📚 Pre-U WJEC Mathematics: Quick Memorisation Guide to Vocabulary & Terminology | Pre-U WJEC 数学：词汇术语速记指南
Pre-U WJEC Mathematics rewards precise language as much as fluent algebra. Marks are lost when a candidate confuses an expression with an equation, a root with a factor, or a stationary point with a point of inflection. This quick guide collects the vocabulary that examiners actually use, pairing every key term with a short Chinese explanation and a memory hook so that revision sticks.
Pre-U WJEC 数学不仅考查运算是否流畅，更考查语言是否精确。把表达式与方程、根与因式、驻点与拐点混为一谈，是考场上最常见的丢分原因。本速记指南汇总考官真正使用的术语，为每个关键英文术语配上一句简短中文解释和一个记忆钩子，让复习记得住、答题用得出。
1. Algebraic Essentials | 代数核心术语
Expression vs equation. on contains an equals sign and states that two expressions are equal, such as 3x + 2 = 11.
表达式与方程。表达式是由数字、变量和运算符号组成的数学短语，不含关系符，如 5x² − 3x + 7；方程则含有等号，表示左右两边相等，如 3x + 2 = 11。速记："expression 无言，equation 有等号定乾坤"。
Identity. An identity holds true for all values of the variable and is usually written with the ≡ symbol. A classic example is (a + b)² ≡ a² + 2ab + b².
恒等式。恒等式对所有变量取值都成立，常用符号 ≡ 表示，比如 (a + b)² ≡ a² + 2ab + b²。联想 identity 像身份证，永远保持真值，三条横线强调"恒等"。
Polynomial and degree. A polynomial is an expression involving only non-negative integer exponents. The degree is determined by the highest exponent; for example, 4x³ − x + 2 is a cubic polynomial of degree 3.
多项式与次数。多项式由非负整数指数构成，最高次项决定多项式的次数。例如 4x³ − x + 2 是三次多项式。poly 意为"多"，nomial 联想到"项"，合起来就是多个项之和。
Term, coefficient and like terms. A term is a single product such as 6x²; the coefficient is its numerical part, here 6. Like terms share the same variable powers and may be collected together.
项、系数与同类项。项是像 6x² 这样的单个乘积；系数是其中的数字部分，此处为 6。同类项的变量与指数完全相同，可以直接合并加减。
Factor theorem. If f(a) = 0, then (x − a) is a factor of f(x).
因式定理。若 f(a) = 0，则 (x − a) 是 f(x) 的因式。速记：根反过来就是因式。
Remainder theorem. When f(x) is divided by (x − a), the remainder is f(a).
余式定理。f(x) 除以 (x − a) 所得的余数就是 f(a)。速记：代入就是余数。两条定理常常在同一题中连用，先试根、再除式。
2. Functions and Graphs | 函数与图像
A function f assigns each input x exactly one output f(x). The domain is the complete set of allowed inputs, while the range is the set of all possible outputs.
函数 f 将每一个输入 x 唯一地映射到一个输出 f(x)。定义域是容许输入的全集，值域是所有可能输出的集合。联想：定义域像入境许可，值域像产出范围。
Inverse function. g: if f(a) = b then f⁻¹(b) = a. Its graph is the reflection of y = f(x) in the line y = x.
反函数。反函数 f⁻¹ 逆转映射关系：若 f(a) = b，则 f⁻¹(b) = a。其图像是 y = f(x) 关于直线 y = x 的镜像。
Composite function. The composite function fg(x) means apply g first, then f. f²(x) often means apply f twice, not square the output — beware of notation!
复合函数。复合函数 fg(x) 表示先作用 g 再作用 f。f²(x) 通常指把 f 连续作用两次，而不是把输出平方——记号陷阱要当心！
One-to-one and many-to-one. A one-to-one function gives a different output for every input, so its inverse is also a function. A many-to-one function has no inverse unless the domain is restricted.
单射与多对一。单射函数对每个输入都给出不同输出，因此其反函数仍是函数。多对一函数必须限制定义域后才能求反函数，否则反函数"一对多"，不再是函数。
Transformations. Translations shift a graph, stretches change its scale, and reflections flip it in an axis. In function notation, f(x) + a is a vertical translation and f(x + a) is a horizontal translation in the opposite direction to the sign.
图像变换。平移移动图像位置，伸缩改变图像比例，反射以坐标轴为镜翻转。用函数记号表示时，f(x) + a 是竖直平移，而 f(x + a) 是水平平移，方向与符号相反。
3. Quadratics, Roots and the Discriminant | 二次式、根与判别式
roots, Δ = 0 gives a repeated root, and Δ < 0 gives no real roots but two complex ones.
对二次方程 ax² + bx + c = 0，判别式为 Δ = b² − 4ac。它的符号决定根的性质：Δ > 0 有两个不同实根，Δ = 0 有重根，Δ < 0 没有实根而是两个复根。
Δ = b² − 4ac x = (−b ± √b² − 4ac) ÷ 2a
Roots, zeroes and solutions. These three words describe the same x-values: where the curve meets the x-axis, and where the corresponding equation equals zero. An examiner may ask for "the roots" in one part and "the solutions" in the next without changing the mathematics.
根、零点与解。这三个词描述同一批 x 值：曲线与 x 轴相交之处，也是方程等于零之处。考官可能前半题问 roots、后半题问 solutions，数学内容完全不变。
Completing the square. metry x = −p, and the sign of a tells you whether the parabola opens upwards or downwards.
配方法。把 ax² + bx + c 写成 a(x + p)² + q，就立刻看出顶点（即 turning point）在 (−p, q)。顶点位于对称轴 x = −p 上，而 a 的正负决定抛物线开口向上还是向下。
Sketch, not plot. A sketch shows the shape and key features — intercepts, vertex, asymptotes — without accurate plotting. Marks are given for labelled intercepts and the correct shape.
sketch 与 plot 的区别。sketch 只要求画出形状与关键特征（截距、顶点、渐近线），不要求精确描点；但截距必须标注，形状必须正确，否则扣分。
4. Sequences, Series and the Binomial Expansion | 数列、级数与二项展开
A sequence is an ordered list of terms; a series is the sum of the terms of a sequence. A progression is a sequence with a recognisable rule, most often arithmetic or geometric.
数列是按顺序排列的一串项；级数是数列各项之和。所谓"数列规律"（progression）指具有可识别法则的数列，最常见的是等差与等比。
Common difference and common ratio. In an arithmetic progression the difference between consecutive terms is constant, denoted d; in a geometric progression the ratio between consecutive terms is constant, denoted r.
公差与公比。等差数列中相邻两项之差恒定，记作 d；等比数列中相邻两项之比恒定，记作 r。d 用减法、r 用除法，这是最快辨别两个数列类型的办法。
uₙ = a + (n − 1)d uₙ = arⁿ⁻¹
Sₙ = n(a + l) ÷ 2 = n[2a + (n − 1)d] ÷ 2 Sₙ = a(1 − rⁿ) ÷ (1 − r), r ≠ 1
Sigma notation. The symbol Σ means "sum of" ctly the geometric series; the numbers below and above Σ are the first and last values of the counter.
Σ 求和记号。符号 Σ 表示"求和"。写作 r = 1 到 n 的 arⁿ⁻¹ 之和，就是等比级数；Σ 下上与上方的数字分别表示计数变量的首值与末值。
Convergent and divergent. An infinite geometric series converges to a finite sum when |r| < 1, with sum a ÷ (1 − r); it diverges when |r| ≥ 1.
收敛与发散。当 |r| < 1 时，无穷等比级数收敛到有限值，和为 a ÷ (1 − r)；当 |r| ≥ 1 时级数发散，没有有限和。这个条件必须写在答案里，否则会丢掉一分。
Binomial coefficients and factorials. The coefficient of the general term is written ⁿCᵣ or (n choose r), and equals n! ÷ [r!(n − r)! (a + b)ⁿ the powers of a decrease while the powers of b increase, and every row of Pascal's triangle supplies the coefficients for n = 0, 1, 2, 3, …
二项式系数与阶乘。通项系数写作 ⁿCᵣ 或"n 选 r"，等于 n! ÷ [r!(n − r)!]。在 (a + b)ⁿ 的展开式中，a 的指数递减、b 的指数递增，而帕斯卡三角形的每一行正好给出 n = 0, 1, 2, 3, … 的系数。
5. Trigonometry Vocabulary | 三角学术语
Radian and degree. ce the full circumference is 2πr, one full turn is 2π radians, so π radians = 180 degrees.
弧度与角度。1 弧度是在圆上取一段与半径等长的弧所对的圆心角。由于整圆周长为 2πr，一整圈为 2π 弧度，故 π 弧度 = 180 度。换算速记：弧度化角度乘 180 ÷ π，角度化弧度乘 π ÷ 180。
Amplitude, period and phase. In y = A sin(bx + c) + d, the amplitude is |A| b or 2π ÷ b, and the phase shift relates to −c ÷ b. The line y = d is the principal axis.
振幅、周期与相位。在 y = A sin(bx + c) + d 中，振幅为 |A|，周期为 360° ÷ b 或 2π ÷ b，相位移动与 −c ÷ b 相关，直线 y = d 是图像的中轴线。
sin²θ + cos²θ ≡ 1 tan θ ≡ sin θ ÷ cos θ
Sine rule and cosine rule. cosine rule, a² = b² + c² − 2bc cos A, handles the side-angle-side and three-sides cases.
正弦定理与余弦定理。正弦定理 a ÷ sin A = b ÷ sin B = c ÷ sin C，适用于已知"一边一角配对"的任意三角形；余弦定理 a² = b² + c² − 2bc cos A，处理"两边夹角"与"已知三边"的情形。口诀：配对角用正弦，夹住角用余弦。
Principal value and CAST diagram. Because trigonometric functions repeat, a calculator only returns the principal value; the CAST diagram, or the symmetry of the graphs, tells you where the other solutions lie in the required interval.
主值与 CAST 图。由于三角函数周期重复，计算器只给出主值；其余解要靠 CAST 图或图像对称性在指定区间内补齐。答题时必须写"在给定区间内"的全部解。
6. Exponentials and Logarithms | 指数与对数
Exponential function. An exponential function has the variable in the exponent, as in y = aˣ, where a > 0. When the base is e ≈ 2.718, the curve y = eˣ has the special property that its gradient at every point equals its own value.
指数函数。指数函数的自变量位于指数上，形如 y = aˣ（a > 0）。当底数为 e ≈ 2.718 时，曲线 y = eˣ 具有特殊性质：每一点的斜率都等于该点的函数值本身。
Logarithm. A logarithm answers the question "what power?". If y = aˣ then x = logₐ y. In particular, ln y means logₑ y and is called the natural logarithm.
对数。对数回答的是"几次方"这个问题。若 y = aˣ，则 x = logₐ y。特别地，ln y 表示 logₑ y，称为自然对数。
ab) = log a + log b log(a ÷ b) = log a − log b log(aⁿ) = n log a
Growth and decay. Exponential growth means the quantity increases by a constant factor in equal time steps; exponential decay means it decreases by a constant factor. Both appear in Pre-U modelling questions with the words "rate of change proportional to the amount present".
增长与衰减。指数增长指在相等时间段内按固定倍数增加；指数衰减指按固定倍数减少。在 Pre-U 建模题中，两者常以"变化率与现有量成正比"这句话出现，看到它就应该写出微分方程形式。
Logarithmic graph. The graph of y = ln x is the reflection of y = eˣ in the line y = x, with a vertical asymptote at x = 0 and domain x > 0.
对数图像。y = ln x 的图像是 y = eˣ 关于直线 y = x 的镜像，在 x = 0 处有竖直渐近线，定义域为 x > 0。两个图像必须会随手画出。
7. Differentiation Terminology | 微分术语
Derivative and gradient function. The derivative of f(x), written f′(x) or dy ÷ dx, is the gradient function: it gives the gradient of the tangent to the curve at each point.
导数与斜率函数。f(x) 的导数记作 f′(x) 或 dy ÷ dx，它就是斜率函数：给出曲线上每一点处切线的斜率。
f′(x) = lim (h → 0) [f(x + h) − f(x)] ÷ h
Differentiation from first principles. This phrase means you must use the limit definition above rather than a shortcut rule. It is a command word, not a suggestion.
从基本原理求导。看到 from first principles，就必须使用上面的极限定义，而不能直接套用求导公式。这是命令词，不是建议。
Chain, product and quotient rules. product rule gives (uv)′ = u′v + uv′. The quotient rule gives (u ÷ v)′ = (u′v − uv′) ÷ v².
链式、乘积与商法则。链式法则处理"函数的函数"，dy ÷ dx = dy ÷ du × du ÷ dx；乘积法则给出 (uv)′ = u′v + uv′；商法则给出 (u ÷ v)′ = (u′v − uv′) ÷ v²。商法则口诀：上导下不导，减去上不导下导，再除以分母平方。
Stationary point, maximum, minimum, point of inflection. nimum, and f″(x) = 0 suggests a possible point of inflection that must be checked further.
驻点、极大、
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