📚 Algebra and Functions: Key Points for GCSE Maths | 代数和函数 考点精讲
Algebra and functions form the backbone of GCSE Mathematics. Mastering these topics means you can manipulate expressions, solve equations, work with sequences, and understand how relationships between variables behave. This article breaks down every essential concept, from simplifying like terms to transforming graphs, giving you clear bilingual explanations to strengthen your understanding.
代数和函数是 GCSE 数学的支柱。掌握这些内容意味着你可以操作表达式、解方程、处理数列，并理解变量之间的关系如何变化。本文分解每一个核心概念，从合并同类项到图像变换，提供清晰的双语解释来强化你的理解。
1. Simplifying Algebraic Expressions | 化简代数式
An algebraic expression contains numbers, variables, and operation symbols. To simplify, we collect like terms — terms that have exactly the same variable parts. For example, 3x and 5x are like terms, but 3x and 3x² are not.
代数表达式包含数字、变量和运算符号。化简时，我们合并同类项——即变量部分完全相同的项。例如，3x 和 5x 是同类项，但 3x 和 3x² 不是。
Always combine coefficients by addition or subtraction while keeping the variable part unchanged. For 4a + 2b - a + 3b, the a-terms give 3a, and the b-terms give 5b, so the simplified expression is 3a + 5b.
始终通过加法或减法合并系数，保持变量部分不变。像 4a + 2b - a + 3b 这样的式子，含 a 的项得到 3a，含 b 的项得到 5b，化简后即为 3a + 5b。
Powers of the same variable are only like terms if the exponents match. x² and 3x² can be combined to 4x², but x² and x cannot be combined.
同一变量的幂只有在指数相同时才是同类项。x² 和 3x² 可以合并为 4x²，但 x² 和 x 不能合并。
2. Expanding and Factorising | 展开与因式分解
ab + ac. With two brackets, use the FOIL method: First, Outer, Inner, Last, then simplify.
展开括号意味着将括号外的项乘以括号内的每一项。对于单括号，a(b + c) = ab + ac。对于双括号，使用 FOIL 方法：先乘首项、外项、内项、末项，然后化简。
When expanding (x + 3)(x + 2), you get x² + 2x + 3x + 6, which simplifies to x² + 5x + 6. Check your work by substituting a value for x.
展开 (x + 3)(x + 2) 时，得到 x² + 2x + 3x + 6，化简为 x² + 5x + 6。可以用一个 x 值代入检验结果。
Factorising is the reverse process — king out the highest common factor. For 6x² + 9x, the HCF is 3x, so 6x² + 9x = 3x(2x + 3).
因式分解是逆过程——将表达式重新写成因式的乘积。先从提取最大公因式开始。对于 6x² + 9x，最大公因式是 3x，所以 6x² + 9x = 3x(2x + 3)。
Factorising quadratics like x² + bx + c involves finding two numbers that multiply to c and add to b. For x² + 7x + 10, the numbers are 2 and 5, giving (x + 2)(x + 5).
对 x² + bx + c 型的二次式因式分解，涉及找到两个相乘得 c、相加得 b 的数。对于 x² + 7x + 10，这两个数是 2 和 5，得到 (x + 2)(x + 5)。
3. Solving Linear Equations | 解一元一次方程
A linear equation has the variable raised to the power of 1. The goal is to isolate the variable by performing inverse operations on both sides of the equation.
一元一次方程中变量的指数为 1。目标是通过在等式两边进行逆运算，将变量分离出来。
For 2x + 5 = 13, subtract 5 from both sides to get 2x = 8, then divide by 2, giving x = 4.
对于 2x + 5 = 13，两边同时减 5 得到 2x = 8，然后除以 2，得到 x = 4。
s: 2x - 2 = 6. Add 2: 2x = 8, so x = 4. Always check your answer in the original equation.
含两边都有变量的方程：3x - 2 = x + 6。两边减 x：2x - 2 = 6。加 2：2x = 8，所以 x = 4。始终要把答案代入原方程检验。
rm by the lowest common denominator. For x/3 + 2 = 5, multiply by 3: x + 6 = 15, so x = 9.
出现分数时，将每一项乘以最小公分母。对于 x/3 + 2 = 5，乘以 3：x + 6 = 15，所以 x = 9。
4. Solving Quadratic Equations | 解一元二次方程
ain solution methods: factorising, completing the square, and using the quadratic formula.
一元二次方程的形式为 ax² + bx + c = 0。有三种主要的解法：因式分解、配方法和使用二次公式。
bracket to zero. For x² - 5x + 6 = 0, factorise to (x - 2)(x - 3) = 0, so x = 2 or x = 3.
如果可以因式分解，令每个括号为零。对于 x² - 5x + 6 = 0，因式分解为 (x - 2)(x - 3) = 0，所以 x = 2 或 x = 3。
When the quadratic does not factorise easily, use the quadratic formula:
x = (-b ± √b² - 4ac) / 2a
当二次式不易因式分解时，使用二次公式：
x = (-b ± √b² - 4ac) / 2a
t: positive → two solutions, zero → one repetitive solution, negative → no real solutions.
判别式 b² - 4ac 告诉你存在多少个实数解：正数 → 两个解，零 → 一个重根，负数 → 没有实数解。
5. Simultaneous Equations | 联立方程组
Simultaneous equations involve two or more unknowns. The two methods are elimination and substitution. Elimination works best when coefficients of one variable can be made equal or opposites.
联立方程组含有两个或更多未知数。两种方法是消元法和代入法。当可以使某个变量的系数相等或互为相反数时，消元法最适用。
the equations: 3x = 9, so x = 3. Substitute back: 3 - y = 2 gives y = 1. Solution: (3, 1).
解 2x + y = 7 和 x - y = 2。两式相加：3x = 9，所以 x = 3。代入：3 - y = 2 得出 y = 1。解为 (3, 1)。
10, substitute the first into the second: x + (2x + 1) = 10 → 3x = 9 → x = 3, then y = 7.
当其中一个方程容易变形时，代入法很有用。若 y = 2x + 1 且 x + y = 10，将第一个代入第二个：x + (2x + 1) = 10 → 3x = 9 → x = 3，然后 y = 7。
Check your solutions in both original equations to avoid errors.
在两个原方程中检验解，以避免错误。
6. Inequalities | 不等式
to). Solving them follows the same rules as equations, with one critical exception: multiplying or dividing by a negative number reverses the inequality sign.
不等式使用 > (大于)、< (小于)、≥ (大于等于) 和 ≤ (小于等于)。解不等式遵循与解方程相同的规则，但有一个关键例外：乘以或除以负数时，不等号方向改变。
≤ 8: divide by -2 and flip the sign → x ≥ -4. This means x can be -4 or any larger number.
解 -2x ≤ 8：除以 -2 并翻转不等号 → x ≥ -4。这意味着 x 可以是 -4 或任何更大的数。
s or by applying operations to all three parts. Add 1: 3 < 3x ≤ 9. Divide by 3: 1 < x ≤ 3.
像 2 < 3x - 1 ≤ 8 这样的组合不等式，可以通过分成两部分求解，或对所有三部分同时进行运算。加 1：3 < 3x ≤ 9。除以 3：1 < x ≤ 3。
open circle for strict inequalities (<, >) and a closed circle for inclusive ones (≤, ≥).
在数轴上表示解时，严格不等式 (<, >) 用空心圆，包含等号的不等式 (≤, ≥) 用实心圆。
7. Sequences and the nth Term | 数列与第 n 项
A sequence is a list of numbers following a rule. Linear sequences have a constant difference between terms. The nth term formula allows you to find any term without listing all previous terms.
数列是按一定规则排列的一系列数字。线性数列的相邻两项之差为常数。第 n 项公式使你可以直接求出任意项，而无需列出之前的所有项。
For the sequence 5, 8, 11, 14, …, the difference is 3, so the nth term is 3n + 2. Check: when n=1, 3×1+2=5, correct.
对于数列 5, 8, 11, 14, …，差为 3，所以第 n 项为 3n + 2。检验：当 n=1 时，3×1+2=5，正确。
Quadratic sequences have a constant second difference. The nth term involves n². If second difference is 4, the n² coefficient is 2. Then adjust for linear part and constant.
二次数列的二阶差为常数。第 n 项包含 n²。若二阶差为 4，则 n² 系数为 2。然后调整线性部分和常数。
Given the sequence 2, 5, 10, 17, 26, … arting point. Compare with n²: 1, 4, 9, 16, 25 → differences are +1, so nth term = n² + 1.
给定数列 2, 5, 10, 17, 26, …，一阶差：3, 5, 7, 9；二阶差 = 2，所以系数为 1，从 n² 开始。与 n² 比较：1, 4, 9, 16, 25 → 差为 +1，所以第 n 项 = n² + 1。
8. Introduction to Functions | 函数入门
A function is a special relationship where each input has exactly one output. You can think of it as a machine that takes an input value, applies a rule, and produces an output.
函数是一种特殊的关系，其中每个输入恰好对应一个输出。你可以把它想象成一台机器，接受输入值，应用规则，然后产生输出。
Functions map elements from the domain (input set) to the range (output set). If a rule assigns two different outputs to the same input, it is not a function.
函数将定义域（输入集）中的元素映射到值域（输出集）。如果同一个输入对应两个不同的输出值，则不是函数。
Common functions include linear functions, quadratic functions, and reciprocal functions. GCSE also covers interpreting functions from graphs and equations.
常见函数包括一次函数、二次函数和反比例函数。GCSE 还涵盖从图像和方程中解读函数。
9. Function Notation and Evaluation | 函数记法与求值
x, multiplies by 2, then adds 3. To evaluate f(4), substitute x = 4: f(4) = 2×4 + 3 = 11.
函数记法使用 f(x) 而非 y。对于 f(x) = 2x + 3，这意味着函数将 x 乘以 2，然后加 3。要求 f(4)，代入 x = 4：f(4) = 2×4 + 3 = 11。
ns apply g first, then f. If f(x) = x² and g(x) = x + 1, then fg(3) = f(g(3)) = f(4) = 16.
复合函数结合了两个函数：fg(x) 表示先应用 g，再应用 f。若 f(x) = x² 且 g(x) = x + 1，则 fg(3) = f(g(3)) = f(4) = 16。
e y = 2x + 5, swap x and y: x = 2y + 5, solve for y: y = (x - 5)/2, so f⁻¹(x) = (x - 5)/2.
反函数逆转运算。对于 f(x) = 2x + 5，求 f⁻¹(x)：写成 y = 2x + 5，交换 x 和 y：x = 2y + 5，解出 y：y = (x - 5)/2，所以 f⁻¹(x) = (x - 5)/2。
10. Graphs of Functions | 函数图像
tions plot as straight lines; y = mx + c where m is the gradient and c is the y-intercept.
函数图像显示了所有点 (x, f(x))。一次函数画出来是一条直线；y = mx + c 其中 m 是斜率，c 是 y 轴截距。
Quadratic functions y = x² produce a parabola — a U-shaped curve. The turning point is either a minimum (U) or a maximum (∩). By finding where the graph crosses the axes, you can sketch it accurately.
二次函数 y = x² 产生抛物线——一条 U 形曲线。转折点可以是最低点（U 形）或最高点（∩ 形）。通过求出图像与坐标轴的交点，可以精确地画出草图。
at GCSE. y = 1/x has two asymptotes: x = 0 and y = 0, with branches in opposite quadrants.
GCSE 中也会出现三次函数图和反比例函数图。y = 1/x 有两条渐近线：x = 0 和 y = 0，图像分支位于相对的象限。
Plot key points — including intersection with the x-axis (roots) and y-axis — then join them with a smooth curve, not straight line segments.
标出关键点——包括与 x 轴的交点（根）和与 y 轴的交点——然后用平滑曲线连接，而不是用折线段。
11. Transformations of Graphs | 图像变换
Graph transformations allow you to move or stretch a function’s graph. Translation f(x) + a shifts the graph vertically: up if a > 0, down if a < 0.
图像变换使你可以移动或拉伸函数图像。平移 f(x) + a 将图像垂直移动：若 a > 0 则上移，若 a < 0 则下移。
f(x + a) translates horizontally: left if a > 0, right if a < 0. Note the sign is opposite to what you might initially think.
f(x + a) 进行水平平移：若 a > 0 则左移，若 a < 0 则右移。注意符号与直觉相反。
Reflection: -f(x) reflects the graph in the x-axis, while f(-x) reflects in the y-axis. For quadratics, these can flip the parabola.
反射：-f(x) 将图像在 x 轴上反射，而 f(-x) 在 y 轴上反射。对于二次函数，这可以翻转抛物线。
Stretches: af(x) stretches vertically by factor a. f(ax) stretches horizontally by factor 1/a, compressing the graph towards the y-axis if a > 1.
拉伸：af(x) 以因子 a 进行垂直拉伸。f(ax) 以因子 1/a 进行水平拉伸，若 a > 1，则将图像向 y 轴压缩。
12. Real-life Graphs and Rates of Change | 实际应用图像与变化率
Graphs model real-world situations: distance-time graphs show motion, where the gradient gives speed. A straight line indicates constant speed; a curve indicates acceleration.
图像用于模拟实际情境：距离-时间图表示运动，其斜率表示速度。直线表示匀速运动；曲线表示加速度。
Velocity-time graphs have gradients representing acceleration and areas under the graph representing distance travelled. Constant acceleration appears as a sloping straight line.
速度-时间图的斜率代表加速度，图下面积代表行驶距离。恒定加速度表现为倾斜的直线。
Conversion graphs and cost graphs are also common. A linear cost graph has a fixed charge (intercept) plus a unit rate (gradient).
换算图和费用图也很常见。线性费用图有一个固定费用（截距）加上一个单价费率（斜率）。
Rates of change are calculated from the gradient of a tangent at a point on a curve, giving the instantaneous rate. Average rate uses the chord between two points.
变化率通过曲线上某点切线的斜率来计算，得出瞬时变化率。平均变化率使用两点之间的弦的斜率。
Interpreting intercepts and slopes in context is a vital exam skill — always state what the gradient and intercept represent.
在具体情境中解读截距和斜率是一项重要的考试技能——始终要说明斜率和截距代表什么。
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