📚 Year 11 Eduqas Further Maths: Essential Glossary & Terminology Guide | Year 11 Eduqas 进阶数学核心词汇术语速记指南
Mastering the precise language of mathematics is vital for success in the Eduqas Further Maths GCSE. This glossary breaks down the key terms you will encounter across algebra, calculus, matrices, vectors and beyond. Each entry is paired with a concise explanation and an example, first in English and then in Chinese, so you can quickly memorise and recall the concepts under exam pressure.
掌握数学的精确语言对于 Eduqas 进阶数学 GCSE 考试的成功至关重要。这份术语表分解了你将在代数、微积分、矩阵、向量等各个领域遇到的关键词汇。每个词条都配有简洁的解释和示例，先英文后中文，让你能够在考试压力下快速记忆和回顾这些概念。
1. Algebraic Expressions and Equations | 代数表达式与方程
An expression is a combination of numbers, variables and operations (e.g. 3x² − 5x + 2), while a term is a single part separated by + or −. The coefficient is the numerical factor multiplying a variable: in 4x³, 4 is the coefficient.
表达式是数字、变量和运算的组合（如 3x² − 5x + 2），而项是由 + 或 − 分隔的单个部分。系数是乘以变量的数字因数：在 4x³ 中，4 是系数。
An equation states that two expressions are equal and is often solved (e.g. 2x + 3 = 7), whereas an identity is true for all values and is written with an equals sign or the triple bar ≡ (e.g. x² − 1 ≡ (x − 1)(x + 1)).
方程表示两个表达式相等，通常需要求解（如 2x + 3 = 7），而恒等式对所有取值都成立，用等号或三横线 ≡ 表示（如 x² − 1 ≡ (x − 1)(x + 1)）。
A quadratic equation has the form ax² + bx + c = 0, a ≠ 0. Its discriminant re are two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.
二次方程具有形式 ax² + bx + c = 0，a ≠ 0。其判别式 Δ = b² − 4ac 决定根的性质。若 Δ > 0，有两个不等实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。
Completing the square rewrites a quadratic x² + bx + c in the form (x + p)² + q, which reveals the turning point. For instance, x² − 6x + 5 becomes (x − 3)² − 4.
配方法将二次式 x² + bx + c 写成 (x + p)² + q 的形式，从而显示出转折点。例如，x² − 6x + 5 变为 (x − 3)² − 4。
2. Functions and Graphs | 函数与图像
A function f is a rule that assigns each input exactly one output. We write f(x) = 2x + 1. The domain is the set of allowed inputs, and the range is the set of possible outputs.
函数 f 是将每个输入精确分配给一个输出的规则。我们写作 f(x) = 2x + 1。定义域是允许输入值的集合，值域是可能输出值的集合。
A composite function ction to the result of another: gf(x) = g(f(x)). For f(x) = x² and g(x) = 2x, gf(x) = 2x².
复合函数是将一个函数应用于另一个函数的结果：gf(x) = g(f(x))。对于 f(x) = x² 和 g(x) = 2x，gf(x) = 2x²。
The inverse function x and y in y = f(x) and solve for y. For f(x) = 3x − 4, the inverse is f⁻¹(x) = (x + 4)/3.
反函数 f⁻¹(x) 反转 f 的作用，所以 f⁻¹(f(x)) = x。要求反函数，在 y = f(x) 中交换 x 和 y 并解出 y。对于 f(x) = 3x − 4，其反函数是 f⁻¹(x) = (x + 4)/3。
The exponential function exp(x) = eˣ perty that its derivative is itself. Exponential growth and decay problems often involve eˣ.
指数函数 exp(x) = eˣ，其中 e ≈ 2.718，其导数等于自身。指数增长和衰减问题常涉及 eˣ。
3. Coordinate Geometry | 坐标几何
The gradient − y₁)/(x₂ − x₁). A positive gradient slopes upwards, a negative gradient slopes downwards.
两点 (x₁, y₁) 和 (x₂, y₂) 之间线段的斜率为 m = (y₂ − y₁)/(x₂ − x₁)。正斜率向上倾斜，负斜率向下倾斜。
The equation of a straight line can be written in the form y = mx + c (m = gradient, c = y‑intercept) or using point‑gradient form y − y₁ = m(x − x₁).
直线方程可以写成 y = mx + c（m 为斜率，c 为 y 轴截距）或使用点斜式 y − y₁ = m(x − x₁)。
The midpoint of a segment joining (x₁, y₁) and (x₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2), and the distance between them is √[(x₂−x₁)² + (y₂−y₁)²].
连接 (x₁, y₁) 和 (x₂, y₂) 的线段中点为 ((x₁+x₂)/2, (y₁+y₂)/2)，两点间的距离为 √[(x₂−x₁)² + (y₂−y₁)²]。
The equation of a circle radius r is (x − a)² + (y − b)² = r². For x² + y² = 25, the centre is (0,0) and radius 5.
圆心为 (a, b)、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。对于 x² + y² = 25，圆心为 (0,0)，半径为 5。
4. Calculus: Differentiation and Integration | 微积分：微分与积分
The derivative f′(x) or dy/dx measures the rate of change or gradient of a function. The power rule states: if f(x) = xⁿ, then f′(x) = n xⁿ⁻¹.
导数 f′(x) 或 dy/dx 衡量函数的变化率或梯度。幂法则：若 f(x) = xⁿ，则 f′(x) = n xⁿ⁻¹。
A stationary point occurs where f′(x) = 0. Its nature is classified by the second derivative s a minimum; if f″(x) < 0 it is a maximum; if f″(x) = 0 it could be a point of inflection.
驻点发生在 f′(x) = 0 处。其性质通过二阶导数 f″(x) 判定：若 f″(x) > 0 则为极小点；若 f″(x) < 0 则为极大点；若 f″(x) = 0 则可能为拐点。
Indefinite integration 1/(n+1)) xⁿ⁺¹ + C, where C is the constant of integration. For example, ∫ 4x³ dx = x⁴ + C.
不定积分是微分的逆运算：∫ xⁿ dx = (1/(n+1)) xⁿ⁺¹ + C，其中 C 为积分常数。例如，∫ 4x³ dx = x⁴ + C。
A definite integral o x = b. It is evaluated by finding the indefinite integral, then substituting the limits.
定积分 ∫ₐᵇ f(x) dx 给出了曲线 y = f(x) 下从 x = a 到 x = b 的精确面积。计算方法是先求出不定积分，再代入上下限。
5. Matrices | 矩阵
A matrix is a rectangular array of numbers. Its order is given as rows × columns. A 2×2 matrix has the form [[a, b], [c, d]].
矩阵是一个矩形数字阵列。其阶数表示为行数 × 列数。一个 2×2 矩阵的形式为 [[a, b], [c, d]]。
The identity matrix I = [[1, 0], [0, 1]] acts like the number 1 in matrix multiplication: A × I = A.
单位矩阵 I = [[1, 0], [0, 1]] 的作用就像矩阵乘法中的数字 1：A × I = A。
The determinant of a 2×2 matrix A = [[a, b], [c, d]] is det A = ad − bc. If det A = 0, the matrix is singular and has no inverse; otherwise it is non‑singular and an inverse exists: A⁻¹ = (1/det A) [[d, −b], [−c, a]].
2×2 矩阵 A = [[a, b], [c, d]] 的行列式为 det A = ad − bc。若 det A = 0，则该矩阵为奇异矩阵，没有逆矩阵；否则为非奇异矩阵，逆矩阵存在：A⁻¹ = (1/det A) [[d, −b], [−c, a]]。
Matrices can represent geometric transformations such as rotations, reflections and enlargements. Applying the matrix to a position vector maps the point to its image.
矩阵可以表示几何变换，如旋转、反射和放大。将矩阵乘以位置向量，可将点映射到它的像。
6. Vectors | 向量
A vector has both magnitude and direction, often written in bold v or as a column vector [[x], [y]]. A scalar is just a number.
向量既有大小又有方向，常用粗体 v 或列向量 [[x], [y]] 表示。标量仅为数字。
The magnitude (length) of a vector v = xi + yj is |v| = √x² + y². A unit vector has magnitude 1 and is often used to indicate direction alone.
向量 v = xi + yj 的模（长度）为 |v| = √x² + y²。单位向量的模为 1，通常仅用于表示方向。
Vector addition is done component‑wise: [[x₁],[y₁]] + [[x₂],[y₂]] = [[x₁+x₂],[y₁+y₂]]. Scalar multiplication stretches or shrinks the vector: k[[x],[y]] = [[kx],[ky]].
向量加法按分量进行：[[x₁],[y₁]] + [[x₂],[y₂]] = [[x₁+x₂],[y₁+y₂]]。标量乘法拉伸或缩短向量：k[[x],[y]] = [[kx],[ky]]。
Vectors are parallel if one is a scalar multiple of the other. Points are collinear if they lie on the same straight line, which can be checked by seeing if vectors between them are parallel.
如果一个向量是另一个向量的标量倍数，则这两个向量平行。如果点位于同一直线上，则这三点共线，这可以通过检查它们之间的向量是否平行来验证。
7. Sequences and Series | 数列与级数
An arithmetic sequence has a constant difference d between consecutive terms. The n‑th term is uₙ = a + (n−1)d, where a is the first term.
等差数列的相邻两项之间有一个常数差 d。第 n 项为 uₙ = a + (n−1)d，其中 a 是首项。
The sum of an arithmetic series with n terms is Sₙ = n/2 (a + l) or Sₙ = n/2 [2a + (n−1)d], where l is the last term.
等差数列前 n 项和 为 Sₙ = n/2 (a + l) 或 Sₙ = n/2 [2a + (n−1)d]，其中 l 为末项。
A geometric sequence between terms: uₙ = a rⁿ⁻¹. The sum of the first n terms is Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1.
等比数列的相邻两项有一常数公比 r：uₙ = a rⁿ⁻¹。前 n 项和为 Sₙ = a(1−rⁿ)/(1−r)，r ≠ 1。
If |r| < 1 in an infinite geometric series, the series converges and the sum to infinity is S∞ = a/(1−r).
若无穷等比数列满足 |r| < 1，则该级数收敛，其无穷和为 S∞ = a/(1−r)。
8. Trigonometry | 三角学
In a right‑angled triangle, sine (sin θ = opposite/hypotenuse), cosine (cos θ = adjacent/hypotenuse) and tangent (tan θ = opposite/adjacent) connect angles to side ratios.
在直角三角形中，正弦 (sin θ = 对边/斜边)、余弦 (cos θ = 邻边/斜边) 和正切 (tan θ = 对边/邻边) 将角度与边长比联系起来。
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