📚 Year 12 AQA Maths: Vocabulary & Terminology Quick Memorisation Guide | AQA 12年级数学词汇术语速记指南
Building a strong command of mathematical language is essential for success in AQA Year 12 Mathematics. This guide breaks down the key vocabulary you will encounter in Pure, Statistics and Mechanics, helping you to recall definitions quickly and use precise terminology in exams.
在AQA 12年级数学课程中，扎实掌握数学语言是取得优异成绩的基础。本指南梳理了纯数、统计和力学中的核心词汇，帮助你快速回忆定义，并在考试中准确运用术语。
1. Algebraic Manipulation & Functions | 代数运算与函数
Expression: A combination of terms using operations such as addition and multiplication, but containing no equals sign. For example, 3x² + 2x − 5.
表达式: 由数字、字母和运算符号组成的式子，不含等号。例如 3x² + 2x − 5。
Equation: A statement that two expressions are equal, involving an equals sign. Solving an equation means finding the value(s) of the unknown that make it true.
方程: 表示两个表达式相等的式子，含有等号。解方程即为找出使等式成立的未知数的值。
Identity: An equality that holds for all values of the variable, often written with the ≡ symbol. For example, (x + 1)² ≡ x² + 2x + 1.
恒等式: 对于变量的所有取值都成立的等式，通常用符号 ≡ 表示。例如 (x + 1)² ≡ x² + 2x + 1。
Function: A relationship where each input (x) gives exactly one output (y), typically written as f(x). The domain is the set of possible inputs; the range is the set of possible outputs.
函数: 每个输入值 x 恰好对应一个输出值 y 的关系，通常记作 f(x)。定义域是所有可能输入值的集合，值域是所有可能输出值的集合。
Composite Function: Applying one function to the result of another, denoted by f(g(x)) or f∘g(x). Work from the inside out when evaluating.
复合函数: 将一个函数的结果作为另一个函数的输入，记作 f(g(x)) 或 f∘g(x)，计算时从内层函数开始。
2. Coordinate Geometry & Graphs | 坐标几何与图像
Gradient ivided by change in x. For points (x₁, y₁) and (x₂, y₂), gradient m = (y₂ − y₁)/(x₂ − x₁).
斜率（梯度）: 表示直线的倾斜程度，等于纵坐标的变化量除以横坐标的变化量。对于点 (x₁, y₁) 和 (x₂, y₂)，斜率 m = (y₂ − y₁)/(x₂ − x₁)。
Equation of a straight line ere m is the gradient and c is the y-intercept. An alternative form is y − y₁ = m(x − x₁).
直线方程: 通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。也可使用点斜式 y − y₁ = m(x − x₁)。
Parallel and perpendicular lines radient. If two lines are perpendicular, the product of their gradients is −1 (m₁m₂ = −1).
平行线与垂直线: 平行线具有相同的斜率。若两条直线互相垂直，则它们斜率的乘积为 −1（m₁m₂ = −1）。
Discriminant 0 gives two distinct real roots, Δ = 0 gives one repeated root, Δ < 0 gives no real roots.
判别式: 对于二次方程 ax² + bx + c = 0，判别式 Δ = b² − 4ac。它揭示了根的情况：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
3. Differentiation | 微分
Derivative: The rate of change of a function, denoted by f'(x) or dy/dx. It gives the gradient of the tangent to a curve at any point.
导数: 函数的变化率，记作 f'(x) 或 dy/dx。它表示曲线上任意一点处切线的斜率。
Differentiation from first principles: The formal definition using limits:
f'(x) = lim (h → 0) [f(x + h) − f(x)] / h
从第一性原理求导: 使用极限的定义式：
f'(x) = lim (h → 0) [f(x + h) − f(x)] / h
Stationary point an be a local maximum, local minimum, or a point of inflection. Use the second derivative to classify.
驻点: 曲线上导数为零的点，可以是极大值点、极小值点或拐点。通常借助二阶导数来判断。
Increasing and decreasing functions: A function is increasing where f'(x) > 0 and decreasing where f'(x) < 0.
增函数与减函数: 在 f'(x) > 0 的区间上函数递增，在 f'(x) < 0 的区间上函数递减。
4. Integration | 积分
Indefinite integral mily of functions. Notation: ∫ f(x) dx = F(x) + C, where C is the constant of integration.
不定积分: 微分的逆运算，得到一族函数。记作 ∫ f(x) dx = F(x) + C，其中 C 为积分常数。
Definite integral x = a to x = b, written as ∫ₐᵇ f(x) dx. Use the fundamental theorem: F(b) − F(a), where F'(x) = f(x).
定积分: 计算曲线与 x 轴之间在 x = a 到 x = b 的有向面积，记作 ∫ₐᵇ f(x) dx。利用微积分基本定理计算：F(b) − F(a)，其中 F'(x) = f(x)。
Area under a curve: For regions below the x-axis, the definite integral gives a negative value; take the absolute value for the actual area. Be careful with curves that cross the axis.
曲线下的面积: 若区域在 x 轴下方，定积分为负值，实际面积应取绝对值。当曲线穿越 x 轴时需分段计算。
Trapezium rule using trapezoids. With n strips of width h, area ≈ ½h (y₀ + 2y₁ + 2y₂ + ... + 2yₙ₋₁ + yₙ).
梯形法则: 一种用梯形近似曲线下面积的数值方法。将区间 n 等分，每份宽 h，面积 ≈ ½h (y₀ + 2y₁ + 2y₂ + ... + 2yₙ₋₁ + yₙ)。
5. Trigonometry | 三角学
Radian 1 radian is the angle subtended by an arc equal in length to the radius. π radians = 180°.
弧度: 基于半径的角度单位：当弧长等于半径时，所对的圆心角为 1 弧度。π 弧度 = 180°。
Sine rule sin C. Use it to find missing sides or angles when you have two angles and any side, or two sides and a non-included angle.
正弦定理: 在任意三角形中，a/sin A = b/sin B = c/sin C。可用于已知两角一边或两边及一个非夹角时求解其他元素。
Cosine rule ² − 2bc cos A. Helpful when you know two sides and the included angle, or all three sides.
余弦定理: 在任意三角形中，a² = b² + c² − 2bc cos A。适用于已知两边及其夹角或三边求角的情形。
Trigonometric identities = 1. Also tanθ = sinθ / cosθ. These are used to simplify expressions and solve equations.
三角恒等式: 基本恒等式为 sin²θ + cos²θ = 1，以及 tanθ = sinθ / cosθ。常用于化简表达式和解方程。
6. Exponentials & Logarithms | 指数与对数
Exponential function: y = aˣ with a > 0 and a ≠ 1; the special case is y = eˣ, where e ≈ 2.71828. Its derivative is itself: d/dx (eˣ) = eˣ.
指数函数: 形如 y = aˣ (a > 0, a ≠ 1) 的函数；特例是 y = eˣ，其中 e ≈ 2.71828。它的导数就是它本身：d/dx (eˣ) = eˣ。
Natural logarithm: The inverse of the exponential function: if y = eˣ en x = ln y. Key properties: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln(aᵏ) = k ln a.
自然对数: 指数函数的反函数：若 y = eˣ，则 x = ln y。关键运算律：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln(aᵏ) = k ln a。
Logarithms in other bases ower to which a must be raised to get x. Change of base formula: logₐ x = logₓ x / logₓ a.
其他底的对数: logₐ x 表示以 a 为底 x 的对数。换底公式：logₐ x = logₓ x / logₓ a。
7. Sequences & Series | 数列与级数
Sequence rdered list of numbers defined by a rule, such as the nth term formula. E.g., uₙ = 3n − 1.
数列: 按照一定规律排列的一列数，常由通项公式定义，如 uₙ = 3n − 1。
Arithmetic progression he nth term is a + (n − 1)d, and the sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d].
等差数列: 相邻两项的差为常数 (d) 的数列。第 n 项为 a + (n − 1)d，前 n 项和 Sₙ = n/2 [2a + (n − 1)d]。
Series: The sum of terms of a sequence, written using sigma notation: Σ uₙ from n=1 to N.
级数: 数列各项的和，常用求和记号表示为：∑ₙ₌₁ᴺ uₙ。
Geometric progression t ratio r. The nth term is arⁿ⁻¹. Sum of first n terms: Sₙ = a(1 − rⁿ)/(1 − r), for r ≠ 1.
等比数列: 每一项乘以固定公比 r 得到后一项的数列。第 n 项为 arⁿ⁻¹。前 n 项和：Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。
8. Binomial Expansion | 二项式展开
Binomial expansion: A method to expand expressions of the form (a + b)ⁿ. For positive integer n, use Pascal's triangle or the formula involving nCr coefficients.
二项式展开: 将形如 (a + b)ⁿ 的式子展开的方法。当 n 为正整数时，可利用帕斯卡三角形或含 nCr 系数的公式。
nCr notation: The binomial coefficient ⁿCᵣ = n! / (r!(n − r)!). It gives the coefficient of the rth term in the expansion of (1 + x)ⁿ.
组合数 nCr: 二项式系数 ⁿCᵣ = n! / (r!(n − r)!)，它给出了 (1 + x)ⁿ 展开中第 r 项的系数。
Expanding (1 + x)ⁿ for any rational n: When n is not a positive integer, the expansion is infinite and valid for |x| < 1: (1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + ...
当 n 为任意有理数时 (1 + x)ⁿ 的展开: 若 n 不是正整数，展开为无穷级数且仅在 |x| < 1 时有效：(1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + ...
9. Statistics Essentials | 统计基础
Population vs sample: A population includes all members of a group; a sample is a subset used to draw conclusions. Random sampling helps avoid bias.
总体与样本: 总体是研究对象的全体，样本是为了推断结论而选取的一部分个体。随机抽样有助于避免偏差。
Measure of central tendency: Mean (sum of data ÷ count), median (middle value), mode (most frequent). Each gives a typical value but is affected differently by outliers.
集中趋势量数: 平均数（数据总和÷个数）、中位数（中间值）、众数（出现次数最多的值）。它们均表示典型值，但受异常值影响的程度不同。
Measures of spread (Q₃ − Q₁), and standard deviation (√ variance). They describe how spread out the data are.
离散程度的量数: 极差（最大值−最小值）、四分位距（Q₃ − Q₁）和标准差（√ 方差）。这些统计量描述了数据的分散程度。
Probability notation: P(A) for the probability of event A. P(A') = 1 − P(A). For independent events, P(A ∩ B) = P(A)P(B); for mutually exclusive events, P(A ∪ B) = P(A) + P(B).
概率记号: P(A) 表示事件 A 的概率。P(A') = 1 − P(A)。独立事件有 P(A ∩ B) = P(A)P(B)；互斥事件有 P(A ∪ B) = P(A) + P(B)。
10. Mechanics Vocabulary | 力学词汇
Displacement: A vector quantity; the distance moved in a given direction. Contrast with distance, which is scalar and has magnitude only.
位移: 矢量，指在给定方向上移动的距离。有别于距离，距离是标量，只有大小。
Velocity and speed e speed is the rate of change of distance (scalar). Average speed = total distance / time.
速度与速率: 速度是位移的变化率（矢量），速率是路程的变化率（标量）。平均速率 = 总路程 / 时间。
Acceleration − u)/t, where u is initial velocity and v is final velocity. Constant acceleration equations (SUVAT) apply when acceleration is uniform.
加速度: 速度的变化率，a = (v − u)/t，其中 u 为初速度，v 为末速度。当加速度恒定时，可使用匀加速运动公式（SUVAT）。
Resultant force gle force that has the same effect as all the original forces acting on a particle. Newton's second law: F = ma, where F is the resultant force.
合力: 产生与所有原作用力相同效果的单一力。牛顿第二定律：F = ma，其中 F 为合力。
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