📚 IB & WJEC Mathematics: Mind Map for Quick Revision | IB与WJEC数学：思维导图速记
This article provides a mind-map approach to quickly memorise key concepts in IB and WJEC Mathematics. From algebra to calculus, these bite-sized points will help you recall formulas and strategies during revision.
本文提供一种思维导图方法，快速记忆IB与WJEC数学的核心概念。从代数到微积分，这些要点将帮助你在复习时快速回忆公式与解题策略。
1. Core Algebra | 核心代数
Always perform operations in the correct order: Brackets, Indices, Division/Multiplication, Addition/Subtraction (BIDMAS). For expansion, carefully distribute each term.
始终按正确顺序运算：括号、指数、乘除、加减。展开时仔细分配每一项。
When factorising trinomials like ax²+bx+c, split the middle term using ac. Remember that x² + (a+b)x + ab = (x+a)(x+b).
因式分解三项式如 ax²+bx+c 时，利用 ac 分解中项。记住 x²+(a+b)x+ab=(x+a)(x+b)。
Simplify rational expressions by cancelling common factors only after factoring numerator and denominator. Watch out for restrictions.
化简有理式时，先分子分母因式分解再约去公因数。注意定义域限制。
2. Functions and Graphs | 函数与图像
Understand function notation f(x). The domain is the set of inputs; the range is outputs. Vertical line test checks if a graph is a function.
理解函数符号 f(x)。定义域是输入集合，值域是输出集合。垂直线检验判断图像是否为函数。
Key transformations: f(x)+a shifts vertically, f(x+a) shifts horizontally, -f(x) reflects in x-axis, f(-x) reflects in y-axis.
关键变换：f(x)+a 垂直平移，f(x+a) 水平平移，-f(x) 关于x轴对称，f(-x) 关于y轴对称。
For inverse functions, swap x and y and solve. Graphically, reflect in y=x line. One-to-one functions have inverses.
反函数：交换x和y后求解。图像上关于y=x对称。一一映射才有反函数。
3. Quadratic Magic | 二次函数要点
Quadratic standard form: f(x)=ax²+bx+c. The discriminant Δ=b²-4ac tells nature of roots: Δ>0 two real, Δ=0 one repeated, Δ<0 no real.
二次函数标准式：f(x)=ax²+bx+c。判别式 Δ=b²-4ac 决定根的性质：Δ>0两个实根，Δ=0一个重根，Δ<0无实根。
Vertex form: f(x)=a(x-h)²+k, vertex at (h,k). Completing the square transforms to vertex form. Axis of symmetry x=h.
顶点式：f(x)=a(x-h)²+k，顶点 (h,k)。配方法可化为顶点式。对称轴 x=h。
Sum and product of roots: if α and β are roots, α+β=-b/a, αβ=c/a. Useful for forming equations.
根的和与积：若根为α和β，则α+β=-b/a，αβ=c/a。可用于构造方程。
4. Exponents and Logarithms | 指数与对数
Exponent laws: aᵐ×aⁿ=aᵐ⁺ⁿ, (aᵐ)ⁿ=aᵐⁿ, a⁻ⁿ=1/aⁿ. For logarithms: logₐ(xy)=logₐx+logₐy, logₐ(x/y)=logₐx-logₐy, logₐ(xⁿ)=n logₐx.
指数律：aᵐ×aⁿ=aᵐ⁺ⁿ，(aᵐ)ⁿ=aᵐⁿ，a⁻ⁿ=1/aⁿ。对数律：logₐ(xy)=logₐx+logₐy，logₐ(x/y)=logₐx-logₐy，logₐ(xⁿ)=n logₐx。
Change of base: logₐb = log b / log a (common log). Natural log ln uses base e. ln(e)=1.
换底公式：logₐb = log b / log a。自然对数 ln 以e为底，ln(e)=1。
Exponential growth/decay: N=N₀eᵏᵗ. Solve using logs. Remember e and ln are inverses.
指数增长/衰减：N=N₀eᵏᵗ。用对数求解。记住e与ln互为反函数。
5. Trigonometry Toolkit | 三角学工具箱
SOHCAHTOA: sinθ=opp/hyp, cosθ=adj/hyp, tanθ=opp/adj. Know exact values for 0°,30°,45°,60°,90°.
基本三角：sinθ=对边/斜边，cosθ=邻边/斜边，tanθ=对边/邻边。熟记特殊角度的精确值。
Unit circle: cosθ=x, sinθ=y. tanθ=sinθ/cosθ. CAST diagram for signs in quadrants.
单位圆：cosθ=x，sinθ=y，tanθ=sinθ/cosθ。用CAST图记忆象限正负号。
Identities to recall: sin²θ+cos²θ=1, 1+tan²θ=sec²θ, 1+cot²θ=cosec²θ. Double angle: sin(2θ)=2sinθcosθ, cos(2θ)=cos²θ-sin²θ.
恒等式：sin²θ+cos²θ=1，1+tan²θ=sec²θ，1+cot²θ=cosec²θ。倍角公式：sin(2θ)=2sinθcosθ, cos(2θ)=cos²θ-sin²θ。
6. Vectors in 2D and 3D | 二维与三维向量
A vector has magnitude and direction. Unit vectors i,j,k. For AB = b – a. Magnitude |v| = √(x²+y²+z²).
向量有大小和方向。单位向量i,j,k。AB = b – a。模 |v| = √(x²+y²+z²)。
Dot product: a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃. Used to find angles and perpendicular (a·b=0).
点积：a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃。用于求角度和判断垂直(a·b=0)。
Vector equation of a line: r = a + tb. a is point on line, b is direction vector. For intersection, set parametric equations equal.
直线向量方程：r = a + tb。a是线上一点，b是方向向量。求交点时联立参数方程。
7. Calculus: Differentiation | 微积分：微分
Derivative f'(x) is rate of change. Power rule: d/dx (xⁿ) = nxⁿ⁻¹. Derivative of constant is zero.
导数f'(x)表示变化率。幂法则：d/dx (xⁿ) = nxⁿ⁻¹。常数的导数为零。
Chain rule: d/dx f(g(x)) = f'(g(x))·g'(x). Product rule: (uv)’ = u’v+uv’. Quotient rule: (u/v)’ = (u’v-uv’)/v².
链式法则：d/dx f(g(x)) = f'(g(x))·g'(x)。乘法法则：(uv)’ = u’v+uv’。除法法则：(u/v)’ = (u’v-uv’)/v²。
Derivatives of special functions: d/dx sinx=cosx, d/dx cosx=-sinx, d/dx eˣ=eˣ, d/dx lnx=1/x. Second derivative f”(x) gives concavity.
特殊函数导数：d/dx sinx=cosx, d/dx cosx=-sinx, d/dx eˣ=eˣ, d/dx lnx=1/x。二阶导数 f”(x) 用于判断凹凸性。
8. Calculus: Integration | 微积分：积分
Integration is reverse of differentiation. Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n≠-1. ∫ 1/x dx = ln|x|+C.
积分是微分的逆运算。幂积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n≠-1。∫ 1/x dx = ln|x|+C。
Definite integral ∫ₐᵇ f(x) dx gives area under curve between a and b. Remember f(x) must be continuous. Add areas for pieces above/below x-axis.
定积分 ∫ₐᵇ f(x) dx 给出曲线下面积。注意函数需连续。若部分在x轴下方，分段取绝对值相加。
Techniques: substitution (u-sub) reverses chain rule. Integration by parts: ∫ u dv = uv – ∫ v du. Know ∫ eˣ sinx dx patterns.
积分技巧：换元法（u替换）逆用链式法则。分部积分：∫ u dv = uv – ∫ v du。熟悉 ∫ eˣ sinx dx 等模式。
9. Sequences and Series | 数列与级数
Arithmetic sequence: uₙ = a + (n-1)d. Sum Sₙ = n/2 [2a + (n-1)d] = n/2 (a+l). Geometric: uₙ = arⁿ⁻¹. Sum Sₙ = a(1-rⁿ)/(1-r), r≠1.
等差数列：uₙ = a + (n-1)d。求和 Sₙ = n/2 [2a + (n-1)d] = n/2 (a+l)。等比数列：uₙ = arⁿ⁻¹。求和 Sₙ = a(1-rⁿ)/(1-r)，r≠1。
Infinite geometric series converges if |r|<1, sum S∞ = a/(1-r). Know Sigma notation Σ and use formulas.
无穷等比级数当 |r|<1 时收敛，和 S∞ = a/(1-r)。掌握Σ求和符号并使用公式。
Binomial expansion: (1+x)ⁿ ≈ 1+nx+ n(n-1)/2! x²+…, valid for |x|<1, n rational. For (a+b)ⁿ, use (a+b)ⁿ = aⁿ(1+b/a)ⁿ.
二项式展开：(1+x)ⁿ ≈ 1+nx+ n(n-1)/2! x²+…，|x|<1，n为有理数。(a+b)ⁿ 可化为 aⁿ(1+b/a)ⁿ 展开。
10. Complex Numbers (HL) | 复数（高级）
Complex number z = a+bi, where i²=-1. Real part a, imaginary part b. Conjugate z̄ = a-bi. Modulus |z|=√(a²+b²).
复数 z = a+bi，i²=-1。实部a，虚部b。共轭 z̄ = a-bi。模 |z|=√(a²+b²)。
Argand diagram: x-axis real, y-axis imaginary. Argument θ = arctan(b/a), adjusting quadrant. Polar form: z = r(cosθ + i sinθ) = re^{iθ}.
阿根图：x轴实部，y轴虚部。辐角 θ = arctan(b/a)，注意象限。极形式：z = r(cosθ+i sinθ)=re^{iθ}。
De Moivre’s theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). Useful for powers and roots. nth roots are equally spaced on circle.
棣莫弗定理：(cosθ+i sinθ)ⁿ = cos(nθ)+i sin(nθ)。用于求幂和根。n次方根均匀分布在圆上。
11. Statistics & Probability | 统计与概率
Mean, median, mode, range, IQR. Standard deviation σ = √(Σ(x-μ)²/N). For sample use n-1.
平均数、中位数、众数、极差、四分位距。标准差 σ = √(Σ(x-μ)²/N)。样本用 n-1。
Probability rules: P(A∪B)=P(A)+P(B)-P(A∩B). Mutually
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