📚 Year 11 WJEC Further Mathematics: Quick Glossary & Terminology Guide | Year 11 WJEC 进阶数学：词汇术语速记指南
This guide covers essential terms you will meet in the WJEC Level 2 Certificate in Further Mathematics (Year 11). Mastering the language of advanced algebra, calculus, matrices and trigonometry helps you recognise question types quickly and communicate your reasoning clearly in the exam. Each term is explained in English and Chinese so you can build both your subject knowledge and your bilingual vocabulary.
本指南涵盖 WJEC Level 2 Certificate in Further Mathematics（Year 11）的核心术语。掌握高等代数、微积分、矩阵和三角学的语言，能让你在考试中快速识别题型并清晰地表达推理过程。每个术语均提供中英文解释，帮助大家同时巩固学科知识和双语词汇。
1. Algebraic Expressions & Polynomials | 代数表达式与多项式
Polynomial: An algebraic expression that contains only non‑negative integer powers of variables, for example 4x³ − 7x + 2. Terms can be added or subtracted, and coefficients must be real numbers.
多项式：只包含变量的非负整数次幂的代数表达式，如 4x³ − 7x + 2。各项可相加或相减，系数必须为实数。
Degree of a polynomial: The highest power of the variable in the polynomial. In 5x⁴ + 3x² − x the degree is 4. The degree of a constant term is zero.
多项式的次数：多项式中变量的最高次幂。在 5x⁴ + 3x² − x 中次数为 4。常数项的次数为零。
Leading coefficient: The coefficient of the term with the highest power. For 3x⁵ − 2x² + 1 the leading coefficient is 3; it determines the end behaviour of the graph.
首项系数：最高次幂项的系数。对于 3x⁵ − 2x² + 1，首项系数为 3；它决定了图像两端的行为。
Factor theorem: omial P(x) if and only if P(a) = 0. This theorem allows you to factorise cubics and higher‑order polynomials efficiently.
因式定理：当且仅当 P(a) = 0 时，(x − a) 是多项式 P(x) 的因式。该定理可高效地对三次及以上多项式进行因式分解。
Remainder theorem: When P(x) is divided by (x − a), the remainder is P(a). It gives a quick way to evaluate remainders without doing full long division.
余数定理：当 P(x) 除以 (x − a) 时，余数为 P(a)。提供了一种无需完整长除法即可求余数的快捷方式。
Polynomial division: A method similar to long division for numbers, used to divide a polynomial by another polynomial of lower or equal degree. Result is expressed as quotient + remainder/divisor.
多项式除法：类似于数字长除法的方法，用于将一个多项式除以另一个次数较低或相等的多项式。结果表示为 商 + 余数/除数。
2. Functions & Graphs | 函数与图像
Function: tten as f(x) = expression. The vertical line test checks if a graph represents a function.
函数：为每个输入恰好分配一个输出的规则。记作 f(x) = 表达式。可通过垂直线检验判断一个图像是否表示函数。
Domain: The set of all possible input values (x‑values) for which the function is defined. In f(x) = √x − 2 the domain is x ≥ 2.
定义域：函数有定义的所有可能输入值（x 值）的集合。在 f(x) = √x − 2 中定义域为 x ≥ 2。
Range: The set of all possible output values (y‑values) the function produces. For f(x) = x², with x real, the range is y ≥ 0.
值域：函数产生的所有可能输出值（y 值）的集合。对于实数 x，f(x) = x² 的值域为 y ≥ 0。
Composite function: : Formed by combining two functions, where the output of one becomes the input of the other. Denoted as fg(x) = f(g(x)). Order matters; usually fg(x) ≠ gf(x).
复合函数：由两个函数组合而成，其中一个函数的输出成为另一个函数的输入。记作 fg(x) = f(g(x))。注意顺序，通常 fg(x) ≠ gf(x)。
Inverse function: , the inverse is a reflection in the line y = x. The domain of f becomes the range of f⁻¹.
反函数：逆向原函数的效果。若 f(x) = y，则 f⁻¹(y) = x。图像上，反函数关于直线 y = x 对称。f 的定义域变为 f⁻¹ 的值域。
3. Differentiation | 微分
Derivative: Measures the instantaneous rate of change of a function. The derivative dy/dx or f'(x) is found by differentiating term‑by‑term using the power rule.
导数：衡量函数的瞬时变化率。通过幂法则逐项求导可得导数 dy/dx 或 f'(x)。
Power rule for differentiation: ultiply by the power and reduce the power by one. This rule works for any real constant n.
微分幂法则：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。乘以指数并将指数减一。该法则适用于任意实数常数 n。
d/dx (xⁿ) = n xⁿ⁻¹
Second derivative: The derivative of the first derivative, written as d²y/dx² or f''(x). It describes the rate of change of the gradient and helps determine the nature of stationary points.
二阶导数：一阶导数的导数，记作 d²y/dx² 或 f''(x)。它描绘梯度的变化率，并用于判定驻点的性质。
Stationary point: ntal. Classify using the second derivative test or a gradient table as a local maximum, local minimum or point of inflection.
驻点：一阶导数为零的点，即 dy/dx = 0。切线为水平。使用二阶导数检验或梯度表可将其分类为局部极大值、局部极小值或拐点。
Tangent and normal: The tangent to a curve at a point has gradient equal to the derivative there. The normal is perpendicular to the tangent; its gradient is the negative reciprocal.
切线与法线：曲线上某点的切线斜率等于该点处的导数值。法线垂直于切线，其斜率为负倒数。
4. Integration | 积分
Indefinite integral: ect to x is (xⁿ⁺¹)/(n+1) + c, where c is the constant of integration. Notation: ∫ f(x) dx.
不定积分：微分的逆过程。xⁿ 对 x 的积分为 (xⁿ⁺¹)/(n+1) + c，其中 c 为积分常数。记作 ∫ f(x) dx。
∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, n ≠ −1
Constant of integration (c): Added when finding an indefinite integral because differentiation of a constant gives zero. It represents any possible vertical shift of the original function.
积分常数 (c)：求不定积分时添加的常数，因为常数微分为零。它表示原函数可能的任意垂直平移。
Definite integral: Evaluates the exact signed area between a curve, the x‑ limits x = a to x = b. Computed as F(b) − F(a) after integrating. The result is a number.
定积分：计算曲线、x 轴及边界 x = a 到 x = b 之间的精确有号面积。积分后代入 F(b) − F(a) 计算。结果是一个数值。
Area under a curve: Given by the definite integral when the curve is above the x‑axis. If the curve dips below the axis, take absolute values or split the integral to obtain the total physical area.
曲线下方面积：当曲线位于 x 轴上方时，可由定积分求出。若曲线部分在轴下方，需取绝对值或分段积分以获得实际总面积。
5. Sequences & Series | 数列与级数
Arithmetic sequence (AP): A list of numbers with a constant difference between consecutive terms. The nth term is a + (n−1)d, where a is the first term and d is the common difference.
等差数列 (AP)：连续项之间具有恒定公差的数列。第 n 项为 a + (n−1)d，其中 a 为首项，d 为公差。
uₙ = a + (n − 1)d
Sum of an arithmetic series: Sₙ = n/2 (a + l), where l is the last term. This formula adds the first n terms of an AP.
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。该公式计算等差数列前 n 项的和。
Geometric sequence (GP): Each term is found by multiplying the previous term by a constant ratio r. The nth term is a rⁿ⁻¹. The ratio r is the geometric progression's common factor.
等比数列 (GP)：每一项由前一项乘以恒定公比 r 得到。第 n 项为 a rⁿ⁻¹。r 是等比数列的公因子。
Binomial expansion: (a + b)ⁿ can be expanded using Pascal's triangle or the nCr formula: (ⁿCᵣ) aⁿ⁻ʳ bʳ. For a positive integer n the expansion has n+1 terms.
二项式展开：(a + b)ⁿ 可利用帕斯卡三角形或 nCr 公式展开：(ⁿCᵣ) aⁿ⁻ʳ bʳ。当 n 为正整数时，展开式共有 n+1 项。
(a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹b + ⁿC₂ aⁿ⁻²b² + … + ⁿCₙ bⁿ
Factorial (n!): The product of all positive integers from n down to 1. For example, 5! = 5×4×3×2×1 = 120. Binomial coefficients use factorials: ⁿCᵣ = n!/(r!(n−r)!).
阶乘 (n!)：从 n 往下到 1 的所有正整数的乘积。如 5! = 5×4×3×2×1 = 120。二项式系数用阶乘表示：ⁿCᵣ = n!/(r!(n−r)!)。
6. Trigonometry | 三角学
Exact value triangles: Special right‑angled triangles help recall exact sine, cosine and tangent of 30°, 45° and 60° without a calculator. They rely on Pythagoras' theorem.
精确值三角形：特殊直角三角形用于记忆 30°、45° 和 60° 的正弦、余弦和正切精确值，无需计算器。它们基于毕达哥拉斯定理。
sin 30° = ½, cos 45° = 1/√2, tan 60° = √3
Sine and cosine rules: (sine rule) and a² = b² + c² − 2bc cos A (cosine rule). Used when triangles are not right‑angled.
正弦定理与余弦定理：适用于任意三角形：a/sin A = b/sin B = c/sin C（正弦定理），以及 a² = b² + c² − 2bc cos A（余弦定理）。用于非直角三角形。
Trigonometric identities: d tan θ = sin θ / cos θ. They are vital for simplifying expressions and solving equations.
三角恒等式：对所有变量值均成立的等式。关键恒等式包括 sin²θ + cos²θ = 1 和 tan θ = sin θ / cos θ。化简表达式和求解方程时至关重要。
Radian measure: lus with trigonometric functions radian mode must be used for derivative formulas to hold.
弧度制：基于弧长的角度度量制，替代度数。π 弧度 = 180°。在含三角函数的微积分中，必须使用弧度制以保证导数公式成立。
7. Exponentials & Logarithms | 指数与对数
Exponential function: A function of the form f(x) = aˣ, often with base a > 0. The most common is eˣ, where e ≈ 2.718. Exponential growth or decay has a constant percentage rate.
指数函数：形如 f(x) = aˣ 的函数，底数 a > 0。最常见的是 eˣ，其中 e ≈ 2.718。指数增长或衰减具有恒定百分比变化率。
Logarithm (log): The inverse of an exponential. If aˣ ogₐ y = x. For example, log₂ 8 = 3 because 2³ = 8. Logs turn multiplication into addition.
对数：指数运算的逆运算。若 aˣ = y，则 logₐ y = x。例如 log₂ 8 = 3，因为 2³ = 8。对数将乘法转化为加法。
Natural logarithm (ln): A logarithm with base e. It is written as ln x. The derivative of ln x is 1/x, and ln eˣ = x. Crucial for calculus with exponentials.
自然对数 (ln)：以 e 为底的对数。记作 ln x。ln x 的导数为 1/x，且 ln eˣ = x。对含指数的微积分运算至关重要。
Laws of logarithms: g b; log (a/b) = log a − log b; log (aⁿ) = n log a. These work for any base, including ln.
对数运算法则：log (ab) = log a + log b；log (a/b) = log a − log b；log (aⁿ) = n log a。适用于任意底数，包括 ln。
Change of base: a allows logarithms with awkward bases to be evaluated on a calculator using base 10 or e.
换底公式：logₐ x = log_b x / log_b a。通过该公式可在计算器上用底数 10 或 e 计算任意底数的对数。
8. Matrices | 矩阵
Matrix: as the form [[a, b], [c, d]]. Matrices are used to represent linear transformations and solve simultaneous equations.
矩阵：按行和列排列的矩形数字阵列。2×2 矩阵形式为 [[a, b], [c, d]]。矩阵用于表示线性变换并求解联立方程组。
M = | a b |
| c d |
Determinant (2×2): For matrix [[a, b], [c, d]], the determinant is ad − bc, denoted det(M) or |M|. If det = 0 the matrix is singular and has no inverse.
行列式 (2×2)：对于矩阵 [[a, b], [c, d]]，行列式为 ad − bc，记作 det(M) 或 |M|。若行列式为零，则该矩阵是奇异矩阵，没有逆矩阵。
Inverse matrix (2×2): [[d, −b], [−c, a]]. Multiplying M by M⁻¹ yields the identity matrix I = [[1, 0], [0, 1]].
逆矩阵 (2×2)：仅当 ad − bc ≠ 0 时存在。M⁻¹ = 1/(ad−bc) [[d, −b], [−c, a]]。M 乘 M⁻¹ 得到单位矩阵 I = [[1, 0], [0, 1]]。
Singular and non-singular: Singular means determinant zero, so no unique inverse. Non‑singular (det ≠ 0) guarantees an inverse and a unique solution when solving matrix equations.
奇异与非奇异矩阵：奇异意味着行列式为零，因此没有唯一的逆矩阵。非奇异 (det ≠ 0) 保证存在逆矩阵，且求解矩阵方程时有唯一解。
Matrix multiplication: [c, d]] × [[e, f], [g, h]] = [[ae+bg, af+bh], [ce+dg, cf+dh]]. Not generally commutative.
矩阵乘法：用第一个矩阵的行乘以第二个矩阵的列。对 2×2，[[a, b], [c, d]] × [[e, f], [g, h]] = [[ae+bg, af+bh], [ce+dg, cf+dh]]。一般不满足交换律。
9. Coordinate Geometry & Vectors | 坐标几何与向量
Gradient of a straight line: asures steepness and direction. For a line given by y = mx + c, m is the coefficient of x.
直线斜率：m = (y₂ − y₁) / (x₂ − x₁)。它衡量倾斜程度和方向。对于直线 y = mx + c，m 是 x 的系数。
Distance between two points: d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Derived from Pythagoras' theorem and essential for calculating lengths in plane geometry.
两点间距离：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。由毕达哥拉斯定理推导，是平面几何中计算长度不可或缺的公式。
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Averages the coordinates and is useful for working with perpendicular bisectors or symmetry.
中点：((x₁+x₂)/2, (y₁+y₂)/2)。求坐标的平均值，可用于处理垂直平分线或对称性问题。
Vector: A quantity with magnitude and direction, often represented as a column matrix [x, y] or in i, j notation. Vectors can be added, subtracted and multiplied by scalars.
向量：具有大小和方向的量，通常表示为列矩阵 [x, y] 或用 i, j 记法。向量可进行加减和数乘运算。
Magnitude of a vector: For v = (x, y), magnitude |v| = √x² + y². This gives the length of the vector and ignores direction.
向量的模：对于 v = (x, y)，模为 |v| = √x² + y²。它给出向量的长度，不考虑方向。
Position vector: A vector that starts at the origin O and ends at a point P. The coordinates of P are the components of the position vector OP.
位置向量：以原点 O 为起点、以点 P 为终点的向量。点 P 的坐标就是位置向量 OP 的分量。
Unit vector: iding a vector by its magnitude. The standard basis vectors are i = (1, 0) and j = (0, 1).
单位向量：模为 1 的向量，通过将向量除以其模得到。标准基向量为 i = (1, 0) 和 j = (0, 1)。
10. Inequalities & Proof | 不等式与证明
Linear inequality: nverse operations, remembering to reverse the sign when multiplying or dividing by a negative.
线性不等式：类似于线性方程，但使用 <, ≤, > 或 ≥ 代替等号。可用逆运算求解，但需注意乘以或除以负数时要反向不等号。
Quadratic inequality: Involves an x² term. Solve by finding critical values (roots) and sketching the parabola, or using a sign table. The solution is often expressed as intervals.
二次不等式：含有 x² 项。通过求临界值（根）并画出抛物线草图或使用符号表来求解。解集常以区间形式表示。
Discriminant: two distinct real roots (Δ > 0), one repeated root (Δ = 0) or no real roots (Δ < 0).
判别式：由 ax² + bx + c = 0 定义的 Δ = b² − 4ac。它揭示根的性质：两个不同实根 (Δ > 0)，一个重根 (Δ = 0) 或无实根 (Δ < 0)。
Proof by counterexample: Disprove a statement by finding just one case where it fails. For example, 'all prime numbers are odd' is disproved by the single counterexample 2.
反例证明：通过找到一个使陈述不成立的例子来证伪。例如“所有质数都是奇数”可被单个反例 2 证伪。
Proof by exhaustion: when the set of cases is small, such as testing all integer possibilities for n = 1, 2, 3.
穷举证明：通过检查有限集合内每一个可能的情况来验证一个陈述。当情况数量较少时适用，如检验 n = 1, 2, 3 的所有整数情形。
Algebraic proof: Uses logical steps and known identities to show a statement is always true. For instance, showing that the sum of two odd integers is even by using 2k+1 representations.
代数证明：运用逻辑步骤和已知恒等式表明一个陈述总是成立。例如，用 2k+1 表示法证明两个奇数之和为偶数。
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