📚 IGCSE Mathematics: Algebra and Functions – Key Topics and Exam Tips | IGCSE 数学：代数与函数考点精讲
Algebra and Functions form the core of the IGCSE Mathematics syllabus. Mastering these topics not only secures a significant portion of the exam marks but also builds essential problem-solving skills. This revision guide walks you through the key concepts, from simplifying expressions to transforming functions, with clear explanations and exam-focused examples.
代数和函数是 IGCSE 数学课程的核心内容。掌握这些专题不仅能够确保考试中的大部分分数，还能培养重要的解题能力。本复习指南将带你梳理从表达式化简到函数变换的关键概念，配以清晰的解释和紧扣考点的示例。
1. Simplifying Algebraic Expressions | 代数表达式化简
To simplify an algebraic expression, collect like terms – terms that have exactly the same variable part. For example, 4x and -2x can be combined, while 3x² and 5x cannot. Always pay attention to signs and follow the order of operations.
化简代数表达式时，需要合并同类项——即变量部分完全相同的项。例如 4x 和 -2x 可以合并，但 3x² 和 5x 不能。务必注意符号并遵循运算顺序。
Example: Simplify 3a + 5b – 2a + 7b.
示例：化简 3a + 5b – 2a + 7b。
(3a – 2a) + (5b + 7b) = a + 12b
- Combine a-terms: 3a and –2a give a.
- 合并 a 项：3a 和 –2a 得到 a。
- Combine b-terms: 5b and 7b give 12b.
- 合并 b 项：5b 和 7b 得到 12b。
ons include multiplication, use the distributive law and then simplify. For instance, 2(3x – 4) + 5x becomes 6x – 8 + 5x = 11x – 8.
当表达式包含乘法时，先使用分配律再化简。例如 2(3x – 4) + 5x 得到 6x – 8 + 5x = 11x – 8。
2. Expanding and Factorising | 展开与因式分解
= ab + ac. For double brackets, use the FOIL method (First, Outer, Inner, Last) or a grid.
展开括号是指将括号外的项乘以括号内的每一项。对于单项括号：a(b + c) = ab + ac。对于双括号，可以使用 FOIL 方法（首、外、内、末）或网格法。
Example: Expand and simplify (x + 4)(x – 3).
示例：展开并化简 (x + 4)(x – 3)。
(x)(x) + (x)(–3) + (4)(x) + (4)(–3) = x² – 3x + 4x – 12 = x² + x – 12
Factorising is the reverse process. Always start by looking for a common factor. For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b.
因式分解是展开的逆过程。首先要寻找公因式。对于形如 x² + bx + c 的二次式，找出两个数，它们的积为 c，和为 b。
Example: Factorise x² + 5x + 6.
示例：因式分解 x² + 5x + 6。
Numbers 2 and 3 multiply to 6 and add to 5, so x² + 5x + 6 = (x + 2)(x + 3).
数 2 和 3 的积为 6，和为 5，因此 x² + 5x + 6 = (x + 2)(x + 3)。
3. Solving Linear Equations | 解一元一次方程
A linear equation has the unknown raised to the power of 1. The goal is to isolate the variable on one side using inverse operations. Whatever you do to one side, you must do to the other to maintain balance.
线性方程中未知数的指数为 1。目标是利用逆运算将变量移到等式一边。对等式一边进行的任何操作，必须在另一边同样进行以保持平衡。
Example: Solve 2x + 7 = 15.
示例：解方程 2x + 7 = 15。
2x + 7 = 15 → subtract 7: 2x = 8 → divide by 2: x = 4
For equations with brackets or fractions, expand or multiply through by the denominator first. Always verify your solution by substitution.
对于含有括号或分数的方程，先展开或乘以分母。一定要通过代入检验解是否正确。
Example: Solve 3(x – 2) = 9.
示例：解方程 3(x – 2) = 9。
Expand: 3x – 6 = 9 → add 6: 3x = 15 → x = 5.
展开：3x – 6 = 9 → 加 6：3x = 15 → x = 5。
4. Solving Quadratic Equations | 解一元二次方程
ods: factorisation, completing the square, and the quadratic formula. In IGCSE, factorisation and the formula are most common.
一元二次方程的形式为 ax² + bx + c = 0。主要有三种解法：因式分解法、配方法和求根公式法。IGCSE 中常用因式分解和求根公式。
When factorising, first set the equation to zero. Then find the two factors. If factorisation is not possible, use the quadratic formula:
因式分解时，先将方程右边设为零，然后找出两个因式。如果不能分解，使用求根公式：
x = [–b ± √b² – 4ac] / (2a)
Example: Solve x² – 5x + 6 = 0 by factorisation.
示例：用因式分解法解 x² – 5x + 6 = 0。
Factors: (x – 2)(x – 3) = 0, so x = 2 or x = 3.
因式分解：(x – 2)(x – 3) = 0，因此 x = 2 或 x = 3。
Always check the discriminant (b² – 4ac) to determine the number of solutions. If it is negative, there are no real roots.
始终检查判别式 (b² – 4ac) 以确定解的个数。如果为负数，则无实数根。
5. Simultaneous Equations | 联立方程组
Simultaneous equations involve two or more unknowns. The two main solving techniques are elimination and substitution. Elimination works well when coefficients of one variable can be matched; substitution isolates one variable first.
联立方程组包含两个或多个未知数。两种主要解法是消元法和代入法。当某个变量的系数可以配平时，消元法效果很好；代入法则先分离出一个变量。
Example: Solve the system: 2x + y = 10 and x – y = 2.
示例：解方程组 2x + y = 10 和 x – y = 2。
Add the equations: (2x + y) + (x – y) = 10 + 2 → 3x = 12 → x = 4
Substitute x = 4 into x – y = 2: 4 – y = 2 → y = 2. Solution: (4, 2).
将 x = 4 代入 x – y = 2：4 – y = 2 → y = 2。解为 (4, 2)。
For non-linear simultaneous equations, such as one linear and one quadratic, substitute the linear expression into the quadratic and solve.
对于非线性联立方程，例如一个一次和一个二次方程，将一次表达式代入二次方程并求解。
6. Inequalities | 不等式
r to solving equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
不等式使用 <、>、≤、≥ 来比较表达式。解不等式类似于解方程，但要注意当乘以或除以一个负数时，必须反转不等号方向。
Example: Solve –2x + 4 > 10.
示例：解不等式 –2x + 4 > 10。
–2x > 6 → divide by –2 and reverse sign: x < –3
for < or > and a closed circle for ≤ or ≥. Shade the region that satisfies the inequality.
在数轴上表示解时，对 < 或 > 用空心圆，对 ≤ 或 ≥ 用实心圆，并涂出满足不等式的区域。
Systems of inequalities define regions on a graph. Shade the region that satisfies all inequalities simultaneously.
不等式组在图形上定义一个区域。涂出同时满足所有不等式的区域。
7. Functions and Notation | 函数与记法
A function relates each input (x) to exactly one output (f(x) or y). The notation f(x) is read as 'f of x'. To evaluate a function, substitute the given number or expression.
函数将每个输入 (x) 对应到唯一输出 (f(x) 或 y)。记法 f(x) 读作 'f of x'。计算函数值时，代入指定的数字或表达式。
Example: If f(x) = 2x² – 3, find f(4).
示例：若 f(x) = 2x² – 3，求 f(4)。
f(4) = 2(4)² – 3 = 2(16) – 3 = 32 – 3 = 29
The domain is the set of all allowed inputs; the range is the set of all possible outputs. Composite functions like f(g(x)) are evaluated from the inside out.
定义域是所有允许的输入集合；值域是所有可能的输出集合。复合函数如 f(g(x)) 从内向外计算。
Inverse functions reverse the original relation. To find f⁻¹(x), swap y and x, then solve for y.
反函数逆转原来的对应关系。要找到 f⁻¹(x)，交换 y 和 x，然后解出 y。
8. Linear Functions and Graphs | 一次函数与图像
is the gradient and c is the y-intercept. The gradient measures steepness: rise over run.
直线方程通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。斜率衡量倾斜程度：垂直变化除以水平变化。
To find the gradient between two points (x₁, y₁) and (x₂, y₂):
求两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率：
m = (y₂ – y₁) / (x₂ – x₁)
ual gradients; perpendicular lines have gradients that are negative reciprocals (product = –1).
平行线斜率相等；垂直线的斜率互为负倒数（乘积为 –1）。
To plot a line, find the y-intercept and use the gradient to determine a second point, or pick two x-values and compute the corresponding y-values.
绘制直线时，先找到 y 截距，再利用斜率确定第二个点，或者选取两个 x 值计算对应的 y 值。
9. Quadratic Graphs and Properties | 二次函数图像与性质
parabola. If a > 0, the parabola opens upwards (U-shaped); if a < 0, it opens downwards (∩-shaped). The vertex is the turning point – minimum for a > 0, maximum for a < 0.
y = ax² + bx + c 的图像是一条抛物线。a > 0 时开口向上（U 形）；a < 0 时开口向下（∩ 形）。顶点是转折点——a > 0 时为最小值点，a < 0 时为最大值点。
The line of symmetry passes through the vertex and is x = –b/(2a). The x-intercepts (roots) are found by solving ax² + bx + c = 0.
对称轴经过顶点，方程式为 x = –b/(2a)。x 轴截距（根）通过解 ax² + bx + c = 0 得到。
Example: Sketch y = x² – 4x + 3.
示例：草绘 y = x² – 4x + 3。
Factorise: (x – 1)(x – 3) = 0, so roots are (1,0) and (3,0). Vertex x-coordinate: –(–4)/(2×1) = 2; y = (2)² – 4(2) + 3 = –1. Vertex (2, –1).
因式分解：(x – 1)(x – 3) = 0，根为 (1,0) 和 (3,0)。顶点 x 坐标：–(–4)/(2×1) = 2；y = (2)² – 4(2) + 3 = –1。顶点 (2, –1)。
10. Transformations of Functions | 函数变换
Given y = f(x), several transformations can be applied to the graph:
对于 y = f(x)，可以对图像进行多种变换：
| Transformation | Effect on graph | 变换 | 图像效果 |
|---|---|---|---|
| f(x) + a | Vertical translation up by a | f(x) + a | 垂直上移 a 个单位 |
| f(x) – a | Vertical translation down by a | f(x) – a | 垂直下移 a 个单位 |
| f(x + a) | Horizontal translation left by a | f(x + a) | 水平左移 a 个单位 |
| f(x – a) | Horizontal translation right by a | f(x – a) | 水平右移 a 个单位 |
| –f(x) | Reflection in the x-axis | –f(x) | 关于 x 轴反射 |
| f(–x) | Reflection in the y-axis | f(–x) | 关于 y 轴反射 |
| af(x) (a > 1) | Vertical stretch by factor a | af(x) (a > 1) | 垂直拉伸 a 倍 |
| f(ax) (a > 1) | Horizontal compression by factor 1/a | f(ax) (a > 1) | 水平压缩为原来的 1/a |
Work systematically through a sequence of transformations when the function is combined. Always perform horizontal shifts and reflections inside the brackets first, then vertical shifts and stretches.
当函数包含组合变换时，要系统地按顺序处理。先进行括号内的水平平移和反射，再进行垂直平移和拉伸。
For example, the graph of y = 2f(x – ng y = f(x) right by 3, stretching vertically by a factor of 2, and then shifting up by 1.
例如，y = 2f(x – 3) + 1 的图像可通过将 y = f(x) 右移 3，垂直拉伸 2 倍，再上移 1 得到。
11. Sequences and the nth Term | 数列与第 n 项
A sequence is a list of numbers following a rule. An arithmetic sequence has a constant difference between terms. The nth term of an arithmetic sequence is:
数列是按照某种规则排列的一列数。等差数列相邻项的差是常数。等差数列的第 n 项公式为：
Uₙ = a + (n – 1)d
ce. For example, the sequence 5, 9, 13, 17,... has a = 5, d = 4, so the nth term is 5 + (n – 1) × 4 = 4n + 1.
其中 a 是首项，d 是公差。例如数列 5, 9, 13, 17,... 有 a = 5，d = 4，因此第 n 项为 5 + (n – 1) × 4 = 4n + 1。
Quadratic sequences have a second difference that is constant. Use the difference method to find the coefficients of n², n and the constant term.
二次数列的二次差是常数。使用差分法确定 n²、n 和常数项的系数。
12. Exam Tips and Common Pitfalls | 备考技巧与常见错误
Always show clear working – marks are awarded for method even if the final answer is wrong. Check your factorisation by expanding. When solving quadratics, never forget to set the equation to zero first. Be careful with negative signs when substituting into formulas.
始终展示清晰的解题步骤——即使最终答案错误，方法也能得分。通过展开检验因式分解。解二次方程时，切勿忘记先将方程设为零。代入公式时注意负号。
In function transformations, the effect of f(x + a) or f(ax) is often the opposite of what many students expect. For graphs, label axes and key points (intercepts, vertex). Always reread the question to ensure you have answered what was asked.
在函数变换中，f(x + a) 或 f(ax) 的效果常与许多学生的直觉相反。作图时标注坐标轴和关键点（截距、顶点）。务必重读题目，确保你回答了所问内容。
Practice past papers regularly and time yourself. The more familiar you are with question styles, the more confident you will be in the exam.
定期练习历年真题并计时。你对题型越熟悉，考试时就越是信心十足。
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