IGCSE OCR Further Maths: Quick Vocabulary & Terminology Guide | IGCSE OCR 进阶数学：词汇术语速记指南

📚 IGCSE OCR Further Maths: Quick Vocabulary & Terminology Guide | IGCSE OCR 进阶数学：词汇术语速记指南

Mastering the terminology of IGCSE OCR Further Maths is half the battle. This guide breaks down the key terms you must know for each topic, pairing clear English definitions with their Chinese equivalents and offering memory hooks to make revision faster and deeper. Whether you are brushing up before an exam or building a solid foundation from the start, these bite-sized bilingual explanations will help you think in both languages and truly understand the concepts behind the symbols.

掌握 IGCSE OCR 进阶数学的术语，就成功了一半。本指南将每个主题的关键词汇提炼出来，用清晰的英文定义和中文释义配对，并提供记忆线索，让复习更高效、理解更深刻。无论是考前速览，还是从零开始打基础，这些中英双语小段解释都能帮助你用两种语言思考，真正读懂符号背后的概念。

1. Functions | 函数

Function: A rule that assigns exactly one output (y) to each input (x). You can test it with a vertical line on its graph — if any vertical line cuts the graph more than once, it is not a function.

函数：一种将每个输入值 (x) 唯一对应到一个输出值 (y) 的规则。你可以用在图像上画竖线来判断 —— 如果任何一条竖线与图像相交多于一次，它就不是函数。

Domain: The set of all allowed input values (x-values) for which the function is defined. Think “d for domain, d for the x-axis”.

定义域：函数有定义的所有允许输入值 (x 值) 的集合。可以联想 “定义域盯住 x 轴”。

Range: The set of all possible output values (y-values) the function actually produces. After you feed the domain through the function, what comes out is the range.

值域：函数实际产生的所有可能输出值 (y 值) 的集合。把定义域输入函数后 “流出来” 的就是值域。

Composite function (f∘g): Applying one function after another: f(g(x)) means do g first, then f. Read as “f of g of x”. The order matters — inside first.

复合函数 (f∘g)：相继应用两个函数：f(g(x)) 表示先做 g，再做 f。读作 “f of g of x”。顺序很重要 —— 先算里面的。

Inverse function f⁻¹: to a. Graphically, it is a reflection in the line y = x. Not all functions have an inverse — the function must be one-to-one.

反函数 f⁻¹：逆转 f 的作用。如果 f 把 a 映射到 b，那么 f⁻¹ 就把 b 映回 a。图像上它是关于直线 y = x 的对称。并非所有函数都有反函数 —— 函数必须是一一对应的。

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2. Quadratic Functions | 二次函数

Quadratic function: al of degree 2, typically written as f(x) = ax² + bx + c (a ≠ 0). Its graph is a parabola.

二次函数：次数为 2 的多项式，通常写作 f(x) = ax² + bx + c (a ≠ 0)。它的图像是抛物线。

Discriminant (Δ = b² − 4ac): s: Δ > 0 → two distinct real roots, Δ = 0 → one repeated real root, Δ < 0 → no real roots.

判别式 (Δ = b² − 4ac)：告诉你二次方程 ax² + bx + c = 0 有多少个实数根：Δ > 0 → 两个不等实根，Δ = 0 → 一个重根，Δ < 0 → 无实根。

Vertex / turning point: e : The maximum or minimum point of the parabola. For y = a(x − h)² + k, the vertex is (h, k). Completing the square reveals it.

顶点 / 转折点：抛物线的最高点或最低点。对于 y = a(x − h)² + k，顶点是 (h, k)。用配方法就能找出来。

Axis of symmetry: e x = h that cuts the parabola into two mirror-image halves. It passes through the vertex.

对称轴：竖直线 x = h，将抛物线分成镜像对称的两半。它穿过顶点。

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3. Equations & Inequalities | 方程与不等式

Root / solution: e that makes the equation true. For f(x) = 0, the roots are the x-intercepts of the graph.

根 / 解：使方程成立的变量值。对于 f(x) = 0，根就是图像与 x 轴的截距。

Inequality: A statement that one expression is greater than, less than, or equal to another. Solving gives a range of values, often shown on a number line.

不等式：表示一个表达式大于、小于或等于另一个表达式的陈述。求解得到的是一个取值范围，常在数轴上表示。

Critical values: The values of x where the related equation equals zero. They divide the number line into intervals for testing signs.

临界值：使相关方程等于零的 x 值。它们将数轴分成几个区间，用来检验符号。

Simultaneous equations: Two (or more) equations that share variables. Solve by substitution, elimination, or graphically — look for intersection points.

联立方程组：共用变量的两个（或多个）方程。可用代入法、消元法或图像法求解 —— 寻找交点。

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4. Polynomials | 多项式

Polynomial: An expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non‑negative integer. Degree is the highest power of x.

多项式：形式为 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ 的表达式，其中 n 为非负整数。次数是 x 的最高次幂。

Factor theorem: (x − a) is a factor of the polynomial f(x). It helps you factorise and find roots quickly.

因式定理：如果 f(a) = 0，那么 (x − a) 就是多项式 f(x) 的因式。它能帮你快速分解因式并找根。

Remainder theorem: When a polynomial f(x) is divided by (x − a), the remainder is f(a). A shortcut for finding remainders without long division.

余式定理：当多项式 f(x) 除以 (x − a) 时，余数等于 f(a)。这是不用长除法就求出余数的捷径。

Cubic polynomial: A polynomial of degree 3, e.g. ax³ + bx² + cx + d. Its graph can have up to two turning points and up to three real roots.

三次多项式：次数为 3 的多项式，如 ax³ + bx² + cx + d。它的图像最多有两个转折点，最多有三个实根。

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5. Exponentials & Logarithms | 指数与对数

Exponential function: A function of the form f(x) = aˣ where a > 0, a ≠ 1. The most important base is e ≈ 2.718; y = eˣ has gradient equal to its own value.

指数函数：形式为 f(x) = aˣ 的函数，其中 a > 0, a ≠ 1。最重要的底数是 e ≈ 2.718；y = eˣ 的梯度等于它自身的值。

Logarithm: The inverse of an exponential. logₐ b = c means a^c = b. “log” without a base usually means base 10; “ln” means natural log, base e.

对数：指数的逆运算。logₐ b = c 意味着 a^c = b。不带底数的 “log” 通常表示以 10 为底；“ln” 表示自然对数，以 e 为底。

Laws of logs: y; log(xⁿ) = n log x. These turn multiplication into addition, making calculations easier.

对数运算法则：log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x。它们把乘法变成加法，使计算更简单。

Exponential growth/decay: Models of the form y = A e^kt. If k > 0, it grows; if k < 0, it decays. Used in population, radioactive decay, and compound interest.

指数增长 / 衰减：模型为 y = A e^kt。若 k > 0 则增长；若 k < 0 则衰减。用于人口、放射性衰变和复利计算。

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6. Trigonometry | 三角学

Sine, cosine, tangent: Ratios of sides in a right‑ ngle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. “SOH CAH TOA” is the classic mnemonic.

正弦、余弦、正切：直角三角形中边的比值：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。“SOH CAH TOA” 是经典记忆法。

Radian measure: °. One radian is the angle subtended by an arc length equal to the radius. Radians are essential for calculus with trig functions.

弧度制：角的另一种度量。π 弧度 = 180°。一弧度是弧长等于半径时所对的圆心角。对三角函数的微积分运算，弧度制必不可少。

Trigonometric identities: sin²θ + cos²θ ≡ 1; tan θ ≡ sin θ / cos θ. These identities are true for all values of θ and are used to simplify expressions and solve equations.

三角恒等式：sin²θ + cos²θ ≡ 1；tan θ ≡ sin θ / cos θ。这些恒等式对所有 θ 都成立，用于化简表达式和解方程。

Ambiguous case (sine rule): ssible solutions (θ and 180° − θ) because sin θ = sin(180° − θ). Always check the context.

模糊情况（正弦定理）：用正弦定理求角时，可能有两个解（θ 和 180° − θ），因为 sin θ = sin(180° − θ)。一定要根据题目条件检验。

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7. Differentiation | 微分

Derivative: The rate of change of a function, denoted f'(x) or dy/dx. It gives the gradient of the tangent at any point on the curve.

导数：函数的变化率，记作 f'(x) 或 dy/dx。它给出了曲线上任意一点切线的斜率。

Power rule: If f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. Bring the power down and reduce the power by one.

幂法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。把指数拿下来，指数减一。

Stationary point: minimum, or point of inflection. Use the second derivative or a sign table to classify it.

驻点：导数为零的点 (dy/dx = 0)。它可能是极大值、极小值或拐点。用二阶导数或符号表来分类。

Chain rule: then dy/dx = (dy/du) × (du/dx). Use it to differentiate composite functions like (3x+2)⁵.

链式法则：如果 y 是 u 的函数，u 是 x 的函数，那么 dy/dx = (dy/du) × (du/dx)。用它来对复合函数如 (3x+2)⁵ 求导。

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8. Integration | 积分

Indefinite integral: The reverse of differentiation. ∫ f(x) dx = F(x) + c, where F'(x) = f(x). The constant c is the constant of integration.

不定积分：微分的逆运算。∫ f(x) dx = F(x) + c，其中 F'(x) = f(x)。常数 c 是积分常数。

Definite integral: ∫ₐᵇ f(x) dx gives the exact area between the curve y = f(x), the x‑axis, and the lines x = a and x = b (taking account of signs).

定积分：∫ₐᵇ f(x) dx 求出曲线 y = f(x) 与 x 轴以及直线 x = a、x = b 之间（考虑正负）的精确面积。

Fundamental theorem of calculus: Connects differentiation and integration: if F' = f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

微积分基本定理：连接微分与积分：如果 F' = f，那么 ∫ₐᵇ f(x) dx = F(b) − F(a)。

Power rule for integration: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, for n ≠ −1. Reverse of the differentiation power rule.

积分的幂法则：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c，n ≠ −1。这是微分幂法则的逆运算。

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9. Vectors | 向量

Vector: A quantity with both magnitude (size) and direction. Often written in column form (x, y) or as xi + yj. Contrast with scalar — a number that has only magnitude.

向量：既有大小又有方向的量。常写作列形式 (x, y) 或 xi + yj。与标量相对 —— 标量只有大小。

Magnitude (modulus): The length of a vector. For a = xi + yj, |a| = √x² + y². Always non‑negative.

模（大小）：向量的长度。对于 a = xi + yj，|a| = √x² + y²。总是非负数。

Unit vector: A vector with magnitude 1. Often used to indicate direction. The unit vector in the direction of a is a/|a|.

单位向量：模为 1 的向量。常用来表示方向。沿向量 a 方向的单位向量是 a/|a|。

Dot product (scalar product): a · b = |a||b| lso calculated as x₁x₂ + y₁y₂ for 2D vectors. If a · b = 0, the vectors are perpendicular.

点积（标量积）：a · b = |a||b| cos θ，在二维中也可按 x₁x₂ + y₁y₂ 计算。若 a · b = 0，则两向量垂直。

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10. Sequences & Series | 数列与级数

Sequence: An ordered list of numbers following a rule. Each number is a term. For example, 2, 5, 8, 11,... is an arithmetic sequence.

数列：按照某一规则排列的一列数。每个数叫一项。例如 2, 5, 8, 11,... 是一个等差数列。

Arithmetic progression (AP): uence where the difference d between consecutive terms is constant. nth term = a + (n−1)d.

等差数列 (AP)：相邻两项的差 d 为常数的数列。第 n 项 = a + (n−1)d。

Geometric progression (GP): where each term is multiplied by a constant ratio r to get the next. nth term = a rⁿ⁻¹. If |r| < 1, the infinite sum converges to a/(1−r).

等比数列 (GP)：每一项乘以常数比 r 得到后一项的数列。第 n 项 = a rⁿ⁻¹。若 |r| < 1，无穷级数收敛于 a/(1−r)。

Sigma notation (Σ): to write a sum. Σ (from k=1 to n) of f(k) means add up all f(k) for integer k from 1 to n.

西格玛记号 (Σ)：一种简洁写和式的方法。Σ (k=1 到 n) f(k) 表示对从 1 到 n 的所有整数 k，把 f(k) 全部加起来。

Convergent series: An infinite series whose sum approaches a finite number. In GP, this happens when |r| < 1. Otherwise it diverges.

收敛级数：和趋向一个有限值的无穷级数。等比级数中，当 |r| < 1 时收敛，否则发散。

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