📚 IGCSE CAIE Further Pure Maths: Vocabulary & Terminology Quick Memory Guide | IGCSE CAIE 进阶数学：词汇术语速记指南
Mastering precise mathematical vocabulary is essential for success in IGCSE CAIE Further Pure Mathematics (Additional Mathematics 0606). This guide presents the key terms of every topic with bilingual explanations, exam-focused definitions and memory tricks, so you can recall a definition in seconds and apply it accurately under timed conditions.
掌握精确的数学词汇是 IGCSE CAIE 进阶数学（附加数学 0606）取得高分的关键。本指南按专题整理核心术语，配以双语解释、考点定义与记忆技巧，帮助你在限时考试中几秒内回忆定义并准确应用。
1. Why Terminology Wins Marks | 术语为何能决定得分
A large share of lost marks in 0606 comes from misreading, not miscalculating. Command words such as show that, hence, find the exact value and express in terms of tell you what form the answer must take. If a question says 'show that', the final line is already given to you, so your working must connect logically to it. If it says 'hence', you are expected to use the previous part, not start again.
0606 失分很大一部分来自读题而非算错。show that（证明）、hence（由此）、find the exact value（求精确值）、express in terms of（用……表示）等指令词决定了答案必须呈现的形式。若题中说 show that，最后的结论已给出，你的推导必须与它严密衔接；若说 hence，你必须使用上一问的结果，而不是另起炉灶。
Three memory routines are especially effective. Dual coding: pair each word with a picture (a parabola for 'vertex', a mirror for 'inverse'). Spaced repetition: review a 10-term glossary on days 1, 3, 7 and 21. Production, not recognition: cover the definitions and write them out; recognising a term is not the same as defining it.
有三种记忆方法特别有效。双重编码：把词汇与图像配对（抛物线对应 vertex，镜子对应 inverse）。间隔重复：在第 1、3、7、21 天复习同一份十个术语的清单。产出而非识别：遮住释义默写定义——能认出术语不等于能写出定义。
2. Functions and Graphs | 函数与图像
A function is a rule that assigns each input x exactly one output y. Picture a 'function machine': one input produces only one output. The domain is the set of all possible inputs, and the range is the set of all possible outputs. A one-to-one function passes the horizontal line test, which guarantees an inverse function f⁻¹(x). A composite function fg(x) means apply g first, then f. The notation f : x ↦ 2x + 1 is read as 'f maps x to 2x + 1'.
函数是把每个输入 x 映射到唯一输出 y 的规则，可想象成"函数机器"：一个输入只产生一个输出。定义域是所有可能输入的集合，值域是所有可能输出的集合。一一函数能通过水平线检验，从而保证存在反函数 f⁻¹(x)。复合函数 fg(x) 表示先作用 g 再作用 f。记号 f : x ↦ 2x + 1 读作"f 把 x 映射到 2x + 1"。
(f⁻¹)⁻¹ = f, and the range of f⁻¹ = the domain of f
Memory aid: 'Domain before Range' alphabetically, d before r, just as x comes before y. For the inverse, 'reflect in y = x' d where a composite is undefined, look for the input that breaks the inner function first.
记忆法：Domain（定义域）与 Range（值域）按字母 d 在 r 前，正如 x 在 y 前。反函数就是"关于 y = x 作反射"——交换 x 与 y 后解出 y。求复合函数无定义的情况时，先找出让内层函数失效的输入值。
3. Quadratic Functions | 二次函数
A quadratic function has the general form f(x) = ax² + bx + c, where a ≠ 0, and its graph is a parabola. The vertex (x) = a(x − h)² + k, gives the vertex (h, k) directly. The parabola is symmetric about the line of symmetry x = −b/(2a). The discriminant 0 gives one repeated real root (the curve is tangent to the x-axis), and Δ < 0 gives no real roots.
二次函数的一般形式是 f(x) = ax² + bx + c（a ≠ 0），图像为抛物线。顶点是转折点，用配方法化为 f(x) = a(x − h)² + k 后可直接读出顶点 (h, k)。抛物线关于对称轴 x = −b/(2a) 对称。判别式 Δ 决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个重根（曲线与 x 轴相切）；Δ < 0 无实根。
Δ = b² − 4ac
| Δ value | Nature of roots | Graph meets x-axis |
| Δ > 0 | two distinct real roots | twice |
| Δ = 0 | one repeated real root | tangent (touches) |
| Δ < 0 | no real roots | never |
Mnemonic: 'the discriminant discriminates' - it separates root cases. Read vertex form as 'h is horizontal shift, k is the kink in the curve'. When a question says 'the line is a tangent to the curve', translate it immediately into Δ = 0.
口诀："判别式会判别"——它把根的情况区分开。顶点式中 h 是水平平移量，k 是曲线转折的纵向位置。题目一旦出现"直线与曲线相切"，立刻翻译为 Δ = 0。
4. Equations, Inequalities and Absolute Values | 方程、不等式与绝对值
An equation asserts equality of two expressions; a root or solution is a value satisfying it. An identity holds for all values of the variable. An inequality uses >, ≥, < or ≤, and its answer is a solution set, often shown on a number line. The modulus (absolute value) |x| is the distance of x from zero, so |x| ≥ 0 always. Translate |x − 3| < 2 as a distance statement: 'the distance from x to 3 is less than 2', giving 1 < x < 5.
方程表示两个表达式相等，使其成立的值为根或解；对变量一切取值都成立的等式称为恒等式。不等式使用 >、≥、<、≤，其答案是一个解集，常用数轴表示。模（绝对值） |x| 表示 x 与零的距离，因此恒有 |x| ≥ 0。把 |x − 3| < 2 读成"x 到 3 的距离小于 2"，即 1 < x < 5。
|A| = a (a > 0) → A = a or A = −a |A| < a → −a < A < a |A| > a → A < −a or A > a
Squaring both sides of |A| = |B| gives A² = B², which is safe. Squaring a modulus inequality or an equation such as |x| = 2x − 1, however, can create an extraneous solution, so always substitute your answers back. For quadratic inequalities, find the roots first, sketch the parabola, then read off the interval: '< 0 inside the roots, > 0 outside the roots' when a > 0.
对 |A| = |B| 两边平方得 A² = B²，这是安全的做法。但对方程如 |x| = 2x − 1 或模不等式平方可能产生增根，因此务必把答案代回原式检验。解二次不等式时，先求根、再画抛物线草图，然后读出区间：当 a > 0 时，"< 0 取两根之间，> 0 取两根之外"。
5. Indices, Surds and Logarithmic Functions | 指数、根式与对数函数
In aⁿ, a is the base and n the index The index laws state aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. A surd is an irrational root such as √2, and rationalising the denominator means removing it, using the conjugate: multiply by (√a − √b)/(√a − √b) for a denominator √a + √b.
在 aⁿ 中，a 是底数，n 是指数（幂）。指数律为：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。根式指 √2 这类无理根；分母有理化就是消除分母中的根号，方法是乘以共轭式：分母为 √a + √b 时，上下同乘 √a − √b。
If a = bˣ, then x is the logarithm n x = log_b a. In other words, logs are the inverse of exponentials, so the graph of y = aˣ − log_b n; log_b (mᵏ) = k log_b m. Powers of e use the natural logarithm: ln x = log_e x.
若 a = bˣ，则 x 叫做以 b 为底 a 的对数，记作 x = log_b a。也就是说，对数是指数的逆运算，因此 y = aˣ 的图像过 (0, 1)，它关于 y = x 的反射图像过 (1, 0)。常用法则：log_b (mn) = log_b m + log_b n；log_b (m/n) = log_b m − log_b n；log_b (mᵏ) = k log_b m。以 e 为底的对数称自然对数，记作 ln x = log_e x。
log_b x = log_a x ÷ log_a b (change of base)
To solve aˣ = b, take logs of both sides: x = ln b ÷ ln a. Watch for the words exact form t log_b x is defined only for x > 0, so always check the domain before cancelling logs.
解 aˣ = b 时两边取对数得 x = ln b ÷ ln a。注意题目中的精确形式（exact form）——要保留 ln 5 而不是写成 1.609。还要记住 log_b x 只在 x > 0 时有定义，因此约去对数前必须先检查定义域。
6. Polynomials: Remainder and Factor Theorem | 多项式：余数定理与因式定理
A polynomial is an expression of the form aₙxⁿ + ... + a₁x + a₀. Its degree is the highest power, the number aₙ is the leading coefficient, and each aᵢ is a coefficient. Long division gives a quotient and a remainder. The remainder theorem states that when f(x) is divided by (x − a), the remainder is f(a). The factor theorem states that (x − a) is a factor of f(x) if and only if f(a) = 0.
多项式形如 aₙxⁿ + … + a₁x + a₀，其次数是最高次幂，aₙ 称为首项系数，各 aᵢ 为系数。长除法的结果包含商式与余式。余数定理指出：f(x) 除以 (x − a) 的余数等于 f(a)。因式定理指出：(x − a) 是 f(x) 的因式，当且仅当 f(a) = 0。
f(x) = (x − a)·Q(x) + f(a)
Two wording traps: 'divided by (x + 2)' means a = −2, and 'find the value of k for which (x − 1) is a factor' ic, once one factor is found, factorise the remaining quadratic to obtain all three roots. 'Show that' is topic almost always expect you to substitute a value and state the conclusion in words.
两个表述陷阱："除以 (x + 2)"意味着 a = −2；"求使 (x − 1) 为因式的 k 值"意味着令 f(1) = 0 求解。三次多项式求出一个因式后，把剩余二次式分解即可得到全部三个根。本专题的"证明题"通常都要求你代入某个值，并用文字明确写出结论。
7. Simultaneous Equations and Algebraic Fractions | 联立方程与代数分式
Simultaneous equations are solved by elimination (adding or subtracting to cancel a variable) or substitution (expressing one variable in terms of the other). When one equation is linear and the other quadratic, substitution always works: rearrange the linear equation, substitute into the quadratic, and solve the resulting quadratic. The number of intersection points of a line and a curve equals the number of real solutions, which is again decided by the discriminant.
联立方程用消元法（相加或相减消去一个未知数）或代入法（用一个未知数表示另一个）求解。当一个方程是一次、另一个是二次时，代入法一定可行：先把一次式整理好，代入二次式，再解所得的二次方程。直线与曲线的交点个数等于实数解的个数，同样由判别式决定。
An algebraic fraction has polynomials in its numerator and denominator. It is proper if the degree of the numerator is lower than that of the denominator, and improper otherwise; an improper fraction must first be divided to produce a polynomial plus a proper fraction. Where a syllabus extends to partial fractions, each distinct linear factor (x − a) receives a constant numerator, and a repeated factor (x − a)² needs two terms.
代数分式的分子与分母都是多项式。若分子次数低于分母次数，称为真分式，否则为假分式；假分式必须先做除法，写成"多项式 + 真分式"。若大纲涉及部分分式，则每个
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