📚 IB & OCR Maths: End-of-Term Revision Checklist | IB & OCR 数学：期末复习提纲
Whether you are studying IB Mathematics: Analysis and Approaches, Applications and Interpretation, or OCR A Level Mathematics, this end-of-term revision guide consolidates the essential topics you need to master. Use it to structure your final review and build confidence for assessments.
无论你正在学习IB数学（分析与方法、应用与解释）还是OCR A Level数学，这份期末复习提纲将重要主题集中在一起，帮助你系统梳理知识，自信迎接评估。
1. Algebra Foundations | 代数基础
Refresh index laws and surds. Simplify expressions using the key rules shown in the table below, and rationalise denominators such as 1/(√a + √b).
复习指数律和根式。运用下表中的核心法则化简表达式，并有理化分母，如 1/(√a + √b)。
| aᵐ × aⁿ = aᵐ⁺ⁿ | Multiply powers, add exponents | 同底数幂相乘，指数相加 |
| (aᵐ)ⁿ = aᵐⁿ | Power of a power | 幂的乘方 |
| a⁻ⁿ = 1/aⁿ | Negative exponent | 负指数 |
| a¹/ⁿ = ⁿ√a | Fractional exponent | 分指数 |
Factorising quadratics and cubics is essential. For IB, be comfortable with the factor theorem and polynomial division. OCR expects fluency in completing the square and discriminant analysis.
二次式和三次式的因式分解至关重要。在IB中，需熟练掌握因式定理和多项式除法。OCR课程则要求熟练运用配方法及判别式分析。
Expand binomials using the binomial theorem: (a + b)ⁿ = Σ_{k=0}ⁿ C(n,k) aⁿ⁻ᵏ bᵏ, where C(n,k) = n!/(k!(n−k)!).
利用二项式定理展开： (a + b)ⁿ = Σ_{k=0}ⁿ C(n,k) aⁿ⁻ᵏ bᵏ，其中 C(n,k) = n!/(k!(n−k)!).
2. Functions and Graphs | 函数与图像
Master the language of functions: domain, range, inverse f⁻¹(x), and composite f(g(x)). Sketch graphs of polynomial, rational, exponential, logarithmic and trigonometric functions. Identify horizontal and vertical asymptotes.
掌握函数的语言：定义域、值域、反函数 f⁻¹(x) 以及复合函数 f(g(x))。熟练绘制多项式函数、有理函数、指数函数、对数函数和三角函数的图像，识别水平和垂直渐近线。
Transformations of graphs: y = f(x) + a (vertical shift), y = f(x + a) (horizontal shift), y = a f(x) (vertical stretch), y = f(ax) (horizontal stretch). Reflections in the axes.
图像变换：y = f(x) + a (竖直平移)，y = f(x + a) (水平平移)，y = a f(x) (竖直伸缩)，y = f(ax) (水平伸缩)；以及关于坐标轴的反射。
For reciprocal and modulus functions, be careful with piecewise definitions. |x| = x for x ≥ 0, −x for x < 0.
倒数函数和绝对值函数需注意分段定义。|x| = x 当 x ≥ 0，−x 当 x < 0。
3. Equations and Inequalities | 方程与不等式
Solve quadratic equations by factorising, completing the square, and the quadratic formula: x = [−b ± √(b² − 4ac)] / 2a. The discriminant Δ = b² − 4ac determines the nature of roots.
解二次方程的方法包括因式分解、配方法和求根公式：x = [−b ± √(b² − 4ac)] / 2a。判别式 Δ = b² − 4ac 决定根的性质。
Simultaneous equations: linear and non‑linear systems. Solve by substitution. Graphically, intersection points correspond to solutions.
联立方程：线性与非线性方程组。用代入法求解。从图像上看，交点坐标即为方程组的解。
Inequalities: solve linear, quadratic and rational inequalities. Represent solutions on a number line or using interval notation. Remember to flip the inequality sign when multiplying/dividing by a negative number.
不等式：解一元线性、二次和有理不等式。用数轴或区间表示法表示解集。注意乘或除以负数时，不等号方向必须改变。
4. Sequences and Series | 数列与级数
Arithmetic sequences: n-th term uₙ = a + (n−1)d, sum of n terms Sₙ = n/2 [2a + (n−1)d]. Geometric sequences: uₙ = arⁿ⁻¹, Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1. Sum to infinity: S∞ = a/(1−r) for |r| < 1.
等差数列：通项 uₙ = a + (n−1)d，前 n 项和 Sₙ = n/2 [2a + (n−1)d]。等比数列：通项 uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a(1−rⁿ)/(1−r)，r ≠ 1。无穷项和：S∞ = a/(1−r)，当 |r| < 1。
Be able to apply sequences to real‑world contexts, such as compound interest, depreciation and exponential growth models.
能将数列应用于实际问题，如复利、折旧和指数增长模型。
Sigma notation: Σ_{i=m}ⁿ f(i). Practise rewriting sums and using standard results for Σ i, Σ i² and Σ i³.
求和符号：Σ_{i=m}ⁿ f(i)。练习改写求和式，并使用 ∑ i, ∑ i² 和 ∑ i³ 的常见公式。
5. Trigonometry | 三角学
Know exact values for sin, cos, tan of 0°, 30°, 45°, 60°, 90° (and radian equivalents). Use the unit circle to extend to all angles and to deduce properties like sin(π − θ) = sin θ.
熟记 0°、30°、45°、60°、90° 的正弦、余弦和正切精确值（以及对应的弧度值）。利用单位圆推广到任意角，并推导性质，如 sin(π − θ) = sin θ。
Key identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ. Double‑angle formulas: sin 2θ = 2 sinθ cosθ, cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
核心恒等式：sin²θ + cos²θ = 1，1 + tan²θ = sec²θ，1 + cot²θ = csc²θ。倍角公式：sin 2θ = 2 sinθ cosθ，cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。
Solving trigonometric equations: find general solutions or solutions in a given interval. Use the CAST diagram or graphical methods to check for all possible solutions.
解三角方程：求通解或给定区间内的解。使用 CAST 图或图像法检查所有可能的解。
Sine and cosine rules: a/sin A = b/sin B = c/sin C (sine rule), a² = b² + c² − 2bc cos A (cosine rule). Area of a triangle = ½ ab sin C.
正弦定理和余弦定理：a/sin A = b/sin B = c/sin C；余弦定理：a² = b² + c² − 2bc cos A。三角形面积 = ½ ab sin C。
6. Exponentials and Logarithms | 指数与对数
Recall that y = aˣ ⇔ x = logₐ y. Natural log: ln x = logₑ x, with e ≈ 2.718. Laws: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln aᵏ = k ln a.
牢记 y = aˣ ⇔ x = logₐ y。自然对数：ln x = logₑ x，其中 e ≈ 2.718。对数运算法则：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln aᵏ = k ln a。
Solve equations where the unknown appears in the power, e.g. 3²ˣ⁺¹ = 5. Take logs of both sides to bring the exponent down linearly.
求解
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