AQA Maths: Quadratic Functions Key Points | AQA 数学：二次函数考点精讲

📚 AQA Maths: Quadratic Functions Key Points | AQA 数学：二次函数考点精讲

Quadratic functions form the backbone of A-Level algebra and appear in almost every exam paper. Mastering their transformations, graphs, and solution techniques is essential for top marks. In this revision guide, we break down every key concept you need, from standard form to inequalities, with clear step-by-step explanations.

二次函数是 A-Level 代数的核心，几乎每份试卷都会出现。掌握其变换、图像与求解技巧是取得高分的必要条件。本考点精讲将逐一梳理从标准式到不等式的每一个关键概念，并配以清晰的步骤讲解。

1. Standard Form and Vertex Form | 标准形式与顶点形式

A quadratic function is any function that can be written as f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. This is called the standard form. The coefficient a determines the direction and width of the parabola.

二次函数是指可以写成 f(x) = ax² + bx + c 的函数，其中 a、b、c 为常数且 a ≠ 0。该形式称为标准形式。系数 a 决定了抛物线的开口方向和宽窄。

An alternative and very useful form is the vertex form: f(x) = a (x − h)² + k, where (h, k) is the vertex of the parabola. Converting between these two forms is a crucial skill, often done by completing the square.

另一种非常实用的形式是顶点式：f(x) = a (x − h)² + k，其中 (h, k) 为抛物线的顶点。通过配方法在两种形式之间进行转换是一项核心技能。

2. Factorisation | 因式分解

Factorising a quadratic expression is the first method to try when solving quadratic equations. If the quadratic can be expressed as (px + q)(rx + s), the solutions are found by setting each bracket to zero.

因式分解是求解二次方程时首选的解法。若二次式可分解为 (px + q)(rx + s)，则令每个括号分别为零即可求得根。

For a monic quadratic (a = 1), such as x² + bx + c, we look for two numbers that multiply to c and add to b. For example, x² + 5x + 6 = (x + 2)(x + 3).

对于首项系数为 1 的二次式，如 x² + bx + c，需寻找两个乘积为 c、和为 b 的数。例如 x² + 5x + 6 = (x + 2)(x + 3)。

When a ≠ 1, we can use the ‘ac method’ or systematic trial. Multiply a and c, then find two numbers that multiply to ac and add to b. Split the middle term and factor by grouping.

当 a ≠ 1 时，可使用“ac 法”或系统试错法。将 a 与 c 相乘，寻找乘积为 ac 且和为 b 的两个数，拆分中项后进行分组分解。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic ax² + bx + c in vertex form a(x − h)² + k. This reveals the vertex coordinates and the parabola’s minimum or maximum value.

配方法将二次式 ax² + bx + c 化为顶点式 a(x − h)² + k，直接展示顶点坐标和抛物线的最大值或最小值。

To complete the square for x² + bx, add and subtract (b/2)². For example, x² + 6x + 5 becomes (x + 3)² − 9 + 5 = (x + 3)² − 4. The vertex is (−3, −4).

对 x² + bx 进行配方时，加上并减去 (b/2)²。例如 x² + 6x + 5 化为 (x + 3)² − 9 + 5 = (x + 3)² − 4，顶点为 (−3, −4)。

When the coefficient a is not 1, first factor out a from the x² and x terms: a(x² + (b/a)x) + c, then complete the square inside the bracket and adjust the constant.

当 a ≠ 1 时，先对前两项提出 a：a(x² + (b/a)x) + c，然后在括号内配方，最后修正常数项。

4. The Quadratic Formula | 二次求根公式

The quadratic formula gives the solutions of any quadratic equation ax² + bx + c = 0, even when factorisation is difficult or impossible.

二次求根公式可用来求解任何二次方程 ax² + bx + c = 0，即便是无法因式分解的情形也适用。

x = [−b ± √(b² − 4ac)] / (2a)

The expression under the square root, b² − 4ac, is called the discriminant and determines the nature of the roots.

根号下的式子 b² − 4ac 称为判别式，它决定了根的性质。

Always ensure the equation is set to zero before identifying a, b, and c. Substitute carefully and simplify the square root when possible.

使用公式前务必将方程整理为零的形式，再确定 a、b、c。代入时需小心，若能化简根式应尽量化简。

5. The Discriminant | 判别式

The discriminant Δ = b² − 4ac tells us how many real solutions a quadratic equation has, without actually solving it.

判别式 Δ = b² − 4ac 无需解方程便可告知二次方程有多少个实数解。

If Δ > 0: two distinct real roots.
If Δ = 0: exactly one real root (a repeated root).
If Δ < 0: no real roots (two complex roots).

若 Δ > 0：有两个不等的实根。
若 Δ = 0：有一个实根（重根）。
若 Δ < 0：无实根（两个共轭复数根）。

In exam questions, you may be asked to find the range of a parameter for which a quadratic has equal roots or no real roots. Set up the discriminant inequality or equation accordingly.

考题中常要求寻找某个参数的范围，使得二次方程具有等根或无实根，这时需相应建立判别式的不等式或等式。

6. Graph and Properties | 图像与性质

The graph of a quadratic function is a parabola. It opens upwards if a > 0 and downwards if a < 0. The axis of symmetry is the vertical line x = −b/(2a), which passes through the vertex.

二次函数的图像是一条抛物线。a > 0 时开口向上，a < 0 时开口向下。对称轴为直线 x = −b/(2a)，经过顶点。

The vertex is the turning point. Its x-coordinate is −b/(2a) and its y-coordinate is f(−b/(2a)). Completing the square gives the vertex directly as (h, k).

顶点即拐点，其横坐标为 −b/(2a)，纵坐标为 f(−b/(2a))。通过配方法可直接得到顶点 (h, k)。

The y-intercept is found by setting x = 0, giving the point (0, c). The x-intercepts are the real roots of the equation ax² + bx + c = 0.

y 轴截距置 x = 0 得，坐标为 (0, c)。x 轴截距为方程 ax² + bx + c = 0 的实根。

7. Solving by Intersection | 利用交点解方程

Solving ax² + bx + c = 0 graphically means finding where the parabola y = ax² + bx + c crosses the x-axis. These points give the real roots.

从图像上看，解 ax² + bx + c = 0 即是寻找抛物线 y = ax² + bx + c 与 x 轴的交点，这些点的横坐标就是实根。

If the parabola does not touch the x-axis (Δ < 0), the equation has no real solutions. If it just touches (Δ = 0), there is a repeated root.

若抛物线不与 x 轴相交（Δ < 0），方程无实数解；若恰好相切（Δ = 0），则有一个重根。

You can also solve equations like x² + 2x − 3 = x + 1 graphically by plotting both sides and finding their intersection points.

对于形如 x² + 2x − 3 = x + 1 的方程，也可以作出左右两函数的图像，寻找其交点来求解。

8. Quadratic Inequalities | 二次不等式

To solve a quadratic inequality such as ax² + bx + c > 0, first solve the corresponding equation to find critical values where the expression equals zero.

解二次不等式如 ax² + bx + c > 0 时，先解对应方程求出临界值（即表达式为零的 x 值）。

Then sketch a quick sign diagram or imagine the parabola. The expression is positive outside the interval between the roots if a > 0, and negative inside. Reverse when a < 0.

然后画出符号表或想象抛物线形状。当 a > 0 时，表达式在两根区间外为正，两根之间为负；a < 0 时正负相反。

Always write the solution set using set notation or inequalities, and be careful with strict vs. non-strict inequalities (e.g., using open or closed brackets in interval notation).

务必用集合或不等式写出解集，并注意严格与不严格不等式的区别（如区间表示时使用开区间或闭区间）。

9. Roots and Coefficients | 根与系数的关系

For the quadratic ax² + bx + c = 0 with roots α and β, the sum of roots is α + β = −b/a, and the product is αβ = c/a. These relations are extremely useful for forming equations or solving symmetric problems.

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，有根的和 α + β = −b/a，根的积 αβ = c/a。这些关系在构造方程或解决对称性问题时非常有用。

You can construct a quadratic with given roots r and s by writing (x − r)(x − s) = 0, which expands to x² − (r + s)x + rs = 0. Multiplying through by any nonzero constant gives an equivalent equation.

若已知两根 r 和 s，可写出 (x − r)(x − s) = 0，展开得 x² − (r + s)x + rs = 0，再乘以任意非零常数即得等价的二次方程。

10. Applications in Modelling | 建模应用

Quadratic functions model many real-world scenarios, such as projectile motion, area problems, and economic profit. The vertex often represents a maximum or minimum value required by the context.

二次函数可用于模拟许多现实场景，如抛体运动、面积问题和经济利润。顶点常代表题目情境下所需要的最大值或最小值。

For a projectile, the height h(t) = −½gt² + v₀t + h₀ is a quadratic in time t. The maximum height occurs at the vertex, and the time of flight can be found by solving h(t) = 0.

对于抛射体，高度 h(t) = −½gt² + v₀t + h₀ 是关于时间 t 的二次函数。最大高度在顶点处取得，飞行时间可解 h(t) = 0 获得。

When optimising, define a suitable variable, express the quantity to be maximised as a quadratic function, then complete the square or use −b/(2a) to find the optimum value.

解决最值问题时，先合理设元，将待优化量表示为二次函数，再通过配方或利用 −b/(2a) 求出最优值。

11. Common Mistakes to Avoid | 常见错误提醒

Forgetting to set the equation to zero before factorising or applying the formula is a frequent error. Always rearrange into ax² + bx + c = 0 first.

忘记先将方程化为零就直接因式分解或代公式是常见错误。务必先整理为 ax² + bx + c = 0。

Sign errors when substituting b into the formula or completing the square are very common. Write down the coefficients clearly, including negative signs, before substituting.

代入公式或配方时符号错误频频发生。代值前应清晰写出包含负号的各个系数。

Misinterpreting the discriminant: Δ = 0 gives one repeated real root, not ‘no solutions’. Also, when solving inequalities, forgetting to reverse the inequality sign when multiplying by a negative number can lead to an incorrect conclusion.

误读判别式：Δ = 0 表示有一个重根，并非“无解”。另外，解不等式时若两边同乘负数，忘记反转不等号方向也会导致错误结论。

12. Exam Tips and Summary | 考试技巧与总结

Always check the question carefully to see which form (factorised, standard, or vertex) is most useful. If the question asks for the turning point, completing the square is often the fastest route.

仔细审题，判断哪种形式（因式、标准或顶点）对解题最有利。若题目要求拐点，配方法往往是最快的途径。

When using the quadratic formula, write out the values of a, b, c explicitly. Simplify the discriminant before taking the square root, and always simplify the final fraction if possible.

使用求根公式时，应明确列出 a、b、c 的值；先化简判别式再开方，最终分式也应尽可能化简。

For inequalities, sketching a quick graph helps avoid sign errors. In modelling questions, check whether the answer makes sense in the real-world context (e.g., negative heights are not allowed).

求解不等式时，快速勾勒图像有助于避免符号错误。在建模题中，要检查答案是否符合实际情境（如高度不能为负）。

Practice, practice, practice: the more quadratic problems you solve, the more instinctive the methods become. Always review your work and look for alternative methods that might save time in the exam.

多加练习：解决的二次问题越多，解题方法就越能运用自如。每次做完习题都要回顾，思考是否有能在考试中节省时间的替代解法。

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AQA Maths: Quadratic Functions Key Points | AQA 数学：二次函数 考点精讲