📚 Year 12 CAIE Mathematics: Glossary Memorisation Guide | Year 12 CAIE 数学：词汇术语速记指南
Mastering the terminology of Year 12 CAIE Mathematics is the first step towards excelling in Pure Mathematics and problem-solving. This guide breaks down key concepts from functions to vectors, pairing definitions with memorable explanations to help you internalise the language of the syllabus quickly and effectively.
掌握 Year 12 CAIE 数学的术语是学好纯数学和解决问题能力的第一步。本指南将函数、向量等核心概念拆解开来，用有助于记忆的讲解搭配定义，帮助你迅速而有效地内化课程纲要中的数学语言。
1. Functions and Transformations | 函数与变换
A function is a rule that assigns each element from the domain exactly one element in the codomain. The domain is the set of all possible input values, while the range is the set of all actual output values.
函数是一种规则，它将定义域中的每个元素恰好分配为陪域中的一个元素。定义域是所有可能输入值的集合，而值域是所有实际输出值的集合。
A one-to-one function maps distinct inputs to distinct outputs, passing the horizontal line test. An inverse function ¹(x), reverses the mapping, and its graph is the reflection of y = f(x) in the line y = x.
单射函数将不同的输入映射为不同的输出，通过水平线测试。反函数记作 f⁻¹(x)，逆转了原来的映射，其图像是 y = f(x) 关于直线 y = x 的反射。
Transformations such as translations, stretches and reflections stretches vertically by factor a. Composite functions, fg(x) or f(g(x)), apply one function after another.
诸如平移、伸缩和反射等变换会改变函数图像。请记住：y = f(x) + a 表示垂直移动；y = f(x + a) 表示水平移动；y = a f(x) 表示垂直伸缩，倍数为 a。复合函数 fg(x) 或 f(g(x)) 指依次应用两个函数。
| English Term | 中文术语 | Quick Hint |
|---|---|---|
| Domain | 定义域 | Input allowed (x) |
| Range | 值域 | Output obtained (y) |
| Composite function | 复合函数 | Chain: f(g(x)) |
2. Coordinate Geometry | 坐标几何
The gradient of a straight line joining two points (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁). The equation of a straight line as y = mx + c, where m is the gradient and c the y-intercept, or using y − y₁ = m(x − x₁).
连接两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 (y₂ − y₁)/(x₂ − x₁)。直线方程可以写成 y = mx + c，其中 m 是斜率，c 是 y 轴截距，或者使用点斜式 y − y₁ = m(x − x₁)。
) and radius r. Completing the square is used to convert the general form to standard. The discriminant of a quadratic b² − 4ac tells the nature of intersection between a line and a curve.
对于圆，标准式 (x − a)² + (y − b)² = r² 指定圆心 (a, b) 和半径 r。通过配方法可将一般式转化为标准式。二次式的判别式 b² − 4ac 可以判断直线与曲线交点的性质。
| English Term | 中文术语 | Memory Aid |
|---|---|---|
| Gradient | 斜率 | Rise over run |
| Perpendicular lines | 垂直直线 | Product of gradients = −1 |
| Midpoint | 中点 | Average coordinates |
3. Sequences and Series | 序列与级数
An arithmetic progression (AP) has a constant common difference d. The nᵗʰ a + (n − 1)d, and the sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d] or n/2 (a + l).
等差数列 (AP) 具有恒定的公差 d。第 n 项为 aₙ = a + (n − 1)d，前 n 项和为 Sₙ = n/2 [2a + (n − 1)d] 或 n/2 (a + l)。
A geometric progression (GP) has a constant common ratio r. The nᵗʰ term is a rⁿ⁻¹, and the sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. If |r| < 1, the sum to infinity exists: S∞ = a/(1 − r).
等比数列 (GP) 具有恒定的公比 r。第 n 项为 a rⁿ⁻¹，前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r) (r ≠ 1)。如果 |r| < 1，则存在无穷项之和：S∞ = a/(1 − r)。
Sigma notation Σ is used to represent series compactly. The upper and lower limits indicate the first and last index. Recognising whether a sequence converges or diverges depends on the behaviour of rⁿ for large n.
西格玛符号 Σ 用于简洁地表示级数。上下限标明了索引起止。判断序列收敛或发散取决于 rⁿ 在 n 很大时的行为。
4. Differentiation | 微分
The derivative f'(x) or dy/dx measures the instantaneous rate of change of a function, represented by the gradient of the tangent. The limit definition is limh→0 [f(x + h) − f(x)]/h.
导数 f'(x) 或 dy/dx 衡量函数的瞬时变化率，由切线的斜率表示。其极限定义为 limh→0 [f(x + h) − f(x)]/h。
Basic rules: power rule d/dx (xⁿ) = n xⁿ⁻¹; constant multiple rule; sum rule. The chain rule differentiates composite functions: dy/dx = (dy/du)(du/dx). The product rule and quotient rule handle products and quotients.
基本法则：幂法则 d/dx (xⁿ) = n xⁿ⁻¹；常数倍法则；和法则。链式法则用于复合函数求导：dy/dx = (dy/du)(du/dx)。积法则和商法则分别处理乘积和商。
The second derivative f''(x) gives the rate of change of the first derivative; it determines concavity and helps classify stationary points as maxima, minima or points of inflection.
二阶导数 f''(x) 表示一阶导数的变化率；它决定凹凸性并帮助将驻点分类为极大值、极小值或拐点。
| Operation | Expression | Key Phrase |
|---|---|---|
| Chain rule | d/dx [f(g(x))] = f'(g(x)) g'(x) | Differentiate outer, then inner |
| Product rule | d/dx (uv) = u'v + uv' | First d second plus second d first |
5. Integration | 积分
Integration is the reverse process of differentiation. The indefinite integral ∫ f(x) dx yields a family of functions plus a constant of integration C. The definite integral ∫ₐᵇ f(x) dx computes the signed area under the curve between limits a and b.
积分是微分的逆过程。不定积分 ∫ f(x) dx 给出一个函数族加上积分常数 C。定积分 ∫ₐᵇ f(x) dx 计算在上下限 a 和 b 之间曲线下的有向面积。
dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ (ax + b)ⁿ dx can be integrated by reverse chain rule. The trapezium rule approximates definite integrals using areas of trapezoids, with formula (h/2)[y₀ + 2(y₁ + ... + yₙ₋₁) + yₙ].
标准模式：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)，通过反向链式法则可积分形如 ∫ (ax + b)ⁿ dx 的式子。梯形法使用梯形的面积近似定积分，公式为 (h/2)[y₀ + 2(y₁ + ... + yₙ₋₁) + yₙ]。
Area between two curves y = f(x) and y = g(x) from a to b is ∫ₐᵇ |f(x) − g(x)| dx. Always check for intersections and split the interval accordingly.
两条曲线 y = f(x) 和 y = g(x) 在 a 到 b 之间的面积为 ∫ₐᵇ |f(x) − g(x)| dx。务必检查交点并相应分段积分。
6. Trigonometry | 三角学
The three primary trigonometric ratios are sine (sin), cosine (cos) and tangent ngle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
三个基本三角比是正弦 (sin)、余弦 (cos) 和正切 (tan)。在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
Exact values for 30°, 45°, 60° can be derived from special triangles. Key identities: sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ. The CAST diagram indicates the sign of each ratio in quadrants I to IV (All, Sin, Tan, Cos positive).
30°、45°、60° 的精确值可以从特殊三角形推导出来。核心恒等式：sin²θ + cos²θ ≡ 1，tan θ ≡ sin θ / cos θ。CAST 图标明了每个比在 I 到 IV 象限的正负号（所有、正弦、正切、余弦为正）。
Radians radians. Trigonometric graphs of sin, cos, tan show periodicity, amplitude and asymptotes.
弧度是角的另一种度量：π 弧度 = 180°。当 θ 以弧度为单位时，弧长 s = rθ，扇形面积 A = ½ r²θ。正弦、余弦、正切的三角函数图像展示周期性、振幅和渐近线。
| Angle (degrees) | Angle (radians) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 30° | π/6 | ½ | √3/2 | 1/√3 |
| 45° | π/4 | 1/√2 | 1/√2 | 1 |
| 60° | π/3 | √3/2 | ½ | √3 |
7. Vectors | 向量
A vector has magnitude and direction, often written as a column vector or in component form ai + bj. The magnitude (length) of vector v = ai + bj is |v| = √a² + b².
向量具有大小和方向，常表示为列向量或分量形式 ai + bj。向量 v = ai + bj 的模（长度）为 |v| = √a² + b²。
A unit vector has magnitude 1; the unit vector in the direction of v is v/|v|. Position vectors locate a point relative to an origin, while direction vectors describe a line's orientation.
单位向量的模为1；沿 v 方向的单位向量为 v/|v|。位置向量确定点相对于原点的位置，而方向向量描述直线的方向。
The dot product (scalar product) of two vectors a and b is a·b = |a||b| cos θ, used to find angles between vectors and test perpendicularity. Vector equation of a line: r = a + td, where a is a point on the line and d the direction vector.
两个向量 a 和 b 的点积（标量积）为 a·b = |a||b| cos θ，用于求向量间夹角并检验垂直性。直线的向量方程：r = a + td，其中 a 是直线上一点，d 是方向向量。
8. Exponentials and Logarithms | 指数与对数
The exponential function eˣ is its own derivative, and its inverse is the natural logarithm ln x. The graph y = eˣ passes through (0,1), while y = ln x passes through (1,0).
指数函数 eˣ 的导数等于其自身，其反函数是自然对数 ln x。y = eˣ 的图像经过点 (0,1)，而 y = ln x 经过点 (1,0)。
ᵃ)ᵇ = eᵃᵇ; ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln aᵇ = b ln a. Relationship: ln(eˣ) = x and eˡⁿ ˣ = x.
核心法则：eᵃ eᵇ = eᵃ⁺ᵇ，(eᵃ)ᵇ = eᵃᵇ；ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln aᵇ = b ln a。关联关系：ln(eˣ) = x 且 eˡⁿ ˣ = x。
When solving exponential equations, taking logarithms of both sides is standard, e.g. aˣ = b → x = logₐ b. Logarithmic scales help linearise exponential growth, often seen in modelling.
解指数方程时，通常对两边取对数，例如 aˣ = b → x = logₐ b。对数尺度有助于将指数增长线性化，常见于建模问题中。
9. Polynomials and Algebraic Fractions | 多项式与代数分式
A polynomial is an expression of the form aₙ xⁿ + ... + a₁ x + a₀. The factor theorem states that (x − p) is a factor of p(x) if p(p) = 0. The remainder theorem: when p(x) is divided by (x − a), the remainder is p(a).
多项式是形如 aₙ xⁿ + ... + a₁ x + a₀ 的表达式。因式定理指出，若 p(p) = 0，则 (x − p) 是 p(x) 的因式。余式定理：当 p(x) 除以 (x − a) 时，余式为 p(a)。
Simplifying algebraic fractions involves factoring numerators and denominators, cancelling common factors, and applying rules for addition, multiplication and division. Partial fractions decompose a complex rational expression into a sum of simpler fractions, crucial for integration.
化简代数分式需要将分子和分母分解因式，约去公因式，并应用加法、乘法和除法的运算法则。部分分式将一个复杂的有理式分解为若干个较简单分式之和，这对积分至关重要。
For improper fractions where the degree of the numerator is at least that of the denominator, perform polynomial long division first to obtain a quotient plus a proper fraction.
对于假分式（分子的次数不低于分母的次数），首先进行多项式长除法，得到一个商式加上一个真分式。
10. Proof and Notation | 证明与符号
Mathematical proof involves logical steps to establish truth. Direct proof starts from known facts and deduces the conclusion. Proof by exhaustion checks all possible cases. Disproof by counterexample provides a single case where the statement fails.
数学证明包含逻辑步骤以确立真理性。直接证明从已知事实出发推导出结论。穷举证明检查所有可能情形。反例证伪提供一个使陈述不成立的个例。
Set notation is common: ℕ natural numbers, ℤ integers, ℚ rationals, ℝ reals. ∈ means 'is an element of', ⊂ denotes subset. The implies sign ⇒ and if and only if ⇔ are essential in logical statements. '∀' means 'for all', '∃' means 'there exists'.
集合符号很常见：ℕ 自然数，ℤ 整数，ℚ 有理数，ℝ 实数。∈ 表示 '属于'，⊂ 表示子集。蕴涵符号 ⇒ 和当且仅当 ⇔ 在逻辑陈述中至关重要。'∀' 意为 '对所有'，'∃' 意为 '存在'。
In Year 12, you'll often use these symbols to state domain restrictions, solve inequalities, and analyse functions. Memorising them early saves confusion in exams.
在 Year 12 课程中，你常会使用这些符号来表述定义域限制、解不等式以及分析函数。尽早记住它们能避免考试时的混淆。
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