📚 IGCSE CCEA Further Mathematics: Key Terminology Memory Guide | IGCSE CCEA 进阶数学：词汇术语速记指南
In CCEA IGCSE Further Mathematics, precise terminology is not just useful for communication - it is frequently tested directly in wording and definitions. Building a quick-recognition vocabulary allows you to interpret questions faster and avoid losing marks through avoidable confusion.
在 CCEA IGCSE 进阶数学中，准确术语不仅有助于表达，还经常直接出现在题干与定义考查中。建立快速识别的词汇库，可以让你更快读懂题目，避免因概念混淆而失分。
This guide expands the core vocabulary of the course into twelve short, memorisable blocks: algebra, factor and remainder theorems, quadratics, functions, graphs, sequences, calculus, matrices and vectors, plus a final block of memory techniques and examiner traps. Every block pairs the English term with its Chinese equivalent and a one-line definition you can recall under pressure.
本指南把课程核心词汇拆成十二个短小、可背诵的模块：代数、因式与余式定理、二次式、函数、图像、数列、微积分、矩阵与向量，最后加上记忆技巧与考场陷阱。每个模块都把英文术语与中文对应词配对，并附一句可在压力下回忆出的定义。
1. Why Terminology Is Marked Directly | 为什么术语会直接得分
CCEA examiners write questions around command words. If a question says "factorise fully", an answer that only takes out a common factor but leaves a bracket that still factorises will lose the final mark. If it says "hence", you are expected to use the previous part, not to start again from scratch. Reading the command word correctly is therefore a mark-scoring skill, not a vocabulary exercise.
CCEA 命题人会围绕"指令词"出题。若题目说 factorise fully（完全因式分解），而答案只提取了公因式、括号内仍可继续分解，就会丢掉最后一分。若题目说 hence（由此），就要求使用上一问的结果，而不是重新开始。因此，正确读取指令词本身就是得分技能，而不只是背单词。
Learn the difference between these high-frequency instructions:
请区分以下高频指令：
- State / Write down - no working required, one line is enough. 直接写出，无需过程。
- Show that - you must reach the given result and display each algebraic step. 必须推导到给定结果，并展示每一步。
- Prove - a logical chain from known facts to the conclusion, with reasons. 用已知事实与理由构成完整逻辑链。
- Hence / Hence or otherwise - the previous part must be used at least once. 至少使用一次上一问的结果。
- Verify - substitute the given value and confirm the statement holds. 代入给定值并验证结论成立。
- Sketch - shape, intercepts and key features matter, but the scale does not. 图形形状、截距与关键特征要准，比例尺不要求。
- Deduce - reach a conclusion using the result just obtained, with minimal extra work. 用刚得到的结果推出结论，不需大量新运算。
Memory aid: underline the command word before you start writing. It costs three seconds and protects several marks.
记忆技巧：动笔前先划出指令词。花三秒，保住好几分。
2. Algebraic Expressions: Terms, Coefficients, Like Terms | 代数表达式：项、系数、同类项
In algebra, a term is a single number, a single variable, or a product of both, such as 7, x, or −4x²y. The coefficient is the numerical factor multiplying the variable part, so in −4x²y the coefficient is −4 and the variable part is x²y. A constant term contains no variable at all.
在代数中，项是单个数字、单个变量，或两者的乘积，例如 7、x 或 −4x²y。系数是乘在变量部分前面的数字因子，所以 −4x²y 的系数是 −4，变量部分是 x²y。常数项不含任何变量。
Like terms can be combined because they have identical variable parts: 3x² and −8x² are like terms, but 3x² and 3x³ are not. The degree of a term is the sum of the indices of its variables; the degree of a polynomial is the highest degree among its terms.
同类项因为变量部分完全相同，可以合并：3x² 与 −8x² 是同类项，但 3x² 与 3x³ 不是。项的次数是各变量指数之和；多项式的次数是其各项中最高的次数。
It is also worth keeping four similar-looking words apart:
还要区分四个长得相近的词：
| Term | A statement true for some values only, e.g. x + 3 = 7. 只对部分值成立的等式。 |
| Identity | True for every value, written with ≡, e.g. (x + 1)² ≡ x² + 2x + 1. 对所有值都成立，用 ≡ 表示。 |
| Formula | A rule linking quantities, e.g. V = πr²h. 连接多个量的规则。 |
| Expression | No equals sign at all, e.g. 5x² − 2x + 1. 不含等号。 |
Memory aid: think of a coefficient as the number 'riding in front' of a variable; like terms wear the same variable 'outfit'.
记忆技巧：把系数看作变量前面 '骑乘' 的数字；同类项穿同样的变量 '外衣'。
3. Expanding and Factorising | 展开与因式分解
Expanding means removing brackets by multiplication, while factorising means re-inserting brackets by taking out common factors or applying standard identities. The two processes are inverses of each other, so a quick check is always to expand your factorised answer back again.
展开指通过乘法去掉括号，因式分解指通过提取公因式或使用标准公式重新加上括号。两者互为逆过程，所以最好的检查方法就是把分解结果重新展开。
Key instructions include expand (remove brackets by multiplying out) and factorise (re-insert brackets by taking out common factors or using identities). Four identities cover almost every question in this section:
关键指令包括展开（通过乘法去括号）和因式分解（通过提取公因式或使用公式重新添加括号）。以下四个恒等式几乎覆盖本节所有题目：
(a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b²
a² − b² = (a + b)(a − b) a³ ± b³ = (a ± b)(a² ∓ ab + b²)
The phrase difference of two squares refers to a² − b², and a perfect square is the result of squaring a bracket. When a quadratic trinomial such as x² + 7x + 12 is factorised, you look for two numbers with product 12 and sum 7, giving (x + 3)(x + 4).
difference of two squares（平方差）指 a² − b²，perfect square（完全平方）指括号平方的结果。分解 x² + 7x + 12 这样的二次三项式时，要找乘积为 12、和为 7 的两个数，得到 (x + 3)(x + 4)。
Common exam trap: 'factorise fully' requires you to repeat the process until no bracket factorises further. For example 2x² − 8 becomes 2(x² − 4) and then 2(x + 2)(x − 2).
常见陷阱：factorise fully 要求反复分解，直到任何括号都无法再分解。例如 2x² − 8 先写成 2(x² − 4)，再写成 2(x + 2)(x − 2)。
4. Factor Theorem and Remainder Theorem | 因式定理与余式定理
e relative: when f(x) is divided by (x − a), the remainder is f(a). Notice that the factor theorem is simply the special case where that remainder equals zero.
因式定理指出，若 f(a) = 0，则 (x − a) 是多项式 f(x) 的因式。余式定理是它的近亲：当 f(x) 除以 (x − a) 时，余数为 f(a)。注意因式定理只是余数为零的特殊情形。
(x + a) means substituting x = −a; and dividing by (ax − b) means substituting x = b ÷ a.
符号约定是最容易失分的地方。除以 (x − a) 就是代入 x = a；除以 (x + a) 就是代入 x = −a；除以 (ax − b) 就是代入 x = b ÷ a。
Related vocabulary you should recognise instantly:
需要一眼认出的相关术语：
- Dividend - the polynomial being divided. 被除式。
- Divisor - what you divide by, such as (x − 2). 除式。
- Quotient - the polynomial result of the division. 商式。
- Remainder - what is left over, of lower degree than the divisor. 余式，次数低于除式。
- Long division and synthetic division - the two written methods for dividing polynomials. 多项式长除法与综合除法。
Memory aid: 'factor means zero remainder'. Say it to yourself each time you see the word factor next to a polynomial.
记忆技巧：'factor 意味着余数为零'。每次在多项式旁边看到 factor 一词，就默念一遍。
5. Quadratic Discriminant and Nature of Roots | 判别式与根的性质
The discriminant of ax² + bx + c is b² − 4ac. It tells you the nature of roots: positive gives two real roots, zero gives one repeated root, negative gives no real roots.
二次式 ax² + bx + c 的判别式是 b² − 4ac。它判断根的性质：正数有两个实根，零有一个重根，负数没有实根。
Examiners often add the word distinct, which means 'different from each other', so a repeated root is real but not distinct. When a question says 'no real roots', it is asking you to set the discriminant less than zero and solve the resulting inequality.
命题人常加 distinct（相异）一词，意为"彼此不同"，所以重根是实根但不是相异实根。题目说 no real roots 时，就要求你令判别式小于零，再解所得不等式。
| b² − 4ac > 0 | Two distinct real roots; the curve crosses the x-axis twice. 两个相异实根，曲线两次穿过 x 轴。 |
| b² − 4ac = 0 | One repeated root; the curve touches the x-axis at its vertex. 一个重根，曲线在顶点处与 x 轴相切。 |
| b² − 4ac < 0 | No real roots; the curve never meets the x-axis. 无实根，曲线与 x 轴不相交。 |
Neighbouring vocabulary you must keep separate: roots or zeros are the x-values where the curve meets the x-axis; solutions are the values satisfying the equation; and the vertex, turning point or stationary point is the maximum or minimum of the curve, found by completing the square.
必须区分的邻近术语：roots 或 zeros 是曲线与 x 轴交点的横坐标；solutions 是满足方程的值；vertex、turning point 或 stationary point 是曲线的最高或最低点，通过 completing the square（配方）求得。
6. Functions: Domain, Range and Notation | 函数：定义域、值域与记号
A function maps each input from the domain to exactly one output in the range. The notation f(x) means the output of function f for input x. The input is also called the argument, and the output is sometimes called the image of the input.
函数将定义域中的每个输入对应到值域中的唯一输出。记号 f(x) 表示函数 f 对输入 x 的输出。输入也叫自变量（argument），输出有时称为输入的像（image）。
The word 'exactly one' is the whole definition: a mapping that sends one input to two different outputs is not a function but a relation. The vertical line test checks this on a graph - if any vertical line meets the curve more than once, it is not a function.
"唯一"二字就是定义的全部：一个输入对应两个不同输出的映射不是函数，只是关系。图像上可用竖线检验法（vertical line test）判断 - 若任一竖线与曲线相交多于一次，就不是函数。
Recognise these mapping types:
请识别以下映射类型：
- One-to-one - each input gives a different output, so the inverse is also a function. 一对一：每个输入对应不同输出，反函数也是函数。
- Many-to-one - several inputs share one output, such as y = x². 多对一：多个输入共享同一输出，如 y = x²。
- One-to-many - not a function at all. 一对多：不是函数。
- Domain and range - the set of allowed inputs and the set of actual outputs. 定义域与值域：允许输入的集合与实际输出的集合。
- Modulus function - gives the size of a number, written |x|. 绝对值函数，写作 |x|。
Memory aid: domain starts with the same letter as 'departure' inputs; range is what you actually receive.
记忆技巧：domain 与"输入出发地"对应；range 是你实际收到的输出结果。
7. Composite and Inverse Functions | 复合函数与反函数
Composite functions apply one function after another: fg(x) means f(g(x)). The function nearest the variable acts first, so fg(x) and gf(x) are generally not the same, and order matters in every calculation.
复合函数是指连续应用两个函数：fg(x) 即 f(g(x))。最靠近变量的函数先作用，因此 fg(x) 与 gf(x) 通常不相等，运算顺序至关重要。
one, which is why an exam question may restrict the domain to x ≥ 0 before asking for f⁻¹.
反函数 f⁻¹(x) 将 f(x) 的效果逆转：交换 x 与 y，然后解出 y。只有当原函数是一对一时才存在反函数，这就是考题有时先限定定义域 x ≥ 0 再要求求 f⁻¹ 的原因。
Three checks that earn method marks:
三个能拿到方法分的检查：
- Check the domain and range of f⁻¹ - they are the range and domain of f swapped. 检查 f⁻¹ 的定义域与值域：它们正是 f 的值域与定义域互换。
- Check the two-way test: ff⁻¹(x) = x and f⁻¹f(x) = x. 双向检验：ff⁻¹(x) = x 且 f⁻¹f(x) = x。
- Check whether the function is self-inverse, meaning f⁻¹(x) = f(x). 检查是否为自反函数，即 f⁻¹(x) = f(x)。
Do not confuse f⁻¹(x) with 1 ÷ f(x). The first is the inverse function, the second is the reciprocal, and mixing them up is a classic error.
不要把 f⁻¹(x) 与 1 ÷ f(x) 混淆。前者是反函数，后者是倒数，二者互混是经典错误。
8. Graphs: Intercepts, Asymptotes and Transformations | 图像：截距、渐近线与变换
mptote is a line the curve approaches but never reaches, such as the x-axis for y = 1 ÷ x.
图像术语直接告诉你该画什么。intercept（截距）是曲线与坐标轴的交点；root 或 zero 是与 x 轴的交点；asymptote（渐近线）是曲线无限接近但永不相交的直线，例如 y = 1 ÷ x 的 x 轴。
A turning point, also called a stationary point, is where the gradient is zero. It may be a maximum, a minimum or a point of inflection, and the first derivative equals zero at each of them.
turning point 又称 stationary point，是梯度为零之处，可能是极大值、极小值或拐点，在这些点上一阶导数都等于零。
Transformation vocabulary is examined with function notation, so learn the five standard forms as a family:
变换术语以函数记号形式考查，请把以下五种标准形式作为一族来记：
| y = f(x) + a | Translation of a units upwards (parallel to the y-axis). 向上平移 a 个单位。 |
| y = f(x + a) | Translation of a units left (opposite to what you expect). 向左平移 a 个单位，方向与直觉相反。 |
| y = af(x) | Vertical stretch with scale factor a. 纵向拉伸，比例因子 a。 |
| y = f(ax) | Horizontal stretch with scale factor 1 ÷ a. 横向拉伸，比例因子 1 ÷ a。 |
| y = −f(x) and y = f(−x) | Reflections in the x-axis and the y-axis respectively. 分别为关于 x 轴与 y 轴的反射。 |
Memory aid: changes inside the bracket affect x and work in reverse; changes outside the bracket affect y and work as written.
记忆技巧：括号内的变化影响 x，且方向相反；括号外的变化影响 y，且方向一致。
9. Sequences, Series and Limits | 数列、级数与极限
uch as uₙ₊₁ = 2uₙ + 3, and the nth term or general term is the formula giving uₙ directly.
sequence（数列）是有序的一列项，series（级数）是这些项的和。recurrence relation（递推关系）用前一项定义后一项，如 uₙ₊₁ = 2uₙ + 3；nth term 或 general term（通项）则是直接给出 uₙ 的公式。
Two families dominate the syllabus, and their vocabulary must never be mixed:
课程中主要涉及两族数列，其术语绝不可混用：
- Arithmetic sequence - constant common difference d between consecutive terms. 等差数列：相邻项之差为常数 d。
- Geometric sequence - constant common ratio r obtained by multiplying. 等比数列：相邻项之比为常数 r。
- Convergent - a sequence or series with a finite limit as n → ∞. 收敛：当 n → ∞ 时数列或级数有有限极限。
- Divergent - it does not settle on a finite value. 发散：不趋于有限值。
- Σ notation - compact way of writing a sum, with lower and upper limits. 求和记号，带上下限。
a + (n − 1)d uₙ = arⁿ⁻¹ Sₙ = (n ÷ 2)(2a + (n − 1)d)
A geometric series converges only when |r| < 1, and its sum to infinity is a ÷ (1 − r). Learn this condition with the formula, because questions often ask you to justify convergence before calculating the sum.
等比级数仅当 |r| < 1 时收敛，其无穷和为 a ÷ (1 − r)。要把这个条件与公式一起记，因为题目常要求先说明收敛理由再求无穷和。
10. Calculus Vocabulary: Differentiation and Integration | 微积分术语：微分与积分
Differentiation finds the gradient or rate of change of a function, and the result is called the derivative, written dy/dx or f'(x). Integration is the reverse process; an indefinite integral includes a constant of integration, while a definite integral is evaluated between two limits.
differentiation（微分）求函数的梯度或变化率，其结果称为 derivative（导数），记作 dy/dx 或 f'(x)。integration（积分）是逆过程；不定积分含积分常数，定积分则在两个限之间求值。
The gradient at a point is also called the instantaneous rate of change, and the second derivative, written d²y/dx² or f''(x), determines whether a stationary point is a maximum or a minimum.
某点处的梯度也叫瞬时变化率；二阶导数记作 d²y/dx² 或 f''(x)，用来判定驻点是极大值还是极小值。
Keep these paired opposites in one flashcard set:
把以下成对的反义词放进同一组卡片：
- Differentiate ↔ Integrate; derivative ↔ integral. 求导 ↔ 积分；导数 ↔ 积分。
- Increasing ↔ Decreasing, tested with the sign of f'(x). 递增 ↔ 递减，用 f'(x) 的符号判断。
- Maximum ↔ Minimum, tested with the sign of f''(x). 极大 ↔ 极小，用 f''(x) 的符号判断。
- Displacement → Velocity → Acceleration, linked by differentiation. 位移 → 速度 → 加速度，通过求导相连。
- Area under a curve - found by a definite integral. 曲线下面积，用定积分求得。
dx (axⁿ) = anxⁿ⁻¹ ∫ axⁿ dx = axⁿ⁺¹ ÷ (n + 1) + c (n ≠ −1)
Exam trap: never forget the constant of integration in an indefinite integral, and never forget to subtract the value at the lower limit in a definite integral.
考场陷阱：不定积分不要漏掉积分常数；定积分不要忘记减去下限处的值。
11. Matrices and Vectors | 矩阵与向量
3 matrix has two rows and three columns. Each number inside is called an element or entry.
matrix（矩阵）是数字组成的矩形阵列，其 order 或 dimension（阶）写作行数 × 列数，因此 2 × 3 矩阵有两行三列。里面的每个数称为 element 或 entry（元素）。
ngle number that tells you whether the matrix has an inverse. If the determinant is zero, the matrix is singular and no inverse exists; otherwise it is non-singular and invertible.
2 × 2 矩阵的 determinant（行列式）是一个数，用来判断该矩阵是否可逆。若行列式为零，矩阵为 singular（奇异），不存在逆矩阵；否则为非奇异、可逆。
det = ad − bc A⁻¹ = (1 ÷ (ad − bc)) × [d −b; −c a]
Core vocabulary list:
核心术语清单：
- Identity matrix I - the matrix equivalent of the number 1. 单位矩阵 I，相当于数字 1。
- Transpose - rows and columns swapped. 转置：行列互换。
- Singular - determinant is zero. 奇异的：行列式为零。
- Column
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