📚 Year 11 CIE Additional Mathematics: Key Terminology Memorization Guide | Year 11 CIE 进阶数学：词汇术语速记指南
Mastering the vocabulary of CIE IGCSE Additional Mathematics is essential for interpreting questions accurately and applying concepts confidently. This guide provides concise definitions of core terms across the syllabus, pairing each English explanation with its Chinese equivalent to accelerate memorisation.
掌握 CIE IGCSE 进阶数学（附加数学）的术语词汇对于准确理解题意、自信应用概念至关重要。本指南提供了大纲中核心术语的简明定义，将每条英文解释与其对应的中文配对，以加速记忆。
1. Functions and Relations | 函数与关系
Function: A rule that assigns each input (x) from the domain to exactly one output (y) in the range. Denoted as f(x).
函数：一种规则，将定义域中的每个输入 (x) 恰好分配给值域中的一个输出 (y)。记为 f(x)。
Domain: The set of all possible input values (x) for which the function is defined.
定义域：函数有定义的所有可能输入值 (x) 的集合。
Range: The set of all output values (y) that the function actually produces.
值域：函数实际产生的所有输出值 (y) 的集合。
One-to-one function: Each y-value corresponds to exactly one x-value; it passes the horizontal line test and has an inverse that is also a function.
单射函数：每个 y 值恰好对应一个 x 值；通过水平线测试，其反函数也是一个函数。
Inverse function f⁻¹(x): Reverses the effect of f(x). The domain of f⁻¹ is the range of f, and vice versa.
反函数 f⁻¹(x)：逆转 f(x) 的作用。f⁻¹ 的定义域是 f 的值域，反之亦然。
Composite function: (f ∘ g)(x) = f(g(x)), also written as fg(x). Apply g first, then f.
复合函数：(f ∘ g)(x) = f(g(x))，也写作 fg(x)。先作用 g，再作用 f。
2. Quadratic Functions and the Discriminant | 二次函数与判别式
Quadratic function: A function of the form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a parabola.
二次函数：形如 f(x) = ax² + bx + c 的函数，其中 a ≠ 0。其图像为抛物线。
Discriminant (Δ): ² − 4ac. It determines the nature of the roots of the quadratic equation ax² + bx + c = 0.
判别式 (Δ)：Δ = b² − 4ac。它决定了二次方程 ax² + bx + c = 0 根的性质。
Δ = b² − 4ac
Nature of roots: Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.
根的性质：若 Δ > 0，有两个不同的实根；若 Δ = 0，有一个重实根；若 Δ < 0，无实根。
Completing the square: Rewriting ax² + bx + c as a(x + p)² + q to find the turning point (−p, q).
配方法：将 ax² + bx + c 改写为 a(x + p)² + q，以求得转折点 (−p, q)。
Vertex form: y = a(x − h)² + k gives a vertex (h, k) and axis of symmetry x = h.
顶点式：y = a(x − h)² + k，顶点为 (h, k)，对称轴为 x = h。
3. Inequalities and Modulus | 不等式与绝对值
Inequality symbols: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to).
不等号：>（大于），<（小于），≥（大于等于），≤（小于等于）。
Solution set: The set of all x-values that satisfy an inequality, often expressed on a number line or using interval notation.
解集：满足不等式的所有 x 值的集合，常表示在数轴上或使用区间记号。
Critical values: The values of x where a rational expression equals zero or is undefined; they split the number line into test intervals.
临界值：使有理式为零或无定义的 x 值；它们将数轴分成需检验的区间。
Modulus (absolute value): |x| gives the distance from x to 0 on the number line. |x| = x if x ≥ 0, and −x if x < 0.
绝对值：|x| 表示数轴上 x 到 0 的距离。|x| = x（当 x ≥ 0），|x| = −x（当 x < 0）。
Modulus equations and inequalities: |x − a| = b leads to x − a = ±b. |x − a| < b ⇔ −b < x − a < b.
绝对值方程和不等式：|x − a| = b 等价于 x − a = ±b。|x − a| < b ⇔ −b < x − a < b。
4. Indices, Surds and Logarithms | 指数、根式与对数
Index (exponent) laws: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.
指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。
Surd: An irrational root, such as √2 or ∛5. Surds are left in root form to maintain exact values.
根式（无理根）：如 √2 或 ∛5 的无理根。保留根号形式以保证精确值。
Rationalising the denominator: Removing a surd from the denominator by multiplying numerator and denominator by the conjugate, e.g., 1/(√a + b) → (√a − b)/(a − b²).
分母有理化：通过分子分母同乘共轭根式消去分母中的根号，例如 1/(√a + b) → (√a − b)/(a − b²)。
Logarithm: logₐ y = x means aˣ = y. The base a must be positive and not equal to 1.
对数：logₐ y = x 表示 aˣ = y。底数 a 必须为正且不等于 1。
Common and natural logs: log₁₀ x is written as lg x; logₑ x is the natural logarithm, written as ln x.
常用对数与自然对数：log₁₀ x 写作 lg x；logₑ x 为自然对数，写作 ln x。
Change of base: logₐ b = (logₓ b) / (logₓ a) for any valid base x.
换底公式：logₐ b = (logₓ b) / (logₓ a)，对任何有效的底 x 都成立。
5. Polynomials and Factor Theorem | 多项式与因式定理
Polynomial: An expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where n is a non‑negative integer.
多项式：形如 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ 的表达式，其中 n 为非负整数。
Degree: The highest power of x in the polynomial. Leading coefficient is the coefficient of that term.
次数：多项式中 x 的最高次幂。首项系数是该次项的系数。
Factor theorem: (x − a) is a factor of polynomial f(x) if and only if f(a) = 0.
因式定理：(x − a) 是多项式 f(x) 的因式当且仅当 f(a) = 0。
Remainder theorem: When f(x) is divided by (x − a), the remainder is f(a).
余式定理：当 f(x) 除以 (x − a) 时，余式为 f(a)。
Long division of polynomials: A method to divide a polynomial by a divisor, yielding a quotient and a remainder.
多项式长除法：用多项式除以除式，得到商式和余式的方法。
6. Binomial Expansion | 二项式展开
Binomial theorem: For positive integer n, (a + b)ⁿ = Σ ᴺCᵣ aⁿ⁻ʳ bʳ, where r goes from 0 to n.
二项式定理：对于正整数 n，(a + b)ⁿ = Σ ᴺCᵣ aⁿ⁻ʳ bʳ，其中 r 从 0 到 n。
Binomial coefficient: ᴺCᵣ = n! / (r!(n − r)!) , also written as (ⁿᵣ). It gives the number of ways to choose r items from n.
二项式系数：ᴺCᵣ = n! / (r!(n − r)!)，也写作 (ⁿᵣ)。表示从 n 个中选取 r 个的组合数。
Pascal's triangle: A triangular array where each number is the sum of the two above. Row n gives coefficients for (a + b)ⁿ.
杨辉（帕斯卡）三角：一个三角形阵列，每个数是上方两数之和。第 n 行给出 (a + b)ⁿ 的系数。
General term: The (r + 1)th term in the expansion is ᴺCᵣ aⁿ⁻ʳ bʳ.
通项：展开中的第 (r + 1) 项为 ᴺCᵣ aⁿ⁻ʳ bʳ。
7. Coordinate Geometry (Straight Lines and Circles) | 坐标几何（直线与圆）
Gradient (slope) of a line: m = (y₂ − y₁) / (x₂ − x₁).
直线的斜率：m = (y₂ − y₁) / (x₂ − x₁)。
Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
中点：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
Distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²].
距离公式：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。
Equation of a straight line: y − y₁ = m(x − x₁) (point‑slope form) or y = mx + c (slope‑intercept form).
直线方程：y − y₁ = m(x − x₁)（点斜式）或 y = mx + c（斜截式）。
Parallel and perpendicular lines: Parallel lines have equal gradients (m₁ = m₂). Perpendicular lines satisfy m₁ × m₂ = −1.
平行与垂直线：平行线斜率相等 (m₁ = m₂)。垂直线满足 m₁ × m₂ = −1。
Equation of a circle: tre (h, k), radius r: (x − h)² + (y − k)² = r². General form: x² + y² + 2gx + 2fy + c = 0.
圆的方程：圆心 (h, k)，半径 r：(x − h)² + (y − k)² = r²。一般式：x² + y² + 2gx + 2fy + c = 0。
Finding centre and radius from general form: Centre (−g, −f), radius = √g² + f² − c.
从一般式求圆心和半径：圆心 (−g, −f)，半径 = √g² + f² − c。
8. Trigonometry | 三角学
Radian measure: subtended at the centre of a circle by an arc equal in length to the radius. π rad = 180°.
弧度制：1 弧度是弧长等于半径的圆弧所对的圆心角。π 弧度 = 180°。
Sine, cosine, tangent: In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ =
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