📚 Core Knowledge Points for IGCSE CIE Additional Mathematics | IGCSE CIE 进阶数学：核心知识点梳理
IGCSE CIE Additional Mathematics (0606) builds on the foundation of IGCSE Mathematics and equips students with essential skills for A Level studies. This article summarises the core knowledge points across the syllabus, including functions, quadratics, equations, logarithms, polynomials, coordinate geometry, binomial theorem, trigonometry, calculus, vectors, permutations and combinations, and sequences. A clear grasp of these topics is vital for exam success.
IGCSE CIE 进阶数学（0606）在 IGCSE 数学的基础上进一步拓展，为学生修读 A Level 课程奠定必要基础。本文梳理教学大纲中的核心知识点，涵盖函数、二次函数、方程、对数、多项式、坐标几何、二项式定理、三角学、微积分、向量、排列组合以及数列。熟练掌握这些主题对考试取得优异成绩至关重要。
1. Functions | 函数
A function f maps each element x in the domain to a unique element f(x) in the range. The domain must be specified; for example, f(x)=√(x−2) requires x ≥ 2. The range is the set of all possible output values. For composite functions such as fg(x), we apply g first then f. An inverse function f⁻¹(x) exists only if the function is one‑one. To find the inverse, swap x and y and solve for y; the graph of an inverse is a reflection of the original graph in the line y = x.
函数 f 将定义域中的每一个元素 x 映射到值域中唯一的 f(x)。定义域需明确指定，例如 f(x)=√(x−2) 要求 x ≥ 2。值域是所有可能的输出值的集合。对于复合函数如 fg(x)，先应用 g 再应用 f。反函数 f⁻¹(x) 仅在函数为一一映射时才存在。求反函数时互换 x 与 y，然后解出 y；反函数的图像是原图像关于直线 y = x 的反射。
Transformations of graphs are also covered. A translation by vector (h,k) sends y = f(x) to y = f(x−h) + k. A stretch parallel to the y‑axis with scale factor a gives y = a f(x); a stretch parallel to the x‑axis with scale factor 1/b gives y = f(bx). Reflections change the sign of either the function or the argument.
图形变换也是考核内容。通过向量 (h,k) 平移，将 y = f(x) 变为 y = f(x−h) + k。平行于 y 轴的伸缩，比例因子为 a，得到 y = a f(x)；平行于 x 轴的伸缩，比例因子为 1/b，得到 y = f(bx)。反射则改变函数或自变量的符号。
2. Quadratic Functions | 二次函数
A quadratic function has the form f(x) = ax² + bx + c, a ≠ 0. Completing the square rewrites it as f(x) = a(x + p)² + q, which reveals the vertex (−p, q) and the axis of symmetry x = −p. The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated real root; Δ < 0 gives no real roots. The graph of a quadratic is a parabola, opening upwards if a > 0 and downwards if a < 0.
二次函数的一般形式为 f(x) = ax² + bx + c，a ≠ 0。通过配方法可将其改写为 f(x) = a(x + p)² + q，由此得出顶点 (−p, q) 和对称轴 x = −p。判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 时有两个不相等的实数根；Δ = 0 时有一个重根；Δ < 0 时无实数根。二次函数的图像是一条抛物线，当 a > 0 时开口向上，a < 0 时开口向下。
Solving quadratic inequalities such as ax² + bx + c > 0 involves sketching the parabola and identifying the regions where the graph lies above or below the x‑axis. The solution set is expressed using interval notation or set notation.
求解二次不等式如 ax² + bx + c > 0，需要画出抛物线的大致形状，找出图像位于 x 轴上方或下方的区间。解集用区间或集合符号表示。
3. Equations, Inequalities and Graphs | 方程、不等式与图像
This topic extends equation solving to cubic and higher‑degree polynomials, as well as equations involving absolute values. For |ax + b| = c, we solve ax + b = c or ax + b = −c. Inequalities like |ax + b| < c translate to −c < ax + b < c. When solving polynomial inequalities, we factorise the polynomial and use a sign table to determine where the expression is positive or negative. Graphs of functions such as y = |f(x)| reflect the parts below the x‑axis upwards, while y = f(|x|) reflects the right‑hand side to the left.
这一主题将方程求解扩展到三次及更高次多项式，以及含绝对值的方程。对于 |ax + b| = c，我们解 ax + b = c 或 ax + b = −c。不等式 |ax + b| < c 等价于 −c < ax + b < c。在求解多项式不等式时，将多项式因式分解，并利用符号表确定表达式为正或负的区间。y = |f(x)| 的图像将 x 轴下方的部分向上翻折，而 y = f(|x|) 的图像则将右侧部分反射到左侧。
4. Indices, Surds and Logarithms | 指数、根式与对数
Laws of indices are fundamental: a^m × aⁿ = a^(m+n), (a^m)ⁿ = a^(mn), a^m ÷ aⁿ = a^(m−n). Surds are simplified using these laws, e.g. √(a²b) = a√b, and rationalising the denominator of expressions like 1/(√a + √b).
指数法则是基础：a^m × aⁿ = a^(m+n), (a^m)ⁿ = a^(mn), a^m ÷ aⁿ = a^(m−n)。根式的化简运用这些法则，例如 √(a²b) = a√b，并对形如 1/(√a + √b) 的分母有理化。
Logarithms are the inverses of exponentials: if a^x = b then x = logₐ b. Key laws: log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n log x. The change‑of‑base formula logₐ b = (log_c b)/(log_c a) is used to evaluate logarithms with any base. Equations involving exponentials or logarithms are solved by applying these laws and isolating the variable.
对数是指数的逆运算：若 a^x = b，则 x = logₐ b。主要法则：log(xy) = log x + log y，log(x/y) = log x − log y，log(xⁿ) = n log x。换底公式 logₐ b = (log_c b)/(log_c a) 用于计算任意底数的对数。解指数方程或对数方程时，运用这些法则并分离变量。
5. Factors of Polynomials and Cubic Equations | 多项式因式与三次方程
The factor theorem states that for a polynomial P(x), if P(a) = 0 then (x − a) is a factor. The remainder theorem gives the remainder when P(x) is divided by (x − a) as P(a). To factorise a cubic expression, we first find one linear factor by testing divisors of the constant term, then use long division or synthetic division to obtain a quadratic factor. The cubic equation can then be solved by setting each factor to zero. Some exam questions provide one root and ask for the remaining roots.
因式定理指出，对多项式 P(x)，若 P(a) = 0，则 (x − a) 为一因式。余数定理给出 P(x) 除以 (x − a) 的余数为 P(a)。要分解一个三次多项式，通常先通过检验常数项的因子找出一个一次因式，然后用长除法或综合除法得到二次因式，再令各因式为零即可求解三次方程。有些考题会给出一个根，要求学生求出所有根。
6. Simultaneous Equations | 联立方程
Simultaneous equations in IGCSE Additional Mathematics typically involve one linear and one quadratic equation, or two equations that require substitution. The substitution method replaces one variable using the linear equation, producing a quadratic in the other variable. For example, from y = 2x + 1 and x² + y² = 10, we substitute to get x² + (2x+1)² = 10 and solve for x. Each x‑value gives a corresponding y‑value. Solutions should be checked, especially when solving intersections of lines and circles, as extraneous solutions can arise.
IGCSE 进阶数学中的联立方程通常包含一个一次方程和一个二次方程，或需要代入求解的两个方程。代入法即利用一次方程将一个变量替换，得到关于另一变量的二次方程。例如，由 y = 2x + 1 和 x² + y² = 10，代入得 x² + (2x+1)² = 10 并求解 x。每个 x 值对应一个 y 值。解出后应加以检验，尤其是在求解直线与圆的交点时，可能会产生多余的解。
7. Coordinate Geometry (Lines & Circles) | 坐标几何（直线与圆）
The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂−x₁)² + (y₂−y₁)²], and the midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). The gradient of a line is m = (y₂−y₁)/(x₂−x₁). Equations of a straight line can be written as y − y₁ = m(x − x₁) or in the general form ax + by + c = 0. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
两点 (x₁, y₁) 与 (x₂, y₂) 间的距离为 √[(x₂−x₁)² + (y₂−y₁)²]，中点为 ((x₁+x₂)/2, (y₁+y₂)/2)。直线的斜率 m = (y₂−y₁)/(x₂−x₁)。直线方程可表示为 y − y₁ = m(x − x₁) 或一般式 ax + by + c = 0。平行线斜率相等；垂直线满足 m₁ × m₂ = −1。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². To find the intersection of a line and a circle, substitute the linear equation into the circle and solve the resulting quadratic. A tangent touches the circle at exactly one point, which occurs when the discriminant of the resulting quadratic is zero. The tangent is perpendicular to the radius at the point of contact.
以 (a, b) 为圆心、半径为 r 的圆的方程为 (x − a)² + (y − b)² = r²。求直线与圆的交点时，将直线方程代入圆的方程，解所得的二次方程即可。当直线为切线时，它与圆恰好交于一点，此时所得二次方程的判别式为零。切线与过切点的半径互相垂直。
8. Binomial Theorem | 二项式定理
The binomial theorem expands (a + b)ⁿ for a positive integer n as Σ (nCr) a^(n−r) b^r, where nCr = n!/(r!(n−r)!). The coefficient of the term a^(n−r) b^r is ⁿCᵣ. Pascal’s triangle can be used to find small‑n coefficients. To find a specific term, such as the term independent of x, identify the value of r that makes the power of x equal to zero. Careful handling of signs is needed when b includes a negative sign.
二项式定理将 (a + b)ⁿ（n 为正整数）展开为 Σ (nCr) a^(n−r) b^r，其中 nCr = n!/(r!(n−r)!)。项 a^(n−r) b^r 的系数为 ⁿCᵣ。对于较小的 n，可以利用杨辉三角求得系数。求某一特定项（如常数项）时，需确定能令 x 的指数为零的 r 值。当 b 包含负号时，要特别注意符号的处理。
9. Trigonometry & Circular Measure | 三角学与弧度制
Angles can be measured in degrees or radians; π rad = 180°. Radian measure simplifies arc length and sector area: arc length s = rθ, sector area A = ½r²θ, where θ is in radians. The six trigonometric ratios are defined for angles of any size using the unit circle. Key identities include tanθ = sinθ/cosθ and sin²θ + cos²θ = 1. Exact values for 0°, 30°, 45°, 60° and 90° (and their radian equivalents) must be memorised.
角度可以用度数或弧度度量；π 弧度 = 180°。弧度制使得弧长与扇形面积公式更为简洁：弧长 s = rθ，扇形面积 A = ½r²θ，其中 θ 以弧度表示。六个三角函数可以利用单位圆推广到任意角。重要的恒等式包括 tanθ = sinθ/cosθ 以及 sin²θ + cos²θ = 1。0°、30°、45°、60° 和 90°（及其弧度表示）的精确值必须熟记。
Trigonometric equations such as 2 sin x = 1 are solved by first finding the principal value and then using the quadrant rules or the graphs of the functions to find all solutions within a given interval. The sine and cosine rules are used to solve non‑right‑angled triangles: a/(sin A) = b/(sin B) = c/(sin C) and a² = b² + c² − 2bc cos A.
求解三角方程（如 2 sin x = 1）时，先求出基本解，再利用象限法则或函数图像找出给定区间内的所有解。正弦定理和余弦定理用于解非直角三角形：a/(sin A) = b/(sin B) = c/(sin C) 及 a² = b² + c² − 2bc cos A。
10. Calculus (Differentiation & Integration) | 微积分（微分与积分）
The derivative of f(x) = axⁿ is f'(x) = naxⁿ⁻¹. The sum, constant multiple, product, quotient, and chain rules are all assessed: (uv)’ = u’v + uv’, (u/v)’ = (u’v − uv’)/v², and if y = f(g(x)) then dy/dx = f'(g(x))·g'(x). These rules allow differentiation of a wide variety of functions, including polynomials, rational expressions, and composite functions such as (3x + 1)⁵.
对 f(x) = axⁿ 求导，得 f'(x) = naxⁿ⁻¹。和差法则、常数倍乘法则、积法则、商法则和链式法则均为考点：(uv)’ = u’v + uv’，(u/v)’ = (u’v − uv’)/v²，若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)。运用这些法则可以对多项式、有理式及复合函数（如 (3x + 1)⁵）进行求导。
Applications of differentiation include finding the gradient of a curve at a point, equations of tangents and normals, identifying stationary points (where f'(x)=0), and determining their nature using the second derivative or a gradient sign test. In kinematics, velocity is the derivative of displacement, and acceleration is the derivative of velocity.
微分的应用包括求曲线上某点的斜率、切线和法线方程、寻找驻点（f'(x)=0），以及利用二阶导数或一阶导数符号变化判断驻点的性质。在运动学中，速度是位移的导数，加速度是速度的导数。
Integration is the reverse of differentiation. The indefinite integral of axⁿ is (axⁿ⁺¹)/(n+1) + c, provided n ≠ −1. The definite integral ∫ₐᵇ f(x) dx calculates the area between the curve y = f(x), the x‑axis, and the lines x = a and x = b. Areas between two curves are found by subtracting the lower curve from the upper curve. Simple integrals of the form (ax+b)ⁿ can be integrated by reversing the chain rule.
积分是微分的逆运算。axⁿ 的不定积分为 (axⁿ⁺¹)/(n+1) + c（n ≠ −1）。定积分 ∫ₐᵇ f(x) dx 用来计算曲线 y = f(x)、x 轴及直线 x = a 和 x = b 所围成的面积。两曲线之间的面积，用上方曲线减去下方曲线后积分求得。形如 (ax+b)ⁿ 的简单函数可通过逆链式法则积分。
11. Vectors | 向量
A vector in two dimensions can be expressed as a column vector (x,y) or in terms of unit vectors i and j: v = xi + yj. The magnitude is |v| = √(x²+y²). Vectors are added geometrically by the triangle law or algebraically by adding components. Scalar multiplication changes the length of a vector, and a negative scalar reverses its direction. Position
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