📚 Core Concepts for CAIE IGCSE Additional Mathematics | CAIE IGCSE 进阶数学核心知识点梳理
CAIE IGCSE Additional Mathematics (0606) builds a strong foundation for advanced study, covering functions, algebra, trigonometry, coordinate geometry, and calculus. This article distills the essential core concepts to help Year 10 students consolidate their understanding and prepare effectively for examinations.
CAIE IGCSE 进阶数学（0606）为高阶学习打下坚实基础，涵盖函数、代数、三角学、坐标几何和微积分。本文凝练核心知识点，帮助十年级学生巩固理解、高效备考。
1. Functions | 函数
A function f maps each input x to exactly one output f(x). The domain is the set of all possible inputs, while the range is the set of all possible outputs. Functions can be one-to-one or many-to-one.
函数 f 将每个输入 x 映射到唯一输出 f(x)。定义域是所有可能输入的集合，值域是所有可能输出的集合。函数可以是一对一或多对一。
pplying f first, then g: gf(x) = g(f(x)). The order matters; fg(x) is generally different.
复合函数 gf(x) 表示先应用 f，再应用 g：gf(x) = g(f(x))。顺序很重要；fg(x) 通常不同。
exists only if f is one-to-one. The graph of f⁻¹ is the reflection of f in the line y = x.
反函数 f⁻¹(x) 逆转 f 的效果。仅当 f 为一对一时存在。f⁻¹ 的图像是 f 关于直线 y = x 的反射。
To find the inverse, write y = f(x), swap x and y, then solve for y.
求反函数时，设 y = f(x)，交换 x 和 y，然后解出 y。
2. Quadratic Functions | 二次函数
re, it can be written as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.
一般式为 f(x) = ax² + bx + c，其中 a ≠ 0。通过配方法，可写成 f(x) = a(x - h)² + k，其中 (h, k) 为抛物线的顶点。
Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.
判别式 Δ = b² - 4ac 决定根的性质：若 Δ > 0，有两个不等实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。
The quadratic formula gives the roots:
二次求根公式为：
x = [-b ± √b² - 4ac] / (2a)
To solve a quadratic inequality, find the critical values from the equation and test intervals. The solution set can be written using set notation or on a number line.
求解二次不等式时，先由方程求出临界值，再检验区间。解集可用集合符号或在数轴上表示。
3. Equations, Inequalities and Graphs | 方程、不等式与图像
Solving equations may involve linear, quadratic, cubic, or simultaneous forms. Graphical solutions are obtained by sketching curves and finding intersection points.
方程求解可能涉及线性、二次、三次或联立形式。图形解法通过绘制曲线并找出交点获得。
When solving |ax + b| = c, consider both ax + b = c and ax + b = -c. For inequalities like |ax + b| > c, split into two separate inequalities without the modulus.
解 |ax + b| = c 时，需考虑 ax + b = c 和 ax + b = -c。对于形如 |ax + b| > c 的不等式，应拆分为两个不含绝对值的不等式。
Modulus functions produce V-shaped graphs. The graph of y = |f(x)| reflects any part of f(x) below the x-axis upwards.
绝对值函数产生 V 形图像。y = |f(x)| 的图像将 f(x) 位于 x 轴下方的部分向上翻折。
4. Indices and Surds | 指数与根式
Key index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a1/n = ⁿ√a.
核心指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，a1/n = ⁿ√a。
Surds are irrational roots. Simplify surds by extracting perfect square factors: √a²b = a√b. Rationalise denominators such as 1/√a by multiplying by √a/√a, or 1/(a + √b) by multiplying by (a - √b)/(a - √b).
根式是无理根。通过提取完全平方因子化简根式：√a²b = a√b。分母有理化如 1/√a 可乘以 √a/√a，或 1/(a + √b) 可乘上 (a - √b)/(a - √b)。
Equations with indices often require writing both sides with the same base, then equating powers.
含指数的方程通常需要将两边写成同底数，再令指数相等。
5. Factors of Polynomials | 多项式因式
The remainder theorem states that when a polynomial p(x) is divided by (x - a), the remainder is p(a).
余数定理指出，多项式 p(x) 除以 (x - a) 时，余数为 p(a)。
The factor theorem follows: if p(a) = 0, then (x - a) is a factor of p(x).
由此得因式定理：若 p(a) = 0，则 (x - a) 是 p(x) 的因式。
To factorise a cubic, find one root by trial, then perform long division or equate coefficients to obtain a quadratic factor, which can be further factorised.
对三次多项式因式分解，先通过试根找到一个根，然后进行长除法或比较系数得到二次因式，再做进一步分解。
6. Logarithmic and Exponential Functions | 对数与指数函数
Logarithms are inverses of exponentials: if aˣ = y, then x = logₐ y. The natural log, ln, uses base e.
对数是指数的逆运算：若 aˣ = y，则 x = logₐ y。自然对数 ln 以 e 为底。
x - logₐ y; logₐ (xⁿ) = n logₐ x. The change-of-base formula is logₐ b = logₓ b / logₓ a.
对数运算法则：logₐ(xy) = logₐ x + logₐ y；logₐ(x/y) = logₐ x - logₐ y；logₐ(xⁿ) = n logₐ x。换底公式为 logₐ b = logₓ b / logₓ a。
ons are solved by taking logs; logarithmic equations by exponentiating. The graph of y = eˣ passes through (0,1) and grows rapidly; y = ln x is its reflection in y = x.
指数方程通过取对数求解；对数方程通过取指数求解。y = eˣ 的图像经过 (0,1) 并快速增长；y = ln x 是其关于 y = x 的反射。
7. Coordinate Geometry | 坐标几何
₁, y₁) and (x₂, y₂) is √[(x₂ - x₁)² + (y₂ - y₁)²]. The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).
两点 (x₁, y₁) 与 (x₂, y₂) 的距离为 √[(x₂ - x₁)² + (y₂ - y₁)²]。中点为 ((x₁+x₂)/2, (y₁+y₂)/2)。
y = mx + c. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = -1.
直线的斜率为 m = (y₂ - y₁)/(x₂ - x₁)。直线方程：y - y₁ = m(x - x₁) 或 y = mx + c。平行线斜率相等；垂直线满足 m₁ × m₂ = -1。
)² = r². The general form is x² + y² + 2gx + 2fy + c = 0, where centre is (-g, -f) and r = √g² + f² - c.
圆心 (a, b)、半径 r 的圆的方程为 (x - a)² + (y - b)² = r²。一般式为 x² + y² + 2gx + 2fy + c = 0，圆心为 (-g, -f)，r = √g² + f² - c。
8. Trigonometry and Circular Measure | 三角学与弧度制
or radians: π rad = 180°. Arc length s = rθ, sector area A = ½ r²θ, where θ is in radians.
角度可用度或弧度度量：π rad = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ，其中 θ 以弧度为单位。
Exact values: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. Corresponding radian equivalents should be memorised.
特殊值：sin 30° = ½，sin 45° = √2/2，sin 60° = √3/2；cos 30° = √3/2，cos 45° = √2/2，cos 60° = ½；tan 30° = 1/√3，tan 45° = 1，tan 60° = √3。对应的弧度值必须熟记。
f sin, cos, and tan show periodic behaviour; sin and cos have period 2π, tan has period π.
基本恒等式：sin²θ + cos²θ = 1，tanθ = sinθ/cosθ。正弦、余弦和正切的图像显示周期性；正弦和余弦的周期为 2π，正切的周期为 π。
Trigonometric equations are solved within a given interval by considering the CAST diagram or the unit circle.
三角方程在给定区间内求解时，可利用 CAST 图或单位圆。
9. Permutations, Combinations and Binomial Expansion | 排列、组合与二项式展开
The number of ways to arrange n distinct items is n! (n factorial). The permutation formula ⁿPᵣ = n!/(n - r)! counts ordered arrangements of r items from n.
将 n 个不同项目排列的方法数为 n!（n 的阶乘）。排列公式 ⁿPᵣ = n!/(n - r)! 计算从 n 个中选取 r 个的有序排列数。
The combination formula ⁿCᵣ = n! / [r!(n - r)!] counts selections where order does not matter. This is also the binomial coefficient.
组合公式 ⁿCᵣ = n! / [r!(n - r)!] 计算不考虑顺序的选择数，它也是二项式系数。
The binomial expansion of (a + b)ⁿ is Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ from r = 0 to n. For (1 + x)ⁿ, the expansion is 1 + nx + n(n-1)/2! x² + ... valid for |x| < 1 when n is not a positive integer.
(a + b)ⁿ 的二项式展开为 Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ，r 从 0 到 n。对于 (1 + x)ⁿ，展开式为 1 + nx + n(n-1)/2! x² + ...，当 n 不是正整数时仅当 |x| < 1 有效。
10. Sequences and Series | 数列与级数
e first n terms is Sₙ = n/2 [2a + (n - 1)d] or Sₙ = n/2 (a + l), where l is the last term.
等差级数具有公差 d：uₙ = a + (n - 1)d。前 n 项和为 Sₙ = n/2 [2a + (n - 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
um of the first n terms is Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1. An infinite GP converges when |r| < 1, and its sum to infinity is S∞ = a/(1 - r).
等比级数具有公比 r：uₙ = arⁿ⁻¹。前 n 项和为 Sₙ = a(1 - rⁿ)/(1 - r)（r ≠ 1）。当 |r| < 1 时，无穷等比级数收敛，其无穷和为 S∞ = a/(1 - r)。
Sigma notation Σ is used to represent sums. Recognise patterns and apply appropriate formulas.
求和符号 Σ 用于表示总和。应识别数列规律并应用相应公式。
11. Vectors in Two Dimensions | 二维向量
A vector has both magnitude and direction. It can be written as a column vector v = (x, y)ᐪ or in component form xi + yj.
向量既有大小又有方向。可写为列向量 v = (x, y)ᐪ 或分量形式 xi + yj。
The magnitude of a vector v is |v| = √x² + y². A unit vector has magnitude 1. The zero vector 0 has magnitude 0.
向量 v 的大小为 |v| = √x² + y²。单位向量大小为 1。零向量 0 的大小为 0。
Vectors are added component-wise. Multiplication by a scalar k changes the magnitude but not the direction (unless k < 0). The position vector of a point P is the vector from the origin O to P.
向量通过分项相加。乘以标量 k 改变大小但不改变方向（除非 k < 0）。点 P 的位置向量是从原点 O 到 P 的向量。
The scalar (dot) product of a and b is a · b = |a||b| cos θ = x₁x₂ + y₁y₂. If a · b = 0, the vectors are perpendicular.
向量 a 与 b 的标量积（点积）为 a · b = |a||b| cos θ = x₁x₂ + y₁y₂。若 a · b = 0，则向量垂直。
12. Differentiation and Integration | 微分与积分
. The derivative of a sum is the sum of derivatives. The second derivative d²y/dx² gives the rate of change of the gradient.
微分给出曲线的斜率。若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。和的导数等于导数的和。二阶导数 d²y/dx² 给出斜率的变化率。
he second derivative test: if d²y/dx² > 0, it is a minimum; if d²y/dx² < 0, it is a maximum; if zero, investigate further.
在驻点处，dy/dx = 0。用二阶导数判别：若 d²y/dx² > 0，则为极小值；若 d²y/dx² < 0，则为极大值；若为零，需进一步分析。
-1. Definite integrals ∫ₐᵇ f(x) dx give the area under the curve between x = a and x = b.
积分是微分的逆运算。∫ xⁿ dx = xⁿ⁺¹/(n+1) + c，n ≠ -1。定积分 ∫ₐᵇ f(x) dx 给出曲线与 x = a 和 x = b 之间的面积。
city or displacement can be found by integration, using initial conditions to find the constant.
运动学将位移 s、速度 v = ds/dt 和加速度 a = dv/dt 联系起来。已知加速度时，可通过积分求得速度或位移，并利用初始条件求出常数。
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