📚 IGCSE CAIE Additional Mathematics: Vocabulary and Terminology Memory Guide | IGCSE CAIE 进阶数学：词汇术语速记指南
In IGCSE CAIE Additional Mathematics, many marks are lost not because the working is hard, but because a key term is misread or a symbol is interpreted incorrectly. This guide collects the most important vocabulary across the syllabus and pairs each term with a quick memory hook. Use it before topic tests and especially before Paper 1 and Paper 2 revision.
在 IGCSE CAIE 进阶数学中，很多失分并不是因为计算过程难，而是因为关键术语被误读或符号被理解错误。本指南汇集了考纲中最重要的词汇，并为每个术语搭配快速记忆提示。建议在单元测验前，尤其是 Paper 1 和 Paper 2 复习前使用。
1. Functions, Domain and Range | 函数、定义域与值域
A function is a rule that assigns each input exactly one output. The domain is the set of all possible input values, usually x-values, while the range is the set of all possible output values, usually y-values. In IGCSE questions, you must state the domain and range using either inequality notation or interval notation.
函数是一种规则，它将每个输入恰好对应一个输出。定义域是所有可能输入值的集合，通常是 x 值；值域是所有可能输出值的集合，通常是 y 值。在 IGCSE 题目中，你必须使用不等式符号或区间符号写出定义域和值域。
| Term | Meaning / 中文含义 | Memory hook / 记忆提示 |
|---|---|---|
| Function | A rule with one output per input / 每个输入只有一个输出 | Think of a vending machine: one button gives one snack. |
| Domain | Set of possible x-values / 可能的 x 值集合 | Domain = x-axis direction. |
| Range | Set of possible y-values / 可能的 y 值集合 | Range = y-axis direction. |
| One-to-one | Each input and output is unique / 每个输入和输出都唯一 | 1 input → 1 output, no sharing. |
| Vertical line test | A graph is a function if any vertical line cuts it once / 垂直线检验 | Vertical line = checks one x at a time. |
2. Composite and Inverse Functions | 复合函数与反函数
A composite function fg(x) means f(g(x)); you apply g first and then f. The inverse function f⁻¹(x) reverses the effect of f. Do not confuse f⁻¹(x) with the reciprocal 1/f(x). The graph of an inverse function is a reflection in the line y = x.
复合函数 fg(x) 表示 f(g(x))，先代入 g，再代入 f。反函数 f⁻¹(x) 会逆转 f 的作用。不要将 f⁻¹(x) 与倒数 1/f(x) 混淆。反函数的图像是原函数关于直线 y = x 的对称图像。
| Term | Meaning / 中文含义 | Memory hook / 记忆提示 |
|---|---|---|
| Composite function | fg(x) = f(g(x)) / 复合函数 | Read fg from right to left. |
| Inverse function | f⁻¹(x) undoes f(x) / 反函数 | Inverse = opposite operation. |
| Domain of inverse | Equals the range of the original function / 反函数定义域等于原函数值域 | Swap domain and range. |
| y = x reflection | Inverse graph is reflected in y = x / 关于 y = x 对称 | Mirror line is the diagonal. |
3. Quadratic Functions and the Discriminant | 二次函数与判别式
A quadratic function has the form ax² + bx + c, where a ≠ 0. The discriminant is Δ = b² − 4ac. Its value tells you how many real roots the equation ax² + bx + c = 0 has: positive gives two distinct real roots, zero gives two equal real roots, and negative gives no real roots.
二次函数的形式为 ax² + bx + c，其中 a ≠ 0。判别式是 Δ = b² − 4ac。它的值可以告诉你方程 ax² + bx + c = 0 有多少个实数根：大于 0 有两个不相等的实根，等于 0 有两个相等的实根，小于 0 没有实根。
Δ = b² − 4ac
| Term | Meaning / 中文含义 | Memory hook / 记忆提示 |
|---|---|---|
| Discriminant | b² − 4ac / 判别式 | Discriminates between root types. |
| Vertex / turning point | Maximum or minimum point / 顶点或转折点 | The graph turns at the vertex. |
| Axis of symmetry | The line x = −b/(2a) / 对称轴 | Cuts the parabola into two mirror halves. |
| Completing the square | Rewriting ax² + bx + c as a(x + p)² + q / 配方法 | Creates a perfect square inside. |
| Roots / zeros | x-values where y = 0 / 根或零点 | Where the graph crosses the x-axis. |
4. Inequalities and Critical Values | 不等式与临界值
To solve a quadratic inequality, first rearrange it to one side and find the critical values where the expression equals zero. Then use a sign diagram or test intervals to decide which regions satisfy the inequality. Always remember that multiplying or dividing by a negative number reverses the inequality sign.
解二次不等式时，先将式子移到一边，并找出使表达式等于零的临界值。然后用符号表或区间测试来判断哪些区域满足不等式。始终记住：乘以或除以负数时，不等号方向要改变。
| Term | Meaning / 中文含义 | Memory hook / 记忆提示 |
|---|---|---|
| Critical value | x-value making the expression zero / 使表达式为零的 x 值 | Critical = boundary between sign regions. |
| Sign diagram | Number line showing positive and negative intervals / 符号图 | Mark + and − on each interval. |
| Strict inequality | Less than < or greater than > / 严格不等式 | Open circle on a number line. |
| Inclusive inequality | ≤ or ≥ / 包含不等式 | Closed circle on a number line. |
5. Indices and Surds | 指数与根式
Index laws control how powers are multiplied, divided, and raised. A surd is an irrational root such as √2 or 3√5. When a surd appears in the denominator, you are usually expected to rationalise the denominator by multiplying both numerator and denominator by the conjugate.
指数法则控制幂的乘法、除法和乘方运算。根式是无理数根，例如 √2 或 3√5。当分母中出现根式时，通常期望通过将分子和分母同时乘以共轭式来对分母有理化。
aᵐ × aⁿ = aᵐ⁺ⁿ | (aᵐ)ⁿ = aᵐⁿ | a⁻ⁿ = 1/aⁿ
| Term | Meaning / 中文含义 | Memory hook / 记忆提示 |
|---|---|---|
| Base | The number being raised to a power / 底数 | Base is the bottom number. |
| Exponent / index | The power to which the base is raised / 指数 | Exponent = how many times it multiplies. |
| Surd | An irrational root like √2 / 根式 | Surd sounds like absurd — cannot be a simple fraction. |
| Rationalising | Removing a surd from the denominator / 分母有理化 | Make the denominator rational. |
| Conjugate | a + b√c and a − b√c / 共轭式 | Same numbers, opposite middle sign. |
6. Logarithmic and Exponential Functions | 对数函数与指数函数
A logarithm answers the question: to what power must the base be raised to get a given number? The equation logₐ y = x is equivalent to aˣ = y. The natural logarithm ln x has base e. Use the log laws to combine or split logarithmic expressions.
对数回答的问题是：底数需要乘以几次才能得到给定数字？方程 logₐ y = x 等价于 aˣ = y。自然对数 ln x 的底数是 e。使用对数法则可以合并或拆分对数表达式。
logₐ(MN) = logₐM + logₐN | logₐ(M/N) = logₐM − logₐN
| Term | Meaning / 中文含义 | Memory hook / 记忆提示 |
|---|---|---|
| Logarithm | Power to which the base is raised / 对数 | Log = power in disguise. |
| Natural logarithm | Log with base e, written ln / 自然对数 | ln is natural because e appears in growth. |
| Change of base | logₐ b = log b / log a / 换底公式 | Divide logs of the same new base. |
| Exponential growth | Quantity increasing by a fixed percentage / 指数增长 | Growth that speeds up like a chain reaction. |
7. Polynomials, Remainder and Factor Theorems | 多项式、余数定理与因式定理
A polynomial is an expression made of powers of x with coefficients. The remainder theorem says that when a polynomial f(x) is divided by (x − a), the remainder is f(a). The factor theorem follows directly: if f(a) = 0, then (x − a) is a factor of f(x). These theorems are used to factorise cubic and higher-degree polynomials.
多项式是由 x 的幂和系数组成的表达式。余数定理指出，当多项式 f(x) 被 (x − a) 除时，余数是 f(a)。因式定理直接推出：如果 f(a) = 0，那么 (x − a) 是 f(x) 的因式。这些定理用于因式分解三次及更高次多项式。
| Term | Meaning / 中文含义 | Memory hook / 记忆提示 |
|---|---|---|
| Degree | Highest power of x / 最高次数 | Degree = largest exponent. |
| Remainder theorem | Remainder after division by (x − a) is f(a) / 余数定理 | Substitute a to find the remainder. |
| Factor theorem | f(a) = 0 means (x − a) is a factor / 因式定理 | Zero remainder creates a factor. |
| Long division | Method for dividing polynomials / 长除法 | Divide, multiply, subtract, bring down. |
8. Circular Measure and Radians | 弧度制与圆度量
Circular measure uses radians instead of degrees. One radian is the angle subtended at the centre of a circle by an arc length equal to the radius. The conversion is π radians = 180°. Arc length and sector area formulas become very clean when the angle θ is measured in radians.
圆度量使用弧度代替角度。一弧度是指圆心角所对的弧长等于半径时的角度。换算关系是 π 弧度 = 180°。当角度 θ 用弧度表示时，弧长和扇形面积公式变得非常简洁。
s = rθ | A = ½ r²θ
| Term | Meaning / 中文含义 | Memory hook /
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