📚 Year 11 CAIE Further Mathematics: Vocabulary & Terminology Quick Guide | CAIE Year 11 进阶数学：词汇术语速记指南
Success in CAIE Year 11 Further Mathematics depends not only on mastering techniques but also on using the correct mathematical language. This guide breaks down the essential vocabulary and terminology you will encounter across the syllabus, giving clear English definitions alongside their Chinese equivalents to help you read questions accurately and present solutions with confidence.
在CAIE Year 11 进阶数学中取得优异成绩不仅需要掌握解题方法，更需要使用准确的数学语言。本文为你整理了贯穿考纲的核心词汇和术语，提供清晰的英文定义和对应的中文解释，帮助你准确理解题意、自信地表达解题过程。
1. Functions and Graphs | 函数与图像
A function maps each input value to exactly one output. Understanding its domain, range, and the notation for inverses and composites is the foundation for working with graphs and transformations.
函数将每个输入值映射到唯一的输出值。理解定义域、值域以及反函数和复合函数的符号，是处理图像和变换的基础。
Function: a rule that assigns exactly one output for every input, often written as f(x).
函数：将每个输入对应唯一输出的规则，通常记作 f(x)。
Domain: the set of all 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) that the function can produce.
值域：函数能产生的所有可能输出值 (y 值) 的集合。
Inverse function f⁻¹: reverses the effect of f; satisfies f(f⁻¹(x)) = x.
反函数 f⁻¹：撤销 f 的作用；满足 f(f⁻¹(x)) = x。
Composite function (f∘g)(x): applying g first, then f; read as 'f composed with g'.
复合函数 (f∘g)(x)：先作用 g，再作用 f；读作 'f 与 g 的复合'。
One-to-one function: each y-value corresponds to exactly one x-value; passes the horizontal line test.
一一映射函数：每个 y 值只对应一个 x 值；通过水平线检验。
Vertical line test: a graphical check to see if a relation is a function; a vertical line must not intersect the graph more than once.
垂直线检验：判断一个关系是否为函数的图像检查；任意垂直线与图形的交点不能超过一个。
Transformation: operations such as translation, reflection, or stretch that alter the position or shape of a graph.
变换：平移、对称、伸缩等改变图像位置或形状的操作。
2. Quadratic Functions and Equations | 二次函数与方程
ppears everywhere. Key terms revolve around its parabolic shape, discriminant, and methods like completing the square and the quadratic formula.
二次函数 y = ax² + bx + c 无处不在。核心术语围绕抛物线的形状、判别式以及配方法和求根公式等方法展开。
Parabola: the U-shaped graph of a quadratic function; opens upwards if a > 0, downwards if a < 0.
抛物线：二次函数的 U 形图像；当 a > 0 时开口向上，a < 0 时开口向下。
Vertex: the turning point (maximum or minimum) of a parabola.
顶点：抛物线的转折点（极大值点或极小值点）。
Axis of symmetry: a vertical line through the vertex that divides the parabola into two mirror images.
对称轴：通过顶点且将抛物线分为对称两部分的垂直线。
Roots (or zeros): the x-values where the graph cuts the x-axis; solutions of f(x) = 0.
根（或零点）：图像与 x 轴交点的横坐标；方程 f(x) = 0 的解。
ives two distinct real roots, Δ = 0 gives one repeated root, Δ < 0 gives no real roots.
判别式 Δ = b² − 4ac：决定根的性质；Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
Completing the square: rewriting ax² + bx + c in the form a(x − h)² + k to find the vertex easily.
配方法：将 ax² + bx + c 改写为 a(x − h)² + k 的形式，方便求出顶点。
Quadratic formula: x = [−b ± √b² − 4ac] / (2a); solves any quadratic equation.
求根公式：x = [−b ± √b² − 4ac] / (2a)；可解任意二次方程。
3. Polynomials and Binomial Expansion | 多项式与二项式展开
Polynomials are expressions of the form aₙxⁿ + ... + a₁x + a₀. The Factor Theorem and Remainder Theorem help factorise, while binomial expansion deals with powers of (a + b)ⁿ.
多项式是形如 aₙxⁿ + ... + a₁x + a₀ 的表达式。因式定理和余数定理用于因式分解，二项式展开则处理 (a + b)ⁿ 的乘方。
Polynomial: an algebraic expression consisting of terms with non-negative integer exponents.
多项式：由指数为非负整数的项组成的代数表达式。
Degree of a polynomial: the highest power of the variable present.
多项式的次数：变量出现的最高次幂。
Coefficient: the constant factor multiplying a term, e.g., in 5x³ the coefficient is 5.
系数：乘在项前面的常数因子，如 5x³ 中系数为 5。
Factor Theorem: (x − a) is a factor of P(x) iff P(a) = 0.
因式定理：(x − a) 是 P(x) 的因式当且仅当 P(a) = 0。
Remainder Theorem: when P(x) is divided by (x − a), the remainder is P(a).
余数定理：P(x) 除以 (x − a) 的余数为 P(a)。
Binomial expansion: (a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ, where ⁿCᵣ = n! / (r!(n−r)!).
二项式展开：(a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ，其中 ⁿCᵣ = n! / (r!(n−r)!).
Pascal's triangle: a triangular array of numbers giving binomial coefficients for small n.
杨辉三角（帕斯卡三角形）：给出较小 n 时二项式系数的三角形数表。
4. Coordinate Geometry and Straight Lines | 坐标几何与直线
This topic connects algebra with geometry through points, lines, and circles. Gradient, distance, midpoint, and equations are the must-know terms.
本主题通过点、直线和圆将代数与几何联系起来。斜率、距离、中点和方程是必须掌握的术语。
Gradient (slope): a measure of steepness, m = (y₂ − y₁) / (x₂ − x₁).
斜率（坡度）：衡量倾斜程度的量，m = (y₂ − y₁) / (x₂ − x₁)。
Distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²] gives the length between two points.
距离公式：d = √[(x₂ − x₁)² + (y₂ − y₁)²] 给出了两点间线段的长度。
Midpoint formula: M = ((x₁+x₂)/2 , (y₁+y₂)/2).
中点公式：M = ((x₁+x₂)/2 , (y₁+y₂)/2)。
ht line: can be written as y = mx + c (slope-intercept) or ax + by + c = 0 (general form).
直线的方程：可以写成 y = mx + c（斜截式）或 ax + by + c = 0（一般式）。
Parallel lines: lines with the same gradient; m₁ = m₂.
平行线：斜率相同的两条直线；m₁ = m₂。
Perpendicular lines: lines whose gradients multiply to −1; m₁ × m₂ = −1.
垂直线：斜率乘积为 −1 的两条直线；m₁ × m₂ = −1。
Circle equation: (x − a)² + (y − b)² = r², centre (a, b) and radius r.
圆的方程：(x − a)² + (y − b)² = r²，圆心 (a, b)，半径 r。
5. Trigonometry and Circular Measure | 三角学与弧度制
Trigonometry extends beyond right-angled triangles to general angles, radians, and the graphs of sine, cosine, and tangent. Circular measure links arc lengths and sector areas to the central angle in radians.
三角学从直角三角形延伸到任意角、弧度以及正弦、余弦和正切的图像。弧度制将弧长和扇形面积与以弧度表示的圆心角联系起来。
nit of angle; 1 radian is the angle subtended by an arc equal to the radius. π rad = 180°.
弧度：角度单位；1 弧度是弧长等于半径时所对的圆心角。π rad = 180°。
Degree: traditional unit of angle; 1 full rotation = 360°.
度：传统的角度单位；一整圈 = 360°。
Arc length: s = rθ, where θ is in radians.
弧长：s = rθ，其中 θ 以弧度计。
Sector area: A = ½ r² θ, again with θ in radians.
扇形面积：A = ½ r² θ，同样 θ 以弧度计。
angle θ, sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent.
正弦、余弦、正切：三角比；对于角 θ，sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。
values: commonly required for 30°, 45°, 60° (π/6, π/4, π/3); e.g., sin30° = ½, tan45° = 1.
特殊精确值：常考查 30°, 45°, 60° (π/6, π/4, π/3)；例如 sin30° = ½, tan45° = 1。
Pythagorean identity: sin²θ + cos²θ = 1.
勾股恒等式：sin²θ + cos²θ = 1。
Amplitude and period: for y = a sin(bx), amplitude = |a|, period = 2π/b.
振幅与周期：对于 y = a sin(bx)，振幅 = |a|，周期 = 2π/b。
6. Vectors in Two Dimensions | 平面向量
Vectors have both magnitude and direction, unlike scalars. Operations include addition, scalar multiplication, and the dot product. Unit vectors i and j provide a standard basis for representation.
向量既有大小又有方向，与标量不同。运算包括加法、数乘和点积。单位向量 i 和 j 为标准表示提供了基。
Vector: a quantity with magnitude and direction, e.g., displacement, velocity. Represented as a, AB, or column form.
向量：既有大小又有方向的量，如位移、速度。表示为 a, AB 或列向量形式。
Magnitude: the length of a vector; for v = (x, y), |v| = √x² + y².
模（大小）：向量的长度；若 v = (x, y)，则 |v| = √x² + y²。
Position vector: a vector from the origin to a point; e.g., point A(a₁, a₂) has position vector a = (a₁, a₂).
位置向量：从原点到某点的向量；例如点 A(a₁, a₂) 的位置向量是 a = (a₁, a₂)。
Unit vector: a vector with magnitude 1 in a given direction; i = (1, 0), j = (0, 1).
单位向量：指定方向上模为 1 的向量；i = (1, 0), j = (0, 1)。
Equal vectors: same magnitude and direction, regardless of their initial point.
相等向量：大小相等且方向相同，与起点位置无关。
Scalar multiplication: multiplying a vector by a constant changes its magnitude (and possibly direction).
数乘：一个向量乘以常数改变其模长（并可能改变方向）。
Dot (scalar) product: a·b = |a||b|cosθ; for column vectors (x₁, y₁)·(x₂, y₂) = x₁x₂ + y₁y₂.
点积（数量积）：a·b = |a||b|cosθ；对列向量 (x₁, y₁)·(x₂, y₂) = x₁x₂ + y₁y₂。
Perpendicular vectors: two vectors are perpendicular if and only if their dot product is zero.
垂直向量：两个向量垂直当且仅当它们的点积为零。
7. Exponential and Logarithmic Functions | 指数与对数函数
Exponential growth and its inverse, the logarithm, appear in many real-world applications. The constant e and natural logarithms are central, along with index laws and the change-of-base formula.
指数增长及其反函数——对数，出现在许多实际应用中。常数 e 和自然对数占据核心地位，同时需要掌握指数律和换底公式。
Exponential function f(x) = aˣ: with a > 0; the special case eˣ has gradient eˣ everywhere.
指数函数 f(x) = aˣ：其中 a > 0；特殊情况 eˣ 处处切线斜率等于自身值。
Logarithm
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