Edexcel Maths: Algebra and Functions - Key Points | Edexcel 数学：代数和函数考点精讲

📚 Edexcel Maths: Algebra and Functions – Key Points | Edexcel 数学：代数和函数考点精讲

Algebra and functions form the backbone of Edexcel A Level Mathematics. From basic algebraic manipulation to advanced function transformations, a solid grasp of these topics is essential for success across pure modules. This revision guide distils all the key concepts, common pitfalls, and exam-style insights you need to master the syllabus efficiently.

代数与函数是 Edexcel A Level 数学的核心支柱。从基础的代数运算到复杂的函数变换，牢固掌握这些知识对于纯数模块的成功至关重要。本考点精讲浓缩了所有关键概念、常见错误以及考试风格洞察，帮助你高效掌握大纲内容。

1. Algebraic Manipulation Basics | 代数基础运算

Mastering the rules of indices, surds, and expanding brackets is non-negotiable. These skills underpin every subsequent topic in the A Level course.

掌握指数律、根式以及展开括号的规则是必不可少的。这些技能是 A Level 课程中每一个后续主题的基础。

For indices, recall that aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ. Negative and fractional indices such as a⁻ⁿ = 1/aⁿ and a^(½) = √a are frequently tested.

关于指数，请牢记 aᵐ × aⁿ = aᵐ⁺ⁿ 及 (aᵐ)ⁿ = aᵐⁿ。负指数和分数指数（例如 a⁻ⁿ = 1/aⁿ 及 a^(½) = √a）是常考内容。

When rationalising denominators with surds, multiply numerator and denominator by the conjugate, e.g. for 1/(√a + √b), use (√a – √b)/(√a – √b).

在对含有根式的分母有理化时，将分子和分母同时乘以共轭根式，例如对于 1/(√a + √b)，使用 (√a – √b)/(√a – √b)。

Expanding triple brackets like (x + a)(x + b)(x + c) often appears; use systematic multiplication or recognise patterns from binomial expansions.

展开三个括号的乘积如 (x + a)(x + b)(x + c) 经常出现；应采用系统乘法或从二项式展开中识别模式。

2. Quadratic Functions | 二次函数

The quadratic function f(x) = ax² + bx + c (a ≠ 0) is central to the entire pure mathematics syllabus. Its graph is a parabola, and its properties are examined in numerous forms.

二次函数 f(x) = ax² + bx + c（a ≠ 0）是整个纯数大纲的核心。它的图像是一条抛物线，其性质以多种形式被考查。

The discriminant, Δ = b² – 4ac, determines the number of real roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.

判别式 Δ = b² – 4ac 决定了实根的数量：Δ > 0 给出两个不等实根，Δ = 0 给出一个重根，而 Δ < 0 则无实根。

Completing the square transforms ax² + bx + c into a(x + p)² + q, where the vertex of the parabola is at (-p, q). This form also reveals the minimum or maximum value of the function.

配方法将 ax² + bx + c 转化为 a(x + p)² + q，此时抛物线的顶点位于 (-p, q)。这一形式还揭示了函数的最小值或最大值。

For factorising quadratics, remember to check for a common factor first. Exam questions often hide factorable expressions within more complex problems.

因式分解二次式时，记得先检查是否有公因子。考试题常常将可分解的表达式隐藏在更复杂的问题中。

3. Solving Equations and Inequalities | 方程与不等式求解

Solving linear and quadratic equations is a prerequisite, but simultaneous equations involving one linear and one quadratic are a typical Edexcel exam feature.

求解一次和二次方程是先决条件，但涉及一个一次方程和一个二次方程的联立方程组是 Edexcel 考试的典型特征。

Substitute the linear equation into the quadratic, rearrange to form a standard quadratic, and solve. Always interpret the solutions in context; not all algebraic solutions may satisfy the original system geometrically.

将线性方程代入二次方程，重新整理为标准二次式，然后求解。始终要在上下文中解释解；并非所有代数解在几何上都满足原方程组。

Inequalities require careful handling of the sense when multiplying or dividing by a negative quantity. For quadratic inequalities such as x² – 5x + 6 > 0, sketch the graph and identify intervals where the curve is above the x-axis.

不等式在处理乘以或除以负数时需要特别注意不等号方向。对于二次不等式如 x² – 5x + 6 > 0，应画出图像并确定曲线位于 x 轴上方的区间。

Critical values from solving the equality give the boundary points; use a sign diagram or the shape of the parabola to write the solution set, often expressed using set notation or intervals.

由解等式得到的临界值给出了边界点；利用符号表或抛物线的形状写出解集，通常使用集合符号或区间表示法。

4. Polynomials and the Factor Theorem | 多项式与因式定理

The Factor Theorem states that for a polynomial p(x), if p(a) = 0 then (x – a) is a factor. This is a powerful tool for factorising cubics and higher-degree polynomials.

因式定理指出，对于多项式 p(x)，若 p(a) = 0，则 (x – a) 是一个因式。这是分解三次及更高次多项式的有力工具。

Use trial and error with factors of the constant term to find the first linear factor, then perform algebraic long division or compare coefficients to find the remaining quadratic factor.

通过用常数项的因子进行试探，找出第一个一次因式，然后进行多项式长除法或比较系数以求出剩下的二次因式。

The Remainder Theorem extends this: when p(x) is divided by (x – a), the remainder is p(a). This often saves time when only a remainder is required.

余式定理是其延伸：当 p(x) 除以 (x – a) 时，余数为 p(a)。当只需计算余数时，这常常能节省时间。

For cubic equations, once fully factorised, solve each linear factor to find all roots. Sketching a cubic using its roots and the sign of the leading coefficient is a common exam requirement.

对于三次方程，一旦完全因式分解，求解每一个一次因式就找到了所有的根。利用根和首项系数的符号来绘制三次函数草图是常见的考试要求。

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

A function is a mapping where every input has exactly one output. The domain is the set of all possible inputs, and the range is the set of all outputs that actually occur.

函数是一种映射，其中每个输入恰好对应一个输出。定义域是所有可能输入的集合，值域是所有实际出现的输出的集合。

Be able to find the maximal domain for functions involving square roots (argument must be ≥ 0) and denominators (must not be zero). Express the domain and range using inequalities or interval notation.

要能够找出涉及平方根（被开方数必须 ≥ 0）和分母（不得为零）的函数的最大定义域。用不等式或区间符号表示定义域和值域。

One-to-one functions are essential for an inverse to exist. The horizontal line test quickly determines whether a function is one-to-one: if any horizontal line cuts the graph more than once, the function is not one-to-one.

一对一的函数是反函数存在的必要条件。水平线检验可快速判断函数是否一一对应：若任何水平线与图像相交多于一次，则该函数不是一一对应。

Set notation such as {x ∈ ℝ : x ≠ 2} or interval notation (-∞, 2) ∪ (2, ∞) must be used accurately. Misuse of brackets is a common mark-losing error.

必须准确使用集合符号如 {x ∈ ℝ : x ≠ 2} 或区间符号 (-∞, 2) ∪ (2, ∞)。括号的误用是常见的失分错误。

6. Composite Functions | 复合函数

A composite function fg(x) means apply g first, then f, so fg(x) = f(g(x)). The order is crucial; gf(x) is generally different from fg(x).

复合函数 fg(x) 意味着先施行 g，再施行 f，因此 fg(x) = f(g(x))。顺序至关重要；gf(x) 通常与 fg(x) 不同。

To find the expression for a composite function, substitute the inner function directly. For example, if f(x) = 2x + 1 and g(x) = x², then fg(x) = 2x² + 1, whereas gf(x) = (2x + 1)².

要找出复合函数的表达式，直接代入内层函数。例如，若 f(x) = 2x + 1 且 g(x) = x²，则 fg(x) = 2x² + 1，而 gf(x) = (2x + 1)²。

The domain of a composite function is determined by both the domain of the inner function and the requirement that the output of the inner function lies in the domain of the outer function. Always check this restriction.

复合函数的定义域由内层函数的定义域以及内层函数的输出必须在外层函数的定义域内这一要求共同决定。始终要检查这一限制。

7. Inverse Functions | 反函数

The inverse function f⁻¹(x) reverses the effect of f(x). To find it, write y = f(x), swap x and y, and then solve for y. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.

反函数 f⁻¹(x) 逆转 f(x) 的作用。要找到它，先写出 y = f(x)，交换 x 和 y，然后解出 y。f⁻¹ 的定义域是 f 的值域，f⁻¹ 的值域是 f 的定义域。

The graph of y = f⁻¹(x) is a reflection of y = f(x) in the line y = x. This geometric property helps visualise whether a function is self-inverse.

y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的反射。这一几何性质有助于直观判断一个函数是否为自反函数。

A common exam task is to restrict the domain of a non-one-to-one function to make it one-to-one, often by choosing x ≥ h for a quadratic where h is the x-coordinate of the vertex.

一个常见的考试任务是限制非一一对应函数的定义域使其成为一一对应，通常对二次函数选择 x ≥ h，其中 h 是顶点的 x 坐标。

8. Modulus Function | 绝对值函数

The modulus function |x| is defined as x when x ≥ 0 and -x when x < 0. Its graph is a V-shape, and it removes any negative sign from the input.

绝对值函数 |x| 定义为：当 x ≥ 0 时取 x，当 x < 0 时取 -x。它的图像呈 V 形，可以消除输入中的任何负号。

Equations such as |ax + b| = c lead to two cases: ax + b = c and ax + b = -c. Always check solutions in the original equation to avoid extraneous answers.

形如 |ax + b| = c 的方程会引出两种情况：ax + b = c 和 ax + b = -c。始终将解代入原方程检验，以避免增根。

Inequalities with modulus, like |2x – 1| < 3, are solved by writing -3 < 2x - 1 < 3 and then solving the resulting compound inequality. For > inequalities, consider the two separate intervals outside the critical values.

含有绝对值的不等式，例如 |2x – 1| < 3，可通过写出 -3 < 2x - 1 < 3 然后求解复合不等式来解决。对于大于形式的不等式，则考虑临界值之外的两个独立区间。

Graphing y = |f(x)| involves reflecting any part of f(x) below the x-axis in the x-axis, while y = f(|x|) reflects the right-hand side of f(x) for x ≥ 0 across the y-axis.

绘制 y = |f(x)| 的图像涉及将 f(x) 在 x 轴下方的部分关于 x 轴进行反射，而 y = f(|x|) 则是将 f(x) 在 x ≥ 0 一侧的图像关于 y 轴进行反射。

9. Graph Transformations | 图像变换

Standard transformations include translations, stretches, and reflections. Know that y = f(x) + a shifts the graph vertically, while y = f(x + a) shifts it horizontally – but note the opposite direction for the argument.

标准变换包括平移、拉伸和反射。须知 y = f(x) + a 使图像垂直平移，而 y = f(x + a) 使图像水平平移——但要注意自变量方向的相反性。

A stretch by factor a parallel to the y-axis is y = a f(x); a stretch by factor 1/a parallel to the x-axis is y = f(ax). Combinations of transformations often require a strict order: horizontal transformations inside the bracket must be applied before vertical ones in examinations unless instructed otherwise.

平行于 y 轴、比例为 a 的拉伸为 y = a f(x)；平行于 x 轴、比例为 1/a 的拉伸为 y = f(ax)。组合变换通常需要严格的顺序：除非另有说明，在考试中应先施行括号内的水平变换，再施行垂直变换。

Be able to describe a sequence of transformations mapping one function to another. For example, mapping y = x² to y = 2(x – 3)² + 1 involves a translation by vector [3, 0], a stretch parallel to y-axis by factor 2, and a translation by [0, 1].

要能描述将一函数映射为另一函数的变换序列。例如，将 y = x² 映射到 y = 2(x – 3)² + 1 涉及一个向 [3, 0] 的平移、一个平行于 y 轴比例为 2 的拉伸，以及一个向 [0, 1] 的平移。

Invariant points occur where points on the graph remain unchanged after a transformation. These are often intersections with axes of reflection or the base of a stretch.

不动点出现在图像上的点经过变换后保持不变的位置。它们通常是与反射轴的交点或拉伸的基点。

10. Algebraic Fractions and Partial Fractions | 代数分式与部分分式

Simplifying rational expressions involves factorising numerator and denominator and cancelling common factors, but always state any restrictions on the variable to avoid division by zero.

化简有理式涉及对分子和分母进行因式分解并约去公因子，但务必要说明对变量的任何限制以避免除以零。

Adding or subtracting algebraic fractions requires a common denominator. For example, when combining 1/(x-1) + 2/(x+2), the LCD is (x-1)(x+2). The resulting numerator often needs expanding and simplifying.

加减代数分式需要公分母。例如，合并 1/(x-1) + 2/(x+2) 时，公分母是 (x-1)(x+2)。所得分子通常需要展开和化简。

Partial fraction decomposition breaks a rational function into simpler fractions. For distinct linear factors, write A/(x-a) + B/(x-b). Solve for constants by equating coefficients or substituting convenient values of x.

部分分式分解将一个有理函数分解为更简单的分式。对于相异的一次因式，写成 A/(x-a) + B/(x-b) 的形式。通过比较系数或代入恰当的 x 值来求解常数。

When a repeated linear factor (x-a)² appears, the decomposition includes A/(x-a) + B/(x-a)². For an improper fraction where the degree of the numerator is ≥ degree of denominator, perform long division first.

当出现重复一次因式 (x-a)² 时，分解中包含 A/(x-a) + B/(x-a)²。对于分子次数大于或等于分母次数的假分式，需先进行长除法。

11. Modelling with Functions | 函数建模

In real-life contexts, functions model relationships such as projectile paths (quadratics), exponential growth, or cost functions. Identifying the correct function type and interpreting constants is key.

在现实情境中，函数用于模拟抛体轨迹（二次函数）、指数增长或成本函数等关系。确定正确的函数类型并解释常数是关键。

For a quadratic model, the vertex gives the maximum or minimum value, which might represent maximum height or minimum cost. The roots often indicate where the model touches the ground or zero profit.

对于二次函数模型，顶点给出了最大值或最小值，这可能代表最大高度或最低成本。根通常指示模型接触地面或利润为零的位置。

Setting up equations from word problems requires defining variables clearly. Always relate the function’s algebraic properties back to the context: for instance, restricting the domain to positive values when dealing with physical quantities like time or length.

根据文字题建立方程需要清晰地定义变量。始终要将函数的代数性质与上下文联系起来：例如，当处理时间或长度等物理量时，将定义域限制为正值。

Differentiation of functions in modelling contexts yields rates of change, but within the pure algebra and functions topic, identifying stationary points via completing the square or symmetry often suffices to optimise a quantity.

尽管在建模中函数的微分能给出变化率，但在纯代数和函数主题内，通常通过配方法或对称性识别驻点就足以优化某个量。

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Edexcel Maths: Algebra and Functions – Key Points | Edexcel 数学：代数和函数 考点精讲