📚 IGCSE Edexcel Maths: Algebra and Functions – Essential Revision | IGCSE Edexcel 数学：代数和函数 考点精讲
Algebra and functions form the backbone of the IGCSE Edexcel Mathematics syllabus. Mastering these topics is essential for success not only in the exam but also in building a solid foundation for further study. This article covers the key concepts—from simplifying expressions and solving equations to understanding function notation, composites, inverses, and graphs. Each section is designed to clarify common pitfalls and highlight typical exam techniques.
代数和函数是 IGCSE Edexcel 数学课程的核心。掌握这些主题不仅对考试成功至关重要，也为进一步学习奠定坚实基础。本文涵盖从化简表达式、解方程到理解函数记法、复合、反函数及图像的关键概念。每个部分旨在澄清常见误区并突出典型考试技巧。
1. Simplifying Algebraic Expressions | 化简代数表达式
Simplifying algebraic expressions involves collecting like terms and using the laws of indices. Like terms have exactly the same variable parts; only their coefficients differ. For example, 3x and 5x are like terms, while 3x and 3x² are not. Index laws are crucial when multiplying or dividing powers: aⁿ × aᵐ = aⁿ⁺ᵐ, aⁿ ÷ aᵐ = aⁿ⁻ᵐ, and (aⁿ)ᵐ = aⁿᵐ.
化简代数表达式包括合并同类项和使用指数运算法则。同类项具有完全相同的变量部分，仅系数不同。例如，3x 和 5x 是同类项，而 3x 和 3x² 不是。乘除幂时指数法则至关重要：aⁿ × aᵐ = aⁿ⁺ᵐ，aⁿ ÷ aᵐ = aⁿ⁻ᵐ，以及 (aⁿ)ᵐ = aⁿᵐ。
A common mistake is to treat addition of powers as multiplication. For instance, a³ × a² = a⁵, but a³ + a² cannot be simplified further. Always check if terms are like before combining. When simplifying expressions with brackets, expand first, then collect like terms.
常见错误是将幂的加法当作乘法处理。例如，a³ × a² = a⁵，但 a³ + a² 无法进一步化简。合并前务必检查项是否同类。化简带括号的表达式时，先展开再合并同类项。
2. Expanding Brackets | 展开括号
Expanding brackets means removing the brackets by multiplying each term inside the bracket by the term outside. For a single bracket, multiply each term: a(b + c) = ab + ac. With two brackets, use the FOIL or grid method: (a + b)(c + d) = ac + ad + bc + bd. Special cases include perfect squares and difference of two squares: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², (a + b)(a − b) = a² − b².
展开括号指通过将括号外的项乘以括号内每一项来移除括号。对于单个括号，乘以每一项：a(b + c) = ab + ac。对于两个括号，使用 FOIL 或网格法：(a + b)(c + d) = ac + ad + bc + bd。特殊情况包括完全平方和平方差：(a + b)² = a² + 2ab + b²，(a − b)² = a² − 2ab + b²，(a + b)(a − b) = a² − b²。
Pay close attention to signs when expanding. A negative sign outside the bracket changes the sign of every term inside. For example, −3(x − 2) = −3x + 6. In the exam, be prepared to expand and then simplify to prepare for further steps, such as solving equations.
展开时务必注意符号。括号外的负号会改变括号内每一项的符号。例如，−3(x − 2) = −3x + 6。考试中应准备好先展开再化简，以便后续步骤（如解方程）使用。
3. Factorising Expressions | 因式分解
Factorising is the reverse of expanding. It involves writing an expression as a product of its factors. The first step is always to look for a common factor. For 4x + 8, the common factor is 4, so 4x + 8 = 4(x + 2). For quadratic expressions like x² + 7x + 12, we need two numbers that multiply to 12 and add to 7: 3 and 4, giving (x + 3)(x + 4). Difference of two squares factorises as a² − b² = (a + b)(a − b).
因式分解是展开的逆运算，将表达式写成因式的乘积。第一步总是寻找公因数。对于 4x + 8，公因数是 4，因此 4x + 8 = 4(x + 2)。对于二次表达式如 x² + 7x + 12，需要两个数乘积为 12 且和为 7：3 和 4，得到 (x + 3)(x + 4)。平方差分解为 a² − b² = (a + b)(a − b)。
Always check your factorisation by expanding to see if you get the original expression. If the quadratic has a coefficient for x² that is not 1, e.g., 2x² + 5x + 3, you can use grouping or the ‘ac method’. Factorising is vital for solving quadratic equations and simplifying algebraic fractions.
始终通过展开检查因式分解是否正确，看是否得到原式。若二次项系数不是 1，如 2x² + 5x + 3，可使用分组或 ‘ac 方法’。因式分解对解二次方程和化简代数分式至关重要。
4. Solving Linear Equations | 解线性方程
Linear equations involve the unknown variable to the power 1. The goal is to isolate the variable using inverse operations while keeping the equation balanced. Perform the same operation on both sides. For 3x + 5 = 17, subtract 5 from both sides to get 3x = 12, then divide by 3 to obtain x = 4. When brackets or fractions are present, clear them first: expand brackets or multiply through by the common denominator.
线性方程中未知数的指数为 1。目标是通过逆运算分离变量，同时保持方程平衡。在等式两边执行相同操作。对于 3x + 5 = 17，两边减 5 得 3x = 12，再除以 3 得 x = 4。当有括号或分母时，先清理它们：展开括号或两边同乘公分母。
Watch for equations with the variable on both sides. Collect variable terms on one side and constants on the other. For example, 2x − 3 = x + 4 becomes x = 7. Always check your solution by substituting back into the original equation. In IGCSE, you may be asked to set up an equation from a word problem and then solve it.
注意变量在等式两边的情况。将变量项移到一边，常数移到另一边。例如，2x − 3 = x + 4 得到 x = 7。始终将解代回原方程检验。IGCSE 中可能会要求根据文字题建立方程并求解。
5. Solving Quadratic Equations | 解二次方程
Quadratic equations have the general form ax² + bx + c = 0 (a ≠ 0). The three main methods are factorising, using the quadratic formula, and completing the square. Factorising is preferred when it is possible: set each factor to zero. For x² − 5x + 6 = 0, factorise to (x − 2)(x − 3) = 0, giving solutions x = 2 and x = 3.
二次方程的一般形式为 ax² + bx + c = 0 (a ≠ 0)。三种主要方法是因式分解、使用二次公式和配方法。可能时优先选用因式分解：令每个因式为零。对于 x² − 5x + 6 = 0，分解为 (x − 2)(x − 3) = 0，得到解 x = 2 和 x = 3。
The quadratic formula works for all quadratics:
x = [−b ± √(b² − 4ac)] / (2a)
The discriminant b² − 4ac determines the nature of the roots: if positive, two distinct real roots; if zero, one repeated real root; if negative, no real roots. Make sure to use brackets when substituting negative numbers into the formula.
二次公式适用于所有二次方程：
x = [−b ± √(b² − 4ac)] / (2a)
判别式 b² − 4ac 决定根的性质：正数则两个不等实根；零则一个重实根；负数则无实根。将负数代入公式时务必使用括号。
6. Inequalities | 不等式
Inequalities work similarly to equations but with one critical difference: multiplying or dividing by a negative number reverses the inequality sign. Solve 2x − 3 > 5 as you would an equation, but if you divide by −2, flip the sign. Answers can be represented on a number line using open or solid circles, and in set notation.
不等式类似方程，但有一个关键区别：乘以或除以负数会反转不等号。像解方程一样解 2x − 3 > 5，但若除以 −2，需反转不等号。答案可用数轴上空心或实心圆点表示，也可用集合记法。
Quadratic inequalities require sketching a graph or using sign analysis. For x² − 4 < 0, factorise to (x − 2)(x + 2) < 0, then determine where the product is negative: between −2 and 2. The solution is −2 < x < 2. Always check boundary values.
二次不等式需要画草图或使用符号分析。对于 x² − 4 < 0，分解为 (x − 2)(x + 2) < 0，然后确定乘积为负的区间：在 −2 和 2 之间。解为 −2 < x < 2。始终检验边界值。
7. Algebraic Fractions | 代数分式
Algebraic fractions are manipulated using the same principles as numerical fractions. To add or subtract, find a common denominator. For 1/x + 1/(x+1), the sum is (x+1 + x) / [x(x+1)] = (2x+1)/[x(x+1)]. Factorise numerators and denominators before multiplying or dividing to cancel common factors.
代数分式的运算原理与数值分式相同。加减时需找公分母。对于 1/x + 1/(x+1)，和为 (x+1 + x) / [x(x+1)] = (2x+1)/[x(x+1)]。乘除前先对分子分母因式分解，以约去公因式。
When solving equations with algebraic fractions, multiply through by the lowest common denominator to clear fractions. Be careful to exclude any x-values that make a denominator zero; these are extraneous solutions. Simplifying complex fractions often involves factorising and cancelling.
解含代数分式的方程时，两边同乘最简公分母以去分母。注意排除任何使分母为零的 x 值，这些是增根。化简复杂分式通常需要因式分解和约分。
8. Introduction to Functions | 函数入门
A function is a rule that maps each input to exactly one output. Function notation f(x) = 2x + 3 means: for input x, multiply by 2 and add 3. Domain is the set of all possible inputs, and range is the set of all possible outputs. IGCSE often specifies the domain as a set of numbers, and you find the range by substituting the smallest and largest x or by considering the function’s behaviour.
函数是将每个输入映射到唯一输出的规则。函数记法 f(x) = 2x + 3 表示：对输入 x，乘以 2 再加 3。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。IGCSE 常指定定义域为一组数，然后通过代入最小和最大 x 或考虑函数行为求值域。
To evaluate a function, replace x with the given value. For f(x) = x² − 1, f(3) = 3² − 1 = 8. Function machines can help visualise the process. Remember, the vertical line test helps identify if a graph represents a function: a vertical line should intersect the graph at most once.
计算函数值时，用给定值替换 x。对于 f(x) = x² − 1，f(3) = 3² − 1 = 8。函数机有助于可视化过程。记住，垂直线检验可判断图像是否为函数：一条垂直线与图像最多有一个交点。
9. Composite Functions | 复合函数
Composite functions combine two functions. fg(x) means apply g first, then f. It is read as ‘f of g of x’. For f(x) = 2x + 1 and g(x) = x², fg(x) = f(g(x)) = f(x²) = 2x² + 1. Note that fg(x) is generally not the same as gf(x). Always work from the innermost function outward.
复合函数结合两个函数。fg(x) 表示先应用 g 再应用 f，读作 ‘f of g of x’。对于 f(x) = 2x + 1 和 g(x) = x²，fg(x) = f(g(x)) = f(x²) = 2x² + 1。注意 fg(x) 通常不同于 gf(x)。始终从最内层函数向外计算。
When finding the domain of a composite function, consider restrictions from both functions. For example, if g(x) = √x and f(x) = 1/(x−2), then fg(x) = 1/(√x − 2). g requires x ≥ 0, and f requires the denominator not zero, so √x ≠ 2, meaning x ≠ 4. The domain is x ≥ 0, x ≠ 4.
求复合函数定义域时需考虑两个函数的限制。例如，若 g(x) = √x 且 f(x) = 1/(x−2)，则 fg(x) = 1/(√x − 2)。g 要求 x ≥ 0，f 要求分母不为零，所以 √x ≠ 2 即 x ≠ 4。定义域为 x ≥ 0，x ≠ 4。
10. Inverse Functions | 反函数
An inverse function reverses the effect of the original function. If f(a) = b, then f⁻¹(b) = a. Not all functions have an inverse; one-to-one functions do. To find the inverse, write y = f(x), swap x and y, and solve for y. For f(x) = 4x − 3, write y = 4x − 3, swap to x = 4y − 3, solve to y = (x + 3)/4, so f⁻¹(x) = (x + 3)/4.
反函数逆转原函数的作用。若 f(a) = b，则 f⁻¹(b) = a。并非所有函数都有反函数；一一对应函数才有。求反函数时，令 y = f(x)，交换 x 和 y，然后解出 y。对于 f(x) = 4x − 3，写 y = 4x − 3，交换得 x = 4y − 3，解得 y = (x + 3)/4，因此 f⁻¹(x) = (x + 3)/4。
The graph of f⁻¹(x) is the reflection of f(x) in the line y = x. The domain of f⁻¹ is the range of f, and vice versa. Always check that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. If the original function is defined for x ≥ 1, then the inverse will have a range of y ≥ 1.
f⁻¹(x) 的图像是 f(x) 关于直线 y = x 的反射。f⁻¹ 的定义域是 f 的值域，反之亦然。始终检验 f(f⁻¹(x)) = x 且 f⁻¹(f(x)) = x。若原函数定义域为 x ≥ 1，则反函数的值域为 y ≥ 1。
11. Function Graphs | 函数图像
Understanding the shapes of basic functions is essential. Linear functions y = mx + c are straight lines with gradient m and y-intercept c. Quadratic functions y = ax² + bx + c are parabolas. If a > 0, a U-shape (minimum); if a < 0, an n-shape (maximum). The vertex can be found by completing the square or using x = −b/(2a).
理解基本函数图像的形状至关重要。线性函数 y = mx + c 是斜率为 m、y 截距为 c 的直线。二次函数 y = ax² + bx + c 是抛物线。若 a > 0，为 U 形（最小值）；若 a < 0，为 n 形（最大值）。顶点可通过配方法或使用 x = −b/(2a) 求得。
Other important graphs include cubic (y = x³ or y = −x³), reciprocal (y = 1/x), and exponential (y = aˣ). The x-intercepts are roots of the equation f(x) = 0. Transformations of functions—such as f(x) + a (vertical translation), f(x + a) (horizontal translation), and −f(x) (reflection in x-axis)—are regularly tested.
其他重要图像包括三次（y = x³ 或 y = −x³）、反比例（y = 1/x）和指数函数（y = aˣ）。x 截距是方程 f(x) = 0 的根。函数变换——如 f(x) + a（垂直平移）、f(x + a)（水平平移）和 −f(x)（关于 x 轴反射）——是常考内容。
When sketching, label any intersections with axes, asymptotes, and turning points. A rough sketch is often sufficient to solve inequality or transformation questions. Make sure you know how the graph of a derivative relates to the original, though calculus may not be covered in this topic; focus on algebraic and functional relationships.
画草图时，标示与坐标轴的交点、渐近线和转折点。简图通常足以解决不等式或变换问题。尽管微积分可能不在本主题内，但要确保了解导数图像与原函数的关系；此处重点应放在代数和函数关系上。
12. Sequences and the nth Term | 数列与第 n 项
Sequences are ordered lists of numbers following a rule. Linear sequences have a constant difference between terms; their nth term is of the form an + b, where a is the common difference. For the sequence 3, 7, 11, 15, …, the difference is 4, so the nth term is 4n − 1 (since 4(1) − 1 = 3).
数列是按规则排列的数字序列。线性数列相邻项差为常数；其第 n 项形如 an + b，其中 a 为公差。对于数列 3, 7, 11, 15, …，公差为 4，因此第 n 项为 4n − 1（因为 4(1) − 1 = 3）。
Quadratic sequences have a second difference that is constant. To find the nth term, compare with the sequence of square numbers. For example, 2, 5, 10, 17, … has second difference 2, and the nth term is n² + 1. The IGCSE also tests using term-to-term rules to generate sequences and finding a term given its position.
二次数列的二次差为常数。求第 n 项时，可与平方数数列比较。例如，2, 5, 10, 17, … 的二次差为 2，第 n 项为 n² + 1。IGCSE 也考查利用逐项规则生成数列，以及根据位置求项。
Recursive definitions express each term based on previous ones. Ensure you can work with subscript notation: uₙ₊₁ = 2uₙ + 3, u₁ = 5. Calculate a specified term by repeated substitution. Always look for a pattern to simplify the process.
递推定义基于前项表示每一项。确保能使用下标记法：uₙ₊₁ = 2uₙ + 3，u₁ = 5。通过重复代入计算指定项。始终寻找规律以简化过程。
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