📚 Year 10 CIE Additional Mathematics: Core Topics Overview | 剑桥IGCSE进阶数学核心知识点梳理
Welcome to your Year 10 CIE Additional Mathematics revision guide. This course builds on the IGCSE Mathematics syllabus and introduces advanced topics such as functions, advanced algebra, trigonometry, and the basics of calculus. Mastering these concepts is crucial for success in A-Level Mathematics and beyond. In this article, we systematically review the core topics you need to know, linking theory with problem-solving techniques.
欢迎来到 Year 10 CIE 进阶数学复习指南。本课程在 IGCSE 数学大纲基础上，引入了函数、高级代数、三角学以及微积分基础。掌握这些概念对于 A-Level 数学及更高层次的学习至关重要。本文将系统梳理你需要掌握的核心知识点，并将理论与解题技巧相结合。
1. Functions | 函数
e set of all possible inputs is the domain, and the set of all outputs is the range. One-to-one functions have unique mappings, and only such functions possess inverses, denoted by f⁻¹(x).
函数 f 将定义域内的每个输入值 x 恰好映射到一个输出值 f(x)，通常记为 f : x → f(x)。所有可能输入值的集合称为定义域，所有输出值的集合称为值域。一一函数具有唯一的映射关系，且只有这样的函数才存在反函数，记作 f⁻¹(x)。
Composite functions combine two functions, such as (f ∘ for y, then verify that the domain and range are also swapped. The modulus function f(x) = |x| that outputs the absolute value, and it is not one-to-one unless the domain is restricted.
复合函数将两个函数组合，例如 (f ∘ g)(x) = f(g(x))，即先作用 g 再作用 f。求反函数时，交换 x 和 y 的位置并解出 y，然后验证定义域与值域是否互换。绝对值函数 f(x) = |x| 是一个分段函数，输出绝对值，除非限制定义域，否则它不是一一函数。
2. Quadratic Functions | 二次函数
arabola, opening upward if a > 0 and downward if a < 0. Key features include the vertex, the axis of symmetry, the y-intercept, and the discriminant.
二次函数的形式为 f(x) = ax² + bx + c，其中 a ≠ 0。其图像是一条抛物线，当 a > 0 时开口向上，当 a < 0 时开口向下。主要特征包括顶点、对称轴、y 轴截距和判别式。
ely, the x-coordinate of the vertex is found with x = -b/(2a). The discriminant Δ = b² - 4ac determines the nature of the roots: if Δ > 0, two distinct real roots; if Δ = 0, one repeated root; if Δ < 0, no real roots.
通过配方法可将二次函数写成顶点式：f(x) = a(x - h)² + k，其中 (h, k) 为顶点坐标，对称轴为 x = h。另外，顶点的 x 坐标也可由 x = -b/(2a) 求得。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。
f(x) = a(x - h)² + k, Δ = b² - 4ac
3. Equations, Inequalities and Graphs | 方程、不等式与图像
n be done by factorising, completing the square, or using the quadratic formula: x = [-b ± √b² - 4ac] / (2a). When solving quadratic inequalities such as ax² + bx + c > 0, we sketch the graph to determine intervals where the function is above the x-axis.
解二次方程可通过因式分解、配方法或使用求根公式：x = [-b ± √b² - 4ac] / (2a)。解二次不等式如 ax² + bx + c > 0 时，可绘制图像来确定函数在 x 轴上方的区间。
For simultaneous equations involving a quadratic and a linear equation, we substitute the linear into the quadratic to form a single equation and solve. The number of intersection points corresponds to the number of solutions. Absolute value inequalities like |ax + b| < c are split into -c < ax + b < c and solved as a compound inequality.
对于包含一个二次方程和一个一次方程的联立方程组，可将线性方程代入二次式得到一元方程并求解。交点个数对应于解的个数。绝对值不等式如 |ax + b| < c 可分解为 -c < ax + b < c 并作为复合不等式求解。
4. Indices and Surds | 指数与根式
ssions containing roots, and we simplify them by rationalising the denominator or applying √a × √b = √ab.
指数法则是基础：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，以及 a⁰ = 1，a⁻ⁿ = 1 / aⁿ。根式是含有根号的表达式，我们通过分母有理化或运用 √a × √b = √ab 进行化简。
When adding or subtracting surds, we combine like terms, e.g., 3√2 + 5√2 = 8√2. Rationalising a denominator with a binomial surd uses the conjugate. Solving surd equations involves isolating the square root, squaring both sides, and checking for extraneous roots.
根式的加减需合并同类项，例如 3√2 + 5√2 = 8√2。对含有二项式根式的分母有理化时，使用共轭式。解根式方程时，需孤立平方根，两边平方，并检验增根。
5. Logarithms | 对数
Logarithms are the inverse of
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