📚 GCSE OCR Maths: Algebra and Functions Revision Notes | GCSE OCR 数学：代数和函数 考点精讲
Welcome to your essential revision guide for the Algebra and Functions topic in GCSE OCR Mathematics. This article breaks down every key concept you need to master, from simplifying expressions and solving quadratics to understanding function notation, transformations, and inverses. Work through each section to build confidence and exam readiness.
欢迎来到 GCSE OCR 数学代数和函数专题的核心复习指南。本文拆解了你需要掌握的每一个关键概念，从化简表达式、解二次方程到理解函数记法、图像变换和反函数。逐一攻破每个小节，为考试树立信心。
1. Algebraic Expressions and Simplification | 代数表达式与化简
Algebraic expressions involve numbers, variables (like x or y), and operations. Simplifying means collecting like terms – terms that have exactly the same variable part. For example, 3x + 5x simplifies to 8x. You must follow the rules of BODMAS and pay attention to signs when adding or subtracting terms.
代数表达式包含数字、变量（如 x 或 y）和运算。化简就是合并同类项——即变量部分完全相同的项。例如，3x + 5x 化简为 8x。合并时必须遵循 BODMAS 规则，并注意加减时的正负号。
When multiplying terms, multiply the coefficients and add the indices for the same base. For instance, 2x² × 3x³ = 6x⁵. Division follows a similar pattern: 12x⁵ ÷ 4x² = 3x³. Always present final answers with positive indices where possible.
在做乘法时，将系数相乘，相同底数的指数相加。例如，2x² × 3x³ = 6x⁵。除法类似：12x⁵ ÷ 4x² = 3x³。最终答案应尽可能用正指数表示。
- Collect like terms: 4a + 3b – 2a + b = 2a + 4b
- Multiply: (x²)(-3x⁴) = -3x⁶
- Divide: 15y⁷ / 5y² = 3y⁵
- 合并同类项：4a + 3b – 2a + b = 2a + 4b
- 乘法：(x²)(-3x⁴) = -3x⁶
- 除法：15y⁷ / 5y² = 3y⁵
2. Expanding and Factorising | 展开与因式分解
Expanding brackets means multiplying each term inside the bracket by the term outside. For a single bracket: a(b + c) = ab + ac. For double brackets, use the FOIL method: (x + p)(x + q) = x² + (p+q)x + pq. Always double-check sign combinations.
展开括号是指将括号外的项乘以括号内的每一项。单项式乘括号：a(b + c) = ab + ac。两个括号相乘时使用 FOIL 方法：(x + p)(x + q) = x² + (p+q)x + pq。务必检查符号组合。
Factorising is the reverse process. Look for a common factor first, then consider quadratic trinomials. For x² + bx + c, find two numbers that multiply to c and add to b. Difference of two squares: a² – b² = (a + b)(a – b) is a crucial pattern tested frequently.
因式分解则是逆过程。先提取公因式，再考虑二次三项式。对于 x² + bx + c，找出乘积为 c、和为 b 的两个数。平方差公式：a² – b² = (a + b)(a – b) 是考试中反复出现的重要模式。
| Type | Example | Factorised Form |
|---|---|---|
| Common factor | 6x² + 9x | 3x(2x + 3) |
| Quadratic trinomial | x² + 5x + 6 | (x + 2)(x + 3) |
| Difference of squares | x² – 25 | (x + 5)(x – 5) |
类型
例子
因式分解形式
- 公因式
- 二次三项式
- 平方差
(Note: table above includes English headers; the following lines repeat in Chinese for clarity as per paired pattern, but we can separate. Actually need to follow: after table, provide Chinese equivalents. Since table can’t have dual language easily inline, I’ll add a short Chinese summary list.)
公因式：6x² + 9x = 3x(2x+3)；二次三项式：x²+5x+6 = (x+2)(x+3)；平方差：x²-25 = (x+5)(x-5)。
3. Solving Quadratic Equations | 解二次方程
Quadratic equations take the form ax² + bx + c = 0. Three main methods are: factorising, completing the square, and the quadratic formula. OCR expects you to apply the most efficient method for a given equation.
二次方程的形式为 ax² + bx + c = 0。三种主要解法是：因式分解法、配方法和公式法。OCR 要求你为给定方程选择最有效的解法。
The quadratic formula is x = [-b ± √(b² – 4ac)] / (2a). Memorise this; it works for any quadratic. The discriminant (b² – 4ac) tells you the nature of roots: positive → two real roots, zero → one repeated root, negative → no real roots.
二次公式为 x = [-b ± √(b² – 4ac)] / (2a)。务必熟记，它适用于任何二次方程。判别式 (b² – 4ac) 揭示了根的性质：正数 → 两个实根，零 → 一个重根，负数 → 无实根。
x = [-b ± √(b² – 4ac)] / (2a)
Always set the equation to zero before solving. If factorising, write (x – r₁)(x – r₂) = 0, then each bracket gives a solution. When completing the square, aim for (x + p)² – q = 0 form.
解题前务必将方程化为等于零的形式。如果用因式分解，写成 (x – r₁)(x – r₂) = 0，然后每个括号得出一个解。配方法则转化为 (x + p)² – q = 0 的形式。
4. Simultaneous Equations | 联立方程
Simultaneous equations involve two (or more) equations that share variables. You will solve linear-linear and linear-quadratic pairs. Methods: elimination for linear systems, and substitution when one is quadratic.
联立方程涉及两个或多个共用变量的方程。你需要会解线性-线性以及线性-二次方程组。方法：线性方程组使用消元法，当一个是二次方程时使用代入法。
For elimination, multiply equations if needed so that the coefficients of one variable match, then add or subtract to eliminate it. For substitution, rearrange the linear equation for one variable and substitute into the quadratic. Always check solutions in both original equations.
用消元法时，如需要可将方程乘以某个数，使得其中一个变量的系数相等，然后相加或相减消去该变量。代入法则将线性方程改写为用其中一个变量表示另一个，代入二次方程。务必把解代回原方程检验。
Example linear-quadratic system: y = 2x + 1 and y = x² – 3x + 5. Set 2x + 1 = x² – 3x + 5, simplify to x² – 5x + 4 = 0, factorise to (x – 1)(x – 4) = 0 giving x = 1, x = 4. Then find corresponding y values: (1,3) and (4,9).
线性-二次方程组例子：y = 2x + 1 和 y = x² – 3x + 5。令 2x+1 = x²-3x+5，化简得 x²-5x+4=0，因式分解为 (x-1)(x-4)=0，解得 x=1 和 x=4。然后求出相应的 y 值：(1,3) 和 (4,9)。
5. Inequalities | 不等式
Inequalities use symbols >, <, ≥, ≤. Solving them is similar to equations but remember: if you multiply or divide by a negative number, reverse the inequality sign. Represent solutions on a number line with open circles for strict inequalities and closed circles for inclusive ones.
不等式使用符号 >、<、≥、≤。解法与方程相似，但切记：乘或除以一个负数时，不等号方向要改变。在数轴上表示解时，严格不等式用空心圈，包含等号用实心圈。
Quadratic inequalities, e.g., x² – 9 > 0, require a sketch graph or sign table. Factorise to (x – 3)(x + 3) > 0, identify critical values x = -3, 3, test intervals. Solution: x < -3 or x > 3. Always express final answers using set notation or inequality notation as requested.
二次不等式如 x² – 9 > 0，需要画草图或使用符号表。因式分解为 (x-3)(x+3) > 0，确定临界值 x = -3 和 3，检验区间。解为 x < -3 或 x > 3。务必按题目要求用集合符号或不等式符号写出最终答案。
For combined inequalities like -3 ≤ 2x + 1 < 5, solve in two parts while keeping the variable in the middle. Subtract 1: -4 ≤ 2x < 4, then divide by 2: -2 ≤ x < 2.
对于联立不等式如 -3 ≤ 2x+1 < 5，保持变量在中间分两步解。先减 1：-4 ≤ 2x < 4，再除以 2：-2 ≤ x < 2。
6. Introducing Functions | 函数引入
A function is a special relationship where each input (x) has exactly one output (y). Think of it as a machine: you put in a value, a rule is applied, and an output is produced. The set of all possible inputs is the domain, and all possible outputs the range.
函数是一种特殊的关系，每个输入 (x) 对应唯一一个输出 (y)。可将其想象成一台机器：你放入一个值，应用某种规则后得到输出。所有可能输入的集合称为定义域，所有可能输出的集合称为值域。
OCR expects you to distinguish functions from other relations by checking if any vertical line cuts the graph more than once (vertical line test). Only graphs that pass this test represent functions.
OCR 要求你通过垂直线检验（即任何一条竖直线与图像相交均不超过一次）来区分函数与其他关系。只有通过此检验的图像才表示函数。
7. Function Notation and Evaluation | 函数记法与求值
Function notation uses f(x) instead of y. For example, f(x) = 2x + 3. To evaluate f(4), substitute 4 for x: f(4) = 2(4) + 3 = 11. Always write the answer with the function name, e.g., f(4) = 11.
函数的记法用 f(x) 代替 y。例如，f(x) = 2x + 3。求 f(4) 时，将 4 代入 x：f(4) = 2(4) + 3 = 11。答案始终要写上函数名，如 f(4) = 11。
You may see piecewise functions: f(x) = { x² for x < 2, 5 for x ≥ 2 }. Evaluate carefully by checking which condition the input satisfies. Domain and range can be given in set notation like {x : x > -1} or interval notation.
你可能会遇到分段函数，如 f(x) = { x² 当 x < 2，5 当 x ≥ 2 }。求值时要仔细检查输入值满足哪个条件。定义域和值域可以用集合符号如 {x : x > -1} 或区间符号表示。
8. Graphs of Functions | 函数图像
Key function graphs to recognise: linear (straight line), quadratic (parabola), cubic (y = x³), reciprocal (y = 1/x), and exponential (y = 2ˣ). Knowing their shapes helps you sketch transformations and solve equations graphically.
需要识别的关键函数图像有：线性（直线）、二次（抛物线）、三次（y = x³）、反比例（y = 1/x）以及指数函数（y = 2ˣ）。熟悉它们的形状有助于画变换图像和用图解法解方程。
For quadratic graphs, find the vertex (completing the square or formula x = -b/(2a)), y-intercept (c), and roots (factorise or formula). For cubic graphs, consider end behaviour: positive leading coefficient → rises to the right; negative → falls to the right.
对于二次函数图像，找出顶点（配方法或公式 x = -b/(2a)）、y 轴截距 (c) 和根（因式分解或公式）。对于三次函数图像，考虑两端走势：首项系数为正 → 右端上升；首项系数为负 → 右端下降。
Use a table of values if you need to plot accurately. OCR may also ask for graphs of trigonometric functions y = sin x, y = cos x, y = tan x within a specified domain – recognise their periodic nature and key angles (0°, 90°, 180°, 270°, 360°).
若需精确绘图，可使用数值表。OCR 可能还会要求绘制指定定义域内的三角函数图像 y = sin x，y = cos x，y = tan x——你需要识别其周期性和关键角度（0°、90°、180°、270°、360°）。
9. Transformations of Functions | 函数变换
Transformations allow you to shift or stretch graphs. There are four main types: translation, stretch (in the x or y direction), reflection. For y = f(x):
变换能让你平移或拉伸图像。主要分为四种类型：平移、沿 x 轴或 y 轴的拉伸、对称。对于 y = f(x)：
| Transformation | Effect on f(x) |
|---|---|
| Translation upwards a units | f(x) + a |
| Translation right a units | f(x – a) |
| Stretch vertically by factor a | a f(x) |
| Stretch horizontally by factor 1/a | f(ax) |
| Reflection in x-axis | -f(x) |
| Reflection in y-axis | f(-x) |
变换类型
对 f(x) 的影响
- 向上平移 a 个单位：f(x) + a
- 向右平移 a 个单位：f(x – a)
- 纵向拉伸为原来的 a 倍：a f(x)
- 横向拉伸为原来的 1/a 倍：f(ax)
- 关于 x 轴反射：-f(x)
- 关于 y 轴反射：f(-x)
Pay attention to order when multiple transformations are applied. Generally, apply horizontal shifts inside the bracket before stretches/reflections, but follow standard precedence: transformations nearest the variable happen first if they affect the same axis.
多个变换同时作用时要注意顺序。通常，括号内的水平变换先于拉伸/反射，但要遵循标准优先级：若影响同一坐标轴，最靠近变量的变换最先发生。
10. Inverse Functions | 反函数
The inverse function f⁻¹(x) reverses the effect of f(x). It maps outputs back to inputs. To find an inverse: write y = f(x), swap x and y, then solve for y. Finally, replace y with f⁻¹(x). The domain of f⁻¹ is the range of f, and vice versa.
反函数 f⁻¹(x) 会逆转 f(x) 的作用，将输出映射回输入。求反函数的步骤：写出 y = f(x)，交换 x 和 y，然后解出 y 的表达式，最后将 y 替换为 f⁻¹(x)。f⁻¹ 的定义域是 f 的值域，反之亦然。
For f(x) = (2x + 3)/5, write y = (2x + 3)/5. Swap: x = (2y + 3)/5. Solve: 5x = 2y + 3 → 2y = 5x – 3 → y = (5x – 3)/2. So f⁻¹(x) = (5x – 3)/2. Check by verifying f(f⁻¹(x)) = x.
以 f(x) = (2x+3)/5 为例：写出 y = (2x+3)/5，交换得 x = (2y+3)/5，解之：5x=2y+3 → 2y=5x-3 → y=(5x-3)/2。因此 f⁻¹(x) = (5x-3)/2。通过验证 f(f⁻¹(x)) = x 来检查。
Not all functions have inverses over their entire domain. You may need to restrict the domain (e.g., for quadratics) to make the function one-to-one. Graphically, a function has an inverse only if it passes the horizontal line test.
并非所有函数在其整个定义域上都有反函数。你可能需要限制定义域（例如对二次函数）使其成为一一对应。从图像上看，只有通过水平线检验的函数才有反函数。
11. Composite Functions | 复合函数
A composite function is the combination of two functions, where the output of one becomes the input of the other. Written as fg(x) or f(g(x)), meaning apply g first, then f. Order is vital: fg(x) is usually different from gf(x).
复合函数是两个函数的结合，其中一个函数的输出成为另一个函数的输入。写作 fg(x) 或 f(g(x))，意思是先应用 g，再应用 f。顺序至关重要：fg(x) 通常与 gf(x) 不同。
Given f(x) = 2x + 1 and g(x) = x² – 3, find fg(x): substitute g(x) into f: f(x² – 3) = 2(x² – 3) + 1 = 2x² – 5. For gf(x): substitute f(x) into g: g(2x+1) = (2x+1)² – 3 = 4x² + 4x + 1 – 3 = 4x² + 4x – 2.
给定 f(x)=2x+1 和 g(x)=x²-3，求 fg(x)：将 g(x) 代入 f：f(x²-3)=2(x²-3)+1=2x²-5。求 gf(x)：将 f(x) 代入 g：g(2x+1)=(2x+1)²-3=4x²+4x+1-3=4x²+4x-2。
You can also evaluate composite functions at a specific value, e.g., fg(4). Calculate g(4) first: 4²-3=13, then f(13)=27. In exam questions, be careful with the notation: f²(x) means f(f(x)), not the square of f(x).
你也可以计算复合函数在特定值处的值，例如 fg(4)。先计算 g(4)：4²-3=13，再计算 f(13)=27。考试中要小心记法：f²(x) 表示 f(f(x))，而不是 f(x) 的平方。
12. Applications and Exam Tips | 应用与考试技巧
Algebra and functions questions often combine multiple skills. You might need to set up an equation from a word problem, solve a quadratic, then interpret the solution in context. Always read the question carefully and underline key information.
代数和函数题目常会结合多种技能。你可能需要从文字题中建立方程，解二次方程，然后在实际情境中解释解的含义。务必仔细读题，勾画出关键信息。
Common pitfalls: forgetting to reverse the inequality sign when multiplying by a negative; misidentifying the order in composite functions; and losing marks by not showing clear steps in expanding or factorising. Show your working – marks are awarded for method.
常见失分点：乘负数时忘记调转不等号方向；复合函数中搞错顺序；因式分解或展开时步骤不清晰而丢分。展现你的解题过程——方法分同样会被赋分。
When sketching graphs, label axes, key points (intercepts, turning points), and asymptotes if relevant. Use a ruler for straight lines. For transformation questions, underline the transformation and apply it systematically to coordinates or to the function’s equation.
画图像时，标注坐标轴、关键点（截距、顶点），以及相关的渐近线。画直线用直尺。对于变换题，在关键词下划线，并系统地将变换应用于坐标或函数方程。
Practice past OCR papers to become familiar with the phrasing and markschemes. The algebra and functions section is heavily weighted, so mastery here boosts your overall grade significantly.
练习历年 OCR 真题，熟悉出题措辞和评分方案。代数和函数部分占分很重，因此掌握好它会极大提升你的总成绩。
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