IGCSE CAIE Additional Mathematics: Core Knowledge Points | IGCSE CAIE 进阶数学：核心知识点梳理

📚 IGCSE CAIE Additional Mathematics: Core Knowledge Points | IGCSE CAIE 进阶数学：核心知识点梳理

This revision guide summarises the most important topics in the CAIE IGCSE Additional Mathematics (0606) syllabus, including functions, quadratics, logarithms, trigonometry, series, vectors and calculus. Use it alongside past papers to build fluency and accuracy.

本文梳理 CAIE IGCSE 进阶数学（0606）的核心考点，涵盖函数、二次函数、对数、三角、数列、向量与微积分等内容。建议配合历年真题进行训练，提高熟练度与准确率。

1. Functions and Inverse Functions | 函数与反函数

A function maps every input x in its domain to exactly one output f(x). In CAIE Additional Mathematics, you must be able to identify one-to-one, many-to-one and inverse functions. The inverse f⁻¹(x) exists only if f is one-to-one.

函数将定义域中的每一个输入 x 映射到唯一输出 f(x)。在 CAIE 进阶数学中，需要会判断一对一、多对一函数以及反函数。只有一对一函数才存在反函数 f⁻¹(x)。

To find an inverse, write y = f(x), swap x and y, then solve for y. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f. For composite functions, remember that fg(x) means apply g first, then f.

求反函数时，先设 y=f(x)，交换 x 与 y，再解出 y。f⁻¹ 的定义域是 f 的值域，f⁻¹ 的值域是 f 的定义域。对于复合函数，fg(x) 表示先作用 g，再作用 f。

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2. Quadratic Functions and the Discriminant | 二次函数与判别式

A quadratic function has the form f(x) = ax² + bx + c, where a ≠ 0. The sign of a controls whether the graph opens upward or downward. Completing the square gives the vertex form f(x) = a(x − h)² + k, where (h, k) is the vertex.

二次函数形式为 f(x)=ax²+bx+c，其中 a≠0。a 的正负决定抛物线开口向上或向下。配方法可将其化为顶点式 f(x)=a(x−h)²+k，其中 (h,k) 为顶点。

The discriminant Δ = b² − 4ac determines the nature of roots: two distinct real roots if Δ > 0, one repeated real root if Δ = 0, and no real roots if Δ < 0. The quadratic formula is given below.

判别式 Δ=b²−4ac 决定方程根的情况：Δ>0 有两个不同实根，Δ=0 有一个重根，Δ<0 无实根。求根公式如下。

x = (−b ± √(b² − 4ac)) / (2a)

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3. Indices, Surds and Logarithms | 指数、根式与对数

Index laws such as aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ and a⁻ⁿ = 1/aⁿ are fundamental. Surds are simplified using √(ab) = √a × √b and by rationalising denominators such as 1/√a → √a/a.

指数法则如 aᵐ×aⁿ=aᵐ⁺ⁿ、(aᵐ)ⁿ=aᵐⁿ 以及 a⁻ⁿ=1/aⁿ 是基础。根式化简常用 √(ab)=√a×√b，并进行分母有理化，例如 1/√a → √a/a。

If aˣ = y, then logₐ y = x. The key log laws are: logₐ x + logₐ y = logₐ(xy), logₐ x − logₐ y = logₐ(x/y), and logₐ xⁿ = n logₐ x. Change of base is logₐ x = log_b x / log_b a.

若 aˣ=y，则 logₐ y=x。核心对数运算法则包括：logₐ x+logₐ y=logₐ(xy)，logₐ x−logₐ y=logₐ(x/y)，logₐ xⁿ=n logₐ x。换底公式为 logₐ x=log_b x/log_b a。

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4. Polynomials and Cubic Equations | 多项式与三次方程

The factor theorem states that if f(a)=0 for a polynomial f(x), then (x − a) is a factor. The remainder theorem gives the remainder when f(x) is divided by (x − c) as f(c).

因式定理指出，若多项式 f(x) 满足 f(a)=0，则 (x−a) 是它的因式。余式定理说明 f(x) 除以 (x−c) 的余数为 f(c)。

A cubic equation may have up to three real roots. Use factorisation or long division to reduce it to a quadratic, then apply the quadratic formula if necessary. Always check for repeated roots and connection to the graph.

三次方程最多有三个实根。可以通过因式分解或长除法将其降为二次方程，再必要时使用求根公式。同时需要关注重根以及与图像之间的关系。

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5. Simultaneous Equations and Inequalities | 联立方程组与不等式

Simultaneous equations often involve one linear and one quadratic equation. Substitute the linear equation into the quadratic to obtain a single quadratic in one variable, then solve and back-substitute to find the paired values.

联立方程组常含一个一次方程和一个二次方程。将一次方程代入二次方程，得到只含一个未知数的二次方程，解出后代回求出对应的另一个未知数。

For quadratic inequalities, sketch the graph of the related quadratic to identify the intervals where the expression is positive or negative. Remember to reverse the inequality sign when multiplying or dividing by a negative number.

解二次不等式时，先画出对应二次函数图像，判断表达式为正或负的区间。注意当乘以或除以负数时，不等号方向要改变。

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6. Circular Measure | 弧度制与圆弧

In circular measure, angles are measured in radians. The arc length of a sector with radius r and angle θ (rad) is s = rθ, and the sector area is A = ½ r²θ.

在弧度制中，角以弧度为单位。半径为 r、圆心角为 θ（弧度）的扇形弧长为 s=rθ，扇形面积为 A=½ r²θ。

s = rθ and A = ½ r²θ

To convert degrees to radians, multiply by π/180. Use radian mode when differentiating or integrating trigonometric functions in this syllabus.

将角度转换为弧度时乘以 π/180。在本课程中，对三角函数进行微分或积分时必须使用弧度制。

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7. Trigonometry and Identities | 三角函数与恒等式

Know the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°. The graphs of sine, cosine and tangent are required, including amplitude, period and transformations.

需要熟记 0°、30°、45°、60°、90° 的 sin、cos、tan 精确值。正弦、余弦和正切图像的特征，包括振幅、周期和变换也必须掌握。

Key identities include sin² θ + cos² θ = 1, tan θ = sin θ / cos θ, and the double angle formulas sin 2θ = 2 sin θ cos θ and cos 2θ = cos² θ − sin² θ.

核心恒等式包括 sin² θ+cos² θ=1、tan θ=sin θ/cos θ，以及倍角公式 sin 2θ=2 sin θ cos θ 和 cos 2θ=cos² θ−sin² θ。

θ	sin θ	cos θ	tan θ
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3

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8. Coordinate Geometry and Linear Graphs | 坐标几何与直线图像

The gradient of a straight line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). The equation of a line can be written as y = mx + c or y − y₁ = m(x − x₁).

经过两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m=(y₂−y₁)/(x₂−x₁)。直线方程可写成 y=mx+c 或 y−y₁=m(x−x₁)。

Parallel lines have equal gradients. Perpendicular lines satisfy m₁ × m₂ = −1. Use the midpoint formula M = ((x₁+x₂)/2

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