📚 Year 10 Cambridge Further Mathematics: Quick Vocabulary and Terminology Memorisation Guide | Year 10 Cambridge 进阶数学：词汇术语速记指南
Mastering the technical language of Further Mathematics is half the battle. This guide groups key terms by topic, explains them with memory hooks, and provides direct Chinese translations to strengthen understanding. Use it alongside your regular revision to lock in terminology that exam questions rely on.
掌握进阶数学的专业术语是成功的一半。本指南按主题整理核心词汇，配合记忆联想法和精准中文翻译，帮助你彻底理解并牢记这些考试中不可或缺的概念。请搭配日常复习使用，让术语内化成第二本能。
1. Functions and Notation | 函数与符号速记
Function f(x): A rule that assigns exactly one output (y) to each input (x). The notation f(x) reads 'f of x'. Picture a machine: you feed in x, the function processes it, and out comes f(x).
函数 f(x)：为每个输入值 x 分配唯一输出值 y 的规则。记号 f(x) 读作“f of x”。可以想象一台机器：放入 x，经过函数处理，吐出 f(x)。
Domain (定义域): The set of all possible x-values for which the function is defined. Remember 'domain' as the 'dominion' where the function rules – think of a king's territory.
定义域：函数有定义的所有可能 x 值的集合。把 'domain' 记为函数统治的“领土”，好比国王的领地，输入值不能越界。
Range (值域): The set of all possible y-values the function can produce. Imagine a mountain 'range' spread across the y-axis; the range describes how far the output can reach.
值域：函数能产生的所有可能 y 值的集合。想象横跨 y 轴的“山脉”，值域描绘了输出值可以延伸到多远。
One-to-one function (单射函数): Each x pairs with a unique y, and each y comes from only one x. These functions pass the horizontal line test and have inverse functions. Think of a perfect dance partner pairing – nobody is left without a match.
单射函数：每个 x 对应唯一的 y，且每个 y 也仅来源于一个 x。这样的函数能通过水平线检验，并且存在反函数。好比舞伴一一配对，无人落单。
Inverse function f⁻¹(x) (x). If f(a) = b, then f⁻¹(b) = a. The superscript ⁻¹ does not mean reciprocal; it signals 'undo'. Imagine pressing Ctrl+Z on a keyboard.
反函数 f⁻¹(x)：逆转 f(x) 的作用。若 f(a) = b，则 f⁻¹(b) = a。上标 ⁻¹ 不表示倒数，而是“撤销”操作。就像键盘上的 Ctrl+Z。
2. Quadratic Functions & Discriminant | 二次函数与判别式
General form ax² + bx + c: A quadratic expression where a ≠ 0. The value of a controls the shape's opening: a > 0 gives a smile (∪), a < 0 gives a frown (∩). Think 'positive people smile'.
一般式 ax² + bx + c：a ≠ 0 的二次表达式。a 的正负决定开口方向：a > 0 呈微笑形 ∪，a < 0 呈皱眉形 ∩。记住“正数微笑”。
Discriminant Δ = b² − 4ac: The single value that reveals the nature of roots. Positive Δ yields two distinct real roots, zero yields one repeated real root, negative gives no real roots. Imagine a detective's verdict: 'two suspects, one or none real'.
判别式 Δ = b² − 4ac：一个数就透露了方程根的性质。Δ 为正 ⇒ 两个相异实根；Δ = 0 ⇒ 一个重实根；Δ 为负 ⇒ 无实根。想象侦探给出结论：“两个、一个或没有真凶（实根）”。
Nature of roots → Δ > 0 : two real, Δ = 0 : one real, Δ < 0 : no real
Vertex / Turning point uare to the form a(x − h)² + k, giving vertex (h, k). The axis of symmetry is x = h. Think 'h for horizontal position of the tip'.
顶点 / 转折点：抛物线的最高或最低点。通过配方法写成 a(x − h)² + k 即可得到顶点 (h, k)，对称轴为 x = h。记住 'h' 定位顶点的水平坐标。
3. Polynomials & Factor Theorem | 多项式与因式定理
Degree of a polynomial: The highest power of x. A cubic has degree 3, a quartic degree 4. The degree dictates the maximum number of roots and the shape's general behaviour. Just count the loftiest exponent.
多项式的次数：x 的最高次幂。三次多项式次数为 3，四次为 4。次数决定了根的最大个数及图像的大致走向。只需找出最高指数即可。
Remainder Theorem: When f(x) is divided by (x − a), the remainder is f(a). This saves you from long division – simply substitute a into f. Remember: 'remainder equals evaluation'.
余式定理：f(x) 除以 (x − a) 时，余式等于 f(a)。这替你省去了长除法的麻烦——直接将 a 代入 f 即可。记住：“余式即为求值”。
Factor Theorem heorem: zero remainder means exact division. To find factors, test f(±1), f(±2) and so on.
因式定理：(x − a) 是 f(x) 的因式，当且仅当 f(a) = 0。由余式定理直接推得：余数为零即整除。找因式时，可试代 f(±1)、f(±2) 等。
4. Exponents & Logarithms | 指数与对数关系巧记
Index laws = aᵐ⁻ⁿ. When multiplying like bases, add the powers; when raising a power, multiply. Think 'same base → operations on exponents step down one level'.
指数运算法则：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ。同底数相乘，指数相加；幂的乘方，指数相乘。记法：“相同底数时，指数运算降一级”。
Logarithm definition: If aʸ = x then logₐ x = y. The logarithm asks: 'To what power must I raise a to get x?' logarithm is an exponent. Memorise the loop: base stays base, output swaps with exponent.
对数定义：若 aʸ = x，则 logₐ x = y。对数在问：“a 的几次方等于 x？”本质上，对数就是一个指数。牢记这个转换循环：底数不变，结果与指数互换位置。
aʸ = x ⇔ logₐ x = y
Log rules + logₐ N, logₐ (M/N) = logₐ M − logₐ N, logₐ (Mᵏ) = k logₐ M. These mirror index laws. The 'product becomes sum' rule helps compress or expand expressions.
对数法则：logₐ (MN) = logₐ M + logₐ N，logₐ (M/N) = logₐ M − logₐ N，logₐ (Mᵏ) = k logₐ M。这与指数法则对应。“乘积变加法”能帮助压缩或展开式子。
5. Coordinate Geometry: Lines & Circles | 坐标几何：直线与圆
Gradient m steepness. m = (y₂ − y₁)/(x₂ − x₁). Positive m slopes upward, negative m slopes downward. 'Rise over run' is the climber's mantra.
斜率 m：衡量直线的倾斜程度。m = (y₂ − y₁)/(x₂ − x₁)。正斜率向上倾斜，负斜率向下倾斜。登山者的口诀：“垂直升高除以水平距离”。
Equation of a straight line -intercept. The form y − y₁ = m(x − x₁) is handy when you know one point and the gradient.
直线方程：y = mx + c 或 y − y₁ = m(x − x₁)。其中 c 为 y 轴截距。已知一点和斜率时，使用点斜式 y − y₁ = m(x − x₁) 非常方便。
Distance and midpoint (x₁, y₁) and (x₂, y₂) is √[(x₂−x₁)² + (y₂−y₁)²]. Midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). Think 'average x, average y' for midpoint and Pythagoras for distance.
距离与中点：两点距离为 √[(x₂−x₁)² + (y₂−y₁)²]。中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。中点就是“x 平均，y 平均”，距离则靠勾股定理。
Equation of a circle ², centre (h, k), radius r. Notice the minus signs. If the equation has (x+3)², the centre's x-coordinate is −3. Always reverse the sign inside brackets to get the centre.
圆的方程：(x − h)² + (y − k)² = r²，圆心 (h, k)，半径为 r。注意括号里的符号。如果出现 (x+3)²，圆心 x 坐标为 −3。取圆心坐标时，括号内的符号要变号。
6. Trigonometric Ratios & Identities | 三角比与恒等式联想
SOH CAH TOA jacent/hypotenuse, tan θ = opposite/adjacent. This simple chant never fails. For any right‑angled triangle, label sides relative to θ and apply the ratios.
SOH CAH TOA：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。这句口诀永不过时。对于任意直角三角形，先相对 θ 标记各边，再套用这些比值。
Exact values for special angles ly. Use the finger method or memorise the table. For instance, sin 30° = 1/2, cos 45° = 1/√2, tan 60° = √3. Picture the 30‑60‑90 and 45‑45‑90 triangles.
特殊角的精确值：0°、30°、45°、60°、90° 频繁出现。可用手指法或直接记忆表格。例如 sin 30° = 1/2，cos 45° = 1/√2，tan 60° = √3。脑中浮现 30‑60‑90 和 45‑45‑90 三角板。
Fundamental identity osec² θ can be derived. Think of sin²+cos² as a Pythagorean diamond that always sums to 1.
基本恒等式：sin² θ + cos² θ = 1。由此可推出 tan² θ + 1 = sec² θ 以及 cot² θ + 1 = cosec² θ。把 sin²+cos² 看作一块毕达哥拉斯钻石，其和恒为 1。
ASTC quadrant rule: All positive in I, Sin positive in II, Tan positive in III, Cos positive in IV. Use the mnemonic 'All Students Take Calculus' or 'Add Sugar To Coffee' to recall which ratios are positive where.
ASTC 象限规则：第一象限 All 为正，第二象限 Sin 为正，第三象限 Tan 为正，第四象限 Cos 为正。用 “All Students Take Calculus” 或 “爱吃甜的孩子爱咖啡” 帮助记忆各象限的正符号。
7. Sequences & Series | 数列与级数术语
Arithmetic progression (AP) h term: uₙ = a + (n−1)d. Sum of first n terms: Sₙ = n/2 [2a + (n−1)d] or Sₙ = n/2 (a + l). 'A' for Arithmetic, think 'Add'.
等差数列 (AP)：每项加上共同的公差 d。通项公式：uₙ = a + (n−1)d。前 n 项和：Sₙ = n/2 [2a + (n−1)d] 或 Sₙ = n/2 (a + l)。'A' 代表 Arithmetic，联想“Add 加”。
Geometric progression (GP) multiplied each time. Nth term: uₙ = a rⁿ⁻¹. Sum of n terms: Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1. 'G' for Geometric, think 'Multiply' (grow by multiplication).
等比数列 (GP)：每项乘以公比 r。通项：uₙ = a rⁿ⁻¹。前 n 项和：Sₙ = a(1−rⁿ)/(1−r)，r ≠ 1。'G' 代表 Geometric，联想“乘 multiply”。
Sum to infinity: For a GP with |r| −r). When the ratio is a proper fraction, the terms shrink to nothing, and the sum approaches a finite limit. Imagine a bouncing ball losing height each bounce.
无穷级数和：当 |r| < 1 时，等比级数收敛，S∞ = a/(1−r)。公比为真分数时，各项逐渐减小至零，总和趋于一个有限值。想象弹跳的球每次弹起高度递减。
8. Differentiation Basics | 微分基础术语
Derivative dy/dx: The instantaneous rate of change of y with respect to x. It gives the gradient of the tangent to a curve. Pronounce dy/dx as 'dy by dx'. Think of it as a speedometer showing how fast y changes for every unit of x.
导数 dy/dx：y 关于 x 的瞬时变化率，表示曲线上切线的斜率。读作 “dy by dx”。就像速度表，显示 y 随每一单位 x 的变化有多快。
Power rule: If f(x) = xⁿ, then f'x) = n xⁿ⁻¹. Bring down the exponent as a coefficient and reduce the power by 1. Remember: 'Multiply by the power and drop the power down one stair'.
幂法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。将指数搬下来作系数，同时指数减 1。口诀：“指数飞下来，方次降一楼”。
d/dx (xⁿ) = n xⁿ⁻¹
Constant and sum rules erivatives. So d/dx (5x³ + 7) = 15x². Constants disappear; terms are treated individually.
常数与和法则：常数的导数为零。和的导数等于导数之和。因此 d/dx (5x³ + 7) = 15x²。常数项在微分中消失；各项分别求导再相加。
Stationary points ima, minima or points of inflection. Use the second derivative or a sign table to classify them. Think 'hills and valleys'.
驻点：出现在 dy/dx = 0 的位置，可能为极大值、极小值或拐点。用二阶导数或符号表来判别类型。把它想成地貌中的“山峰与山谷”。
9. Integration & Area | 积分与面积记忆
Indefinite integral: The reverse process of differentiation. ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, provided n ≠ −1. The '+c' is the constant of integration. Think 'antiderivative' – working backwards from a gradient to the original function.
不定积分：微分的逆过程。∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c，其中 n ≠ −1。'+c' 是积分常数。想成“反导数”——从斜率倒推回原函数。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + c
Definite integral: ∫ₐᵇ f(x) dx computes the signed area between the curve and the x‑axis from x = a to x = b. Evaluate F(b) − F(a), where F is any antiderivative of f. The 'signed' means area below the axis counts as negative.
定积分
Published by TutorHao | Year 10 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图