📚 GCSE WJEC Further Maths: Key Terminology & Quick Memorisation Guide | GCSE WJEC进阶数学：关键术语速记指南
Mastering the technical vocabulary of WJEC GCSE Further Mathematics is just as important as practising the sums. This bilingual glossary pinpoints the most essential terms from algebra to matrices, provides clear definitions, and offers memory hooks to help you retain them. Use it alongside your revision to build precision and confidence for the exam.
掌握 WJEC GCSE 进阶数学的专业术语与练习计算同等重要。这份双语术语指南覆盖从代数到矩阵的核心词汇，提供清晰定义和记忆技巧，帮助你在复习中牢牢记住它们。配合使用可提升答题的准确性和自信。
1. Algebraic Terminology & Polynomials | 代数术语与多项式
Variable – a symbol, typically x, y or t, that represents an unknown or changeable quantity.
变量 – 通常用 x、y 或 t 等符号表示未知或可变的量。
Coefficient – the numerical factor multiplying a variable in a term. In 7x³, 7 is the coefficient.
系数 – 项中乘以变量的数字因数。在 7x³ 中，7 是系数。
Constant – a term with a fixed value; no variable is attached. For example, +5 is a constant.
常数 – 具有固定值的项，不包含变量。例如 +5 是常数。
Polynomial – an expression built from variables and constants using addition, subtraction and multiplication, with non‑negative integer exponents. A quadratic is a polynomial of degree 2.
多项式 – 由变量和常数通过加、减、乘运算组成的表达式，指数为非负整数。二次式是次数为 2 的多项式。
f(x) = 2x³ + 5x − 7 is a cubic polynomial (degree 3).
f(x) = 2x³ + 5x − 7 是一个三次多项式（次数为 3）。
Factor theorem – if f(a) = 0, then (x − a) is a factor of f(x). Remember: “Factor ⇔ Zero”.
因式定理 – 若 f(a) = 0，则 (x − a) 是 f(x) 的因式。记忆：“因式等价于零点”。
Remainder theorem – when f(x) is divided by (x − a), the remainder is f(a).
余数定理 – 当 f(x) 除以 (x − a) 时，余数为 f(a)。
Discriminant – for a quadratic ax² + bx + c, the discriminant Δ = b² − 4ac determines the nature of roots.
判别式 – 对于二次式 ax² + bx + c，判别式 Δ = b² − 4ac 决定了根的性质。
2. Functions & Graphs | 函数与图像
Domain – the set of all possible input values (x) for which a 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 possible output values (f(x)) the function can produce.
值域 – 函数能够产生的所有可能输出值（f(x)）的集合。
Composite function – combining two functions where the output of one becomes the input of another: fg(x) = f(g(x)). Read “f of g of x”.
复合函数 – 将一个函数的输出作为另一个函数的输入：fg(x) = f(g(x))，读作“f of g of x”。
Inverse function – reverses a function, denoted f⁻¹(x). Graphically it reflects f(x) in the line y = x.
反函数 – 逆转原函数的函数，记作 f⁻¹(x)。其图像是 f(x) 关于直线 y = x 的反射。
Modulus function – written |x|, it gives the absolute value (distance from zero), so |−3| = 3.
绝对值函数 – 写作 |x|，返回非负值（到零的距离），因此 |−3| = 3。
Transformation of graphs – f(x + a) shifts left by a; f(x) + a shifts up; −f(x) reflects in x‑axis; f(−x) reflects in y‑axis.
图像变换 – f(x + a) 向左平移 a 个单位；f(x) + a 向上平移；−f(x) 关于 x 轴反射；f(−x) 关于 y 轴反射。
3. Coordinate Geometry & Circles | 坐标几何与圆
Gradient (slope) – m = Δy/Δx = (y₂ − y₁)/(x₂ − x₁). Rise over run.
斜率（坡度） – m = Δy/Δx = (y₂ − y₁)/(x₂ − x₁)，即垂直变化除以水平变化。
Midpoint – the point halfway between two coordinates: ((x₁+x₂)/2, (y₁+y₂)/2).
中点 – 两点之间的中点坐标：((x₁+x₂)/2, (y₁+y₂)/2)。
Distance formula – distance = √[(x₂ − x₁)² + (y₂ − y₁)²], derived from Pythagoras.
距离公式 – 距离 = √[(x₂ − x₁)² + (y₂ − y₁)²]，由勾股定理导出。
Equation of a circle – centre (a, b) radius r: (x − a)² + (y − b)² = r². The expanded form x² + y² + 2gx + 2fy + c = 0 gives centre (−g, −f) and radius √(g² + f² − c).
圆的方程 – 圆心 (a, b)、半径 r：(x − a)² + (y − b)² = r²。一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (−g, −f)，半径 √(g² + f² − c)。
Perpendicular bisector – a line passing through the midpoint of a segment at a right angle, used in circle geometry.
垂直平分线 – 通过线段中点且与之垂直的直线，常用于圆的几何问题。
4. Differentiation | 微分
Derivative – the rate of change of a function; denoted f'(x) or dy/dx. It gives the gradient of the tangent to a curve.
导数 – 函数的变化率；记为 f'(x) 或 dy/dx。它给出了曲线切线的斜率。
Power rule – for f(x) = xⁿ, f'(x) = n xⁿ⁻¹. “Bring down the power, reduce the power by one.”
幂法则 – 对于 f(x) = xⁿ，f'(x) = n xⁿ⁻¹。“把指数拿下来，指数减一”。
Stationary point – occurs where dy/dx = 0. It can be a local maximum, local minimum or point of inflection.
驻点 – 发生在 dy/dx = 0 处。可以是局部极大点、局部极小点或拐点。
Second derivative – d²y/dx² tests the concavity. If d²y/dx² > 0, the stationary point is a minimum; if d²y/dx² < 0, it is a maximum.
二阶导数 – d²y/dx² 检验凹凸性。若 d²y/dx² > 0，驻点为极小点；若 d²y/dx² < 0，则为极大点。
Tangent and normal – the tangent’s gradient is dy/dx at a point; the normal is perpendicular, so its gradient is −1/(dy/dx).
切线与法线 – 切线在某点的斜率为 dy/dx；法线与之垂直，因此其斜率为 −1/(dy/dx)。
Increasing/decreasing function – f'(x) > 0 means f is increasing; f'(x) < 0 means f is decreasing.
递增/递减函数 – f'(x) > 0 表示函数递增；f'(x) < 0 表示函数递减。
5. Integration | 积分
Indefinite integral – the reverse of differentiation, ∫ f'(x) dx = f(x) + c. The “+ c” is the constant of integration.
不定积分 – 微分的逆运算，∫ f'(x) dx = f(x) + c。“+ c” 是积分常数。
Power rule for integration – ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, provided n ≠ −1.
幂函数积分法则 – ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c，其中 n ≠ −1。
Definite integral – ∫ₐᵇ f(x) dx calculates the area under the curve between x = a and x = b. Evaluate F(b) − F(a) where F'(x) = f(x).
定积分 – ∫ₐᵇ f(x) dx 计算曲线在 x = a 与 x = b 之间的下方面积。先求原函数 F(x)，再计算 F(b) − F(a)。
Trapezium rule – a numerical method for approximating a definite integral by splitting the area into trapezia. The formula: (h/2)[y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b − a)/n.
梯形法则 – 一种将区域分割成梯形以近似计算定积分的数值方法。公式为 (h/2)[y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 h = (b − a)/n。
6. Sequences & Series | 数列与级数
Arithmetic progression (AP) – a sequence where the difference between terms is constant, called the common difference d. The nᵗʰ term: uₙ = a + (n − 1)d.
等差数列 (AP) – 相邻项之差为常数的数列，该常数称为公差 d。第 n 项：uₙ = a + (n − 1)d。
Sum of an AP – Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l) where l is the last term.
等差数列求和 – Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
Geometric progression (GP) – a sequence with a constant common ratio r between terms. uₙ = a rⁿ⁻¹.
等比数列 (GP) – 相邻项之比为常数 r 的数列。第 n 项：uₙ = a rⁿ⁻¹。
Sum of a GP – for |r| < 1, sum to infinity: S∞ = a/(1 − r). For finite terms, Sₙ = a(1 − rⁿ)/(1 − r).
等比数列求和 – 当 |r| < 1 时，无穷项和为 S∞ = a/(1 − r)。有限项和为 Sₙ = a(1 − rⁿ)/(1 − r)。
Sigma notation (Σ) – compactly writes a series. For example, Σ (from k=1 to n) of k² means 1² + 2² + … + n².
连加符 (Σ) – 简洁地表示级数。例如 Σ (k=1 到 n) k² 代表 1² + 2² + … + n²。
7. Trigonometry & Radian Measure | 三角学与弧度制
Radian – an alternative unit for angles where π radians = 180°. 1 rad ≈ 57.3°. Used in calculus to simplify derivatives of trig functions.
弧度 – 角度的一种单位，π 弧度 = 180°。1 弧度约等于 57.3°。在微积分中使用可简化三角函数的导数。
Sine rule – a/sin A = b/sin B = c/sin C. Used for non‑right‑angled triangles. Memory: “side over sine of opposite angle”.
正弦定理 – a/sin A = b/sin B = c/sin C。用于非直角三角形。记忆：“边除以对角的正弦”。
Cosine rule – a² = b² + c² − 2bc cos A. Useful for finding a side given two sides and the included angle, or an angle given three sides.
余弦定理 – a² = b² + c² − 2bc cos A。已知两边及夹角求第三边，或已知三边求角时使用。
Area of a triangle – ½ ab sin C. The area is half the product of two sides times the sine of the included angle.
三角形面积公式 – ½ ab sin C。面积等于两边乘积乘以夹角正弦的一半。
Key identity – sin²θ + cos²θ = 1. Leads to tan²θ + 1 = sec²θ in further work, but for GCSE the first is crucial.
核心恒等式 – sin²θ + cos²θ = 1。由此可导出 tan²θ + 1 = sec²θ，但 GCSE 阶段重点掌握第一个。
Solving trig equations – always check the interval (e.g. 0 ≤ θ ≤ 360°) and use the CAST diagram or graph to find all solutions.
解三角方程 – 始终检查给定区间（如 0 ≤ θ ≤ 360°），并使用 CAST 图或图像找出所有解。
8. Exponentials & Logarithms | 指数与对数
Exponential function – f(x) = aˣ where a > 0. For a > 1 the function shows exponential growth; for 0 < a < 1 it shows decay.
指数函数 – f(x) = aˣ，其中 a > 0。当 a > 1 时呈指数增长；0 < a < 1 时呈指数衰减。
Natural exponential – eˣ, where e ≈ 2.71828, is the unique function whose derivative equals itself.
自然指数函数 – eˣ 中 e ≈ 2.71828，是唯一导数等于自身的函数。
Logarithm – the inverse of exponentiation. logₐb = c means aᶜ = b. In GCSE we focus on log₁₀ and logₑ (ln).
对数 – 指数运算的逆运算。logₐb = c 意味着 aᶜ = b。GCSE 主要使用 log₁₀ 和自然对数 ln。
Laws of logs – logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx − logₐy; logₐ(xⁿ) = n logₐx. Memory: “multiplication becomes addition inside logs”.
对数运算法则 – logₐ(xy) = logₐx + logₐy；logₐ(x/y) = logₐx − logₐy；logₐ(xⁿ) = n logₐx。记忆：“乘法在对数里变成加法”。
Change of base – logₐb = (log_c b)/(log_c a). Useful when the base is not available on your calculator.
换底公式 – logₐb = (log_c b)/(log_c a)。当计算器无法直接计算某底数时使用。
9. Vectors | 向量
Vector – a quantity with magnitude (size) and direction, represented by a column vector or bold letter. A scalar has only magnitude.
向量 – 有大小和方向的量，用列向量或粗体字母表示。标量仅有大小。
Magnitude (modulus) – the length of a vector. For vector v = (x, y), |v| = √(x² + y²).
模（大小） – 向量的长度。对于向量 v = (x, y)，|v| = √(x² + y²)。
Unit vector – a vector with magnitude 1. To obtain a unit vector in the direction of v, divide v by |v|.
单位向量 – 长度为 1 的向量。要得到 v 方向上的单位向量，将 v 除以其模。
Position vector – starts from the origin O and gives the location of a point. The vector from A to B is OB − OA = b − a.
位置向量 – 起点为原点 O，给出点的位置。从 A 到 B 的向量为 OB − OA = b − a。
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的标量倍，则两向量平行。
Resultant vector – the sum of two or more vectors. Addition is done head‑to‑tail or by adding components.
合向量 – 两个
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