📚 IGCSE WJEC Further Maths: Vocabulary Quick Revision Guide | IGCSE WJEC 进阶数学：词汇术语速记指南
Mastering the terminology of IGCSE WJEC Further Mathematics is the first step towards solving complex problems with confidence. This guide breaks down key terms across all major topics, offering memory aids and bilingual explanations to help you internalise concepts quickly.
掌握IGCSE WJEC进阶数学的术语，是自信解决复杂问题的第一步。本指南分解了所有主要主题中的关键术语，提供记忆辅助和中英双语解释，帮助你快速内化概念。
1. Algebra & Polynomials | 代数与多项式
In Further Maths, algebra extends beyond solving linear equations to manipulating polynomials, using the remainder theorem and factorising cubic expressions. Remember that a 'coefficient' is the numerical factor in front of a variable.
在进阶数学中，代数不止于求解线性方程，还要操作多项式、运用余式定理和因式分解三次式。记住：‘系数’是变量前面的数字因子。
A 'polynomial' is an expression like 3x³ + 2x² − x + 5, where terms have non-negative integer exponents. The 'degree' is the highest power of the variable — here it is 3.
‘多项式’是像 3x³ + 2x² − x + 5 这样的表达式，其各项指数为非负整数。‘次数’是变量的最高幂——这里是 3。
The 'Remainder Theorem' states that when a polynomial f(x) is divided by (x − a), the remainder is f(a). This allows quick evaluation without long division.
‘余式定理’指出，当一个多项式 f(x) 除以 (x − a) 时，余数为 f(a)。这可以快速求值，无需长除。
The 'Factor Theorem' is a special case: if f(a) = 0, then (x − a) is a factor of f(x).
‘因式定理’是特例：如果 f(a) = 0，那么 (x − a) 是 f(x) 的一个因式。
| English Term | 中文术语 | Memory Tip |
|---|---|---|
| Coefficient | 系数 | Co- (together) + efficient, a number acting with a variable |
| Monic polynomial | 首一多项式 | Leading coefficient is 1 |
| Root / Zero | 根 / 零点 | Value making the polynomial equal zero |
2. Equations & Inequalities | 方程与不等式
The 'discriminant' for ax²+bx+c=0. Δ>0: two distinct real roots; Δ=0: one repeated root; Δ<0: no real roots.
‘判别式’ (Δ = b² − 4ac) 决定二次方程 ax²+bx+c=0 的根的性质。Δ>0：两不等实根；Δ=0：一个重根；Δ<0：无实根。
s >, <, ≥, ≤. When multiplying or dividing by a negative number, flip the inequality sign.
不等式使用 >, <, ≥, ≤ 符号。当乘以或除以负数时，要反转不等号方向。
A 'strict inequality' excludes the boundary value (e.g., x > 2), whereas a 'non-strict' one includes it (x ≥ 2). Used with interval notation like (2, 5] means 2 < x ≤ 5.
‘严格不等式’不包含边界值（例如 x > 2），而‘非严格’不等式包含（x ≥ 2）。区间记号如 (2, 5] 表示 2 < x ≤ 5。
| English Term | 中文术语 | Memory Tip |
|---|---|---|
| Root / Solution | 根 / 解 | A value satisfying the equation |
| Simultaneous equations | 联立方程 | Two or more equations solved together |
3. Functions & Graphs | 函数与图像
The 'domain' is the set of all possible input values (x), while the 'range' is the set of all possible output values (y). Think of the domain as what you can put into the function without breaking it.
‘定义域’是所有可能输入值 (x) 的集合，而‘值域’是所有可能输出值 (y) 的集合。可将定义域视为可以安全放进函数里的值。
An 'inverse function' eflection in the line y = x. Only one-one functions have inverses that are also functions.
‘反函数’ f⁻¹(x) 逆转 f(x) 的作用。图像上，它是关于直线 y = x 的反射。只有一一函数才有函数形式的反函数。
The 'modulus function' |x| gives the absolute value, turning negative inputs into positive. Graphs of modulus functions often show a sharp V-shape where negative parts are reflected upwards.
‘模函数’ |x| 给出绝对值，将负数输入变为正数。模函数图像通常在负数部分向上反射，呈现 V 形。
| English Term | 中文术语 | Memory Tip |
|---|---|---|
| Composite function (fg) | 复合函数 | Apply g, then f |
4. Exponentials & Logarithms | 指数与对数
A 'logarithm' is the inverse of exponentiation. If aˣ = N, then logₐN = x. The base a must be positive and not equal to 1.
‘对数’是指数运算的逆运算。如果 aˣ = N，那么 logₐN = x。底数 a 必须为正且不等于 1。
ral logarithm, ln, uses base e ≈ 2.718. It appears in calculus because the derivative of eˣ is eˣ, and the derivative of ln x is 1/x.
自然对数 ln 以 e ≈ 2.718 为底。它出现在微积分中，因为 eˣ 的导数是 eˣ，ln x 的导数是 1/x。
logₐx − logₐy; logₐ(xⁿ) = n logₐx. These help condense or expand logarithmic expressions.
关键对数定律：logₐ(xy) = logₐx + logₐy；logₐ(x/y) = logₐx − logₐy；logₐ(xⁿ) = n logₐx。这些帮助压缩或展开对数表达式。
| English Term | 中文术语 |
|---|---|
| Exponential growth | 指数增长 |
| Change of base | 换底公式 |
5. Trigonometry | 三角学
A 'radian' ength to the radius. π rad = 180°. Radian measure simplifies calculus with trig functions.
‘弧度’是半径长度对应的圆弧所对的圆心角。π rad = 180°。弧度制简化了三角函数的微积分运算。
The CAST diagram helps remember where sine, cosine and tangent are positive. Quadrants: All (I), Sine (II), Tangent (III), Cosine (IV).
CAST 图帮助记忆正弦、余弦和正切在哪些象限为正。象限：全正 (I)，正弦 (II)，正切 (III)，余弦 (IV)。
tanθ = sinθ / cosθ. For reciprocal functions, secθ = 1/cosθ, cscθ = 1/sinθ, cotθ = 1/tanθ.
关键恒等式：sin²θ + cos²θ = 1，tanθ = sinθ / cosθ。倒数函数：secθ = 1/cosθ，cscθ = 1/sinθ，cotθ = 1/tanθ。
