IB & WJEC Maths: Last-Minute Revision Notes | IB 与 WJEC 数学：考前冲刺笔记

📚 IB & WJEC Maths: Last-Minute Revision Notes | IB 与 WJEC 数学：考前冲刺笔记

This last‑minute revision guide brings together the most essential concepts, formulas, and problem‑solving strategies for students tackling IB and WJEC Mathematics. Whether you are sitting the Analysis & Approaches or Applications & Interpretation papers, or preparing for WJEC pure and applied units, these notes highlight what you must know before walking into the exam hall. Every section pairs core theory with practical tips, helping you avoid common mistakes and use your time efficiently.

这份考前冲刺笔记汇总了 IB 与 WJEC 数学中最核心的概念、公式和解题策略。无论你参加的是 IB 分析与方法、应用与解释考试，还是 WJEC 纯数及应用单元，这些要点都能帮助你快速回顾必考内容。每个部分都将基础理论与实用技巧相结合，帮你避开易错点，高效利用最后的时间。

1. Core Algebra and Equations | 核心代数与方程

Mastering algebra is non‑negotiable: almost every exam question relies on your ability to manipulate expressions and solve equations fluently. Start by ensuring you can factorise quadratics by inspection, by grouping, and by using the quadratic formula x = [−b ± √(b² − 4ac)] / 2a. Remember that the discriminant Δ = b² − 4ac tells you how many real roots exist — when Δ > 0 there are two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 means no real roots. This is particularly important for hidden quadratics where a substitution such as y = x² or y = eˣ reduces a more complicated equation to a quadratic in y.

代数基本功至关重要：几乎每道考题都需要你熟练地变形和求解方程。确保你能够通过观察、分组以及求根公式 x = [−b ± √(b² − 4ac)] / 2a 来分解二次式。记住判别式 Δ = b² − 4ac 决定了实根的数量——Δ > 0 时有两个不等实根，Δ = 0 时有一个重根，Δ < 0 时无实根。这对于隐藏的二次方程尤为重要，例如通过代换 y = x² 或 y = eˣ 将复杂方程转化为关于 y 的二次方程。

Simultaneous equations appear in both linear and non‑linear forms. Always sketch a quick mental picture — one linear and one quadratic can produce zero, one, or two intersection points. When solving algebraically, isolate one variable from the linear equation and substitute it into the non‑linear one. On the WJEC specification, you will also see older‑style questions on completing the square, which is the gateway to finding the vertex of a parabola: x² + bx + c = (x + b/2)² − (b/2)² + c.

联立方程既有线性也有非线性形式。先在脑中快速画出草图——一条直线和一个二次函数可能产生零个、一个或两个交点。代数求解时，从线性方程中分离出一个变量，再代入非线性方程。在 WJEC 大纲中，你还会遇到传统的配方法题目，它也是求抛物线顶点的钥匙：x² + bx + c = (x + b/2)² − (b/2)² + c。

2. Functions and Graphs | 函数与图像

A function must map each input to exactly one output — use the vertical line test on a graph to check this. The domain is the set of allowed inputs, and the range is the set of possible outputs. For composite functions f∘g(x) = f(g(x)), the domain of f∘g is restricted by both the domain of g and the requirement that g(x) lies in the domain of f. Always work from the inside out. Inverse functions f⁻¹(x) exist only when f is one‑to‑one; to find the inverse, write y = f(x), swap x and y, then solve for y. The graphs of f and f⁻¹ are reflections in the line y = x.

函数必须将每个输入映射到唯一的输出——可在图像上用垂直线检验。定义域是允许的输入集合，值域是可能的输出集合。对于复合函数 f∘g(x) = f(g(x))，f∘g 的定义域既要受到 g 定义域的限制，还要求 g(x) 落在 f 的定义域内。永远从内向外逐步检查。反函数 f⁻¹(x) 仅在 f 是一一映射时才存在；求反函数时写出 y = f(x)，交换 x 与 y，再解出 y。f 与 f⁻¹ 的图像关于直线 y = x 对称。

Graph transformations often appear in combination. The order matters: when an expression involves both a horizontal shift and a horizontal stretch, factorise the coefficient of x first. For example, starting from f(x), the graph of f(2x + 6) should be rewritten as f(2(x + 3)). This tells you to shift left by 3 and then compress horizontally by a factor of 2. Remember these key rules:

图像变换常常组合出现。顺序至关重要：当表达式中同时含有水平平移和水平伸缩时，先提取 x 的系数。例如，从 f(x) 出发，f(2x + 6) 应改写为 f(2(x + 3))。这表示先向左平移 3 个单位，再水平压缩为原来的二分之一。记住这些关键规则：

f(x) + a → vertical translation up by a | 向上垂直平移 a 个单位
f(x + a) → horizontal translation left by a | 向左水平平移 a 个单位
a f(x) → vertical stretch by factor a | 垂直伸缩 a 倍
f(a x) → horizontal stretch by factor 1/a | 水平伸缩 1/a 倍
−f(x) → reflection in the x‑axis | 关于 x 轴反射
f(−x) → reflection in the y‑axis | 关于 y 轴反射

3. Trigonometry Essentials | 三角学基础

Exact trigonometric values for 30°, 45°, and 60° (π/6, π/4, π/3) must be memorised, as they are heavily tested without a calculator. Use the two special triangles: the isosceles right triangle with legs 1, 1, hypotenuse √2, and the half‑equilateral triangle with sides 1, √3, 2. From these you can quickly read off sin, cos, and tan. Know the signs in each quadrant — “All Students Take Calculus” reminds you which functions are positive: All in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4.

必须熟记 30°、45° 和 60°（π/6、π/4、π/3）的精确三角值，这些是不允许使用计算器的常见考点。利用两个特殊三角形：边长为 1, 1, 斜边 √2 的等腰直角三角形，以及边长为 1, √3, 2 的半个等边三角形。从中可以直接得出 sin、cos、tan 值。掌握各象限的符号规律——“All Students Take Calculus”告诉你哪一象限的哪个函数为正：第一象限全正，第二象限正弦为正，第三象限正切为正，第四象限余弦为正。

Radians are the default angle measure in calculus. The conversion π radians = 180° gives arc length s = rθ and sector area A = ½ r²θ. For solving trigonometric equations, always draw a quick unit‑circle sketch. After using the inverse function to find a principal value, locate all other solutions within the required interval by considering symmetry. For equations of the form a sin θ + b cos θ = c, use the harmonic form R sin(θ ± α) or R cos(θ ± α), where R = √(a² + b²) and α = arctan(b/a) with the quadrant adjusted.

弧度是微积分中默认的角度度量。π 弧度 = 180°，由此得到弧长公式 s = rθ 和扇形面积公式 A = ½ r²θ。解三角方程时，务必快速画出单位圆草图。利用反三角函数求出主值后，通过对称性找到指定区间内的所有其他解。对于形如 a sin θ + b cos θ = c 的方程，使用谐波形式 R sin(θ ± α) 或 R cos(θ ± α)，其中 R = √(a² + b²)，α = arctan(b/a) 并调整象限。

4. Exponentials and Logarithms | 指数与对数

The natural exponential function y = eˣ has the unique property that its derivative is itself. Its inverse, the natural logarithm y = ln x, is defined only for x > 0. The laws of logarithms must become second nature:

自然指数函数 y = eˣ 具有导数等于自身的独特性质。它的反函数是自然对数 y = ln x，其定义域仅为 x > 0。对数运算法则必须化为本能：

ln (ab) = ln a + ln b
ln (a/b) = ln a − ln b
ln (aᵏ) = k ln a
eˡⁿ ᵃ = a and ln (eᵃ) = a

Many exam questions require you to change a model such as y = a bˣ into linear form. Taking natural logs of both sides gives ln y = ln a + x ln b, so plotting ln y against x yields a straight line with gradient ln b and intercept ln a. Similarly, for a power law y = a xⁿ, taking logs on both axes gives ln y = ln a + n ln x, so the graph of ln y against ln x has gradient n.

许多考题要求你将 y = a bˣ 这类模型转化为线性形式。两边取自然对数得到 ln y = ln a + x ln b，因此以 ln y 对 x 作图会得到一条斜率为 ln b、截距为 ln a 的直线。类似地，对于幂律 y = a xⁿ，双对数变换给出 ln y = ln a + n ln x，因此 ln y 对 ln x 图像的斜率为 n。

When solving exponential equations, use the strategy “take logs of both sides” as soon as the unknown is in the exponent. Be careful: ln (a + b) is not ln a + ln b — this is one of the most common errors on the exam. Always isolate the exponential term before taking logarithms.

当求解指数方程时，一旦未知数出现在指数上，就使用“两边取对数”的策略。注意：ln (a + b) 不等于 ln a + ln b——这是考试中最常见的错误之一。在取对数之前，务必先将指数项隔离开来。

5. Differentiation Techniques | 微分技巧

The derivative gives the instantaneous rate of change, or the gradient of the tangent. For polynomials, use the power rule: d/dx (xⁿ) = n xⁿ⁻¹. This extends to rational exponents and roots: √x = x¹/², so its derivative is ½ x⁻¹/². The chain rule handles composite functions: d/dx [f(g(x))] = f'(g(x)) · g'(x). Think “differentiate the outer function, leaving the inside unchanged, then multiply by the derivative of the inside.” The product rule states d/dx (u v) = u’ v + u v’, and the quotient rule: d/dx (u/v) = (u’ v − u v’) / v².

导数表示瞬时变化率，也就是切线的斜率。对于多项式，使用幂法则：d/dx (xⁿ) = n xⁿ⁻¹。这也可以推广到有理指数和根式：√x = x¹/²，其导数为 ½ x⁻¹/²。链式法则处理复合函数：d/dx [f(g(x))] = f'(g(x)) · g'(x)。理解为“对外层函数求导，内部保持不变，再乘以内部函数的导数”。乘积法则为 d/dx (u v) = u’ v + u v’；商法则为 d/dx (u/v) = (u’ v − u v’) / v²。

Special derivatives you must know: d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x. If using radians, these hold as written; if the question accidentally uses degrees, convert first. The second derivative f”(x) tells you about the rate of change of the gradient, which determines concavity and helps classify stationary points as local maxima or minima.

需要记住的特殊导数：d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x。如果采用弧度制，上述公式直接成立；如果题目误用了度数，务必先转换。二阶导数 f”(x) 表示斜率的变化率，它决定曲线的凹凸性，并帮助将驻点分类为局部极大值或极小值。

6. Integration Methods | 积分方法

Integration is the reverse of differentiation. The basic power rule for integration is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1. The special case ∫ 1/x dx = ln |x| + C is often tested. Definite integrals ∫ₐᵇ f(x) dx calculate the exact area between the curve and the x‑axis from x = a to x = b; remember that areas below the axis contribute a negative value unless you split the interval.

积分是微分的逆运算。基本的幂积分法则为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ −1。特殊情形 ∫ 1/x dx = ln |x| + C 经常出现。定积分 ∫ₐᵇ f(x) dx 计算从 x = a 到 x = b 曲线与 x 轴之间的准确面积；注意位于 x 轴下方的部分会产生负值，除非你将区间分割开来。

Integration by substitution is the chain rule in reverse. When a factor of the derivative of an inner function is present, let u = inner function. For definite integrals, remember to change the limits to u‑values before substituting back — this avoids having to rewrite the answer in terms of x. Integration by parts, ∫ u dv = u v − ∫ v du, is used for products where one function becomes simpler when differentiated (like ln x) and the other can be integrated easily (like xⁿ, eˣ, sin x). Choose u using the LIATE rule: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.

换元积分法是链式法则的逆过程。当被积函数中含有内层函数导数的因子时，令 u = 内层函数。对于定积分，记住在代入之前将积分限换成 u 值——这可以避免再换回 x 的麻烦。分部积分法 ∫ u dv = u v − ∫ v du 适用于乘积形式，其中一个函数求导后变得更简单（比如 ln x），另一个容易积分（如 xⁿ、eˣ、sin x）。按照 LIATE 规则选择 u：对数函数、反三角函数、代数函数、三角函数、指数函数。

7. Sequences and Series | 数列与级数

Arithmetic sequences have a constant difference d. The nth term is uₙ = a + (n−1)d, and the sum of the first n terms is Sₙ = n/2 [2a + (n−1)d] or n/2 (a + l) where l is the last term. Geometric sequences have a constant ratio r. The nth term is uₙ = a rⁿ⁻¹, and the sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r), valid for r ≠ 1. An infinite geometric series converges to a sum S∞ = a/(1 − r) provided |r| < 1. Many modelling questions involve compound interest or population growth, where you need to recognise whether the pattern is arithmetic or geometric.

等差数列的公差 d 恒定。第 n 项为 uₙ = a + (n−1)d，前 n 项和为 Sₙ = n/2 [2a + (n−1)d] 或 n/2 (a + l)，其中 l 为末项。等比数列的公比 r 恒定。第 n 项为 uₙ = a rⁿ⁻¹，前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)，当 r ≠ 1 时成立。当 |r| < 1 时，无穷等比级数收敛，和为 S∞ = a/(1 − r)。许多建模问题涉及复利或人口增长，你需要识别出数列模式是等差还是等比。

The sigma notation Σ needs careful attention to the starting index. The sum Σₖ₌₁ⁿ (a k + b) can be split into a Σ k + Σ b. You must know that Σ k = n(n+1)/2 and Σ k² = n(n+1)(2n+1)/6. These formulas sometimes appear in WJEC summation questions or in IB as part of a proof by induction. Speaking of induction: the four key steps are: prove true for n = 1, assume true for n = k, prove true for n = k + 1 using the assumption, and write a concluding statement.

Σ 求和符号需要仔细注意起始下标。Σₖ₌₁ⁿ (a k + b) 可拆分为 a Σ k + Σ b。必须掌握 Σ k = n(n+1)/2 和 Σ k² = n(n+1)(2n+1)/6。这些公式有时会出现在 WJEC 求和问题中，或在 IB 中作为数学归纳法证明的一部分。关于归纳法：四个关键步骤为：证明 n = 1 时成立，假设 n = k 时成立，利用假设证明 n = k + 1 时成立，最后写出结论语句。

8. Vectors in 2D and 3D | 二维与三维向量

A vector has both magnitude and direction. In component form v = a i + b j (+ c k), the magnitude is |v| = √(a² + b² + c²). The scalar product of two vectors a · b = |a||b| cos θ is used to find the angle between vectors and to test perpendicularity: a · b = 0 ⇔ vectors are perpendicular. In pure geometry, this helps with problems about triangles and quadrilaterals. In 3D, the scalar product is a · b = a₁b₁ + a₂b₂ + a₃b₃.

向量既有大小又有方向。在分量形式 v = a i + b j (+ c k) 中，模长 |v| = √(a² + b² + c²)。向量的点积 a · b = |a||b| cos θ 用于求向量间的夹角以及检验垂直关系：a · b = 0 ⇔ 向量互相垂直。在纯几何问题中，这有助于处理三角形和四边形。在三维情形下，点积 a · b = a₁b₁ + a₂b₂ + a₃b₃。

Vector equations of a line are essential. The form r = a + t b gives the position of any point on the line, where a is a fixed point on the line and b is a direction vector. To check whether a point lies on a line, see if its position vector satisfies the equation for some t. The shortest distance from a point to a line can be found by constructing a perpendicular. For two lines, they may intersect, be parallel, or be skew (in 3D). Finding the intersection means solving the three component equations simultaneously.

直线的向量方程至关重要。形式 r = a + t b 表示直线上任意一点的位置，其中 a 是直线上的一个固定点，b 是一个方向向量。要检查某点是否在直线上，只需看其位置向量是否对某个 t 满足方程。点到直线的最短距离可以通过构造垂线来求。对于两条直线，它们可能相交、平行或异面（在三维中）。求交点意味着同时求解三个分量方程。

9. Probability Fundamentals | 概率基础

Probability measures how likely an event is, always between 0 and 1. For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B). For independent events, P(A ∩ B) = P(A) × P(B). Conditional probability is given by P(A | B) = P(A ∩ B) / P(B). Tree diagrams are the safest tool for multi‑stage experiments: multiply along branches and add between branches. Always check that the probabilities on each set of branches sum to 1. Venn diagrams and two‑way tables are powerful for organising overlapping events, especially when dealing with complex “given that” wording.

概率衡量事件发生的可能性，取值总是在 0 到 1 之间。对于互斥事件 A 和 B，P(A ∪ B) = P(A) + P(B)。对于独立事件，P(A ∩ B) = P(A) × P(B)。条件概率公式为 P(A | B) = P(A ∩ B) / P(B)。树形图是多阶段试验最安全的工具：沿分支相乘，不同分支相加。务必检查每一层分支的概率之和为 1。韦恩图和双向表是组织重叠事件的强有力工具，尤其当面对复杂的“已知……条件下”的措辞时。

For IB students, Bayes’ theorem can appear: P(A | B) = [P(B | A) P(A)] / P(B). It is often easier to solve such problems with a tree diagram turned around, or by filling in a contingency table rather than plugging numbers into the formula blindly. The WJEC specification places more emphasis on discrete distributions like the binomial and the use of the formula P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ. In either course, recognise the words “exactly”, “at least”, and “more than” and translate them into correct probability statements.

对于 IB 学生，贝叶斯定理可能出现：P(A | B) = [P(B | A) P(A)] / P(B)。通常，用反向树形图或填列联表来求解比死板套公式更容易。WJEC 大纲更强调二项分布等离散分布以及公式 P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ 的使用。无论哪个课程，都要识别“恰好”、“至少”、“多于”等措辞，并将其转化为正确的概率表述。

10. Statistical Distributions | 统计分布

The binomial distribution B(n, p) models the number of successes in n independent trials, each with probability p. The mean is E(X) = np, and the variance is Var(X) = np(1−p). Always check the four conditions: fixed number of trials, two possible outcomes per trial, constant probability, and independence. The normal distribution N(μ, σ²) is continuous and symmetric. To find probabilities, standardise using Z = (X − μ)/σ and use the standard normal table. For inverse normal questions, draw a diagram and shade the known area, then work backwards through the table.

二项分布 B(n, p) 模拟了 n 次独立试验中成功的次数，每次成功的概率为 p。均值为 E(X) = np，方差为 Var(X) = np(1−p)。务必检查四个条件：试验次数固定、每次试验只有两种结果、概率不变、试验互相独立。正态分布 N(μ, σ²) 是连续且对称的。求概率时，先标准化 Z = (X − μ)/σ，再查标准正态分布表。对于反向正态问题，画出草图并涂上已知面积，然后通过查表反向求解。

When approximating a binomial with a normal distribution (IB and some WJEC units), check that both np and n(1−p) are greater than 5 or 10. Apply a continuity correction: P(X ≤ k) becomes P(X < k + 0.5) under the normal approximation. For the sample mean distribution, the central limit theorem says that for a large enough sample size, X̄ is approximately normally distributed with mean μ and standard error σ/√n, regardless of the original population shape.

当用正态分布近似二项分布时（IB 和部分 WJEC 单元），需验证 np 和 n(1−p) 均大于 5 或 10。应用连续性修正：P(X ≤ k) 在正态近似下变为 P(X < k + 0.5)。对于样本均值分布，中心极限定理表明，当样本容量足够大时，X̄ 近似服从正态分布，均值为 μ，标准误为 σ/√n，无论原始总体的分布形态如何。

Distribution \| 分布	Mean / 均值	Variance / 方差	When to use / 使用场景
Binomial B(n,p)	np	np(1−p)	Count of successes in fixed trials / 固定试验中成功次数
Normal N(μ,σ²)	μ	σ²	Continuous measurements, naturally occurring data / 连续测量数据，自然现象数据
Sample Mean X̄	μ	σ²/n	Means of samples, CLT applies / 样本均值，中心极限定理适用

The final minutes before the exam should be spent on recognising question types rather than learning new content. Scan through past papers and note the command terms: “write down” implies no working is needed, “find” and “hence” signal that you should use the previous part, and “show that” requires a fully reasoned argument, even if you know the result. Manage your time so you attempt the high‑mark questions early while your mind is fresh, but never leave any part completely blank — a diagram or a correct formula can score method marks.

考试前的最后几分钟应花在识别题型而非学习新内容上。浏览历年真题，注意指令词：“write down”意味着不需要步骤，“find”和“hence”提示应使用前一问的结果，“show that”则要求完整的推理过程，即使你已经知道结论。合理分配时间，趁头脑清醒时尽快攻下高分大题，但绝不要让任何小题空着——一个草图或正确公式就可能得到方法分。

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IB & WJEC Maths: Last-Minute Revision Notes | IB 与 WJEC 数学：考前冲刺笔记