📚 GCSE CAIE Further Maths: Vocabulary & Terminology Quick Memory Guide | GCSE CAIE 进阶数学：词汇术语速记指南
Mastering the terminology in GCSE CAIE Further Maths (Additional Mathematics) is half the battle. This guide breaks down key terms with memory hooks, word origins and analogies, pairing every English explanation with its Chinese equivalent so you can move between languages and concepts fluidly. Use it alongside your past papers to sharpen your exam responses and eliminate those "I know it but can't name it" moments.
掌握 GCSE CAIE 进阶数学（附加数学）中的术语，等于赢下了一半的战役。这份指南借助记忆钩子、词源和类比，将每个核心术语的英文解释与中文配对讲解，让你能够在语言和概念之间自如切换。配合真题使用，能有效提升答题的精准度，告别“心里明白但说不出来”的困境。
1. Algebra Fundamentals | 代数基础
Variable, Constant, Coefficient – A variable (often x, y) can change, a constant stays fixed, and a coefficient is the number multiplying a variable. Think: the coefficient "co-operates" with the variable to scale it. In 5x², 5 is the coefficient, x is the variable and the exponent 2 is a constant power.
变量、常数、系数 – 变量（常用 x, y）可以取不同的值，常数固定不变，系数则是乘法中与变量相乘的数字。联想：系数（coefficient）中的 "co-" 表示共同作用，它和变量一起决定了项的大小。在 5x² 中，5 是系数，x 是变量，指数 2 是一个不变的幂。
Expression, Equation, Identity – pressions are equal and is usually solved for specific values (3x + 2 = 14). An identity (≡) is true for all values of the variable. Mnemonic: an identity is "identical" on both sides.
表达式、方程、恒等式 – 表达式是没有等号的符号组合（如 3x + 2）。方程则断言两个表达式相等，通常针对特定值求解（如 3x + 2 = 14）。恒等式（用 ≡ 表示）对于变量的所有取值都成立。记住：恒等式的两边始终“恒等”。
Polynomial, Degree, Like Terms – A polynomial is a sum of terms with non‑negative integer powers of x. Its degree is the highest power present. Like terms have exactly the same variable part and can be combined. The word "polynomial" contains "poly" (many) and "nomial" (term) – many terms.
多项式、次数、同类项 – 多项式是若干项的和，每一项中 x 的指数都是非负整数。多项式的次数指的是最高次幂。同类项拥有完全相同的字母部分，可以进行合并。"polynomial" 中 "poly" 表示“多”，"nomial" 表示“项”，即“多项”。
Factorising, Completing the Square, Discriminant – x + p)² + q. The discriminant Δ = b² - 4ac determines the nature of roots. Think of Δ as a "discriminating" judge: positive → two real roots, zero → one repeated root, negative → no real roots.
因式分解、配方法、判别式 – 因式分解将二次式改写为两个一次因式的乘积。配方法把 x² + bx 转化为 (x + p)² + q 的形式。判别式 Δ = b² - 4ac 判定根的性质。可以想象 Δ 是一位“辨别是非”的法官：正值 → 两个不等实根，零 → 一个重根，负值 → 无实根。
2. Inequalities & Graphs | 不等式与图像
Strict vs Inclusive Inequalities – inequality uses < or > and excludes the boundary (think of a strict parent who doesn' the boundary. On a number line, strict gives an open circle, inclusive gives a closed dot.
严格不等式与带等号不等式 – 严格不等式使用 < 或 >，不包括边界（想象一位严格的家长，不允许触碰界限）。带等号的不等式使用 ≤ 或 ≥，边界也包含在内。在数轴上，严格不等式用空心圆点表示，带等号的用实心圆点。
Critical Values & Interval Notation – Critical values are the solutions of the related equation that split the number line into regions. Test a value from each interval to see where the inequality holds. Intervals are often written as, for example, x < 2 or x > 5.
临界值与区间表示法 – 临界值是将原不等式对应的方程的解，它们把数轴分割成若干区间。在每个区间内选取一个测试值，判断不等式是否成立。区间常用不等式符号直接表示，例如 x < 2 或 x > 5。
Quadratic Inequalities & Graphs – Sketch the quadratic curve. The regions where the curve is above or below the x‑axis give the solution. Key rule: if the coefficient of x² is positive, the curve is ∪‑shaped; if negative, ∩‑shaped. Always start by finding where the curve cuts the axis.
二次不等式与图像 – 先画出二次函数的草图。曲线在 x 轴上方或下方的区域即为解集。关键规律：若 x² 系数为正，曲线呈 ∪ 形；若为负，呈 ∩ 形。解题时一定从求曲线与 x 轴的交点入手。
Modulus: |x| = a → x = ±a
Modulus (Absolute Value) – The modulus |x| gives the distance from zero, always non‑negative. Equations like |2x - 1| = 5 split into two cases: 2x - 1 = 5 or 2x - 1 = -5. Visualise the V‑shape graph of y = |f(x)| to solve inequalities.
模（绝对值） – 模 |x| 表示到零点的距离，总是非负值。形如 |2x - 1| = 5 的方程需拆分为两种情况：2x - 1 = 5 或 2x - 1 = -5。求解模不等式时，可以借助 y = |f(x)| 的 V 形图像帮助判断。
3. Functions & Transformations | 函数与变换
Domain, Range, One‑to‑one, Many‑to‑one – The domain is the set of allowed inputs (x‑values), the range is the set of possible outputs (y‑values). A one‑to‑one function sends each x to a unique y; a many‑to‑one function can send different x to the same y. Only one‑to‑one functions have inverses that are also functions.
定义域、值域、一一映射、多对一映射 – 定义域是所有允许的输入值（x 值）的集合，值域是所有可能的输出值（y 值）的集合。一一映射的每个 x 对应唯一的 y；多对一映射则允许多个不同的 x 对应同一个 y。只有一一映射才存在函数意义上的反函数。
Inverse Function f⁻¹(x) – reflection in the line y = x. To find an inverse, swap x and y and solve for y. Mnemonic: "inverse" does the opposite – think of pressing undo.
反函数 f⁻¹(x) – 反函数将原函数的映射反转；若 f(2)=5，则 f⁻¹(5)=2。图像上，反函数是原函数关于直线 y = x 的反射。求反函数的步骤是交换 x 与 y 的位置，然后解出 y。记忆窍门：反函数就是“撤销”操作。
Composite Functions – gf(x) means apply f first, then g. Order matters! gf(x) is read as "g of f of x". Always work from the inside out, like putting on socks (f) then shoes (g).
复合函数 – gf(x) 表示先执行 f，再执行 g。顺序至关重要！gf(x) 读作 "g of f of x"。始终由内向外计算，就像先穿袜子 (f)，再穿鞋子 (g)。
Translations & Stretches – f(x) + a moves the graph vertically, f(x + a) moves it horizontally (opposite direction: +a shifts left). Stretches: af(x) stretches vertically by factor a; f(ax) stretches horizontally by factor 1/a. Remember: inside the bracket affects x and does the opposite.
平移与伸缩 – f(x) + a 将图像垂直移动，f(x + a) 则水平移动（注意方向相反：+a 向左移）。伸缩变换中，af(x) 是在竖直方向上伸缩，伸缩因子为 a；f(ax) 是在水平方向上伸缩，伸缩因子为 1/a。记住：括号内部影响的是 x，且方向往往相反。
4. Exponentials & Logarithms | 指数与对数
Exponent, Base, Power – e exponent or index, and the whole expression is a power. Exponential functions like y = aˣ have the variable in the exponent. The graph of y = aˣ (a>1) rises rapidly; think of "exponential growth" in headlines.
指数、底数、幂 – 在 aⁿ 中，a 称为底数，n 称为指数，整个表达式叫做幂。指数函数如 y = aˣ 的特点是变量出现在指数位置。y = aˣ（a>1）的图像迅猛增长，就像新闻里常说的“指数级增长”。
Logarithms – The Inverse of Exponents – The statement logₐ N = x means aˣ = N. Common log (log₁₀) has base 10; natural log (ln) has base e ≈ 2.718. The word "logarithm" comes from Greek "logos" (ratio) and "arithmos" (number) – it counts the number of times a base is multiplied.
对数——指数的逆运算 – logₐ N = x 的含义是 aˣ = N。常用对数 (log₁₀) 以 10 为底；自然对数 (ln) 以 e ≈ 2.718 为底。"logarithm" 一词源于希腊语的“比率”和“数字”——它实质上是在统计底数被乘的次数。
Key laws: logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx - logₐy; logₐ(xⁿ) = n logₐx
Laws of Logarithms Memory Trick – Multiplying inside the log turns into adding outside; division inside becomes subtraction outside; powers inside jump to the front as multipliers. Visualise the log as a shrinking machine that pushes exponents down to coefficients.
对数律记忆技巧 – 对数内部做乘法，外部变成加法；内部做除法，外部变成减法；内部有乘方，指数会跳到前面变成系数。可以想象对数是一台“压缩器”，把指数压缩成系数。 Published by TutorHao | GCSE 进阶数学 Revision Series | aleveler.com
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