📚 GCSE AQA Further Maths: Vocabulary and Terminology Quick Reference Guide | GCSE AQA 进阶数学：词汇术语速记指南
Welcome to your quick-reference guide for essential vocabulary and terminology in AQA GCSE Further Maths. Mastering the precise language of advanced algebra, calculus, matrices and geometry is crucial for tackling multi-step problems and communicating solutions clearly. In each section you will find key terms paired with concise English definitions followed by their Chinese equivalents, making it easy to memorise and recall the concepts you need in the exam.
欢迎使用AQA GCSE进阶数学核心词汇术语速记指南。掌握代数、微积分、矩阵与几何的精确用语对解决多步问题与清晰表达解题过程至关重要。本文每一节都将列出核心术语，先提供简明的英文释义，紧接着是中文对应释义，帮助你轻松记忆并在考试中快速调用。
1. Number and Surds | 数与根式
Surd: An irrational number expressed exactly in root form, such as √2 or ∛5, which cannot be simplified to a whole number or fraction.
根式：以根号形式精确表示的无理数，如√2或∛5，无法简化为整数或分数。
Rationalise the denominator: The process of rewriting a fraction so that the denominator contains no surds, e.g. 1/√2 becomes √2/2 by multiplying top and bottom by √2.
分母有理化：将分母中含有根式的分数改写，使分母不含根式，如1/√2通过分子分母同乘√2化为√2/2。
Index (or exponent): The small raised number indicating how many times a base is multiplied by itself; in 3⁴, 4 is the index.
指数：表示底数自乘次数的小型上标数字，如3⁴中的4。
Base: The number that is raised to a power in an expression like an, where a is the base.
底数：在指数表达式中被乘方运算的数字，如an中的a。
Law of indices: multiplication: am × an = am+n. When multiplying like bases, add the indices.
指数乘法法则：am × an = am+n。同底数幂相乘，指数相加。
Law of indices: power of a power: (am)n = am×n. The indices are multiplied when a power is raised to another power.
指数的幂法则：(am)n = am×n。幂的乘方时，指数相乘。
Fractional index: An index expressed as a fraction, where the numerator is a power and the denominator is a root; a½ = √a, a⅔ = ∛(a²).
分数指数：以分数形式出现的指数，分子为幂，分母为根指数；如a½ = √a，a⅔ = ∛(a²)。
Negative index: Indicates a reciprocal; a−n = 1 / an, moving the term to the denominator.
负指数：表示倒数；a−n = 1 / an，将该项移至分母。
2. Algebraic Manipulation | 代数运算
Like terms: Terms that contain exactly the same variable parts raised to the same powers; only like terms can be added or subtracted.
同类项：含有完全相同变量部分及其指数的项；只有同类项才能进行加减合并。
Expanding (brackets): Multiplying each term inside a bracket by the factor outside, e.g. a(b + c) = ab + ac.
去括号（展开）：将括号外的因子乘以括号内的每一项，如a(b + c) = ab + ac。
Factorising: The reverse of expanding; writing an expression as a product of its factors, usually by finding the highest common factor or using special identities.
因式分解：展开的逆运算；将表达式写成因子的乘积，通常先提取最大公因式或运用乘法公式。
Linear equation n equation of degree 1, typically in the form ax + b = 0, which graphs as a straight line.
线性方程：最高次数为1的方程，一般形式为ax + b = 0，其图像是一条直线。
Simultaneous equations: A set of two or more equations involving the same unknowns, solved together to find values that satisfy all equations.
联立方程：一组含有相同未知数的方程，需要同时求解满足所有方程的解。
Elimination method: Solving simultaneous equations by adding or subtracting equations to remove one variable, making it possible to solve for the other.
消元法：通过将方程相加或相减消去一个变量，从而可先解出另一个未知数的联立方程求解方法。
Substitution method: Solving by rearranging one equation for one variable and substituting that expression into the other equation.
代入法：将其中一个方程变形为一个变量的表达式，再代入另一个方程求解的方法。
3. Quadratic Functions and Expressions | 二次函数与表达式
Quadratic expression: An algebraic expression of degree 2, typically ax² + bx + c, where a ≠ 0.
二次表达式：最高次数为2的代数式，一般形式为ax² + bx + c，其中a ≠ 0。
Completing the square in the form a(x + p)² + q to reveal its vertex or solve equations; e.g. x² + 6x = (x + 3)² – 9.
配方法：将二次式变形成a(x + p)² + q的形式来显示顶点或解方程；如x² + 6x = (x + 3)² – 9。
Discriminant: The expression b² – 4ac from the quadratic formula, which determines the number and type of roots (real/distinct, repeated, or complex).
判别式：二次公式根号下的b² – 4ac，用于判定二次方程根的数量与类型（相异实根、重根或无实根）。
Quadratic formula: x = (–b ± √b² – 4ac) / 2a; gives the solutions of ax² + bx + c = 0.
二次公式：x = (–b ± √b² – 4ac) / 2a，用于求ax² + bx + c = 0的解。
Roots of a quadratic: The values of x where the quadratic equals zero; graphically these are the x-intercepts.
二次方程的根：使二次式等于零的x值；图像上为抛物线与x轴的交点。
Vertex (turning point): The highest or lowest point on the parabola y = ax² + bx + c; coordinates given by (–b/(2a), f(–b/(2a))).
顶点（转折点）：抛物线y = ax² + bx + c的最高点或最低点；坐标可用(–b/(2a), f(–b/(2a)))表示。
Axis of symmetry ine passing through the vertex, dividing the parabola into two mirror images; equation x = –b/(2a).
对称轴：过顶点并平分抛物线的垂直线，使两侧镜像对称；方程为x = –b/(2a)。
4. Polynomials and Factor Theorem | 多项式与因式定理
Polynomial: An algebraic expression consisting of terms in powers of a variable, e.g. 2x³ – 5x + 7; named by degree: linear, quadratic, cubic, quartic.
多项式：由变量的非负整数次幂项组成的代数式，如2x³ – 5x + 7；按次数称为一次、二次、三次、四次多项式等。
Degree of a polynomial: The highest power of the variable in the polynomial; in 4x⁵ + 2x², the degree is 5.
多项式的次数：多项式中变量的最高指数；在4x⁵ + 2x²中，次数为5。
Coefficient: The numerical factor multiplying a variable or power of a variable; in 3x², 3 is the coefficient.
系数：变量或其幂次前面的数字因数；在3x²中，3是系数。
Factor theorem: For a polynomial f(x), (x – a) is a factor if and only if f(a) = 0. Used to factorise cubic and higher polynomials.
因式定理：对于多项式f(x)，(x – a)是其因式的充要条件是f(a) = 0。常用于三次及以上多项式的因式分解。
Remainder theorem: When a polynomial f(x) is divided by (x – a), the remainder is f(a). Useful for quick evaluation without full division.
余式定理：多项式f(x)除以(x – a)时，余数为f(a)。无需完整除法即可快速求值。
Cubic expression: A polynomial of degree 3, general form ax³ + bx² + cx + d; can be factorised using the factor theorem.
三次式：次数为3的多项式，一般形式为ax³ + bx² + cx + d；可借助因式定理分解。
Quotient and divisor: In polynomial division, the divisor is the polynomial you divide by, and the quotient is the result after division, excluding the remainder.
商与除式：在多项式除法中，除式是用于除的式子，商是除法结果中除余式外的部分。
5. Binomial Expansion | 二项式展开
Binomial: An algebraic expression containing two terms, such as (a + b) or (x – 3). The expansion (. . .)ⁿ for positive integer n follows a pattern of coefficients.
二项式：含有两项的代数式，如(a + b)或(x – 3)。其正整数次幂( . . . )ⁿ的展开遵循特定系数模式。
Pascal's triangle: A triangular array of numbers where each number is the sum of the two above; gives the coefficients for (a + b)ⁿ expansions.
帕斯卡三角形：一种数字三角形，每个数是上方两数之和；提供了(a + b)ⁿ展开的各项系数。
Binomial coefficient (nCr): The number of ways to choose r items from n, written as nCr or C(n,r); used in the general term of a binomial expansion.
二项式系数（nCr）：从n个中选取r个的组合数，记作nCr或C(n,r)；用于二项展开式的通项。
Factorial: The product of all positive integers up to n, denoted n!; e.g. 4! = 4 × 3 × 2 × 1 = 24. Used in nCr = n! / [r!(n–r)!].
阶乘：所有小于等于n的正整数的乘积，记作n!；如4! = 24。算式nCr = n! / [r!(n–r)!]中用到。
General term of expansion: In (a + b)ⁿ, the (r+1)th term is nCr × aⁿ⁻ʳ × bʳ, where r goes from 0 to n.
展开通项：在(a + b)ⁿ展开中，第(r+1)项为nCr × aⁿ⁻ʳ × bʳ，r取0到n。
Coefficient of a specific term: For a given power, find the appropriate r and compute nCr × (coefficient of a)n–r × (coefficient of b)r.
特定项的系数：对某指定次幂，确定相应r并计算nCr × (a的系数)n–r × (b的系数)r。
6. Coordinate Geometry: Lines and Circles | 坐标几何：直线与圆
Gradient (slope): The steepness of a line, usually m; m = (y₂ – y₁)/(x₂ – x₁) between two points.
斜率：直线的倾斜程度，通常记为m；两点间斜率m = (y₂ – y₁)/(x₂ – x₁)。
y-intercept: The point where a line crosses the y-axis; for y = mx + c, c is the y-intercept.
y轴截距：直线与y轴的交点；在y = mx + c中，c即为y轴截距。
Midpoint: The point exactly halfway between two given points; coordinates: ((x₁+x₂)/2, (y₁+y₂)/2).
中点：两点间正中央的点；坐标为((x₁+x₂)/2, (y₁+y₂)/2)。
Perpendicular bisector: A line that is perpendicular to a segment and passes through its midpoint; its gradient is the negative reciprocal of the segment's gradient.
垂直平分线：与线段垂直且通过该线段中点的直线；其斜率是该线段斜率的负倒数。
Equation of a circle (centre origin): x² + y² = r², where (0,0) is the centre and r is the radius.
圆心在原点的圆方程：x² + y² = r²，(0,0)为圆心，r为半径。
Equation of a circle (general): (x – a)² + (y – b)² = r², centre (a,b), radius r.
一般位置圆方程：(x – a)² + (y – b)² = r²，圆心(a,b)，半径为r。
Tangent to a circle: A straight line that touches the circle at exactly one point; the radius to that point is perpendicular to the tangent.
圆的切线：与圆仅相交于一点的直线；该点处的半径垂直于切线。
7. Functions and Graph Transformations | 函数与图像变换
Function that assigns each input (x) exactly one output (y); often written as f(x) = an expression.
函数：一种将每个输入x对应唯一输出y的规则；常记作f(x) = 表达式。
Domain: The complete set of possible input values (x-values) for which the function is defined.
定义域：函数有定义的所有可能输入值（x值）的集合。
Range: The set of all possible output values (y-values) produced by the function over its domain.
值域：函数在定义域内产生的所有可能输出值（y值）的集合。
Composite function: One function applied inside another; fg(x) = f(g(x)), meaning first apply g, then f.
复合函数：一个函数作用于另一个函数内部；fg(x) = f(g(x))，即先执行g函数，再执行f函数。
Inverse function ct of f, denoted f⁻¹(x); it satisfies f(f⁻¹(x)) = x, and graphs reflect in the line y = x.
反函数：能将f的作用逆转的函数，记作f⁻¹(x)；满足f(f⁻¹(x)) = x，其图像与y = x对称。
Translation (transformation) graph vertically or horizontally: y = f(x) + a moves it a units up; y = f(x + a) moves it –a units left.
平移变换：垂直或水平移动图像：y = f(x) + a 上移a个单位；y = f(x + a) 左移a个单位（因内加左移）。
Reflection: Flipping the graph: y = –f(x) reflects in the x-axis; y = f(–x) reflects in the y-axis.
反射变换：翻转图像：y = –f(x) 关于x轴反射；y = f(–x) 关于y轴反射。
Stretch = a f(x) stretches vertically by factor a; y = f(bx) stretches horizontally by factor 1/b.
拉伸变换：缩放图像：y = a f(x) 垂直方向拉伸a倍；y = f(bx) 水平方向拉伸1/b倍。
8. Trigonometry | 三角学
Sine, cosine, tangent ngle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
正弦、余弦、正切：直角三角形中的三角函数比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
SOHCAHTOA: A mnemonic for the three trigonometric
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