GCSE WJEC Mathematics: Algebra and Functions Exam Focus | GCSE WJEC 数学：代数和函数 考点精讲

📚 GCSE WJEC Mathematics: Algebra and Functions Exam Focus | GCSE WJEC 数学：代数和函数 考点精讲

Algebra and functions form the backbone of the GCSE WJEC Mathematics syllabus, covering everything from simple expansions to quadratic graphs and function notation. This article breaks down each topic into clear, bilingual explanations to help you master the key concepts and avoid common pitfalls in your exam.

代数和函数是 GCSE WJEC 数学课程的核心，从简单的展开到二次函数图像和函数符号都有涉及。本文用清晰的双语解释逐一分解各个知识点，帮助你掌握关键概念并在考试中回避常见错误。

1. Algebraic Expressions and Simplifying | 代数表达式与化简

An algebraic expression combines numbers, variables and operations. Simplifying means collecting like terms — terms with the same variable and power. For example, 3a + 5a simplifies to 8a, while 2x² + 3x² gives 5x². Always keep the sign in front of each term and remember that variables with different powers are not like terms.

代数表达式由数字、变量和运算组成。化简是指合并同类项——即变量和指数都相同的项。例如，3a + 5a 化简为 8a，2x² + 3x² 得 5x²。始终保留各项前面的符号，并记住不同次数的变量不属于同类项。

2. Expanding and Factorising | 展开与因式分解

Expanding removes brackets using the distributive law: a(b + c) = ab + ac. Double brackets such as (x + 3)(x + 2) expand to x² + 5x + 6. Factorising is the reverse process — taking out the highest common factor or writing a quadratic as a product of two binomials. For WJEC, you must be confident with both simple and harder factorisations, including difference of two squares: a² – b² = (a – b)(a + b).

展开是利用分配律去掉括号：a(b + c) = ab + ac。像 (x + 3)(x + 2) 这样的双括号展开后得到 x² + 5x + 6。因式分解是逆过程——提取最大公因数，或者将二次式写成两个一次式的乘积。在 WJEC 考试中，你需要熟练掌握简单和较难的因式分解，包括平方差公式：a² – b² = (a – b)(a + b)。

3. Linear Equations | 线性方程

A linear equation involves a variable to the power 1. The goal is to isolate the unknown using inverse operations. Always do the same to both sides. For equations with brackets, expand first. For fractions, multiply through by the common denominator. Checking your solution by substitution is a quick way to avoid mistakes.

线性方程中的未知数次数为 1。目标是用逆运算将未知数单独解出。等式两边要始终进行相同的操作。含有括号的方程，先展开。含有分数的方程，两边乘以公分母。代入检查解是避免错误的快捷方法。

3(x – 4) = 2x + 5 → x = 17

4. Inequalities | 不等式

Inequalities are solved like equations, but remember the golden rule: when multiplying or dividing by a negative number, you must reverse the inequality sign. Solutions are often shown on a number line with open or closed circles. Integer solutions to inequalities are common in WJEC questions, so list them systematically.

解不等式的方法与解方程类似，但切记黄金法则：当乘以或除以负数时，必须反转不等号。解集通常用数轴表示，空心圆和实心圆加以区分。整数解在 WJEC 考题中很常见，因此要有条理地列出所有整数解。

-2x + 1 ≤ 5 → x ≥ -2

5. Simultaneous Equations | 联立方程组

When two equations share two unknowns, they can be solved by elimination or substitution. In elimination, align the equations and add or subtract to cancel one variable. Substitution is useful when one equation gives a variable explicitly. WJEC often includes word problems that require setting up simultaneous equations before solving.

当两个方程含有两个相同的未知数时，可以用消元法或代入法求解。消元法是对齐方程后相加或相减，消去一个变量。代入法则在其中一个方程已明确表示出一个变量时使用。WJEC 常考应用题，需要先列出联立方程再求解。

• Elimination example: 2x + y = 7 and x – y = 2 → adding gives 3x = 9, so x = 3, then y = 1.
• 消元法示例：2x + y = 7 与 x – y = 2 → 相加得 3x = 9，所以 x = 3，进而 y = 1。

6. Quadratic Expressions and Equations | 二次表达式与方程

Quadratic expressions have the form ax² + bx + c. Solving ax² + bx + c = 0 can be done by factorising, using the quadratic formula, or completing the square. The WJEC exam expects fluency in factorising monic and non‑monic quadratics. Always set the equation to zero first. The quadratic formula is given on the formula sheet, but you must apply it correctly.

二次表达式的一般形式为 ax² + bx + c。解二次方程 ax² + bx + c = 0 可以通过因式分解、公式法或配方法完成。WJEC 考试要求熟练分解首项系数为 1 和非 1 的二次三项式。务必先将方程设为等于零。公式表会给出求根公式，但你必须正确运用。

x = [ -b ± √(b² – 4ac) ] ÷ (2a)

7. Functions and Function Notation | 函数与函数记号

f(x) is read as ‘f of x’ and represents an output for a given input x. Evaluating f(3) means replacing x with 3. Composite functions such as fg(x) mean apply g first, then f. Inverse functions f⁻¹(x) reverse the operation of f(x). WJEC questions often involve simple algebraic manipulation to find the inverse and then check with f(f⁻¹(x)) = x.

f(x) 读作“f of x”，表示对于给定输入 x 的输出。求 f(3) 就是将 x 替换为 3。复合函数 fg(x) 表示先代入 g 再代入 f。反函数 f⁻¹(x) 是 f(x) 运算的逆过程。WJEC 的题目经常要求简单的代数变形求出反函数，然后用 f(f⁻¹(x)) = x 检验。

8. Graphs of Functions | 函数图像

Linear functions produce straight lines (y = mx + c), while quadratic functions give parabolas (y = ax² + bx + c). Sketching requires identifying key features: intercepts, turning points, and line of symmetry for quadratics. Cubic and reciprocal graphs are also part of the WJEC specification. Recognising the shape from the equation is a common short‑answer question.

线性函数画出直线 (y = mx + c)，二次函数画出抛物线 (y = ax² + bx + c)。作图需要识别关键特征：截距、二次函数的顶点和对称轴。三次函数和反比例函数图像也属于 WJEC 考试范围。根据方程判断图像形状是常见的简答题。

Vertex form: y = a(x – h)² + k

9. Algebraic Fractions | 代数分式

Simplifying algebraic fractions involves factorising numerators and denominators, then cancelling common factors. When adding or subtracting, find a common denominator. Multiplication is factorising and cancelling, while division is multiplying by the reciprocal. Always state restrictions on x that make the denominator zero.

化简代数分式需要先将分子分母因式分解，然后约去公因式。加减运算时，先通分找到公分母。乘法是因式分解后约分，除法是乘以倒数。始终要注明使分母为零的 x 的限制值。

Operation 运算	Key step 关键步骤
Addition / Subtraction	Find the lowest common denominator
Multiplication	Factorise and cancel before multiplying
Division	Multiply by the reciprocal

10. Sequences | 数列

Linear sequences have a constant first difference. The nth term is of the form an + b, where a is the common difference and b is the zero‑th term. Quadratic sequences show a constant second difference; their nth term is an² + bn + c. WJEC may ask you to find a specific term, write the nth term, or recognise special sequences like triangular numbers.

线性数列的一阶差分为常数。第 n 项公式形如 an + b，其中 a 为公差，b 为零项。二次数列的二阶差分为常数，其第 n 项为 an² + bn + c。WJEC 可能会要求你求特定项、写出第 n 项或识别特殊数列，如三角形数。

Linear nth term: uₙ = 3n + 2

Quadratic nth term: uₙ = n² + n + 1

11. Forming and Solving Equations from Word Problems | 列方程解应用题

Word problems test your ability to translate real‑world situations into algebraic equations or inequalities. Assign a letter to the unknown quantity, build an expression using the given information, and then solve. Common contexts include ages, perimeter, area, money, and simple kinematics. Always interpret your solution back into the context and check it makes sense.

应用题考查你将现实情境转化为代数方程或不等式的能力。用字母表示未知量，利用已知信息建立表达式，然后求解。常见背景包括年龄、周长、面积、金钱和简单运动学。解出后务必把答案放回原情景检验，确认合理。

• “Three times a number added to 5 gives 20” → 3x + 5 = 20
• “某数的三倍加上5等于20” → 3x + 5 = 20

12. Exam Tips and Common Mistakes | 考试技巧与常见错误

Always show your working — WJEC awards method marks even if the final answer is wrong. When expanding, watch the signs; a common mistake is forgetting to multiply the negative term. In equations, check for extraneous solutions when squaring. For functions, bracket the input carefully when substituting negative numbers. Double‑check factorising by expanding mentally. Manage your time so you can review your answers.

务必展示解题步骤——WJEC 即使最终答案错了也会给过程分。展开时要留意符号；一个常见错误是忘记乘负项。解方程时，平方后要检查增根。求函数值时，代入负数要小心使用括号。因式分解后可以通过心算展开来检查。合理分配时间，以便复查答案。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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