📚 Year 11 WJEC Further Maths: Key Terms Memorisation Guide | Year 11 WJEC 进阶数学：词汇术语速记指南
Mastering mathematical terminology is essential for tackling WJEC Further Maths at Year 11. This guide presents key terms with clear definitions in both English and Chinese to accelerate memorisation and deepen understanding.
掌握数学术语对于攻克 Year 11 WJEC 进阶数学至关重要。本指南提供关键术语及清晰定义，中英对照，助你加速记忆，深化理解。
1. Algebraic Manipulation | 代数操作
Term: a single number, variable, or product of numbers and variables, separated by + or − in an expression.
项：表达式中被加号或减号分隔的单个数字、变量或数字与变量的乘积。
Expression: a combination of terms linked by operators, without an equals sign (e.g., 3x² + 2x − 5).
表达式：由运算符号连接项的组合，不含等号（如 3x² + 2x − 5）。
Equation atement that two expressions are equal, always contains an equals sign (e.g., 2x + 3 = 7).
方程：声明两个表达式相等的陈述，总含等号（如 2x + 3 = 7）。
Identity: an equation that holds true for all values of the variable, often denoted by '≡' (e.g., (x+1)² ≡ x² + 2x + 1).
恒等式：对变量所有取值都成立的方程，常用符号 '≡' 表示（如 (x+1)² ≡ x² + 2x + 1）。
Coefficient: the numerical factor multiplying a variable in a term (in 5xy, 5 is the coefficient).
系数：项中乘以变量的数字因子（在 5xy 中，5 是系数）。
2. Quadratics and Polynomials | 二次式与多项式
Quadratic expression: a polynomial of degree 2, standard form ax² + bx + c, where a ≠ 0.
二次表达式：次数为 2 的多项式，标准形式 ax² + bx + c，a ≠ 0。
Discriminant: the value D = b² − 4ac that determines the nature of the roots of a quadratic equation.
判别式：值 D = b² − 4ac，用于判断二次方程根的性质。
Factorising: rewriting an expression as a product of its factors (e.g., x² − 9 = (x+3)(x−3)).
因式分解：将表达式写成其因式乘积（如 x² − 9 = (x+3)(x−3)）。
Completing the square: converting a quadratic into the form a(x + p)² + q to find the vertex or solve equations.
配方法：将二次式转化为 a(x + p)² + q 的形式，用于求顶点或解方程。
3. Inequalities | 不等式
Inequality: a relation comparing two expressions using signs <, >, ≤, ≥ (e.g., 3x + 1 > 7).
不等式：使用符号 <、>、≤、≥ 比较两个表达式的关系（如 3x + 1 > 7）。
Solution set: the range of values that satisfy an inequality, often shown on a number line or using interval notation.
解集：满足不等式的值的范围，常用数轴或区间记号表示。
Region: in graphical inequalities, the shaded area on a coordinate plane representing all points that satisfy the inequality.
区域：在图形不等式中，坐标平面上代表所有满足不等式点的阴影区。
4. Functions and Graphs | 函数与图像
Function: a rule that assigns each input (x) exactly one output (y), often written f(x).
函数：为每个输入 (x) 指定唯一输出 (y) 的规则，常写作 f(x)。
Domain: the set of all possible input values (x) for which the function is defined.
定义域：函数所有可能输入值 (x) 的集合。
Range: the set of all output values (y) that the function produces.
值域：函数产生的所有输出值 (y) 的集合。
Composite function: a function formed by applying one function to the output of another, denoted f(g(x)).
复合函数：将一个函数应用于另一个函数输出所形成的函数，记作 f(g(x))。
Inverse function at reverses the effect of the original function, denoted f⁻¹(x), satisfying f(f⁻¹(x)) = x.
反函数：逆转原函数效果的函数，记作 f⁻¹(x)，满足 f(f⁻¹(x)) = x。
5. Coordinate Geometry | 坐标几何
Gradient: the steepness of a line, calculated as change in y divided by change in x (m = Δy/Δx).
斜率：直线的倾斜程度，计算为 y 的变化除以 x 的变化 (m = Δy/Δx)。
Midpoint: the point halfway between two points, ( (x₁+x₂)/2, (y₁+y₂)/2 ).
中点：两点间正中间的点，坐标为 ( (x₁+x₂)/2, (y₁+y₂)/2 )。
Equation of a straight line: commonly y = mx + c or y − y₁ = m(x − x₁), where m is gradient, c is y-intercept.
直线方程：常见形式 y = mx + c 或 y − y₁ = m(x − x₁)，其中 m 为斜率，c 为 y 截距。
Equation of a circle: centre (a, b), radius r: (x − a)² + (y − b)² = r².
圆的方程：圆心 (a, b)，半径 r： (x − a)² + (y − b)² = r²。
6. Introduction to Calculus | 微积分入门
Derivative: the instantaneous rate of change of a function, also the gradient of the tangent, denoted f'(x) or dy/dx.
导数：函数的瞬时变化率，也是切线的斜率，记作 f'(x) 或 dy/dx。
Differentiation from first principles: using the limit definition (f(x+h) − f(x))/h as h → 0 to find the derivative.
从基本原理求导：使用极限定义 (f(x+h) − f(x))/h (当 h → 0) 求导。
Stationary point curve where the gradient is zero (dy/dx = 0), could be a maximum, minimum, or inflection.
驻点：曲线上斜率为零的点 (dy/dx = 0)，可能是极大值、极小值或拐点。
Second derivative: the derivative of the derivative, f''(x) or d²y/dx²; used to determine concavity and nature of stationary points.
二阶导数：导数的导数，记作 f''(x) 或 d²y/dx²；用于判断凹凸性和驻点性质。
7. Sequences and Series | 数列与级数
Sequence: an ordered list of numbers following a rule (e.g., 2, 5, 8, ...). Term-to-term rule.
数列：按规则排列的有序数字列表（如 2, 5, 8, ...），有逐项规则。
Arithmetic sequence: a sequence with a constant common difference d; nth term: a + (n−1)d.
等差数列：有恒定公差 d 的数列；第 n 项：a + (n−1)d。
Geometric sequence: a sequence with a constant common ratio r; nth term: arⁿ⁻¹.
等比数列：有恒定公比 r 的数列；第 n 项：arⁿ⁻¹。
Series: the sum of terms of a sequence, often expressed using sigma notation Σ.
级数：数列各项的和，常用求和符号 Σ 表示。
8. Trigonometry | 三角学
Sine, cosine, tangent: the three basic trigonometric ratios: sinθ = opp/hyp, cosθ = adj/hyp, tanθ = opp/adj.
正弦、余弦、正切：三个基本三角比：sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。
Radian: an alternative angle measure where π radians = 180°; used extensively in calculus.
弧度：另一种角度度量，π 弧度 = 180°；微积分中广泛使用。
Trigonometric identity: an equation involving trig ratios that is true for all values, e.g., sin²θ + cos²θ ≡ 1.
三角恒等式：对所有值都成立的含三角比的等式，如 sin²θ + cos²θ ≡ 1。
Sine rule, cosine rule: used for non-right triangles: a/sinA = b/sinB = c/sinC; a² = b² + c² − 2bc cosA.
正弦定理、余弦定理：用于非直角三角形：a/sinA = b/sinB = c/sinC；a² = b² + c² − 2bc cosA。
9. Matrices | 矩阵
Matrix: a rectangular array of numbers in rows and columns; order is m × n (row × column).
矩阵：由数字排列成的矩形阵列；阶数为 m × n（行 × 列）。
Element: an individual number in a matrix, identified by row i, column j, often aᵢⱼ.
元素：矩阵中的单个数字，由第 i 行第 j 列定位，常表示为 aᵢⱼ。
Matrix addition/subtraction: performed element-wise for matrices of the same order.
矩阵加减法：对同阶矩阵逐元素进行。
Matrix multiplication: product AB defined when columns of A = rows of B; entry cᵢⱼ is the dot product of row i of A and column j of B.
矩阵乘法：当 A 的列数等于 B 的行数时定义 AB 乘积；元素 cᵢⱼ 是 A 第 i 行与 B 第 j 列的点积。
10. Vectors | 向量
Vector: a quantity having both magnitude and direction, represented as a column (x y) or in i, j notation.
向量：既有大小又有方向的量，用列向量 (x y) 或 i、j 记号表示。
Magnitude: the length of a vector, for v = (x y) it is √x² + y².
模：向量的长度，对 v = (x y)，为 √x² + y²。
Position vector: a vector from the origin to a point; the position vector of point A is often OA.
位置向量：从原点到某点的向量；点 A 的位置向量常记作 OA。
Resultant: the sum of two or more vectors, found by the triangle law or component addition.
合向量：两个或多个向量的和，由三角形法则或分量相加求得。
11. Proof and Reasoning | 证明与推理
Conjecture: a statement believed to be true but not yet proven, based on pattern observation.
猜想：基于模式观察，相信为真但尚未证明的陈述。
Proof by deduction: using logical steps and known facts to show a statement must be true.
演绎证明：使用逻辑步骤和已知事实，说明一个陈述必然为真。
Counterexample: an example that disproves a conjecture; a single counterexample is sufficient.
反例：驳倒猜想的例子；一个反例就足够。
Implication symbol ⇒: means 'implies'; used in proof chains. Conversely, ⇐ 'is implied by'; ⇔ 'if and only if'.
蕴涵符号 ⇒：表示 '推出'，用于证明链；反之 ⇐ 表示 '由...推出'；⇔ 表示 '当且仅当'。
Exhaustion: a method of proof testing all possible cases, often used when there is a limited number.
穷举法：验证所有可能情况的证明方法，常用于情况数量有限时。
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