📚 Year 10 Eduqas Further Mathematics: Glossary Quick Guide | Year 10 Eduqas 进阶数学：词汇术语速记指南
This quick-reference glossary is designed to help Year 10 students master the essential terminology of the Eduqas GCSE Further Mathematics course. Each key term is defined clearly in English and followed immediately by its Chinese equivalent, helping bilingual learners build confidence in both languages while preparing for exams. From algebraic expressions to vectors and calculus, this guide covers the vocabulary you will encounter across all major topics.
这份速记术语指南专为 Year 10 学生掌握 Eduqas GCSE 进阶数学的核心词汇而设计。每个关键术语先用英文明确定义，紧接着给出对应的中文解释，帮助双语学习者在备考时用两种语言建立信心。从代数表达式到向量和微积分，本指南涵盖了你将在所有主要课题中遇到的词汇。
1. Algebraic Expressions and Terms | 代数表达式与项
Polynomial: An algebraic expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non‑negative integer exponents. For example, 4x³ – 2x² + x – 7 is a polynomial in x.
多项式：由变量和系数组成的代数表达式，只涉及加、减、乘和非负整数指数。例如，4x³ – 2x² + x – 7 是一个关于 x 的多项式。
Term: A single number, variable, or product of numbers and variables separated by addition or subtraction. In 5x² + 3x – 8, the terms are 5x², 3x and –8.
项：由加号或减号分隔的单个数字、变量或数字与变量的乘积。在 5x² + 3x – 8 中，项分别为 5x²、3x 和 –8。
Coefficient: The numerical factor multiplying a variable in a term. In the term 7y⁴, the coefficient is 7.
系数：项中乘以变量的数值因子。在项 7y⁴ 中，系数为 7。
Like terms: Terms that contain exactly the same variables raised to the same powers. The coefficients may differ. 2ab² and –5ab² are like terms, but 3a²b is not.
同类项：包含完全相同变量且指数相同的项。系数可以不同。2ab² 和 –5ab² 是同类项，但 3a²b 不是。
Degree of a polynomial: The highest power of the variable in the polynomial. The polynomial 6x⁴ + x³ – 9 has degree 4.
多项式的次数：多项式中变量的最高次幂。多项式 6x⁴ + x³ – 9 的次数为 4。
2. Quadratic Equations and the Discriminant | 二次方程与判别式
Quadratic equation e : An equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
二次方程：可写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。
Discriminant: The expression Δ = b² – 4ac, which determines the nature of the roots of a quadratic equation.
Δ = b² – 4ac
判别式：表达式 Δ = b² – 4ac，它决定了二次方程根的性质。
Nature of roots Δ = 0, it has one repeated real root; if Δ < 0, it has no real roots (two complex roots).
根的性质：若 Δ > 0，方程有两个不相等的实数根；若 Δ = 0，有一个重根；若 Δ < 0，没有实数根（有两个复数根）。
Completing the square: A technique for rewriting a quadratic expression in the form a(x + p)² + q. It is often used to solve quadratics or find the vertex of a parabola.
配方法：将二次表达式改写成 a(x + p)² + q 形式的方法。常用于解二次方程或求抛物线的顶点。
Quadratic formula: The formula giving the solutions of ax² + bx + c = 0.
x = [ –b ± √b² – 4ac ] / (2a)
求根公式：给出 ax² + bx + c = 0 的解的公式。
3. Inequalities and Simultaneous Equations | 不等式与联立方程组
Linear inequality: An inequality involving a linear expression, such as 3x – 5 > 7. Solutions are often represented on a number line or in interval notation.
一元一次不等式：涉及线性表达式的不等式，如 3x – 5 > 7。解通常用数轴或区间表示法表示。
Quadratic inequality: An inequality like x² – set is found by first solving the corresponding quadratic equation and testing intervals.
二次不等式：形如 x² – 4x + 3 ≤ 0 的不等式。通过先解对应的二次方程并检验区间来求其解集。
Simultaneous equations: A set of equations with multiple variables that are all satisfied by the same values. In Further Maths, this often includes one linear and one quadratic equation solved by substitution.
联立方程组：一组包含多个变量的方程，这些方程由相同的变量值同时满足。在进阶数学中，常包括一个线性方程和一个二次方程，通过代入法求解。
Feasible region: The area on a graph that satisfies a system of inequalities. It is bounded by the lines representing the equalities.
可行区域：图上满足一组不等式的区域。它由表示等式的直线围成。
4. Functions and Graph Transformations | 函数与图像变换
Function h input (x) has exactly one output, typically written as f(x). For example, f(x) = 2x + 3.
函数：每个输入（x）恰好对应一个输出的关系，通常记作 f(x)。例如 f(x) = 2x + 3。
Domain: The set of all possible input values (x) for which a function is defined. For f(x) = √x, the domain is x ≥ 0.
定义域：使函数有定义的所有可能输入值（x）的集合。对于 f(x) = √x，定义域为 x ≥ 0。
Range : The set of all possible output values the function can produce. For f(x) = x², the range is f(x) ≥ 0.
值域：函数可以产生的所有可能的输出值的集合。对于 f(x) = x²，值域为 f(x) ≥ 0。
Composite function ion created by applying one function to the result of another, denoted by fg(x) = f(g(x)).
复合函数：将一个函数应用于另一个函数的结果所创建的函数，记作 fg(x) = f(g(x))。
Inverse function inal function. It satisfies f(f⁻¹(x)) = x and its graph is a reflection in the line y = x.
反函数：记作 f⁻¹(x) 的函数，它逆转原函数的作用。它满足 f(f⁻¹(x)) = x，且其图像是关于直线 y = x 的反射。
Graph translations = f(x) + a translates the graph up by a units; y = f(x + a) translates it left by a units.
图像平移：将图像在不改变形状的情况下进行水平或垂直移动的变换。y = f(x) + a 将图像向上平移 a 个单位；y = f(x + a) 将图像向左平移 a 个单位。
Graph reflections: y = –f(x) reflects the graph in the x‑axis, while y = f(–x) reflects it in the y‑axis.
图像反射：y = –f(x) 是将图像关于 x 轴反射，而 y = f(–x) 是关于 y 轴反射。
5. Differentiation | 微分
Derivative: The rate of change of a function with respect to x. It is written as f'(x) or dy/dx and represents the gradient of the tangent to the curve at a point.
导数：函数关于 x 的变化率。记作 f'(x) 或 dy/dx，它代表曲线上某点切线的斜率。
Power rule for differentiation: For f(x) = xⁿ, the derivative is f'(x) = n xⁿ⁻¹. Multiply by the power and reduce the power by 1.
d/dx (xⁿ) = n xⁿ⁻¹
幂函数求导法则：对于 f(x) = xⁿ，导数为 f'(x) = n xⁿ⁻¹。乘以指数并将指数减 1。
Tangent and normal: A tangent is a straight line that touches a curve at a point with the same gradient as the curve there. A normal is perpendicular to the tangent at that point.
切线与法线：切线是与曲线在某点相切且与该点曲线斜率相同的直线。法线是在该点与切线垂直的直线。
Stationary points: Points where f' , local minimum or point of inflection. They are found by setting the first derivative to zero.
驻点：满足 f'(x) = 0 的点，可能是局部极大值、局部极小值或拐点。通过令一阶导数为零求得。
Increasing and decreasing functions: A function is increasing where f'(x) > 0, and decreasing where f'(x) < 0.
增函数与减函数：在 f'(x) > 0 的区间函数递增，在 f'(x) < 0 的区间函数递减。
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