📚 Year 10 Cambridge Additional Mathematics: Core Topics Overview | 剑桥 Year 10 进阶数学：核心知识点梳理
Welcome to this comprehensive overview of the core topics in the Year 10 Cambridge IGCSE Additional Mathematics (0606) syllabus. Mastering these fundamentals will not only boost your exam confidence but also lay a solid foundation for further study in mathematics and sciences. Let's walk through each key area, highlighting essential concepts, notation, and techniques.
欢迎阅读这篇 Year 10 剑桥 IGCSE 进阶数学 (0606) 核心知识点梳理。掌握这些基础内容不仅能提升你的考试信心，也能为后续的数学与科学学习打下扎实的基础。下面我们逐一梳理每个关键领域，突出基本概念、记号与解题方法。
1. Functions | 函数
is a rule that assigns each input (x) exactly one output (y). We write f(x) = ... or f: x ↦ ; the range is the set of all y-values the function actually produces. For example, f(x) = √x has domain x ≥ 0 and range y ≥ 0.
函数是一种法则，将每个输入值 x 对应到唯一的输出值 y。我们记作 f(x) 或 f: x ↦ f(x)。定义域是所有可能 x 值的集合，值域是函数实际输出的所有 y 值的集合。例如 f(x) = √x 的定义域为 x ≥ 0，值域为 y ≥ 0。
formed by applying one function after another: gf(x) = g(f(x)). Pay attention to the order – main of gf consists of x-values in the domain of f for which f(x) lies in the domain of g.
复合函数是将两个函数依次作用：gf(x) = g(f(x))。注意顺序，fg(x) 通常不同。gf 的定义域由那些在 f 定义域内且使得 f(x) 落入 g 定义域的 x 组成。
An inverse function f⁻¹(x) 'undoes' f. It only exists if f is a one‑ swap x and y, and solve for y. The graph of f⁻¹ is the reflection of f in the line y = x.
反函数 f⁻¹(x) 的作用是逆转 f。仅当 f 是一一映射时反函数才存在。求反函数时，令 y = f(x)，交换 x 与 y 后解出 y。反函数的图像是原函数关于直线 y = x 的对称图形。
2. Quadratic Functions | 二次函数
x + c, with a ≠ 0. By completing the square we obtain the vertex form a(x + p)² + q. The vertex (turning point) is (−p, q). If a > 0, the graph is ∪‑shaped (minimum point); if a < 0, it is ∩‑shaped (maximum point).
一般二次函数为 f(x) = ax² + bx + c，a ≠ 0。通过配方法可化为顶点式 a(x + p)² + q。顶点（转折点）坐标为 (−p, q)。若 a > 0，图像呈 ∪ 形（极小值）；若 a < 0，呈 ∩ 形（极大值）。
ct real roots when Δ > 0, one repeated real root when Δ = 0, and no real roots when Δ < 0.
Δ = b² − 4ac
判别式 Δ = b² − 4ac 揭示根的性质。Δ > 0 时有两个相异实根，Δ = 0 时有一个重根，Δ < 0 时无实根。
Solving quadratic inequalities such as ax² + bx + c > 0 can be done by sketching the parabola or using a sign table. Always check where the graph lies above or below the x-axis.
解二次不等式如 ax² + bx + c > 0 可借助抛物线草图或符号表。始终要判断图像在 x 轴上方还是下方。
3. Equations, Inequalities and Graphs | 方程、不等式与图像
Simultaneous equations, one linear and one quadratic, are solved by substitution. Always substitute the linear expression into the quadratic, solve for one variable, then find the corresponding values of the other.
求解一个线性方程与一个二次方程联立的方程组时使用代入法。将线性表达式代入二次方程，解出一个变量，再求另一个变量的对应值。
Equations involving absolute (modulus) functions, such as |ax + b| ases: ax + b = c or ax + b = −c. Graphs of modulus functions help visualise solutions; y = |f(x)| reflects negative parts of f above the x‑axis.
含绝对值的方程如 |ax + b| = c 可分为两种情况：ax + b = c 或 ax + b = −c。绝对值函数的图像有助于直观理解解的情况；y = |f(x)| 将 f 的负值部分翻折到 x 轴上方。
For inequalities like |x − a| < b, rewrite as −b < x − a < b. On a graph, the solution corresponds to the interval where the modulus graph is below the horizontal line.
对于不等式 |x − a| < b，可重写为 −b < x − a < b。在图形上，解集对应绝对值图像低于水平线的区间。
4. Indices and Surds | 指数与根式
Index laws are indispensable: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = ¹⁄ₐⁿ, and a1/n = ⁿ√a. These rules apply for any rational exponents.
指数法则是必不可少的：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = ¹⁄ₐⁿ，以及 a1/n = ⁿ√a。这些规则适用于任意有理指数。
Surds are irrational roots like √2. Simplify surds by extracting square factors: √a²b = a√b. Rationalise denominators, for example 1/(√a + √b) multiply numerator and denominator by (√a − √b).
根式是指如 √2 这样的无理根。化简根式时提取平方因子：√a²b = a√b。分母有理化，例如 1/(√a + √b) 的分子分母同乘以 (√a − √b)。
5. Logarithmic and Exponential Functions | 对数与指数函数
The exponential function y = aˣ (a > 0, a ≠ 1) and the logarithmic function y = logₐ x are inverses. Thus logₐ aˣ = x and alogₐ x = x. The natural exponential y = eˣ and natural logarithm y = ln x follow the same relationship.
指数函数 y = aˣ（a > 0, a ≠ 1）与对数函数 y = logₐ x 互为反函数。因此 logₐ aˣ = x 且 alogₐ x = x。自然指数 y = eˣ 与自然对数 y = ln x 也具有同样的关系。
ₐ M. The change-of-base formula is logₐ b = logₓ b / logₓ a, often used with base 10 or e.
对数运算律：logₐ (MN) = logₐ M + logₐ N，logₐ (M/N) = logₐ M − logₐ N，logₐ (Mᵏ) = k logₐ M。换底公式为 logₐ b = logₓ b / logₓ a，常以 10 或 e 为底。
ial and logarithmic equations are solved by taking logs or exponentiating. For example, 3²ˣ⁺¹ = 5 ⇒ (2x+1)ln3 = ln5, then solve for x.
解指数或对数方程时常取对数或化为指数形式。例如 3²ˣ⁺¹ = 5 ⇒ (2x+1)ln3 = ln5，进而解出 x。
6. Polynomials | 多项式
he remainder is P(a). The Factor Theorem says (x − a) is a factor if and only if P(a) = 0.
n 次多项式 P(x) 除以一次因式 (x − a) 可使用长除法或综合除法。余数定理指出余数为 P(a)。因式定理表明 (x − a) 是因式当且仅当 P(a) = 0。
Use the Factor Theorem to factorise cubic or quartic expressions and solve polynomial equations. Once one factor is found, reduce the degree and repeat. Always fully factorise over the real numbers.
运用因式定理对三次或四次多项式进行因式分解并解方程。找到一个因式后降低次数继续分解。最终要在实数范围内完全分解。
7. Coordinate Geometry | 坐标几何
− y₁)²]. The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). The gradient of AB is (y₂ − y₁)/(x₂ − x₁).
两点 A(x₁, y₁) 与 B(x₂, y₂) 的距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]。中点为 ((x₁+x₂)/2, (y₁+y₂)/2)。AB 的斜率为 (y₂ − y₁)/(x₂ − x₁)。
y = mx + c. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
直线方程：y − y₁ = m(x − x₁) 或 y = mx + c。平行直线斜率相等；互相垂直的直线满足 m₁ × m₂ = −1。
Completing the square helps convert the general form x² + y² + 2gx + 2fy + c = 0 to centre‑radius form.
圆心 (a, b)、半径 r 的圆的方程为 (x − a)² + (y − b)² = r²。利用配方法可将一般式 x² + y² + 2gx + 2fy + c = 0 化为标准式。
8. Trigonometry | 三角学
alues of sin, cos, tan for 0°, 30°, 45°, 60°, 90° (and radian equivalents). The unit circle defines trigonometric functions for all angles and reveals their periodic nature.
角度可用度数或弧度表示，π 弧度 = 180°。熟记 0°、30°、45°、60°、90°（及对应弧度）的正弦、余弦、正切精确值。单位圆的定义将三角函数推广到任意角，并体现出周期性。
1 + cot²θ = cosec²θ. Use these to simplify expressions and solve trigonometric equations.
核心恒等式：sin²θ + cos²θ = 1，tanθ = sinθ/cosθ，以及 1 + tan²θ = sec²θ、1 + cot²θ = cosec²θ。运用它们化简三角式并解三角方程。
equations like 2 sin x = 1 for 0 ≤ x ≤ 2π requires reference to the graph or CAST diagram.
绘制 y = a sin(bx) + c、y = a cos(bx) + c 及 y = a tan(bx) + c 的图像，理解振幅、周期与垂直平移。解形如 2 sin x = 1（0 ≤ x ≤ 2π）的方程时需借助图像或 CAST 图。
9. Sequences and Series | 数列与级数
the first n terms: Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l) where l is the last term.
等差数列的首项为 a，公差为 d。第 n 项 uₙ = a + (n − 1)d。前 n 项和 Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
ratio r. The nth term is uₙ = arⁿ⁻¹. Sum of the first n terms: Sₙ = a(1 − rⁿ)/(1 − r) for |r| ≠ 1. The sum to infinity S∞ = a/(1 − r) exists only when |r| < 1.
等比数列的首项为 a，公比为 r。第 n 项 uₙ = arⁿ⁻¹。前 n 项和当 |r| ≠ 1 时为 Sₙ = a(1 − rⁿ)/(1 − r)。无穷等比级数的和 S∞ = a/(1 − r) 仅在 |r| < 1 时存在。
Also be able to use Σ notation, and solve problems where two terms or sums are given, then find a and d (or r).
还要能使用 Σ 记号，并解决已知某两项或部分和求首项和公差（或公比）的问题。
10. Differentiation | 微分
gives the gradient of a curve. The derivative of y with respect to x is written dy/dx or f'(x). For y = xⁿ, dy/dx = n xⁿ⁻¹. This power rule extends to any rational exponent.
d/dx (xⁿ) = n xⁿ⁻¹
微分用来求曲线的斜率。y 对 x 的导数记作 dy/dx 或 f'(x)。对于 y = xⁿ，dy/dx = n xⁿ⁻¹，该幂法则适用于任何有理指数。
term. The derivative of a constant is zero. For products or quotients, the product rule and quotient rule are used later in the syllabus; at this stage, stick to simple polynomials and sums.
若 y = axⁿ + bxᵐ，可逐项求导。常数的导数为零。乘积或商的求导产品会在后续学习乘积法则与商法则，现阶段主要掌握简单多项式及和差的求导。
The gradient of the tangent at a point is dy/dx. The equation of the tangent can be written using y −
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