📚 Year 10 Cambridge Additional Mathematics: Essential Vocabulary and Memory Aids | Year 10 剑桥进阶数学：词汇术语速记指南
Mastering the vocabulary of Cambridge IGCSE Additional Mathematics is half the battle won. When you decode the language of functions, calculus, and trigonometry, the concepts become clearer and problem-solving feels more natural. This guide breaks down key terms into memorable chunks, linking them to root words, visual hooks, and exam-ready examples.
掌握剑桥 IGCSE 进阶数学的词汇，就等于打赢了一半的仗。当你破解了函数、微积分和三角函数的语言密码，概念会变得更清晰，解题也会更得心应手。这份指南将核心术语拆解成易记的片段，通过词根、图像挂钩和备考范例帮你牢牢记住它们。
1. Core Function Terminology | 核心函数术语
The word 'function' comes from a Latin root meaning 'to perform'. Think of a function as a machine: you feed it an input (x), it performs a rule, and produces exactly one output (f(x) or y). Domain is simply 'where you can go' – all possible inputs. Range is 'how far you can reach' – all possible outputs.
'函数'一词源于拉丁语'执行'的意思。把函数想象成一台机器：投入一个输入值 (x)，它按照规则执行，只产生一个输出值 (f(x) 或 y)。定义域就是'你能走多远'——所有可能的输入值。值域就是'你能触及多远'——所有可能的输出值。
- Mapping: Think of a street directory mapping every address to a location. A function maps every element in the domain to a unique element in the codomain.
- 映射：想象一本街道指南把每个地址对应到一个位置。函数把定义域中的每一个元素都映射到陪域中的一个唯一元素。
- One-to-one function: No two friends share the same phone number; each input gives a distinct output. Passes both vertical and horizontal line tests.
- 单射函数：没有两个朋友共用一个电话号码；每个输入产生不同的输出。通过竖线和水平线检验。
- Composite function (f∘g): 'Outer function of inner function' – apply g first, then f. Pronounce the little circle as 'of'.
- 复合函数 (f∘g)：'外部函数包含内部函数'——先作用 g，再作用 f。把那个小圆圈读作'圈'。
To remember inverse function he original function: if f(a)=b, then f⁻¹(b)=a. The graph is a reflection in the line y=x.
要记住 反函数 (f⁻¹)，想象倒带键。它把原函数撤销：如果 f(a)=b, 那么 f⁻¹(b)=a。它的图像关于直线 y=x 反射。
2. Quadratic Expressions and Roots | 二次表达式与根
Quadratic comes from Latin 'quadratus' meaning 'square', because the highest power is x². The discriminant Δ = b² − 4ac 'discriminates' or tells you the nature of the roots. If it' , roots are real and distinct; zero gives equal real roots; negative yields no real roots.
二次源自拉丁语 'quadratus'，意为'平方'，因为最高次幂是 x²。判别式 Δ = b² − 4ac 起到'判别'作用，告诉你根的性质。正数：两个不等实根；零：相等实根；负数：没有实根。
- Roots/zeros : Where the graph crosses the x-axis. Think tree roots going into the ground. Solutions to f(x)=0.
- 根/零点：图像与 x 轴的交点。想象树根扎入地下。方程 f(x)=0 的解。
- Completing the square: Transforming ax²+bx+c into a(x+p)²+q. Imagine geometrically completing an incomplete square tile.
- 配方法：把 ax²+bx+c 化为 a(x+p)²+q。几何上想象补全一块正方形瓷砖。
- Vertex: The turning point of a parabola. For y=a(x−h)²+k, the vertex is (h,k) – the 'peak' or 'valley'.
- 顶点：抛物线的转折点。对于 y=a(x−h)²+k，顶点是 (h,k)——'山峰'或'谷底'。
If ax² + bx + c = 0, sum of roots = −b/a, product = c/a
如果 ax² + bx + c = 0，根的和 = −b/a，根的积 = c/a
3. Exponential and Logarithmic Lingo | 指数与对数行话
Exponent comes from Latin 'exponere' – to put forth. It shows how many times the base 'puts forth' itself as a factor. Logarithm is just a fancy word for 'power'. The statement logₐ y = x is asking: 'To what power must I raise a to get y?' Think 'log is a slow snail looking for the exponent.'
指数来自拉丁语 'exponere'——提出。它表示底数'提出'自身作为因子的次数。对数不过是'幂'的另一种说法。logₐ y = x 在问：'把 a 提升到几次方才能得到 y？' 联想：'对数是一只慢蜗牛，在寻找指数。'
- Base: Foundation of the power. In 2³, 2 is the base. The same base must match when manipulating logarithms.
- 底数：幂的基础。在 2³ 中，2 是底数。对数运算时必须保证底数一致。
- Natural log (ln): Log with base e ≈ 2.718. Just remember 'ln is natural, like leaves on a tree'.
- 自然对数 (ln)：以 e ≈ 2.718 为底的对数。记住'ln 很自然，就像树上的叶子'。
logₐ (MN) = logₐ M + logₐ N
logₐ (M/N) = logₐ M − logₐ N
logₐ (Mⁿ) = n logₐ M
4. Trigonometric Terms | 三角函数术语
The word trigonometry literally means 'triangle measuring' site over Hypotenuse; Cosine = Adjacent over Hypotenuse; Tangent = Opposite over Adjacent. 'SOH CAH TOA' sounds like a mountain climber's chant on a triangular peak.
三角学字面意思是'三角形测量'。记住 SOH CAH TOA：正弦 = 对边/斜边；余弦 = 邻边/斜边；正切 = 对边/邻边。'SOH CAH TOA' 听起来像登山者在三角形山峰上的号子。
- Radian : An angle measure where the arc length equals the radius. π radians = 180°. Imagine wrapping the radius around the circle.
- 弧度：一种角度度量，当弧长等于半径时的角。π 弧度 = 180°。想象把半径沿着圆周缠绕。
- Amplitude: The height of a wave from the midline. For y = A sin(x), |A| is the amplitude. Think 'A' for altitude.
- 振幅：波浪偏离中线的最大高度。对于 y = A sin(x)，|A| 是振幅。把 A 想到'Altitude（高度）'。
- Period: The length of one full cycle. For y = sin(bx), period = 2π/|b|. Period is the time 'repeating again' – 'per' means through.
- 周期：完成一个完整循环的长度。对于 y = sin(bx)，周期 = 2π/|b|。周期意味着'再次重复'的时间。
- Phase shift: Horizontal translation of the graph. y = sin(x − c) shifts by c units right. 'Phase' suggests a different stage.
- 相移：图像的水平平移。y = sin(x − c) 向右平移 c 个单位。'相'暗示不同的阶段。
5. Differentiation Decoded | 微分术语解密
Differentiation tells you the rate of change or slope. The derivative f'(x) is the 'slope function'. The process is 'finding the derivative'. Think of a speedometer: the derivative of position is velocity. Key word: 'gradient of the tangent'.
微分告诉你变化率或斜率。导数 f'(x) 是'斜率函数'。过程就是'求导'。想象汽车速度表：位置的导数是速度。关键词：'切线的斜率'。
- Power rule: Bring down the power, reduce power by one. For xⁿ, derivative = n xⁿ⁻¹. Say aloud: 'power drops like a bomb, exponent then minus one'.
- 幂法则：把指数拿下来，指数减一。xⁿ 的导数为 n xⁿ⁻¹。念出来：'指数跳下投炸弹，然后自减一。'
- Stationary point: Where f'(x)=0, the tangent is horizontal. Could be maximum, minimum, or inflection. The curve 'stops' rising/falling momentarily.
- 驻点：f'(x)=0 的地方，切线水平。可能是极大值、极小值或拐点。曲线暂时'停住'上升或下降。
- Second derivative f''(x): The derivative of the derivative; tells you the rate of change of slope – concavity. f''(x) > 0 means smile (convex), < 0 means frown (concave).
- 二阶导数 f''(x)：导数的导数；表示斜率的变化率——凹凸性。f''(x) > 0 是笑脸（凹向上），< 0 是哭脸（凹向下）。
If y = xⁿ, dy/dx = n xⁿ⁻¹
如果 y = xⁿ，dy/dx = n xⁿ⁻¹
6. Integration Essentials | 积分基础词汇
Integration is anti-differentiation – gives you the area under a curve or a family of curves. The symbol ∫ is an elongated S for 'sum', because integration sums infinitely thin slices.
积分是微分的逆运算——求导的逆过程。它给出曲线下的面积或一族曲线。符号 ∫ 是一个拉长的 S，代表'求和'，因为积分是对无穷薄的切片求和。
- Indefinite integral: No limits; includes the constant of integration '+ C'. Because differentiating a constant gives zero, we add C to account for any lost constant.
- 不定积分：没有上下限，包含积分常数 '+ C'。因为常数求导为零，我们加 C 来追溯那个失去的常数。
- Definite integral: Has upper and lower limits; yields a number (area). The result is F(b) − F(a). Think of fixed boundaries.
- 定积分：有上下限，结果是一个数字（面积）。结果为 F(b) − F(a)。想象固定的边界。
- Power rule for integration: Add one to the power, divide by the new power. ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ −1. 'Up one, divide by the new exponent'.
- 积分的幂法则：指数加一，再除以新的指数。∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ −1。'指数加一，除以新幂'。
7. Vectors in Plain Words | 向量通俗讲解
A vector has both magnitude (size) and direction. Think of an arrow. Scalar has only magnitude (like speed), while vector is velocity. Displacement is a vector, distance is scalar. The notation a or AB with an arrow shows the journey from A to B.
向量既有大小又有方向。想象一支箭。标量只有大小（比如速率），而向量是速度。位移是向量，路程是标量。符号 a 或带箭头的 AB 表示从 A 到 B 的行程。
- Magnitude |a|: Length of the vector. Use Pythagoras for components: √x²+y². 'How long the arrow is'.
- 模 |a|：向量的长度。用勾股定理计算分量：√x²+y²。'箭有多长'。
- Unit vector: A vector of length 1 in a specific direction. Denoted with a hat, like î. Divide a vector by its magnitude to get unit vector.
- 单位向量：长度为 1 的向量，指向特定方向。用小帽子表示，如 î。将向量除以其模得到单位向量。
- Collinear vectors: Vectors lying on the same straight line. One is a scalar multiple of the other. 'Co-linear' = shared line.
- 共线向量：在同一条直线上的向量。一个是另一个的标量倍数。'共线'就是共享一条线。
- Position vector: The vector from the origin O to a point P. Usually written as p.
- 位置向量：从原点 O 到点 P 的向量。通常写作 p。
8. Sequences and Series | 数列与级数
A sequence is a list of numbers in order; a series is their sum. 'Arithmetic progression' (AP) adds a constant difference d. 'Geometric progression' Arithmetic = Addition (both start with A); Geometric = multiplication (common ratio, think 'growth').
数列是按顺序排列的一列数；级数是它们的和。'等差数列' (AP) 每次加上常数公差 d。'等比数列' (GP) 每次乘以常数公比 r。记住：Arithmetic = Addition (都以 A 开头)；Geometric = 乘法（公比，联想'增长'）。
- nth term of AP: a+(n−1)d. First term a, common difference d. Step by step you add d.
- 等差数列的通项：a+(n−1)d。首项 a，公差 d。一步一步加 d。
- Sum of AP: Sn = n/2 [2a + (n−1)d] or n/2 (first+last). Pair up first and last terms for easy sum.
- 等差数列求和：Sn = n/2 [2a + (n−1)d] 或 n/2 (首项+末项)。将首末配对求和。
- nth term of GP: a rⁿ⁻¹. Common ratio r multiplies repeatedly.
- 等比数列通项：a rⁿ⁻¹。公比 r 重复相乘。
- Sum to infinity for |r| < 1: S∞ = a/(1−r). The series 'converges' because terms get tiny.
- 无穷等比数列求和 当 |r| < 1：S∞ = a/(1−r)。级数'收敛'因为项越来越小。
9. Permutations and Combinations | 排列与组合
Permutation: 'Order matters'. Think of arranging books on a shelf. Combination: 'Order does not matter'. Think of picking a team: ABC is same as CBA. 'Permutation' has 'perm' like permanent – fixed order matters. 'Combination' like a combo meal, items just grouped together.
排列：'顺序重要'。想象把书排列在书架上。组合：'顺序不重要'。想象挑选一支队伍：ABC 和 CBA 是同一个组。'排列' 有 '排' 字，强调排队顺序。'组合' 像套餐组合，物品只是放在一起。
- Factorial n! = n×(n−1)×...×1. The number of ways to arrange n distinct items in a line. 'Shout out the countdown!'
- 阶乘 n! = n×(n−1)×...×1。将 n 个不同物品排成一行的方式数。'倒数大喊!'
- Permutation formula nPr = n!/(n−r)! (r items from n, order matters).
- 排列公式 nPr = n!/(n−r)!（从 n 个中选 r 个，顺序重要）。
- Combination formula nCr = n!/((n−r)! r!) (order ignored). Sometimes written as binomial coefficient (n choose r).
- 组合公式 nCr = n!/((n−r)! r!)（顺序忽略）。有时写作二项式系数 (n 选 r)。
10. Set Notation and Logic | 集合符号与逻辑
Sets are collections, curly braces {}. 'Element of' ∈ looks like an epsilon (e for element). 'Not an element' ∉. The universal set ξ is the entire collection under consideration. Visualise Venn diagrams as coloured blobs.
集合就是一堆东西，用大括号 {} 表示。'属于' ∈ 像 epsilon (e 代表元素)。'不属于' ∉。全集 ξ 是当前考虑的所有元素。用文氏图彩色泡泡来形象化。
- Subset A ⊆ B: all elements of A are also in B. 'A is inside B'. Proper subset ⊂ means strictly smaller.
- 子集 A ⊆ B：A 的所有元素都在 B 中。'A 在 B 里面'。真子集 ⊂ 表示严格较小。
- Union ∪: elements in A OR B (or both). The cup holds everything from both.
- 并集 ∪：在 A 或 B（或两者）中的元素。杯子 ∪ 盛下两者。
- Intersection ∩: elements common to both A AND B. Think of where the circles cross.
- 交集 ∩：A 和 B 的共同部分。想象圆圈交叉处。
- Complement A': elements not in A. 'Complement' completes the set.
- 补集 A'：不在 A 中的元素。'补'集补全了集合。
11. Graph Transformations | 图像变换
Transformations change the graph's position or shape. Think in terms of puppet strings: f(x) + a moves up; f(x+a) shifts left (counterintuitively!). 'Inside the bracket changes x opposite direction'. Stretches: a f(x) stretches vertically by factor a; f(ax) compresses horizontally by factor 1/a.
变换改变图像的位置或形状。用提线木偶来想：f(x) + a 向上移动；f(x+a) 向左移动（反直觉！）。'括号里面改变 x，方向相反'。伸缩：a f(x) 垂直拉伸 a 倍；f(ax) 水平压缩到 1/a 倍。
- Translation: Sliding without turning. Vector (p,q) means shift p right, q up.
- 平移：无旋转的滑动。向量 (p,q) 表示右移 p，上移 q。
- Reflection: Mirror image. −f(x) reflects in x-axis; f(−x) reflects in y-axis. Think minus sign flips.
- 反射：镜像。−f(x) 关于 x 轴反射；f(−x) 关于 y 轴反射。想负号翻转。
- Stretch vs compression: Stretch pulls away from axis; compression pushes towards axis.
- 拉伸 vs 压缩：拉伸远离坐标轴；压缩靠近坐标轴。
If k > 1: f(kx) compresses horizontally; if 0 < k < 1, it stretches horizontally.
如果 k > 1：f(kx) 水平压缩；如果 0 < k < 1，水平拉伸。
12. Inequality Language | 不等式语言
lity uses < or > (no equality). Inclusive inequality uses ≤ or ≥ , including the boundary. 'Less than' rhymes with 'left on number line' – smaller numbers are left.
不等式使用 <（小于）、>（大于）这样的符号。小口诀：尖角指向较小的量。严格不等式用 < 或 >（不含等于）。含等号不等式用 ≤ 或 ≥，包括边界。'小于'在数轴上指向左边——较小的数字在左边。
- Linear inequality: Solved like an equation, but if you multiply/divide by a negative, reverse the inequality sign. 'Negative flips the arrow'.
- 线性不等式：像方程一样解，但如果乘或除以负数，要反转不等号。'负数翻转箭头'。
- Quadratic inequality: Sketch the parabola. Identify where it is above or below x-axis based on the sign.
- 二次不等式：画抛物线草图。根据符号判断哪里在 x 轴上方或下方。
- Region: Shaded area representing all solutions. Test a point to decide which side to shade.
- 区域：表示所有解的阴影区域。测试一点决定涂哪一边。
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