IGCSE Cambridge Additional Mathematics: Vocabulary & Terminology Quick Memory Guide | IGCSE Cambridge进阶数学：词汇术语速记指南

📚 IGCSE Cambridge Additional Mathematics: Vocabulary & Terminology Quick Memory Guide | IGCSE Cambridge进阶数学：词汇术语速记指南

Mastering the language of Additional Mathematics is half the battle. This guide covers the essential terms you will encounter in the Cambridge IGCSE 0606 syllabus, with simple definitions and memory tricks to help you recall them fast. Each term appears in English first, followed by its Chinese equivalent and explanation.

掌握进阶数学的语言是成功的一半。本篇速记指南覆盖了Cambridge IGCSE 0606考纲中的核心术语，每个术语均配有简明定义与记忆窍门，帮你快速记牢。先英文后中文，一一对应，便于对照学习。

1. Functions | 函数

Function: A special relation where each input (x) has exactly one output (y). Think of a vending machine – one button gives one drink.

函数：一种特殊的对应关系，每个输入（x）有且只有一个输出（y）。想象自动售货机——按一个按钮只出一罐饮料。

Domain: All possible input values (x-values) that the function can accept. Memory tip: 'Domain' starts with 'Do' – ask 'What can x do?'

定义域：函数可以接受的所有输入值（x值）。记忆窍门：'Domain'以'Do'开头——问'x能做什么？'

Range: All possible output values (y-values) the function can produce. 'Range' rhymes with 'change' – outputs change as x varies.

值域：函数可以产生的所有输出值（y值）。'Range'与'change'押韵——当x变化时，输出也变化。

One-to-one function: Each y-value comes from exactly one x-value. Passes the horizontal line test. Think of strict ID – one person, one passport.

一一映射函数：每个y值仅由一个x值得到，通过水平线检验。想象严格的身份证——一人一证。

Inverse function f⁻¹: Reverses the effect of f. Swap x and y and solve for y. The '-1' is not an exponent – it signals 'undo'. Graph is a reflection in y = x.

反函数 f⁻¹：将f的作用逆转。交换x与y并解出y。'-1'不是指数——它表示'撤销'操作。图像关于直线y = x对称。

Composite function gf(x): First apply f, then g. Read from right to left: g after f. Think 'g follows f'.

复合函数 gf(x)：先作用f，再作用g。从右往左读：f之后是g。记住'g跟着f'。

Modulus function |x|: Gives the absolute value or distance from zero. Always non-negative. 'Modulus' makes everything positive, like a mood lifter.

绝对值函数 |x|：给出数值的绝对值或到零点的距离。结果总是非负的。'Modulus'让一切变正，像情绪振奋器。

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2. Quadratic Functions | 二次函数

General form ax² + bx + c: a, b, c are constants, a ≠ 0. The graph is a parabola. 'a' affects whether it smiles (a > 0) or frowns (a < 0).

一般式 ax² + bx + c：a, b, c为常数，a ≠ 0。图像为抛物线。a决定开口向上（a > 0）还是向下（a < 0）。

Vertex form a(x – h)² + k: (h, k) is the turning point. 'h' for horizontal shift, 'k' for vertical. 'Vertex' – think 'vertex = peak or valley'.

顶点式 a(x – h)² + k：(h, k)为顶点。h代表水平移位，k代表垂直移位。'Vertex'——记住'顶点即峰顶或谷底'。

Discriminant Δ = b² – 4ac of roots. Δ > 0: two real distinct roots; Δ = 0: one repeated root; Δ < 0: no real roots. 'Discriminant discriminates root types.'

判别式 Δ = b² – 4ac：揭示根的性质。Δ > 0：两个不等实根；Δ = 0：一个重根；Δ < 0：无实根。'判别式判别根的种类'。

Completing the square: Rewrite ax² + bx + c → a(x + p)² + q. Helps find vertex and solve quadratics. 'Complete the square to find the peak—perfect square reveals the vertex.'

配方法：将二次式写成a(x + p)² + q的形式。用于求顶点和解二次方程。'配平方找顶点——完全平方揭顶点'。

Roots / solutions / zeros : x-values where the graph crosses the x-axis (y = 0). Solving ax² + bx + c = 0. All three words mean the same.

根 / 解 / 零点：图像与x轴交点的x值（y = 0）。解方程ax² + bx + c = 0。三个词同义。

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3. Equations and Inequalities | 方程与不等式

Linear equation: Degree 1, e.g. 2x + 3 = 7. Straight line when graphed. 'Linear = line'.

线性方程：次数为1，如2x + 3 = 7。图像为直线。'Linear即直线'。

Quadratic inequality: e.g. x² – 5x + 6 > 0. Solve by sketching the parabola, not just algebra. 'Sketch it to catch the region.'

二次不等式：如x² – 5x + 6 > 0。通过画抛物线草图求解，不可只靠代数。'画图捕捉区域'。

Interval notation: Uses brackets: (a, b) open interval, [a, b] closed. Round brackets think 'open arms', square brackets 'closed box'.

区间表示法：圆括号( )表示开区间，方括号[ ]表示闭区间。圆括号像张开的双臂，方括号如关闭的盒子。

Set notation { x | condition }: "Set of all x such that...". The vertical bar means 'such that'. Curly braces enclose the group.

集合表示法 { x | 条件 }：'满足条件的全体x的集合'。竖线表示'满足'。花括号包裹整个集合。

Critical values: The x-values where the inequality changes from true to false, usually where the expression equals zero. They 'critically' split the number line.

临界值：使不等式真值发生转变的x值，通常为表达式等于零的点。它们'关键地'划分了数轴。

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4. Polynomials | 多项式

Polynomial P(x): Sum of terms aₙxⁿ + ... + a₁x + a₀. Non-negative integer exponents only. 'Poly' means many, 'nomial' means term.

多项式 P(x)：形如aₙxⁿ + … + a₁x + a₀的和，指数仅限非负整数。'Poly'意为多，'nomial'意为项。

Degree of a polynomial: Highest power of x present. The degree determines the maximum number of turning points (degree – 1).

多项式的次数：x的最高次幂。最大转向点的个数为（次数 – 1）。

Remainder Theorem: When P(x) is divided by (x – a), the remainder is P(a). Plug a into P – quick remainder without long division.

余数定理：P(x)除以(x – a)的余数为P(a)。代入a即可快速得到余数，无需长除法。

Factor Theorem: If P(a) = 0, then (x – a) is a factor. Converse of remainder theorem – zero remainder means factor found.

因式定理：若P(a) = 0，则(x – a)为因式。余数定理的逆定理——余数为零即得因式。

Binomial expansion (1 + x)ⁿ: For positive integer n, terms follow Pascal's triangle. Expansion with nCr coefficients. 'Binomial = two terms raised to power n.'

二项式展开 (1 + x)ⁿ：n为正整数时，各项系数遵循帕斯卡三角。系数为组合数nCr。'二项式 = 两项的n次幂'。

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5. Simultaneous Equations | 联立方程

Linear simultaneous equations ax + by = c and dx + ey = f. Solve by elimination or substitution. Where two lines cross.

线性联立方程：两个方程如ax + by = c和dx + ey = f。用消元法或代入法求解，即两条直线的交点。

One linear, one quadratic: Substitute linear into quadratic to form a single quadratic equation. Two intersections possible, one or none.

一个线性、一个二次：将线性方程代入二次方程，得到一个二次方程。交点可能有两个、一个或没有。

Elimination method: Add/subtract equations to cancel one variable. 'Eliminate to isolate – make a variable vanish.'

消元法：将方程相加或相减消去一个变量。'消元以孤立——让一个变量消失'。

Substitution method: Express one variable in terms of the other, then replace. 'Substitute to reduce unknowns.'

代入法：用一个变量表示另一个变量，然后代入。'代入以减少未知数'。

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6. Logarithmic and Exponential Functions | 对数与指数函数

Exponent / index: In aⁿ, n is the exponent. 'Exponent exposes the power.'

指数：aⁿ中，n为指数。'指数揭示了幂次'。

Logarithm logₐ x = y: Means aʸ = x. 'Log is the exponent you need.' Memory: 'log base a of x asks: a to what power gives x?'

对数 logₐ x = y：意为aʸ = x。'对数即所需的指数'。记住：'以a为底x的对数问：a的几次幂等于x？'

Natural logarithm ln x: log with base e ≈ 2.718. 'ln' stands for 'log natural' – think 'natural base e'.

自然对数 ln x：以e ≈ 2.718为底的对数。'ln'即'自然对数'——想'自然底数e'。

Exponential function eˣ: The inverse of ln x. Graph always above x-axis, passes through (0,1). 'Exponential explodes upwards.'

指数函数 eˣ：ln x的反函数。图像始终在x轴上方，过(0,1)。'指数函数向上爆炸'。

Laws of logs: logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ (xⁿ) = n logₐ x. 'Logs turn multiplication into addition – a calculator's trick before calculators.'

对数运算律：logₐ (xy) = logₐ x + logₐ y, logₐ (x/y) = logₐ x – logₐ y, logₐ (xⁿ) = n logₐ x。'对数将乘变加——计算器诞生前的计算技巧'。

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7. Straight-Line Graphs | 直线图像

Gradient m: Rise over run, (y₂ – y₁)/(x₂ – x₁). Measures steepness. 'Mountains have steep gradients – so do lines.'

斜率 m：纵差除以横差，(y₂ – y₁)/(x₂ – x₁)。衡量倾斜度。'山有陡坡——直线亦然'。

y = mx + c: Slope-intercept form. m = gradient, c = y-intercept. 'c' is where the line cuts the y-axis – easy to see.

y = mx + c：斜截式。m为斜率，c为y轴截距。'c'是直线与y轴交点的y坐标一目了然。

Converting to linear form: If y = abˣ, take logs: ln y = x ln b + ln a. Plot ln y against x to get a straight line. 'Log-trick turns curves into straight lines.'

化为线性形式：若y = abˣ，取对数得ln y = x ln b + ln a。以ln y对x作图即得直线。'对数技巧化曲为直'。

Parallel lines: Same gradient m₁ = m₂. Never meet. 'Parallel = equal m.'

平行线：斜率相同m₁ = m₂。永不相交。'平行即等m'。

Perpendicular lines: Gradients multiply to –1: m₁ × m₂ = –1. 'Perpendicular = negative reciprocal slopes.'

垂直线：斜率乘积为–1：m₁ × m₂ = –1。'垂直即负倒数斜率'。

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8. Trigonometry | 三角学

sin θ, cos θ, tan θ: Ratios of sides in a right-angled triangle. 'SOH CAH TOA': Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

sin θ, cos θ, tan θ：直角三角形中边的比值。'SOH CAH TOA'口诀：正弦=对边/斜边，余弦=邻边/斜边，正切=对边/邻边。

Sine rule n-right triangles when given two angles and one side (AAS) or two sides and a non-included angle (SSA). 'Sine rule pairs sides with opposite angles.'

正弦定理：a/sin A = b/sin B = c/sin C。用于非直角三角形，已知两角一边或两边一对角。'正弦定理将边与对角配对'。

Cosine rule: a² = b² + c² – 2bc cos A. Used when given three sides (SSS) or two sides and the included angle (SAS). 'Cosine rule – when you have the 'included' angle.'

余弦定理：a² = b² + c² – 2bc cos A。用于已知三边或两边及夹角。'余弦定理——当你有'夹角'时使用'。

Radian measure = 180°. Real number measure of angle. Arc length = rθ, sector area = ½ r²θ (θ in radians). 'Radian relates angle to arc – it's the distance rolled along the circumference.'

弧度制：π rad = 180°。角度的实数度量。弧长 = rθ，扇形面积 = ½ r²θ（θ以弧度计）。'弧度将角度与弧联系——即沿圆周滚过的距离'。

Trigonometric identities: sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ. 'The fundamental identity ties sin and cos in a perfect circle.'

三角恒等式：sin²θ + cos²θ ≡ 1，tan θ ≡ sin θ / cos θ。'基本恒等式将正弦与余弦系于完美圆上'。

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9. Differentiation | 微分

Derivative f'(x) or dy/dx: Measures rate of change, gradient of tangent. 'Differentiation = finding how fast y changes as x ticks forward.'

导数 f'(x) 或 dy/dx：衡量变化率，即切线斜率。'微分 = 探寻y随x前进的速度'。

Power rule: d/dx (xⁿ) = n xⁿ⁻¹. 'Bring down the power and cut the power by one – the easiest rule.'

幂法则：d/dx (xⁿ) = n xⁿ⁻¹。'把指数搬下来，指数减一——最简单的规则'。

Chain rule: d/dx [f(g(x))] = f'(g(x))·g'(x). Differentiate outside, times derivative of inside. 'Chain links outer and inner functions.'

链式法则：d/dx [f(g(x))] = f'(g(x))·g'(x)。先对外层求导，乘以内层导数。'链子连接外函数与内函数'。

Product rule: d/dx (uv) = u'v + uv'. 'First times diff second plus second times diff first.' Think: 'u dance v plus v dance u.'

积法则：d/dx (uv) = u'v + uv'。'第一个乘第二个的导数加第二个乘第一个的导数'。口诀：'u跳v加v跳u'。

Quotient rule: d/dx (u/v) = (u'v – uv')/v². 'Low d high minus high d low, draw the line and square below.'

商法则：d/dx (u/v) = (u'v – uv')/v²。'分母乘分子导数减分子乘分母导数，除以分母平方'。

Stationary points mum, minimum or point of inflection. Second derivative test: d²y/dx² < 0 → max, > 0 → min.

驻点：dy/dx = 0的点。可能是极大值、极小值或拐点。二阶导数检验：d²y/dx² < 0 → 极大，> 0 → 极小。

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10. Integration | 积分

Indefinite integral ∫ f(x) dx = F(x) + C: Reverse of differentiation – add one to exponent, divide by new exponent. '+ C' is the constant of integration.

不定积分 ∫ f(x) dx = F(x) + C：微分的逆运算——指数加一，除以新指数。'+ C'为积分常数。

Definite integral ∫ₐᵇ f(x) dx: Gives the area under the curve from x = a to b. Compute F(b) – F(a). 'Definite = a specific numeric area.'

定积分 ∫ₐᵇ f(x) dx：给出x = a到b之间曲线下的面积。计算F(b) – F(a)。'定积分 = 具体的数值面积'。

Area between curves: ∫ₐᵇ [f(x) – g(x)] dx, where f is above g. 'Top minus bottom, integrate along x.'

曲线间的面积：∫ₐᵇ [f(x) – g(x)] dx，其中f在上，g在下。'上减下，沿x积分'。

Kinematics application Velocity v = ds/dt, acceleration a = dv/dt. Integration reverses: s = ∫ v dt, v = ∫ a dt. 'Derivative connects position→velocity→acceleration; integral leads back.'

运动学应用：速度v = ds/dt，加速度a = dv/dt。积分逆推：位移s = ∫ v dt，速度v = ∫ a dt。'微分连接位置→速度→加速度；积分带我们返回'。

Power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ –1. 'Add one to exponent, divide by new exponent – the anti-derivative jingle.'

积分幂法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ –1。'指数加一，除以新指数——反导数顺口溜'。

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11. Vectors | 向量

Vector notation: Bold v, or column form (x y)ᵀ, or unit vectors i, j. Vectors have magnitude and direction, unlike scalars. 'Vector = magnitude + direction – think of an arrow.'

向量表示：粗体v，或列向量 (x y)ᵀ，或单位向量i, j。向量有大小和方向，区别于标量。'向量 = 大小 + 方向——想象一支箭'。

Magnitude |v|: Length of vector, found by Pythagoras: √x² + y². 'Magnitude is the distance the arrow flies.'

模 |v|：向量的长度，用勾股定理求：√x² + y²。'模即箭飞行的距离'。

Unit vector: Vector of length 1, e.g. v/|v|. 'Unit vector gives direction without distance.'

单位向量：长度为1的向量，如v/|v|。'单位向量只给方向，不涉距离'。

Position vector: Vector from origin O to point P, written as OP⃗ or p. 'Position fixes where you are relative to origin.'

位置向量：从原点O到点P的向量，记作OP⃗或p。'位置向量确定你相对于原点的位置'。

Scalar product / Dot product: a·b = |a||b| os θ = x₁x₂ + y₁y₂. Used to find angle between vectors or show perpendicularity (dot = 0). 'Dot product multiplies magnitudes and cos of angle between.'

数量积 / 点积：a·b = |a||b| cos θ = x₁x₂ + y₁y₂。用于求两向量夹角或证明垂直（点积为0）。'点积 = 大小相乘再乘夹角的余弦'。

Collinear vectors: Vectors that are parallel and share a common point, so they lie on same line. Check if one vector is a scalar multiple of the other.

共线向量：平行且拥有公共点的向量，因此位于同一直线上。检验一向量是否为另一向量的标量倍数。

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