Algebra and Functions for Edexcel A-Level Mathematics | A-Level Edexcel 数学：代数和函数考点精讲

📚 Algebra and Functions for Edexcel A-Level Mathematics | A-Level Edexcel 数学：代数和函数考点精讲

Algebra and functions form the bedrock of the Edexcel A-Level Mathematics course. Mastering these topics is essential for success across Pure Mathematics. This article unpacks key concepts, common pitfalls, and exam strategies for every subtopic, from indices and quadratics to functions and graph transformations. Carefully crafted bilingual explanations ensure you grasp the theory and exam technique in depth.

代数和函数是 Edexcel A-Level 数学课程的基石。掌握这些内容对于纯数考试的成功至关重要。本文深入解读从指数、二次方程到函数与图像变换等各子主题的核心概念、常见错误及考试策略。精心编写的中英双语讲解，助你透彻理解理论与应试技巧。

1. Laws of Indices and Surds | 指数律与根式

Indices (exponents) and surds are fundamental for simplifying algebraic expressions. You must be fluent in applying index laws for multiplication, division, powers and roots, including fractional and negative exponents.

指数（幂）和根式是化简代数表达式的基础。你必须熟练掌握乘、除、乘方、开方以及分数和负指数等指数定律的运用。

Product of powers: aᵐ × aⁿ = aᵐ⁺ⁿ

同底数幂相乘：aᵐ × aⁿ = aᵐ⁺ⁿ

Quotient of powers: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)

同底数幂相除：aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)

Power of a power: (aᵐ)ⁿ = aᵐⁿ

幂的乘方：(aᵐ)ⁿ = aᵐⁿ

Negative exponent: a⁻ⁿ = 1 / aⁿ

负指数：a⁻ⁿ = 1 / aⁿ

Fractional exponent: a^(¹/ⁿ) = ⁿ√a, and a^(ᵐ/ⁿ) = (ⁿ√a)ᵐ or ⁿ√(aᵐ)

分数指数：a^(¹/ⁿ) = ⁿ√a，且 a^(ᵐ/ⁿ) = (ⁿ√a)ᵐ 或 ⁿ√(aᵐ)

Rationalising denominators often uses (a + √b)(a – √b) = a² – b. Always simplify surds fully; for example √50 = 5√2.

分母有理化常用 (a + √b)(a – √b) = a² – b。务必完全化简根式；例如 √50 = 5√2。

2. Quadratic Functions and the Discriminant | 二次函数与判别式

A quadratic function f(x) = ax² + bx + c (a ≠ 0) graphs as a parabola. Completing the square rewrites it as a(x + p)² + q, revealing the vertex (–p, q) and axis of symmetry x = –p.

二次函数 f(x) = ax² + bx + c (a ≠ 0) 的图像为抛物线。通过配方法可写成 a(x + p)² + q 的形式，从而得出顶点 (–p, q) 和对称轴 x = –p。

The discriminant is given by:

判别式由下式给出：

Δ = b² – 4ac

When Δ > 0, two distinct real roots; Δ = 0, one repeated real root; Δ < 0, no real roots (and the graph does not cross the x‑axis). This determines intersection with the x‑axis and existence of real factors.

当 Δ > 0 时，有两个相异实根；Δ = 0，有一个重实根；Δ < 0，无实根（因而图像不与 x 轴相交）。这决定了与 x 轴的交点情况以及是否存在实因式。

Quadratic inequalities such as ax² + bx + c > 0 are solved by sketching the parabola and identifying where it lies above or below the x‑axis. Always check the sign of a.

二次不等式如 ax² + bx + c > 0 可通过绘制抛物线草图并确定其位于 x 轴上方或下方的区间来求解。务必注意 a 的符号。

3. Polynomial Division, Factor and Remainder Theorems | 多项式除法、因式与余数定理

For a polynomial p(x), the factor theorem states: if p(a) = 0 then (x – a) is a factor. The remainder theorem: when p(x) is divided by (x – a), the remainder is p(a).

对于多项式 p(x)，因式定理指出：若 p(a) = 0，则 (x – a) 是一个因式。余数定理指出：当 p(x) 除以 (x – a) 时，余数为 p(a)。

Algebraic long division is used to divide a polynomial by a linear or quadratic divisor. Always write 0x² for missing terms to keep alignment.

多项式长除法用于除以一次或二次除式。务必在缺失的幂次项处补写 0x² 以保持对齐。

When factorising cubics, first find one linear factor by testing factors of the constant term using the factor theorem. Then use division or comparing coefficients to obtain the quadratic factor.

因式分解三次式时，先用因式定理检验常数项的因子，找到一个一次因式。然后用除法或对比系数法得到二次因式。

Example: For p(x) = 2x³ – 3x² – 3x + 2, test x = 1: p(1) = 2 – 3 – 3 + 2 = –2 ≠ 0; x = –1 gives p(–1) = –2 – 3 + 3 + 2 = 0, so (x + 1) is a factor. Proceed to divide.

示例：对于 p(x) = 2x³ – 3x² – 3x + 2，检验 x = 1：p(1) = 2 – 3 – 3 + 2 = –2 ≠ 0；x = –1 得 p(–1) = –2 – 3 + 3 + 2 = 0，故 (x + 1) 是一个因式。继续进行除法。

4. Manipulating Algebraic Fractions | 代数分式的化简

Algebraic fractions are simplified by factorising numerators and denominators and cancelling common factors. Always state restrictions on x where the denominator becomes zero.

代数分式通过因式分解分子和分母并约去公因式来化简。务必写出分母为零时 x 的限制条件。

Addition and subtraction require a common denominator. For example:

加减运算需要通分。例如：

1/(x – 2) + 3/(x + 1) = [1·(x + 1) + 3·(x – 2)] / [(x – 2)(x + 1)] = (4x – 5)/[(x – 2)(x + 1)]

When multiplying, multiply numerators and denominators; when dividing, multiply by the reciprocal. Always leave your answer in fully factorised form.

乘法时分子分母分别相乘；除法时乘以倒数。答案务必保留最简因式分解形式。

Be aware that a common mistake is to cancel terms instead of factors: (x + a)/(x + b) cannot be simplified unless a = b.

一个常见错误是将项而非因式约分：(x + a)/(x + b) 不能化简，除非 a = b。

5. Functions: Domain, Range and Mapping | 函数：定义域、值域与映射

A function f : x → f(x) associates each input (x) with exactly one output. The domain is the set of all allowed inputs; the range is the set of all possible outputs. Always consider natural restrictions, such as denominators ≠ 0 and inside square roots ≥ 0.

函数 f : x → f(x) 将每个输入 (x) 与唯一输出对应。定义域是所有允许输入值的集合；值域是所有可能输出值的集合。务必考虑自然限制，如分母不为零，平方根下非负。

For a graph, the domain is read along the x‑axis, the range along the y‑axis. When finding the range of a quadratic, complete the square to identify the vertex.

对于图像，定义域沿 x 轴读取，值域沿 y 轴读取。求二次函数的值域时，应通过配方法确定顶点。

Not every relation is a function; use the vertical line test. A one‑to‑one function passes the horizontal line test. Edexcel often asks you to restrict a domain to make a function invertible.

并非每个关系都是函数；用竖直线检验。一一函数可通过水平线检验。Edexcel 常要求限制定义域使函数可逆。

6. Composite Functions | 复合函数

A composite function applies one function after another. gf(x) means apply f first, then g: gf(x) = g(f(x)). The order matters; generally gf(x) ≠ fg(x).

复合函数指将一个函数作用在另一个函数的结果上。gf(x) 表示先作用 f，再作用 g：gf(x) = g(f(x))。顺序很重要；通常 gf(x) ≠ fg(x)。

For gf(x) to exist, the range of f must be a subset of the domain of g. In exam questions, you may need to find the domain of a composite function by considering the inner function’s domain and the outer function’s restrictions.

要使 gf(x) 存在，f 的值域必须是 g 定义域的子集。考试中，你可能需要同时考虑内层函数的定义域和外层函数的限制，从而找出复合函数的定义域。

Example: f(x) = 2x + 1, g(x) = x². Then fg(x) = 2x² + 1, domain all real numbers; gf(x) = (2x + 1)², domain all real numbers. However if g(x) = √x, domains become restricted.

示例：f(x) = 2x + 1, g(x) = x²。则 fg(x) = 2x² + 1，定义域为全体实数；gf(x) = (2x + 1)²，定义域为全体实数。但若 g(x) = √x，定义域便会受限。

7. Inverse Functions | 反函数

An inverse function f⁻¹(x) undoes the effect of f. For a function to have an inverse, it must be one‑to‑one. Graphically, y = f⁻¹(x) is the reflection of y = f(x) in the line y = x.

反函数 f⁻¹(x) 可撤销 f 的作用。函数必须有反函数的前提是一一映射。图形上，y = f⁻¹(x) 是由 y = f(x) 关于直线 y = x 反射得到的。

To find the inverse algebraically, swap x and y in the equation y = f(x) and solve for y. Then state the domain of f⁻¹, which is the range of f.

代数上求反函数时，将方程 y = f(x) 中的 x 和 y 互换，然后解出 y。最后写出 f⁻¹ 的定义域，即 f 的值域。

Example: f(x) = (x – 3)/(x + 1), x ≠ –1. Let y = (x – 3)/(x + 1), swap to get x = (y – 3)/(y + 1) and solve: x(y + 1) = y – 3 → xy + x = y – 3 → xy – y = –x – 3 → y(x – 1) = –x – 3 → y = (–x – 3)/(x – 1). So f⁻¹(x) = (–x – 3)/(x – 1) with domain x ≠ 1.

示例：f(x) = (x – 3)/(x + 1), x ≠ –1。设 y = (x – 3)/(x + 1)，互换得 x = (y – 3)/(y + 1)，求解：x(y + 1) = y – 3 → xy + x = y – 3 → xy – y = –x – 3 → y(x – 1) = –x – 3 → y = (–x – 3)/(x – 1)。因此 f⁻¹(x) = (–x – 3)/(x – 1)，定义域 x ≠ 1。

8. Graph Transformations | 图像变换

Transformations allow you to sketch related graphs quickly. The key mappings are:

通过变换可以快速绘制相关图像。关键对应关系如下：

f(x) + a: vertical translation by a units (up if a > 0).

f(x) + a：垂直平移 a 个单位（a > 0 时向上）。

f(x + a): horizontal translation by –a units (to the left if a > 0).

f(x + a)：水平平移 –a 个单位（a > 0 时向左）。

a f(x): vertical stretch by scale factor a (a > 1 makes it taller).

a f(x)：垂直伸缩，伸缩因子为 a（a > 1 时变高）。

f(ax): horizontal stretch by scale factor 1/a (a > 1 squashes the graph).

f(ax)：水平伸缩，伸缩因子为 1/a（a > 1 时图像被压缩）。

–f(x): reflection in the x‑axis.

–f(x)：关于 x 轴的反射。

f(–x): reflection in the y‑axis.

f(–x)：关于 y 轴的反射。

Always apply

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Algebra and Functions for Edexcel A-Level Mathematics | A-Level Edexcel 数学：代数和函数 考点精讲