) + a translates vertically, y = f(x + a) translates horizontally, y = a f(x) is a vertical stretch, y = f(ax) is a horizontal stretch. Reflections take the form y = −f(x) and y = f(−x). Be prepared to combine multiple transformations and to apply them to any parent function.
图像变换必须内化成直觉：y = f(x) + a 为垂直平移，y = f(x + a) 为水平平移，y = a f(x) 为垂直伸缩，y = f(ax) 为水平伸缩；y = −f(x) 和 y = f(−x) 分别表示关于 x 轴和 y 轴的反射。要能够对一个母函数进行多重变换的组合。
Composite functions are written as (g ∘ f)(x) = g(f(x)). Always check that the range of the inner function lies within the domain of the outer function. Questions often ask you to find the domain of a composite function from first principles.
复合函数写作 (g ∘ f)(x) = g(f(x))，务必检查内层函数的值域是否在外层函数的定义域内。题目常要求从定义出发求复合函数的定义域。
3. Exponentials and Logarithms | 指数与对数
Logarithms are the inverse of exponentials. The fundamental laws are: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, and logₐ(xⁿ) = n logₐx. The natural logarithm ln x is the logarithm with base e, where ln(eˣ) = x and e^(ln x) = x.
对数是指数的逆运算。基本运算法则包括：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，logₐ(xⁿ) = n logₐx。自然对数 ln x 是以 e 为底的对数，满足 ln(eˣ) = x 且 e^(ln x) = x。
Exponential growth and decay models are almost always of the form N = N₀ e^(kt). Know that if k > 0 the quantity grows, and if k < 0 it decays. The half‑life or doubling time is found by solving the equation with N/N₀ = 0.5 or 2 respectively.
指数增长和衰减模型通常形式为 N = N₀ e^(kt)，其中 k>0 表示增长，k<0 表示衰减。半衰期或倍增时间可通过令 N/N₀ 等于 0.5 或 2 求得。
logₐ b = (ln b) / (ln a) (change of base)
When solving exponential equations, take logarithms of both sides and apply the power rule. For equations like 2ˣ = 5, use x = ln 5 / ln 2. In modelling, you may need to linearise the data using logarithms.
解指数方程时两边取对数并运用幂次法则。例如 2ˣ = 5，得到 x = ln 5 / ln 2。在建模题中，你可能需要利用对数进行数据线性化。
4. Trigonometry | 三角学
Memorise exact trigonometric values for 0°, 30°, 45°, 60°, 90° (and their radian equivalents) for sin, cos, and tan. Use the unit circle and the CAST diagram to find all solutions of trigonometric equations within a given interval.
熟记 0°、30°、45°、60°、90°（及其弧度制对应值）的正弦、余弦和正切精确值。善用单位圆和 CAST 图来求给定区间内三角方程的所有解。
tan θ = sin θ / cos θ, sin²θ + cos²θ = 1
Be fluent in applying the compound‑angle and double‑angle formulas: sin(A ± B), cos(A ± B), tan(A ± B), sin 2A = 2 sin A cos A, cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A. These are indispensable for proving identities and solving more complex equations.
熟练运用和角公式与倍角公式：sin(A ± B)、cos(A ± B)、tan(A ± B)，以及 sin 2A = 2 sin A cos A，cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A。这些公式在证明恒等式和求解稍复杂的方程时必不可少。
For non‑right‑angled triangles, the Sine Rule (a / sin A = b / sin B = c / sin C) and the Cosine Rule (a² = b² + c² − 2bc cos A) are essential. Area of a triangle can be calculated using ½ ab sin C. Radian measure gives arc length l = rθ and sector area A = ½ r²θ.
对于非直角三角形，正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 a² = b² + c² − 2bc cos A 至关重要。三角形面积可用 ½ ab sin C 计算。弧度制下弧长 l = rθ，扇形面积 A = ½ r²θ。
5. Differentiation | 微分
Differentiation from first principles calculates the gradient of a tangent as a limit: f'(x) = limₕ→₀ [f(x + h) − f(x)] / h. Though you may not be required to use it for every function, you must understand the concept and be able to apply it to simple polynomials.
从第一原理出发的微分将切线斜率定义为极限：f'(x) = limₕ→₀ [f(x + h) − f(x)] / h。即使不必对所有函数都用第一原理求导，也必须理解这一概念并能对简单的多项式进行推导。
Learn the standard derivatives by heart:
| f(x) | f'(x) | 备注 |
|---|---|---|
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