IGCSE Edexcel Mathematics Revision: Algebra & Functions | IGCSE Edexcel 数学复习：代数与函数

📚 IGCSE Edexcel Mathematics Revision: Algebra & Functions | IGCSE Edexcel 数学复习：代数与函数

Algebra and functions form the backbone of the IGCSE Edexcel Mathematics syllabus. This revision guide covers the essential techniques you need to master, from manipulating expressions to sketching and transforming graphs, with exam-style tips throughout.

代数与函数是 IGCSE Edexcel 数学大纲的核心内容。本复习指南涵盖了从表达式运算到函数作图与图像变换的关键技巧，并附有贯穿全文的考试型提示，助你高效备考。

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1. Algebraic Manipulation | 代数运算

Expanding brackets, factorising expressions, and simplifying algebraic fractions are fundamental skills. You must be confident expanding products such as (a + b)(c + d), and identifying common factors, difference of two squares, and quadratic factorisations.

去括号、因式分解和化简分式是基础技能。你必须熟练掌握 (a + b)(c + d) 这类乘积的展开，以及提取公因式、识别平方差公式和二次三项式的因式分解。

• Expanding: (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15

  展开：(x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15

• Difference of two squares: a² − b² = (a + b)(a − b), e.g. 4x² − 9 = (2x + 3)(2x − 3)

  平方差公式：a² − b² = (a + b)(a − b)，例如 4x² − 9 = (2x + 3)(2x − 3)

• Factorising quadratics: x² + 7x + 12 = (x + 3)(x + 4)

  二次三项式因式分解：x² + 7x + 12 = (x + 3)(x + 4)

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

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2. Solving Linear and Quadratic Equations | 解一元一次与一元二次方程

Linear equations require balancing both sides, while quadratic equations can be solved by factorisation, completing the square, or using the quadratic formula. Always check for extraneous roots when squaring or multiplying by variables.

一元一次方程需要保持两边平衡，而一元二次方程可通过因式分解、配方法或求根公式求解。当平方或乘以变量时，务必检查是否产生增根。

For quadratics ax² + bx + c = 0, the formula is:

对于 ax² + bx + c = 0，求根公式为：

x = (−b ± √(b² − 4ac)) ÷ (2a)

The discriminant Δ = b² − 4ac tells us:
If Δ > 0 → two distinct real roots; Δ = 0 → one repeated root; Δ < 0 → no real roots.

判别式 Δ = b² − 4ac 告诉我们：
若 Δ > 0 → 有两个不等实根；Δ = 0 → 有一个重根；Δ < 0 → 无实根。

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3. Simultaneous Equations | 联立方程

Two linear equations can be solved by elimination or substitution. When one equation is linear and the other is quadratic, use substitution to obtain a quadratic in one variable, then solve.

两个线性方程可用消元法或代入法求解。当一个方程是线性、另一个是二次时，用代入法得到一个一元二次方程，再求解。

Example | 示例: Solve y = x + 1 and y = x² − 3x + 2.

例：解方程组 y = x + 1 和 y = x² − 3x + 2。

Substitute: x + 1 = x² − 3x + 2 → x² − 4x + 1 = 0

代入得：x + 1 = x² − 3x + 2 → x² − 4x + 1 = 0

x = (4 ± √(16 − 4)) ÷ 2 = 2 ± √3

Then find corresponding y values. This method is essential in coordinate geometry.

再代入求出对应的 y 值。此方法在坐标几何中至关重要。

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4. Inequalities | 不等式

Solving linear inequalities is similar to equations, but multiplying or dividing by a negative number reverses the inequality sign. Quadratic inequalities are solved by finding critical values and testing intervals.

解线性不等式与解方程类似，但乘以或除以负数时需反转不等号。二次不等式需先求临界值，再测试区间。

• Linear: 2x − 3 > 7 → 2x > 10 → x > 5

  线性：2x − 3 > 7 → 2x > 10 → x > 5

• Quadratic: x² − 5x + 6 < 0 → (x − 2)(x − 3) < 0 → 2 < x < 3

  二次：x² − 5x + 6 < 0 → (x − 2)(x − 3) < 0 → 2 < x < 3

Graphical representation on a number line uses open circles for strict inequalities and closed circles for inclusive ones.

在数轴上表示时，严格不等式用空心圆，包含端点用实心圆。

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5. Factor Theorem and Remainder Theorem | 因式定理与余数定理

For a polynomial f(x), the remainder when f(x) is divided by (x − a) is f(a). If f(a) = 0, then (x − a) is a factor. This allows factorisation of cubic and higher-degree polynomials.

对于多项式 f(x)，除以 (x − a) 的余数为 f(a)。若 f(a) = 0，则 (x − a) 是它的一个因式。这使我们能够分解三次及更高次多项式。

Example | 示例: Show (x − 2) is a factor of f(x) = x³ − 3x² + 4.

示例：证明 (x − 2) 是 f(x) = x³ − 3x² + 4 的因式。

f(2) = 8 − 12 + 4 = 0 → (x − 2) is a factor. Dividing gives f(x) = (x − 2)(x² − x − 2) = (x − 2)²(x + 1).

f(2) = 8 − 12 + 4 = 0 → (x − 2) 是因式。用除法得 f(x) = (x − 2)(x² − x − 2) = (x − 2)²(x + 1)。

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6. Functions: Notation and Evaluation | 函数：记号与求值

A function maps each input to exactly one output. We write f(x) = … to define the rule. To evaluate f(a), substitute x = a into the expression. The domain is the set of allowed inputs; the range is the set of outputs.

函数将每个输入映射到唯一的输出。我们用 f(x) = … 来定义规则。求 f(a) 只需将 x = a 代入表达式。定义域是允许输入的集合，值域是输出结果的集合。

• f(x) = x² − 3 → f(2) = 4 − 3 = 1

  f(x) = x² − 3 → f(2) = 4 − 3 = 1

• Domain restriction: f(x) = 1/x is undefined at x = 0

  定义域限制：f(x) = 1/x 在 x = 0 处无定义

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7. Composite Functions | 复合函数

The composite function fg(x) means apply g first, then apply f to the result, i.e. fg(x) = f(g(x)). Order matters! fg is generally not equal to gf.

复合函数 fg(x) 表示先对 x 应用 g，再将结果应用到 f，即 fg(x) = f(g(x))。顺序很重要！fg 通常不等于 gf。

Example | 示例: f(x) = 2x + 1, g(x) = x².

示例：f(x) = 2x + 1，g(x) = x²。

fg(x) = f(x²) = 2x² + 1; gf(x) = g(2x + 1) = (2x + 1)²

fg(x) = f(x²) = 2x² + 1；gf(x) = g(2x + 1) = (2x + 1)²

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8. Inverse Functions | 反函数

The inverse function reverses the mapping of f. Write f⁻¹(x) by rearranging y = f(x) to make x the subject, then swap x and y. The graph of f⁻¹ is the reflection of f in the line y = x.

反函数将 f 的映射过程逆转。要写出 f⁻¹(x)，先将 y = f(x) 变形为 x 的表达式，再交换 x 与 y。f⁻¹ 的图像是 f 关于直线 y = x 的镜像。

Example | 示例: f(x) = 3x − 2.

示例：f(x) = 3x − 2。

y = 3x − 2 → x = (y + 2) ÷ 3 → f⁻¹(x) = (x + 2) ÷ 3

Check: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

验证：f(f⁻¹(x)) = x 且 f⁻¹(f(x)) = x。

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9. Graph Sketching: Linear and Quadratic | 图像绘制：一次与二次函数

A linear graph y = mx + c has slope m and y-intercept c. A quadratic graph y = ax² + bx + c is a parabola. For sketching, find the y-intercept, roots (if any), and the turning point using completing the square or symmetry.

一次函数 y = mx + c 的斜率为 m，y 截距为 c。二次函数 y = ax² + bx + c 的图像是抛物线。画图时需要找到 y 截距、根（如果有）以及通过配方法或对称性求得的顶点。

For a quadratic written as y = a(x − p)² + q, the vertex is (p, q). When a > 0 the parabola opens upward; a < 0 opens downward.

二次函数写成 y = a(x − p)² + q 时，顶点为 (p, q)。当 a > 0 时抛物线开口向上；a < 0 时开口向下。

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10. Transformations of Graphs | 图像变换

Shifts, reflections, and stretches of graphs correspond to simple changes in the function’s equation.

图像的平移、反射和伸缩对应函数方程的简单变化。

Transformation | 变换	Function | 函数
Translation by (a, 0) | 沿 x 轴平移 a 个单位	y = f(x − a)
Translation by (0, b) | 沿 y 轴平移 b 个单位	y = f(x) + b
Reflection in x-axis | 关于 x 轴对称	y = −f(x)
Reflection in y-axis | 关于 y 轴对称	y = f(−x)
Vertical stretch | 纵向伸缩	y = k f(x)
Horizontal stretch | 横向伸缩	y = f(kx)

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11. Sequences and nth Term | 数列与第 n 项

Linear sequences have a constant difference d. The nth term is a + (n − 1)d. Quadratic sequences have a constant second difference and are written as an² + bn + c.

等差数列的公差 d 恒定，第 n 项为 a + (n − 1)d。二次数列的二阶差恒定，可写成 an² + bn + c。

Example | 示例: Sequence 3, 7, 11, 15 — first difference 4, so nth term = 4n − 1.

示例：数列 3, 7, 11, 15 — 首差为 4，所以第 n 项 = 4n − 1。

For quadratic sequences, find half of the second difference as the coefficient of n², then adjust with a linear term.

对于二次数列，取二阶差的一半作为 n² 的系数，再用线性项进行调整。

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

Many students lose marks on algebraic fraction simplification, forgetting to reverse inequality signs, and confusing composite function order. Here are key revision reminders.

许多学生在代数分式化简、忘记反转不等式符号以及混淆复合函数顺序上丢分。以下是一些重点复习提醒。

• Always simplify fractional answers and cancel common factors before multiplying.

  在乘法前务必化简分式答案并约去公因式。

• When solving equations with fractions, multiply through by the common denominator first.

  解含分式的方程时，先乘以公分母，再去分母。

• For graph transformations, apply translations after reflections/stretches.

  对于图像变换，先进行反射/伸缩，再进行平移。

• Check your inverse function by composing it with the original function.

  通过将反函数与原函数复合来检查是否正确。

• Read questions carefully: “write down” means no working required; “show that” requires clear reasoning.

  仔细读题：”write down” 表示无需过程；”show that” 要求清晰推理。

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