📚 IB WJEC Mathematics: End-of-Term Revision Guide | IB WJEC 数学：期末复习提纲
This comprehensive revision guide covers the essential topics needed for end-of-term assessments in the IB Diploma Programme Mathematics course, with reference to the WJEC specification structure where relevant. Whether you are studying Analysis and Approaches or Applications and Interpretation, the core skills outlined here will help you consolidate key concepts, avoid common pitfalls, and approach your exams with confidence. Use this guide alongside your class notes, past paper questions, and textbook exercises to build a robust understanding of mathematics.
本综合复习提纲涵盖了 IB 文凭课程数学期末评估所需的核心主题，并在相关处参考了 WJEC 规范的结构。无论你修读的是分析与方法还是应用与解释课程，这里梳理的基本技能都将帮助你巩固关键概念、避开常见陷阱，并自信地应对考试。请将此提纲与课堂笔记、历年真题和教材练习结合使用，以构建坚实的数学理解。
1. Algebraic Fundamentals and Equation Solving | 代数基础与方程求解
Mastering algebraic manipulation is the foundation of all higher mathematics. Ensure you can confidently expand brackets, factorise quadratic and cubic expressions, and simplify rational expressions. When solving equations, always check for extraneous solutions, particularly in rational and radical equations. The quadratic formula x = [-b ± √(b² – 4ac)] / 2a must be second nature, and you should recognise the discriminant Δ = b² – 4ac to determine the nature of roots.
掌握代数运算是所有高等数学的基础。务必能熟练展开括号、对二次和三次表达式进行因式分解，以及化简有理式。解方程时，要时刻检查增根，尤其是在有理方程和根式方程中。求根公式 x = [-b ± √(b² – 4ac)] / 2a 必须内化为本能，并应能识别判别式 Δ = b² – 4ac 以判断根的性质。
Systems of linear equations can be solved by substitution, elimination, or using matrices (Gaussian elimination). For two equations with two unknowns, interpret the solution as the intersection point of two lines. If the lines are parallel and distinct, there is no solution; if they coincide, there are infinitely many solutions. In the WJEC context, you may be asked to set up equations from word problems involving cost, mixtures, or geometry, so translate carefully.
线性方程组可通过代入法、消元法或矩阵（高斯消元法）求解。对于两个方程两个未知数的情况，可将解理解为两条直线的交点。若两线平行且不重合，则无解；若重合，则有无穷多解。在 WJEC 情境中，你可能需要根据涉及成本、混合物或几何的应用题列出方程，因此需仔细转化。
For quadratic inequalities such as x² – 5x + 6 > 0, sketch the parabola to identify intervals where the y-value is positive. Always express the solution using interval notation or set-builder notation. Polynomial inequalities of higher degree can be tackled by finding critical values and testing intervals on a sign chart.
对于二次不等式，例如 x² – 5x + 6 > 0，应画出抛物线草图来确定 y 值为正的区间。始终使用区间表示法或集合构造表示法表达解。更高次的多项式不等式可通过寻求临界值并在符号表上测试区间来解决。
2. Functions and Graph Transformations | 函数与图像变换
A function is a rule that maps each input to exactly one output. Understand domain and range, and be able to find the inverse of a function f⁻¹(x) by swapping x and y and solving for y. The graph of f⁻¹(x) is the reflection of f(x) in the line y = x. A function must be one-to-one over its domain to have an inverse; if not, restrict the domain appropriately.
函数是将每个输入映射到唯一输出的规则。要理解定义域和值域，并能通过交换 x 和 y 并解出 y 来求逆函数 f⁻¹(x)。逆函数的图像是原函数关于直线 y = x 的对称图形。函数在其定义域上必须一一对应才有逆函数；若不是，则需适当限定定义域。
Transformations of graphs include translations, stretches, and reflections. Given y = f(x), the transformation y = a·f(b(x – h)) + k applies a vertical stretch factor a, a horizontal stretch factor 1/b, a horizontal translation h, and a vertical translation k. Remember that changes inside the function argument affect x-direction oppositely: f(x + 2) shifts left, f(2x) compresses horizontally.
图像变换包括平移、伸缩和对称变换。对于 y = f(x)，变换 y = a·f(b(x – h)) + k 施加了垂直伸缩因子 a、水平伸缩因子 1/b、水平平移 h 和垂直平移 k。记住：函数内部的变换对 x 方向的影响相反：f(x + 2) 向左平移，f(2x) 水平压缩。
Piecewise functions are defined by different rules on different intervals. Evaluate them by first determining which rule applies. The modulus function |x| is an important example, giving the distance from zero. Equations involving modulus, such as |2x – 1| = 3, require considering both the positive and negative case. Graphically, |f(x)| reflects any parts below the x-axis upward.
分段函数在不同区间上由不同规则定义。求值时需先确定适用哪条规则。绝对值函数 |x| 是一个重要的例子，它给出到零的距离。含有绝对值的方程，如 |2x – 1| = 3，需要考虑正负两种情形。在图像上，|f(x)| 会将 x 轴下方的部分向上翻折。
3. Trigonometry and Circular Functions | 三角学与圆函数
Begin with the right-triangle definitions of sine, cosine, and tangent: SOH CAH TOA. For any angle, use the unit circle to extend these definitions. Know the exact values for 0°, 30°, 45°, 60°, 90° and their radian equivalents. The Pythagorean identity sin²θ + cos²θ = 1 links the functions and can be used to derive others, such as 1 + tan²θ = sec²θ.
从正弦、余弦、正切的直角三角形定义开始：SOH CAH TOA。对于任意角，使用单位圆来推广这些定义。熟记 0°、30°、45°、60°、90° 及其弧度对应值的精确值。勾股恒等式 sin²θ + cos²θ = 1 将各函数联系起来，并可用于推导其他等式，如 1 + tan²θ = sec²θ。
Sine and cosine rules are essential for non-right-angled triangles. The sine rule: a/sin A = b/sin B = c/sin C. The cosine rule: a² = b² + c² – 2bc cos A. Use the sine rule for two angles and a side, or two sides and a non-included angle (be cautious of the ambiguous case). The cosine rule is best for three sides or two sides and the included angle.
正弦定理和余弦定理对非直角三角形至关重要。正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² – 2bc cos A。在两角一边或两边一对角（小心歧义情形）时使用正弦定理；三边或两边夹一角时用余弦定理最佳。
Trigonometric equations like sin x = 0.5 have infinitely many solutions; use the quadrant diagram or graph to find all solutions within a specified interval. Factorising, such as 2sin²x – sin x – 1 = 0, often reduces to solving basic trigonometric equations. Double-angle formulas, e.g., sin 2θ = 2 sin θ cos θ, are tested frequently, especially in calculus and proof contexts.
诸如 sin x = 0.5 的三角方程有无穷多解；使用象限图或图像找出给定区间内的所有解。因式分解，例如 2sin²x – sin x – 1 = 0，常归结为解基本三角方程。倍角公式，例如 sin 2θ = 2 sin θ cos θ，经常在考试中出现，尤其在微积分和证明题中。
4. Exponentials and Logarithms | 指数函数与对数函数
Exponential functions of the form f(x) = a·bˣ model growth and decay. The natural exponential function eˣ has base e ≈ 2.718. Its inverse is the natural logarithm ln x, defined for x > 0. The key relationship is e^(ln x) = x and ln(eˣ) = x. When solving exponential equations, take the natural log of both sides, e.g., 3ˣ = 7 becomes x ln 3 = ln 7.
形如 f(x) = a·bˣ 的指数函数可模拟增长与衰减。自然指数函数 eˣ 以 e ≈ 2.718 为底数，其反函数为自然对数 ln x，定义域为 x > 0。关键关系是 e^(ln x) = x 且 ln(eˣ) = x。解指数方程时，两边取自然对数，如 3ˣ = 7 变为 x ln 3 = ln 7。
Logarithm laws are crucial: logₐ(xy) = logₐ x + logₐ y; logₐ(x/y) = logₐ x – logₐ y; logₐ(xⁿ) = n logₐ x. Change of base formula: logₐ x = log_b x / log_b a. These allow you to manipulate expressions and solve logarithmic equations. Always check that solutions keep arguments positive, otherwise they are extraneous.
对数运算法则至关重要：logₐ(xy) = logₐ x + logₐ y；logₐ(x/y) = logₐ x – logₐ y；logₐ(xⁿ) = n logₐ x。换底公式：logₐ x = log_b x / log_b a。这些公式可助你处理表达式并解对数方程。务必检验所得解是否使真数保持正数，否则便是增根。
Graphs of exponential functions pass through (0, a) and have a horizontal asymptote y = c. Logarithmic graphs have a vertical asymptote x = 0 and pass through (1, 0). Transformations apply as usual. In applied contexts, exponential models often describe population growth, radioactive decay, and compound interest. The half-life or doubling time can be found using logarithms.
指数函数的图像经过 (0, a)，并有一条水平渐近线 y = c。对数函数的图像有一条垂直渐近线 x = 0，并经过 (1, 0)。常规的变换同样适用。在应用背景中，指数模型常用来描述人口增长、放射性衰变和复利。半衰期或倍增时间可借助对数求得。
5. Introduction to Differential Calculus | 微积分导论
Differentiation finds the instantaneous rate of change or gradient of a curve. The derivative of f(x) is denoted f'(x) or dy/dx. From first principles, f'(x) = lim(h→0) [f(x+h) – f(x)] / h. The power rule states that for f(x) = xⁿ, f'(x) = n·xⁿ⁻¹. This extends to sums, differences, and constant multiples.
微分用于求瞬时变化率或曲线的斜率。函数 f(x) 的导数记作 f'(x) 或 dy/dx。根据第一原理，f'(x) = lim(h→0) [f(x+h) – f(x)] / h。幂法则指出，若 f(x) = xⁿ，则 f'(x) = n·xⁿ⁻¹。这一法则可推广至和、差以及常数倍情形。
Common derivatives include: d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x. The chain rule handles composite functions: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x). The product rule: (uv)’ = u’v + uv’. The quotient rule: (u/v)’ = (u’v – uv’) / v².
基本的导数公式包括：d/dx(sin x) = cos x，d/dx(cos x) = -sin x，d/dx(eˣ) = eˣ，d/dx(ln x) = 1/x。链式法则处理复合函数：若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)。积的法则：(uv)’ = u’v + uv’。商的法则：(u/v)’ = (u’v – uv’) / v²。
The derivative helps find tangents and normals to curves. The gradient of the tangent at x = a is f'(a); the normal is perpendicular, so its gradient is -1/f'(a). Use y – y₁ = m(x – x₁) to form equations. Stationary points occur where f'(x) = 0; classify them as maxima, minima, or points of inflection using the second derivative or a sign table.
导数有助于求曲线的切线和法线。在 x = a 处切线的斜率为 f'(a)；法线与之垂直，故其斜率为 -1/f'(a)。利用 y – y₁ = m(x – x₁) 得出方程。稳定点出现在 f'(x) = 0 处；可利用二阶导数或符号表将其分类为极大值点、极小值点或拐点。
6. Integral Calculus and Area | 积分与面积
Indefinite integration reverses differentiation. The general power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ -1. Remember the integral of 1/x is ln|x| + C. Basic trigonometric integrals: ∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C. Exponential: ∫ eˣ dx = eˣ + C. Always add the constant of integration C for indefinite integrals.
不定积分是微分的逆运算。一般幂法则：当 n ≠ -1 时，∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。注意 1/x 的积分为 ln|x| + C。基本三角函数的积分：∫ sin x dx = -cos x + C，∫ cos x dx = sin x + C。指数函数：∫ eˣ dx = eˣ + C。不定积分务必加积分常数 C。
Definite integrals compute the net area between a curve and the x-axis. The Fundamental Theorem of Calculus: ∫ₐᵇ f(x) dx = F(b) – F(a), where F is an antiderivative. When the curve lies below the x-axis, the integral gives a negative value. To find total area, split the interval at roots and take absolute values of each segment’s integral.
定积分计算曲线与 x 轴之间的净面积。微积分基本定理：∫ₐᵇ f(x) dx = F(b) – F(a)，其中 F 是一个原函数。当曲线位于 x 轴下方时，积分值为负。要求总面积，需以零点为界划分区间，并将各段积分取绝对值加总。
Area between two curves y = f(x) and y = g(x) from a to b is ∫ₐᵇ [f(x) – g(x)] dx, provided f(x) ≥ g(x). If the graphs intersect, split the integral. Kinematics applications link displacement, velocity, and acceleration: v(t) = ds/dt, a(t) = dv/dt, and s(t) = ∫ v dt. Pay attention to initial conditions to find specific constants.
两条曲线 y = f(x) 与 y = g(x) 在 a 到 b 之间所围面积是 ∫ₐᵇ [f(x) – g(x)] dx，前提是 f(x) ≥ g(x)。若图像相交，需拆分积分。运动学应用将位移、速度和加速度联系起来：v(t) = ds/dt，a(t) = dv/dt，s(t) = ∫ v dt。注意利用初始条件确定具体常数。
7. Vectors in Two and Three Dimensions | 平面与空间向量
A vector has both magnitude and direction, represented by a column matrix or using i, j (and k in 3D) notation. Add and subtract vectors component-wise. Scalar multiplication stretches the vector. The magnitude of vector v = (x, y) is |v| = √(x² + y²). A unit vector in the direction of v is v / |v|.
向量既有大小又有方向，可用列矩阵或 i、j（三维中加 k）记号表示。向量的加法与减法逐分量进行，标量乘法则拉伸向量。向量 v = (x, y) 的模为 |v| = √(x² + y²)。沿 v 方向的单位向量为 v / |v|。
The scalar (dot) product: a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. It is used to find the angle between vectors and test perpendicularity (a·b = 0). The vector (cross) product applies in 3D and yields a vector perpendicular to both. Its magnitude gives the area of a parallelogram.
标量积（点积）：a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃，用于求向量夹角和检验垂直性（a·b = 0）。向量积（叉积）适用于三维，所得向量垂直于两向量所在平面，其模等于平行四边形面积。
Vector equations of lines: r = a + t d, where a is a position vector, d is a direction vector, and t is a scalar parameter. In 2D, convert to Cartesian by eliminating t. For intersections, equate vectors and solve. In plane geometry, use vectors to prove properties like mid-points, parallel lines, and collinearity.
直线的向量方程：r = a + t d，其中 a 是位置向量，d 是方向向量，t 是标量参数。在二维中，消去 t 可转化为笛卡尔方程。求交点时令向量相等并求解。在平面几何中，要会用向量证明中点、平行和共线等性质。
8. Probability and Statistical Distributions | 概率与统计分布
Probability measures the likelihood of events on a scale from 0 to 1. For combined events: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Mutually exclusive events have P(A ∩ B) = 0; independent events satisfy P(A ∩ B) = P(A)·P(B). Conditional probability P(A|B) = P(A ∩ B) / P(B). Use tree diagrams or Venn diagrams to organise information.
概率以 0 到 1 的尺度衡量事件的可能性。对于组合事件：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。互斥事件有 P(A ∩ B) = 0；独立事件满足 P(A ∩ B) = P(A)·P(B)。条件概率 P(A|B) = P(A ∩ B) / P(B)。使用树形图或韦恩图来整理信息。
The binomial distribution models the number of successes in n independent trials, each with probability p. Notation: X ~ B(n, p). The probability of exactly r successes is C(n, r)·p^r·(1-p)^(n-r). Mean μ = np, variance σ² = np(1-p). For the normal distribution X ~ N(μ, σ²), standardise using Z = (X – μ)/σ and use the Z-table for probabilities.
二项分布描述 n 次独立试验中成功的次数，每次成功概率为 p。记作 X ~ B(n, p)。恰好 r 次成功的概率为 C(n, r)·p^r·(1-p)^(n-r)。均值 μ = np，方差 σ² = np(1-p)。对于正态分布 X ~ N(μ, σ²)，使用 Z = (X – μ)/σ 进行标准化，并查 Z 值表求概率。
Be able to calculate the mean, median, variance, and standard deviation for both discrete and grouped continuous data. The interquartile range (IQR = Q₃ – Q₁) measures spread and is used to identify outliers. Cumulative frequency graphs and box plots are useful for comparing data sets. Understand the concept of correlation and the line of best fit (linear regression).
要会计算离散数据和分组连续数据的均值、中位数、方差和标准差。四分位距（IQR = Q₃ – Q₁）衡量离散程度，可用于识别异常值。累积频数图和箱线图有助于比较数据集。理解相关性概念和最佳拟合线（线性回归）。
9. Sequences, Series, and Proof | 数列、级数与证明
Arithmetic sequences have a constant difference d: uₙ = a + (n-1)d. The sum of the first n terms is Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l), where l is the last term. Geometric sequences have a constant ratio r: uₙ = a·rⁿ⁻¹. The sum Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. An infinite geometric series converges to a/(1-r) if |r| < 1.
等差数列的公差 d 为常数：uₙ = a + (n-1)d。前 n 项和为 Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l)，其中 l 为末项。等比数列的公比 r 为常数：uₙ = a·rⁿ⁻¹。当 r ≠ 1 时，其和 Sₙ = a(1 – rⁿ)/(1 – r)。当 |r| < 1 时，无穷等比级数收敛于 a/(1-r)。
Sigma notation (Σ) compacts sums. Be able to extract terms and use standard formulas for Σi, Σi². Proof by induction is a typical IB requirement: prove a statement true for n = 1, assume true for n = k, then show it follows for n = k+1. This is often used for divisibility, series sums, or matrix powers. Clearly write the inductive hypothesis and conclusion.
西格玛记号（Σ）可简洁地表示求和。要能展开各项，并使用 Σi、Σi² 的标准公式。数学归纳法是 IB 课程的典型考点：先证明 n = 1 时命题成立，假设 n = k 时成立，再推出 n = k+1 时成立。此法常用于整除性、级数求和或矩阵乘方问题。要清晰写出归纳假设和结论。
Other proof techniques include direct proof, contradiction, and contrapositive. For example, prove √2 is irrational by contradiction. In examination settings, small algebraic proofs and trigonometric identities are common. Always state what is given and what you need to prove, and provide logical steps connecting them.
其他证明方法包括直接证明、反证法和逆否命题法。例如，用反证法证明 √2 是无理数。在考试中，常出现小型代数证明和三角恒等式证明。务必写明已知条件与待证结论，并给出将它们联系起来的逻辑步骤。
10. Complex Numbers (HL Extension) | 复数（HL 拓展）
Complex numbers are written in the form z = a + bi, where i² = -1. The real part is a, the imaginary part is b. The complex conjugate z* = a – bi is used in division and to find modulus: |z| = √(a² + b²). The Argand diagram plots complex numbers as points, with the x-axis as real part and y-axis as imaginary part.
复数写作 z = a + bi，其中 i² = -1。实部为 a，虚部为 b。共轭复数 z* = a – bi 可用于除法和求模：|z| = √(a² + b²)。阿尔岡图将复数绘成点，x 轴表实部，y 轴表虚部。
Polar form: z = r(cos θ + i sin θ) = r cis θ, where r = |z| and θ is the argument (angle). Multiplication and division are simpler in polar form: multiply moduli, add arguments. De Moivre’s theorem states (r cis θ)ⁿ = rⁿ cis(nθ). Use it to find powers and roots of complex numbers. The nth roots of unity are equally spaced on the unit circle.
极坐标形式：z = r(cos θ + i sin θ) = r cis θ，其中 r = |z|，θ 为幅角。在极坐标形式下，乘法和除法更为简便：模相乘，幅角相加。棣莫弗定理指出 (r cis θ)ⁿ = rⁿ cis(nθ)，可用来求复数的乘方和根。n 次单位根均匀分布在单位圆上。
Solving polynomial equations may yield complex roots, which always occur in conjugate pairs if coefficients are real. Use the quadratic formula when discriminant is negative, giving roots of the form p ± qi. The fundamental theorem of algebra guarantees n roots for an nth-degree polynomial (counting multiplicity). Transformations like rotation and dilation in the Argand plane correspond to multiplication by a complex number.
解多项式方程可能产生复数根，若系数为实数，复根总是以共轭对的形式出现。当判别式为负时使用求根公式，得出 p ± qi 形式的根。代数基本定理确保 n 次多项式有 n 个根（计重数）。在阿尔岡平面上的旋转和缩放变换对应于乘以某个复数。
11. Exam Technique and Common Pitfalls | 考试技巧与常见陷阱
Start each session by reading the question carefully, underlining command terms such as ‘hence’, ‘find’, ‘show that’, ‘exact value’, or ‘to 3 significant figures’. Display your working clearly — even if the final answer is incorrect, method marks may be awarded. In show that questions, derive the required result logically and avoid assuming what you need to prove.
拿到试卷后仔细读题，在指令词下画线，如“由此”“求”“证明”“精确值”或“保留三位有效数字”。清晰展示解题过程——即便最终答案有误，仍可获得方法分。在证明题中，要逻辑推导出所需结果，避免提前假设要证结论成立。
Manage your time: do not spend too long on a single sub-question. If stuck, move on and return later. Use the reading time to plan which questions to attempt first. For calculator papers, ensure your device is in the correct mode (degree/radian) and that you know how to store values to avoid rounding errors. In non-calculator papers, practise mental arithmetic and fraction manipulation.
合理安排时间：不要在单个小问上耗时过久。若遇困难，先跳过，稍后再回做。利用阅题时间规划答题顺序。对于允许使用计算器的考卷，确保计算器处于正确模式（角度/弧度），并知道如何存储数值以避免舍入误差。在无计算器考卷中，要练习心算和分数运算。
Common mistakes include: forgetting to consider ± when square rooting, mixing up derivative and integral of trig functions, misapplying the chain rule, ignoring domain restrictions, and missing units in applied problems. Double-check your algebraic signs and ensure that your answer is reasonable in the context of the problem. Write legibly and label graphs and diagrams with scales.
常见错误包括：开平方时忘记 ± 号、混淆三角函数的导数和积分、误用链式法则、忽略定义域限制以及应用题中遗漏单位。反复检查代数符号是否无误，并确保解答在题目情境中合理。书写工整，为图像和示意图标注刻度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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