IGCSE CCEA Mathematics: Algebra and Functions Key Points | IGCSE CCEA 数学：代数和函数考点精讲

📚 IGCSE CCEA Mathematics: Algebra and Functions Key Points | IGCSE CCEA 数学：代数和函数考点精讲

This comprehensive revision guide covers the essential Algebra and Functions topics for the IGCSE CCEA Mathematics examination. It walks you through key concepts, worked examples, and exam-style tips to support your preparation.

这份全面的复习指南涵盖了 IGCSE CCEA 数学考试中代数和函数的重要主题。它将引导您掌握关键概念、例题解析以及考试风格的技巧，为您的备考提供支持。

1. Algebraic Expressions and Basic Terminology | 代数表达式与基本术语

An algebraic expression is formed using numbers, variables (letters representing unknown values), and operation symbols. Each part of an expression is called a term, and a coefficient is the number factor of a term that contains a variable.

代数表达式由数字、变量（代表未知值的字母）和运算符号构成。表达式中的每一部分称为项，系数是含有变量的项的数字因数。

Expression: 5x³ – 2x² + 7x – 9

In the term 5x³, 5 is the coefficient, x is the variable, and 3 is the exponent. Constant terms, like -9, have no variable part. Understanding this terminology is the foundation for all algebraic manipulation.

在项 5x³ 中，5 是系数，x 是变量，3 是指数。常数项（如 -9）没有变量部分。理解这些术语是所有代数运算的基础。

2. Simplifying and Collecting Like Terms | 化简与合并同类项

To simplify an expression, collect ‘like terms’ — terms that have exactly the same variable and the same exponent. Only the coefficients are combined.

要化简一个表达式，需要合并“同类项”——即变量和指数都完全相同的项。只将系数进行合并。

Example: Simplify 3a + 5b – a + 2b.

示例：化简 3a + 5b – a + 2b。

3a – a = 2a, 5b + 2b = 7b, so answer is 2a + 7b

Always check the signs in front of each term. Simplifying reduces the expression to its most compact form without changing its value.

始终检查每一项前面的符号。化简能将表达式化为最紧凑的形式而不改变其值。

3. Expanding Brackets | 括号展开

Expanding brackets involves multiplying each term inside the bracket by the term outside. For two binomials, use the distributive property (FOIL: First, Outer, Inner, Last) to ensure all products are included.

展开括号是用括号外的项乘以括号内的每一项。对于两个二项式，使用分配律（首、外、内、末）确保所有乘积都被包括。

Single bracket: 2(3x – 4) = 6x – 8.

单项式括号：2(3x – 4) = 6x – 8。

Double brackets: (x + 2)(x – 5) = x² – 5x + 2x – 10 = x² – 3x – 10.

双括号：(x + 2)(x – 5) = x² – 5x + 2x – 10 = x² – 3x – 10。

Remember to simplify by collecting like terms after expansion. This skill is essential for factorisation and solving equations.

记住在展开后合并同类项进行化简。这项技能对于因式分解和解方程至关重要。

4. Factorising Algebraic Expressions | 因式分解代数表达式

Factorising is the reverse of expanding. It involves writing an expression as a product of its factors. Start by looking for a common factor in all terms, then consider special patterns like the difference of two squares.

因式分解是展开的逆过程，即将表达式写成因式的乘积。首先查找所有项的公因式，然后考虑特殊模式，如平方差。

Common factor: 6x² + 9x = 3x(2x + 3).

公因式：6x² + 9x = 3x(2x + 3)。

Difference of squares: x² – 16 = (x + 4)(x – 4).

平方差：x² – 16 = (x + 4)(x – 4)。

Quadratic trinomial: x² + 5x + 6, find two numbers that multiply to 6 and add to 5 → (x + 2)(x + 3).

二次三项式：x² + 5x + 6，找到两个数乘积为 6 且和为 5 → (x + 2)(x + 3)。

Regular practice with factorising builds fluency for solving quadratic equations quickly.

经常练习因式分解可提高熟练度，从而快速解二次方程。

5. Solving Linear Equations | 解线性方程

A linear equation in one variable can be solved by isolating the variable using inverse operations. Perform the same operation on both sides of the equation to maintain balance.

一元线性方程可以通过逆运算将变量分离来求解。在方程两边同时进行相同运算以保持平衡。

Solve 2x + 3 = 11:

解 2x + 3 = 11：

Subtract 3 from both sides: 2x = 8
Divide both sides by 2: x = 4

两边减3：2x = 8
两边除以2：x = 4

Equations with brackets should be expanded first. Equations with fractions can be cleared by multiplying by the lowest common denominator.

带有括号的方程应首先展开。带有分数的方程可乘以最小公分母来消去分母。

6. Solving Simultaneous Equations | 解联立方程

Simultaneous equations can be solved by elimination or substitution. The elimination method adds or subtracts equations to remove one variable. The substitution method rearranges one equation to express one variable in terms of the other.

联立方程可用消元法或代入法求解。消元法是通过加减方程消去一个变量。代入法是重新整理其中一个方程，将一个变量用另一个变量表示。

Elimination example:

消元法示例：

2x + y = 7, x – y = 2. Adding gives 3x = 9, so x = 3. Substitute back: 3 – y = 2 → y = 1.

2x + y = 7, x – y = 2。相加得 3x = 9，故 x = 3。回代：3 – y = 2 → y = 1。

Substitution example:

代入法示例：

y = 2x + 1 and 3x + y = 16. Substitute y into second equation: 3x + (2x + 1) = 16 → 5x + 1 = 16 → x = 3, y = 7.

y = 2x + 1 和 3x + y = 16。将 y 代入第二个方程：3x + (2x + 1) = 16 → 5x + 1 = 16 → x = 3, y = 7。

7. Solving Quadratic Equations | 解二次方程

Quadratic equations of the form ax² + bx + c = 0 can be solved by factorising, using the quadratic formula, or completing the square. Factorising is the quickest method when the trinomial factorises easily.

形如 ax² + bx + c = 0 的二次方程可通过因式分解、使用二次公式或配方法来求解。当三项式容易分解时，因式分解是最快的方法。

Factorising: x² – x – 6 = 0 → (x – 3)(x + 2) = 0, so x = 3 or x = -2.

因式分解：x² – x – 6 = 0 → (x – 3)(x + 2) = 0，故 x = 3 或 x = -2。

The quadratic formula works for all quadratics:

二次公式适用于所有二次方程：

x = [ -b ± √(b² – 4ac) ] / (2a)

Always set the equation to zero before factorising or applying the formula. Discriminant b² – 4ac indicates the nature of roots.

在因式分解或应用公式之前，务必将方程设为零。判别式 b² – 4ac 指示根的性质。

8. Inequalities | 不等式

Inequalities compare two expressions using symbols <, >, ≤, ≥. Solving them is similar to solving equations, but remember: multiplying or dividing by a negative number reverses the inequality sign.

不等式使用符号 <, >, ≤, ≥ 来比较两个表达式。求解不等式与解方程类似，但请记住：乘以或除以负数时，不等号方向要改变。

Solve -2x < 8: divide by -2 and reverse sign → x > -4.

解 -2x < 8：除以 -2 并反转符号 → x > -4。

Inequalities can be represented on a number line with open or closed circles. A closed circle (●) means the value is included (≤ or ≥); an open circle (○) means it is not (< or >).

不等式可以在数轴上用空心或实心圆圈表示。实心圆（●）表示包含该值（≤ 或 ≥）；空心圆（○）表示不包含（< 或 >）。

9. Functions and Notation | 函数与记号

A function is a rule that assigns exactly one output to each input. Function notation f(x) reads ‘f of x’, where x is the input and f(x) is the output. The domain is the set of possible inputs; the range is the set of possible outputs.

函数是一种规则，为每个输入指定唯一的输出。函数记号 f(x) 读作“f of x”，其中 x 是输入，f(x) 是输出。定义域是可能的输入集合；值域是可能的输出集合。

For f(x) = 2x + 3, f(4) = 2(4) + 3 = 11. A function can be thought of as a machine: you input a number, the machine applies the rule, and outputs a new number.

对于 f(x) = 2x + 3，f(4) = 2(4) + 3 = 11。可以将函数想象成一台机器：输入一个数字，机器应用规则，输出一个新数字。

The vertical line test helps identify whether a graph represents a function.

垂直线测试有助于判断一个图像是否表示一个函数。

10. Composite Functions | 复合函数

The composition of two functions means applying one function to the result of another. The notation fg(x) means f(g(x)) — first apply g, then apply f to the result. Order matters.

两个函数的复合是指将一个函数应用于另一个函数的结果。记号 fg(x) 表示 f(g(x))——先应用 g，再将 f 应用于结果。顺序很重要。

If f(x) = 3x + 1 and g(x) = x², then:

若 f(x) = 3x + 1 且 g(x) = x²，则：

fg(x) = f(g(x))	f(g(x)) = 3(x²) + 1 = 3x² + 1
gf(x) = g(f(x))	g(f(x)) = (3x + 1)² = 9x² + 6x + 1

Note that fg(x) is generally not equal to gf(x). Composite functions are often tested with evaluation at a specific value, e.g., fg(2).

注意 fg(x) 通常不等于 gf(x)。复合函数常以特定值求值的形式考查，例如 fg(2)。

11. Inverse Functions | 反函数

The inverse function, denoted f⁻¹(x), reverses the effect of the original function. To find an inverse, swap x and y in the equation y = f(x) and then solve for y. The inverse exists only if the function is one-to-one.

反函数，记作 f⁻¹(x)，逆转原函数的效果。要找到反函数，在方程 y = f(x) 中交换 x 和 y，然后解出 y。只有一一对应的函数才存在反函数。

Find f⁻¹(x) for f(x) = 2x + 3: Write y = 2x + 3 → swap → x = 2y + 3 → solve → y = (x – 3)/2, so f⁻¹(x) = (x – 3)/2.

求 f(x) = 2x + 3 的反函数：写出 y = 2x + 3 → 交换 → x = 2y + 3 → 求解 → y = (x – 3)/2，因此 f⁻¹(x) = (x – 3)/2。

The graph of an inverse function is a reflection of the original graph in the line y = x. Check your inverse by verifying f(f⁻¹(x)) = x.

反函数的图像是原函数图像关于直线 y = x 的反射。通过验证 f(f⁻¹(x)) = x 来检验您的反函数。

12. Graphs of Functions | 函数图像

The graph of a linear function is a straight line with equation y = mx + c, where m is the gradient and c is the y-intercept. Quadratic functions y = ax² + bx + c produce parabolas; if a > 0, it opens upward, and if a < 0, it opens downward.

线性函数的图像是一条直线，方程为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。二次函数 y = ax² + bx + c 产生抛物线；若 a > 0，开口向上；若 a < 0，开口向下。

To sketch a graph, create a table of values by choosing several x-values, computing the corresponding y-values, and plotting the points. Key features include intercepts, turning points, and symmetry.

要绘制草图，先选取若干 x 值构成数值表，计算对应的 y 值，然后描点。关键特征包括截距、转折点和对称性。

For y = x² – 4x + 3, roots are x = 1 and x = 3; y-intercept is (0,3); turning point (vertex) at (2, -1). Plot these and join smoothly.

对于 y = x² – 4x + 3，根为 x = 1 和 x = 3；y 轴截距为 (0,3)；转折点（顶点）在 (2, -1)。描出这些点并平滑连接。

Recognising the shape and position of graphs helps solve equations graphically and understand function behaviour.

识别图像的形状和位置有助于通过图像解方程并理解函数性质。

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IGCSE CCEA Mathematics: Algebra and Functions Key Points | IGCSE CCEA 数学：代数和函数 考点精讲