📚 IGCSE AQA Maths: Algebra and Functions Key Points | IGCSE AQA 数学：代数和函数 考点精讲
Algebra and functions form the bedrock of the IGCSE AQA Mathematics syllabus. Mastering these topics means you can confidently manipulate expressions, solve a wide range of equations, interpret graphs, and understand how one quantity depends on another. This guide revisits the essential concepts, worked examples, and common pitfalls that every student should know.
代数和函数是 IGCSE AQA 数学课程的基础。掌握这些主题意味着你能够自信地处理表达式、求解各类方程、解读图像并理解变量之间的依赖关系。本指南回顾了每个学生都必须掌握的核心概念、典型示例和常见失分点。
1. Simplifying Algebraic Expressions | 化简代数表达式
To simplify an algebraic expression, combine like terms — those with exactly the same variable(s) raised to the same power(s). For example, 3x and 5x are like terms, but 3x and 5x² are not.
化简代数表达式时，需要合并同类项——即含有完全相同字母部分且对应指数也相同的项。例如，3x 和 5x 是同类项，而 3x 和 5x² 则不是。
When adding or subtracting terms, only the coefficients (the numbers in front of the variables) are combined. The variable part stays unchanged. For instance, 7a − 2a + 4a simplifies to 9a.
进行加减运算时，只合并系数（变量前面的数字），字母部分保持不变。例如 7a − 2a + 4a 化简为 9a。
You may also need to deal with parentheses. If a minus sign sits in front of a bracket, every sign inside the bracket flips when the bracket is removed: −(2x − 3) becomes −2x + 3.
你还需要处理括号。如果括号前是负号，去掉括号时括号内每一项的符号都要改变：−(2x − 3) 变为 −2x + 3。
2. Expanding and Factorising | 展开与因式分解
Expanding means multiplying out brackets. A single term outside a bracket multiplies every term inside: 3(2x + 5) = 6x + 15. When you have two brackets, like (x + 2)(x + 3), multiply each term in the first bracket by each term in the second — often remembered as FOIL (First, Outer, Inner, Last): x² + 3x + 2x + 6 = x² + 5x + 6.
展开即去括号。用括号外的一个单项式乘以括号内的每一项：3(2x + 5) = 6x + 15。当出现两个括号时，如 (x + 2)(x + 3)，将第一个括号中的每一项分别乘以第二个括号中的每一项——常记为 FOIL（首、外、内、尾）：x² + 3x + 2x + 6 = x² + 5x + 6。
Factorising is the reverse of expanding. Look for a common factor first: 6x² + 9x = 3x(2x + 3). For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b: x² + 7x + 10 = (x + 2)(x + 5).
因式分解是展开的逆过程。首先提取公因式：6x² + 9x = 3x(2x + 3)。对于形如 x² + bx + c 的二次式，找到两个数，其积为 c，其和为 b：x² + 7x + 10 = (x + 2)(x + 5)。
Remember the difference of two squares: a² − b² = (a + b)(a − b). For instance, x² − 25 = (x + 5)(x − 5). This pattern is frequently tested.
记住平方差公式：a² − b² = (a + b)(a − b)。例如 x² − 25 = (x + 5)(x − 5)。这一模式经常考查。
3. Solving Linear Equations | 解线性方程
A linear equation has the variable raised to the power 1. To solve it, use inverse operations to isolate the variable on one side. Always do the same to both sides to keep the equation balanced.
线性方程中变量的指数为 1。求解时，使用逆运算将变量单独留在方程的一边。切记两边必须同时进行相同的运算以保持方程平衡。
Example: 5x − 7 = 3x + 9. First, collect x‑terms on one side (−3x from both sides gives 2x − 7 = 9). Then add 7 to both sides: 2x = 16. Finally divide by 2: x = 8.
示例：5x − 7 = 3x + 9。首先将含 x 的项移到同一边（两边同时减 3x 得 2x − 7 = 9），然后两边加 7：2x = 16，最后除以 2：x = 8。
If an equation contains fractions, multiply every term by the lowest common denominator to clear them. For (x+2)/3 = (x−1)/2, multiply by 6: 2(x+2) = 3(x−1). Expand and solve: 2x + 4 = 3x − 3 → x = 7.
若方程中含有分数，可将每一项乘以最简公分母以消去分母。例如 (x+2)/3 = (x−1)/2，乘以 6 得 2(x+2) = 3(x−1)。展开并求解：2x + 4 = 3x − 3 → x = 7。
4. Solving Quadratic Equations | 解二次方程
A quadratic equation is of the form ax² + bx + c = 0 (a ≠ 0). The three main solving methods are factorising, using the quadratic formula, and completing the square.
二次方程的一般形式为 ax² + bx + c = 0 (a ≠ 0)。三种主要解法是：因式分解法、使用求根公式以及配方法。
If the quadratic factorises, set each factor to zero. For x² − x − 6 = 0, factorise as (x − 3)(x + 2) = 0, giving x = 3 or x = −2.
如果二次式可以因式分解，则令每个因式为零。对于 x² − x − 6 = 0，因式分解为 (x − 3)(x + 2) = 0，得 x = 3 或 x = −2。
If factorising is not straightforward, use the quadratic formula:
x = (−b ± √(b² − 4ac)) / 2a
若不易因式分解，可使用求根公式：
x = (−b ± √(b² − 4ac)) / 2a
The expression under the square root, b² − 4ac, is called the discriminant. It tells you how many real solutions exist: positive → two distinct roots, zero → one repeated root, negative → no real roots.
根号下的式子 b² − 4ac 称为判别式。它决定实数解的个数：正值 → 两个不等实根，零 → 一个重根，负值 → 无实数根。
5. Simultaneous Equations | 联立方程组
Simultaneous equations involve two (or more) unknowns linked by two (or more) equations. At IGCSE, you mainly solve pairs of linear equations, or one linear and one quadratic.
联立方程组包含两个（或更多）未知数，并由两个（或更多）方程关联。在 IGCSE 阶段，主要考查两个线性方程，或一个线性与一个二次方程的组合。
For two linear equations, elimination is efficient. Align the equations vertically, then add or subtract to cancel one unknown. Example: 3x + 2y = 12 and 5x − 2y = 4. Adding gives 8x = 16 → x = 2. Substitute back to find y = 3.
对于两个线性方程，消元法较为高效。将方程竖排对齐，然后相加或相减以消去一个未知数。示例：3x + 2y = 12 和 5x − 2y = 4。相加得 8x = 16 → x = 2。代回原方程求得 y = 3。
Substitution is used when one equation is linear and the other is quadratic. From the linear equation, express one variable in terms of the other, then substitute into the quadratic. Solve the resulting quadratic in one unknown.
当其中一个方程为线性、另一个为二次时，使用代入法。从线性方程中用一个未知数表示另一个未知数，然后代入二次方程。求解得到的一元二次方程。
6. Inequalities | 不等式
Inequalities compare expressions using <, >, ≤, ≥. Solving them follows similar rules to equations, with one critical exception: when multiplying or dividing by a negative number, reverse the inequality sign.
不等式使用 <、>、≤、≥ 符号比较表达式。求解规则与方程相似，但有一个关键例外：当两边同时乘以或除以一个负数时，不等号方向要改变。
Example: −2x < 8. Divide both sides by −2, so x > −4 (sign reversed).
示例：−2x < 8。两边除以 −2，得 x > −4（不等号反向）。
For a compound inequality like −3 ≤ 2x + 1 < 5, solve the two parts simultaneously: subtract 1 from all three parts to get −4 ≤ 2x < 4, then divide by 2 to obtain −2 ≤ x < 2. Represent this on a number line with a solid dot at −2 and an open dot at 2.
对于复合不等式如 −3 ≤ 2x + 1 < 5，同时求解三部分：三部分同减 1 得 −4 ≤ 2x < 4，再同除以 2 得 −2 ≤ x < 2。在数轴上表示时，−2 处用实心圆点，2 处用空心圆点。
7. Functions and Function Notation | 函数与函数记号
A function is a rule that assigns exactly one output to each input. Function notation, f(x), reads “f of x”. For f(x) = 3x − 4, f(2) means replace x with 2: 3(2) − 4 = 2.
函数是一种对应法则，每个输入值对应唯一一个输出值。函数记号 f(x) 读作“f of x”。若 f(x) = 3x − 4，则 f(2) 表示将 x 替换为 2：3(2) − 4 = 2。
The domain is the set of possible input values; the range is the set of possible outputs. For f(x) = √x, the domain is x ≥ 0, because you cannot take the square root of a negative number in the real number system.
定义域是所有可能输入值的集合；值域是所有可能输出值的集合。对于 f(x) = √x，定义域为 x ≥ 0，因为在实数范围内不能对负数开平方。
You may be asked to find an inverse function, f⁻¹(x). Write y = f(x), swap x and y, then solve for y. For f(x) = 2x + 3, y = 2x + 3 → x = 2y + 3 → y = (x − 3)/2, so f⁻¹(x) = (x − 3)/2.
你可能需要求反函数 f⁻¹(x)。令 y = f(x)，交换 x 和 y，然后解出 y。例如 f(x) = 2x + 3，y = 2x + 3 → x = 2y + 3 → y = (x − 3)/2，因此 f⁻¹(x) = (x − 3)/2。
8. Graphs of Functions | 函数图像
Plotting a function’s graph helps visualise its behaviour. Linear functions (y = mx + c) produce straight lines with gradient m and y‑intercept c. Quadratic functions (y = x²) give parabolas — ∪‑shaped when the coefficient of x² is positive, ∩‑shaped when negative.
绘制函数图像有助于直观理解其性质。一次函数 (y = mx + c) 的图像为直线，斜率为 m，y 轴截距为 c。二次函数 (y = x²) 的图像为抛物线——x² 系数为正时开口向上呈 ∪ 形，为负时开口向下呈 ∩ 形。
To sketch a graph, identify key features: intercepts, turning points, and asymptotes. For quadratic graphs, complete the square to find the vertex. For reciprocal graphs like y = 1/x, note that the curve has two branches and asymptotes at x = 0 and y = 0.
绘制函数草图时，需识别关键特征：截距、顶点和渐近线。对于二次函数图像，可通过配方法求顶点。对于反比例函数图像如 y = 1/x，注意图像有两支分支，且渐近线为 x = 0 和 y = 0。
Cubic graphs (y = x³) and other higher-degree polynomials may have several turning points. Always check where the graph crosses the axes and the general shape for large positive and negative x.
三次函数图像 (y = x³) 及其他更高次多项式的图像可能有多个转向点。务必检查图像与坐标轴的交点，以及 x → ±∞ 时图像的走向。
9. Transforming Graphs | 图像变换
Graphs can be transformed by shifting, stretching, or reflecting. These transformations correspond to changes in the function’s equation.
图像可以通过平移、伸缩或反射进行变换。这些变换对应函数表达式中的相应变化。
A translation by vector (a, b) means shifting the graph a units horizontally and b units vertically. For f(x), replacing x by (x − a) translates the graph a units to the right. Adding b to the whole function, f(x) + b, moves it up by b units.
平移向量 (a, b) 表示将图像水平移动 a 个单位、垂直移动 b 个单位。对于 f(x)，将 x 替换为 (x − a) 使图像向右平移 a 个单位。整个函数加 b，即 f(x) + b，图像向上平移 b 个单位。
Stretches: f(kx) compresses the graph horizontally by a factor of 1/k; kf(x) stretches it vertically by a factor of k. Reflections: −f(x) reflects in the x‑axis, f(−x) reflects in the y‑axis.
伸缩变换：f(kx) 使图像水平压缩为原来的 1/k；kf(x) 使图像垂直拉伸为原来的 k 倍。反射变换：−f(x) 关于 x 轴对称，f(−x) 关于 y 轴对称。
When describing a transformation, specify the direction and the amount. For instance, “stretch vertically by factor 2” or “reflect in the y‑axis”. Always link the change back to the function’s equation.
描述变换时，要指明方向和幅度。例如，“沿垂直方向拉伸为原来的 2 倍”或“关于 y 轴反射”。始终将变化与函数表达式联系起来。
10. Sequences | 数列
A sequence is an ordered list of numbers. Linear sequences change by a constant difference from one term to the next. The nth term of a linear sequence is of the form an + b, where a is the common difference.
数列是一组按序排列的数。线性数列相邻两项的差恒定。其第 n 项公式形如 an + b，其中 a 为公差。
Example: sequence 5, 8, 11, 14, … has common difference 3, so nth term = 3n + 2 (because the zero‑th term would be 2). Use the formula to find any term, e.g. the 20th term = 3(20) + 2 = 62.
示例：数列 5, 8, 11, 14, … 公差为 3，故第 n 项 = 3n + 2（因为第零项为 2）。可用此公式求任意项，如第 20 项 = 3(20) + 2 = 62。
Quadratic sequences have a constant second difference. Their nth term is of the form an² + bn + c. To find a, b, c, compare the second difference to 2a, then solve simultaneous equations.
二次数列的二级差分（二阶差）为常数。其第 n 项公式形如 an² + bn + c。求 a、b、c 时，先利用二阶差等于 2a 求出 a，再通过解方程组确定 b 和 c。
Recognising whether a sequence is linear or quadratic is a common exam skill. Always check the first and second differences carefully.
判断数列是线性还是二次是常见的考试技能。务必仔细检查一阶差分和二阶差分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply