📚 Year 11 AQA Further Maths: Quick Guide to Key Vocabulary & Terminology | Year 11 AQA 进阶数学：词汇术语速记指南
Mastering the vocabulary of AQA Level 2 Certificate in Further Mathematics (8365) is the first step to exam success. This guide covers the essential terms you need to know—from algebra and calculus to matrices and vectors—helping you recall definitions quickly and use them accurately in context.
掌握 AQA 进阶数学（8365）的词汇是考试成功的第一步。本指南涵盖了你需要知道的关键术语——从代数、微积分到矩阵和向量——帮助你快速回忆定义并在语境中准确使用。
1. Core Algebraic Concepts | 核心代数概念
Polynomial: An algebraic expression consisting of variables and coefficients, involving only non-negative integer powers of x, e.g. 3x² − 5x + 7.
多项式：由变量和系数构成的代数表达式，其中 x 的指数均为非负整数，例如 3x² − 5x + 7。
Degree (of a polynomial): The highest power of the variable in a polynomial. In 4x³ + 2x − 1, the degree is 3.
次数（多项式）：多项式中变量的最高次幂。在 4x³ + 2x − 1 中，次数为 3。
Leading coefficient: The coefficient of the term with the highest degree. For 2x⁴ − 3x² + x, the leading coefficient is 2.
首项系数：次数最高项的系数。对于 2x⁴ − 3x² + x，首项系数为 2。
Factor theorem: If f(a) = 0 for a polynomial f(x), then (x − a) is a factor of f(x).
因式定理：对于多项式 f(x)，若 f(a) = 0，则 (x − a) 是 f(x) 的一个因式。
Remainder theorem: When a polynomial f(x) is divided by (x − a), the remainder is equal to f(a).
余数定理：当多项式 f(x) 除以 (x − a) 时，余数等于 f(a)。
Root (or zero): A value of x that makes the polynomial equal to zero. It is the solution to f(x) = 0.
根（或零点）：使多项式等于零的 x 值，即方程 f(x) = 0 的解。
2. Binomial Expansion | 二项式展开
Binomial: An algebraic expression with exactly two terms, such as (x + y) or (3a − 2b).
二项式：恰有两个项的代数表达式，例如 (x + y) 或 (3a − 2b)。
Binomial expansion sum of terms using binomial coefficients. For example, (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴.
二项式展开：利用二项式系数将 (a + b)ⁿ 表示为各项之和，如 (1 + x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴。
Binomial coefficient: The number of ways to choose r items from n, written as ⁿCᵣ or (ⁿᵣ). It appears as the coefficient in the expansion.
二项式系数：从 n 个物品中选取 r 个的方法数，记作 ⁿCᵣ 或 (ⁿᵣ)，在展开式中作为系数出现。
Pascal's triangle: A triangular array where each number is the sum of the two directly above it, providing binomial coefficients for small n.
帕斯卡三角形：一个三角形数阵，每个数等于其上方两数之和，为较小的 n 提供二项式系数。
Factorial (n!): The product of all positive integers from 1 to n. Used in the formula ⁿCᵣ = n! / [r!(n − r)!].
阶乘 (n!)：从 1 到 n 所有正整数的乘积，用于公式 ⁿCᵣ = n! / [r!(n − r)!]。
Term independent of x: The term in a binomial expansion that does not contain the variable x (the constant term).
与 x 无关的项：二项展开式中不含变量 x 的项（常数项）。
3. Functions and Transformations | 函数与变换
Function e that assigns exactly one output f(x) for every input x. Often written as f : x → 2x + 3.
函数：为每个输入 x 指定唯一输出 f(x) 的规则，常写作 f : x → 2x + 3。
Domain: The set of all possible input values (x) for which the function is defined.
定义域：函数有定义的所有可能输入值 (x) 的集合。
Range: The set of all output values f(x) obtained from the domain.
值域：由定义域获得的所有输出值 f(x) 的集合。
Composite function tion formed by applying one function to the result of another, written as fg(x) = f(g(x)).
复合函数：将一个函数作用于另一个函数的结果所构成的函数，记作 fg(x) = f(g(x))。
Inverse function f⁻¹ erses the effect of f, so that f⁻¹(f(x)) = x. Its graph is a reflection in the line y = x.
反函数 f⁻¹：逆操作 f 的函数，满足 f⁻¹(f(x)) = x，其图像关于直线 y = x 对称。
Transformation: An operation that moves or resizes a graph, including translation, stretch, and reflection.
变换：移动或缩放图像的操作，包括平移、伸缩和反射。
Modulus function |x|: Gives the absolute value of x, its distance from zero. The graph is V-shaped.
绝对值函数 |x|：给出 x 的绝对值，即到零的距离，图像呈 V 形。
4. Coordinate Geometry of Curves | 曲线的坐标几何
Gradient of a chord: The slope of a straight line connecting two points on a curve; an approximation of the gradient of the curve.
弦的梯度：连接曲线上两点的直线斜率，用于近似曲线的梯度。
Tangent: A straight line that touches a curve at exactly one point, having the same gradient as the curve at that point.
切线：与曲线恰好交于一点且在该点与曲线具有相同梯度的直线。
Normal: A straight line perpendicular to the tangent at the point of contact. If the tangent's gradient is m, the normal's gradient is −1/m.
法线：垂直于切线的直线。若切线斜率为 m，则法线斜率为 −1/m。
Stationary point re the gradient is zero (dy/dx = 0). It can be a maximum, minimum, or point of inflection.
驻点：曲线上梯度为零 (dy/dx = 0) 的点，可以是极大值、极小值或拐点。
Turning point: A stationary point that is a local maximum or minimum.
转折点：作为局部极大值或局部极小值的驻点。
Point of inflection: A point where the curve changes concavity, and where dy/dx = 0 may also occur.
拐点：曲线改变凹凸性的点，此处也可能满足 dy/dx = 0。
5. Calculus: Differentiation & Integration | 微积分：微分与积分
Derivative: A function that gives the gradient of a curve y = f(x) at any point, denoted f'(x) or dy/dx.
导数：给出曲线 y = f(x) 在任意点梯度的函数，记作 f'(x) 或 dy/dx。
Differentiation from first principles: The limit of the average gradient as h → 0: f'(x) = lim(h→0) [f(x+h) − f(x)] / h.
基于第一原理的求导：当 h → 0 时平均梯度的极限：f'(x) = lim(h→0) [f(x+h) − f(x)] / h。
Power rule: For f(x) = xⁿ, the derivative is f'(x) = n xⁿ⁻¹. Extends to negative and fractional n.
幂函数法则：对于 f(x) = xⁿ，导数为 f'(x) = n xⁿ⁻¹，适用于负指数和分数指数。
Chain rule: Used to differentiate a composite function: if y = g(f(x)) then dy/dx = g'(f(x)) × f'(x).
链式法则：用于求复合函数的导数：若 y = g(f(x))，则 dy/dx = g'(f(x)) × f'(x)。
Product rule: If y = u v, where u and v are functions of x, then dy/dx = u' v + u v'.
乘法法则：若 y = u v，其中 u 和 v 均为 x 的函数，则 dy/dx = u' v + u v'。
Quotient rule: If y = u / v, then dy/dx = (v u' − u v') / v².
除法法则：若 y = u / v，则 dy/dx = (v u' − u v') / v²。
Indefinite integral ferentiation, represented as ∫ f(x) dx = F(x) + C, where C is the constant of integration.
不定积分：微分的逆运算，记为 ∫ f(x) dx = F(x) + C，其中 C 为积分常数。
Definite integral: The integral function between two limits a and b, giving the exact area under the curve when f(x) ≥ 0.
定积分：函数在上下限 a 和 b 之间的积分，当 f(x) ≥ 0 时给出曲线下的精确面积。
6. Matrices | 矩阵
Matrix: A rectangular array of numbers arranged in rows and columns. The order is 'rows × columns'.
矩阵：由行和列排列的数字矩形阵列。阶数表示为“行 × 列”。
Element: Each individual number in a matrix, often referred to by its row and column position, e.g. a₁₂.
元素：矩阵中的每个单独的数字，通常通过其行、列位置表示，如 a₁₂。
Square matrix: A matrix with the same number of rows and columns (n × n).
方阵：行数与列数相等 (n × n) 的矩阵。
Identity matrix (I): A square matrix with 1s on the main diagonal and 0s elsewhere. It acts like the number 1 in matrix multiplication.
单位矩阵 (I)：主对角线为 1，其余位置为 0 的方阵，在矩阵乘法中的作用如同数字 1。
Determinant (det A) 2×2 matrix A = [[a, b], [c, d]], det A = ad − bc. A matrix with determinant 0 is singular.
行列式 (det A)：对于 2×2 矩阵 A = [[a, b], [c, d]]，det A = ad − bc。行列式为 0 的矩阵称为奇异矩阵。
Inverse matrix A⁻¹: A matrix such that A A⁻¹ = I. For a 2×2 matrix, A⁻¹ = 1/(ad − bc) [[d, −b], [−c, a]].
逆矩阵 A⁻¹：满足 A A⁻¹ = I 的矩阵。对于 2×2 矩阵，A⁻¹ = 1/(ad − bc) [[d, −b], [−c, a]]。
Transpose (Aᵀ): A matrix obtained by swapping rows and columns. If A is m×n, Aᵀ is n×m.
转置矩阵 (Aᵀ)：通过交换行与列得到的矩阵。若 A 为 m×n，则 Aᵀ 为 n×m。
Singular vs non-singular: A singular matrix has determinant 0 and no inverse; a non-singular matrix has a non-zero determinant and an inverse exists.
奇异与非奇异：奇异矩阵的行列式为 0，无逆矩阵；非奇异矩阵的行列式非零，逆矩阵存在。
7. Vectors | 向量
Vector: A quantity that has both magnitude and direction, often written as a column [x, y] or in i, j notation.
向量：既有大小又有方向的量，通常表示为列向量 [x, y] 或使用 i, j 分量。
Scalar: A quantity with magnitude only (e.g. speed, mass). Multiplying a vector by a scalar stretches it.
标量：仅有大小的量（如速率、质量）。将向量乘以标量会使其伸缩。
Magnitude (|v|): The length of a vector. For v = a i + b j, |v| = √a² + b².
模长 (|v|)：向量的长度。对于 v = a i + b j，|v| = √a² + b²。
Unit vector: A vector with magnitude 1. The standard unit vectors are i = [1, 0] and j = [0, 1].
单位向量：模长为 1 的向量。标准单位向量为 i = [1, 0] 和 j = [0, 1]。
Position vector: A vector that gives the position of a point relative to the origin.
位置向量：表示某点相对于原点位置的向量。
Direction vector: A vector that describes the direction of a line. Any scalar multiple gives the same direction.
方向向量：描述直线方向的向量，其任意标量倍均表示相同方向。
Parallel vectors: Two vectors are parallel if one is a scalar multiple of the other.
平行向量：若一个向量是另一个向量的标量倍，则它们平行。
Equal vectors: Vectors that have the same magnitude and direction, regardless of their initial point.
相等向量：具有相同大小和方向的向量，与起点无关。
8. Trigonometry | 三角学
Trigonometric ratios enuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent for a right-angled triangle.
三角比：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 trigonometric values Standard angles (30°, 45°, 60°, 90°) have exact surd values, e.g. sin 30° = 1/2, cos 45° = √2/2.
精确三角函数值：标准角度（30°、45°、60°、90°）有精确的根式值，例如 sin 30° = 1/2，cos 45° = √2/2。
Radian radians = 180°. One radian is the angle subtended by an arc equal in length to the radius.
弧度：一种角度度量单位，π 弧度 = 180°。当弧长等于半径时所对应的圆心角为 1 弧度。
Arc length: In a circle of radius r, the length of an arc with angle θ (radians) is s = rθ.
弧长：在半径为 r 的圆中，角度为 θ（弧度）的弧长 s = rθ。
Sector area: The area enclosed by two radii and an arc, given by A = ½ r² θ (θ in radians).
扇形面积：由两条半径和一段弧围成的面积，公式为 A = ½ r² θ（θ 以弧度计）。
Fundamental identity: sin² θ + cos² θ ≡ 1, which can be rearranged to express one in terms of the other.
基本恒等式：sin² θ + cos² θ ≡ 1，可通过变形表达其中一个函数。
Solving trig equations ge that satisfy an equation like sin x = 0.5, often using the unit circle or CAST diagram.
解三角方程：在给定范围内找出所有满足如 sin x = 0.5 的 x 值，常借助单位圆或 CAST 图。
9. Inequalities and Regions | 不等式与区域
Linear inequality: An inequality involving a linear expression, e.g. 2x − 3 < 7. Solve by treating it like an equation, reversing inequality sign if multiplying or dividing by a negative.
一元一次不等式：涉及一次式的不等式，例如 2x − 3 < 7。解法与方程类似，乘除负数时不等号方向改变。
Quadratic inequality: An inequality such as x² − 4x + 3 > 0. Solved by finding critical values from the corresponding quadratic equation and testing regions.
二次不等式：如 x² − 4x + 3 > 0 的不等式。通过求解对应二次方程得到临界值，再分区检验来解。
Critical values: The roots of the equation formed by replacing the inequality sign with equality; they divide the number line into intervals.
临界值：将不等号换为等号后形成的方程的根，它们将数轴分成若干区间。
Graphical inequality region re inequalities. The boundary is dotted for strict inequalities (<, >) and solid for ≤, ≥.
不等式图示区域：在坐标系中表示满足一个或多个不等式的点集 (x, y) 的区域。严格不等式边界用虚线，包含等号用实线。
Feasible region: The overlapping area that satisfies all given inequalities in a linear programming problem.
可行区域：在线性规划问题中满足所有给定不等式的重叠区域。
10. Sequences and Series | 数列与级数
Sequence: An ordered list of numbers, e.g. 3, 7, 11, 15, ... defined by a rule for the nth term.
数列：按一定顺序排列的一列数，例如 3, 7, 11, 15, ...，由通项公式定义。
nth term: A formula that allows direct calculation of any term in a sequence, such as uₙ = 2n + 1.
通项（第 n 项）：能够直接计算数列中任意项的公式，例如 uₙ = 2n + 1。
Arithmetic progression (AP) e where the difference d between consecutive terms is constant, e.g. 2, 5, 8, ... (d = 3).
等差数列：相邻两项的差 d 为常数的数列，例如 2, 5, 8, ... (d = 3)。
Geometric progression (GP) uence where the ratio r between consecutive terms is constant, e.g. 3, 6, 12, ... (r = 2).
等比数列：相邻两项的比值 r 为常数的数列，例如 3, 6, 12, ... (r = 2)。
Sum to infinity (S∞): For a GP with |r| < 1, the sum of all terms approaches a fixed value S∞ = a / (1 − r).
无穷和 (S∞)：对于 |r| < 1 的等比数列，所有项的和趋于一个固定值 S∞ = a / (1 − r)。
Series: The sum of terms of a sequence. Notation Σ (sigma) is often used to denote a sum.
级数：数列各项的和。常使用 Σ（西格玛）符号表示求和。
Recurrence relation n : An equation that defines a term of a sequence using previous terms, e.g. uₙ₊₁ = 2uₙ − 3 with u₁ = 5.
递推关系：利用前项定义数列某项的等式，例如 uₙ₊₁ = 2uₙ − 3，已知 u₁ = 5。
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