Quadratic Functions for AQA A-Level | A-Level AQA 数学：二次函数考点精讲

📚 Quadratic Functions for AQA A-Level | A-Level AQA 数学：二次函数考点精讲

Quadratic functions lie at the very heart of A-Level mathematics. Under the AQA specification, you are expected to manipulate quadratics fluently in all three common forms, interpret their graphs, solve equations and inequalities, and apply your skills to modelling problems. This article walks you through every essential technique, from completing the square to using the discriminant, with careful attention to the wording of AQA exam questions.

二次函数是 A-Level 数学的核心内容。在 AQA 考试大纲中，你需要熟练掌握二次函数的三种常见形式、解读图像、解方程与不等式，并将这些技能应用于建模问题。本文带你梳理所有必备技巧，从配方法到判别式的运用，并特别关注 AQA 真题的提问方式。

1. The Three Standard Forms | 三种标准形式

A quadratic function can be written in three principal ways. The general form is f(x) = ax² + bx + c, where a, b and c are real numbers and a ≠ 0. This form immediately gives the y-intercept at (0, c) and is the starting point for algebraic manipulation.

二次函数有三种主要的表示方式。一般式为 f(x) = ax² + bx + c，其中 a、b、c 为实数且 a ≠ 0。这种形式直接给出 y 轴截距 (0, c)，也是代数变形的起点。

The factorised form f(x) = a(x − p)(x − q) reveals the roots (or zeros) of the equation f(x) = 0, which are x = p and x = q, provided the quadratic factorises neatly over the integers. In AQA exams, you often factorise to solve equations or sketch graphs.

因式分解形式 f(x) = a(x − p)(x − q) 展示了方程 f(x) = 0 的根（或零点）x = p 和 x = q，前提是二次式可在整数范围内分解。在 AQA 考试中，你经常会通过因式分解来解方程或画草图。

The completed square form f(x) = a(x − h)² + k gives the coordinates of the vertex (h, k) directly. This is particularly useful for finding the minimum or maximum value of the function and for solving equations where factorisation is not straightforward.

配方式 f(x) = a(x − h)² + k 直接给出顶点坐标 (h, k)。这在寻找函数的最小值或最大值时特别有用，也用于在不易因式分解时解方程。

2. Completing the Square | 配方法

Completing the square transforms a quadratic from general form to vertex form. For a monic quadratic x² + bx, you add and subtract (b/2)² to write it as (x + b/2)² − (b/2)². With a leading coefficient a ≠ 1, factor out a from the first two terms first.

配方法将二次式从一般式转化为顶点式。对于首一二次式 x² + bx，通过加上并减去 (b/2)²，可写成 (x + b/2)² − (b/2)²。若首项系数 a ≠ 1，则需先从含有 x 的前两项中提取 a。

For example, to express 2x² − 12x + 5 in the form a(x − h)² + k, write: 2(x² − 6x) + 5 = 2[(x − 3)² − 9] + 5 = 2(x − 3)² − 18 + 5 = 2(x − 3)² − 13. Hence the vertex is (3, −13) and the minimum value is −13.

例如，将 2x² − 12x + 5 转化为 a(x − h)² + k 的形式：先写 2(x² − 6x) + 5 = 2[(x − 3)² − 9] + 5 = 2(x − 3)² − 18 + 5 = 2(x − 3)² − 13。因此顶点为 (3, −13)，最小值为 −13。

Completing the square is also the method used to derive the quadratic formula, and it is the key to finding the maximum or minimum output of a quadratic model in optimisation problems.

配方法也是推导求根公式的基础，并且是在优化问题中寻找二次模型最大值或最小值输出的关键方法。

3. The Quadratic Formula and the Discriminant | 求根公式与判别式

For any quadratic ax² + bx + c = 0, the solutions are given by the formula x = [−b ± √(b² − 4ac)] / (2a). The expression under the square root, D = b² − 4ac, is called the discriminant. Its value determines the nature of the roots.

对于任何二次方程 ax² + bx + c = 0，其解由公式 x = [−b ± √(b² − 4ac)] / (2a) 给出。根号下的表达式 D = b² − 4ac 称为判别式。其值决定了根的性质。

If D > 0, the equation has two distinct real roots. If D = 0, there is one repeated real root (or the quadratic is a perfect square). If D < 0, there are no real roots, but two complex conjugate roots, which AQA may test in the context of quadratic equations having no real solutions.

若 D > 0，方程有两个不相等的实根。若 D = 0，则有一个重根（或二次式是完全平方式）。若 D < 0，则没有实根，只有一对共轭复根，AQA 有时会在“二次方程无实数解”的情境下进行考查。

AQA exam questions frequently ask you to “find the range of values of k for which the equation has two distinct real roots” or “has no real solutions”. This requires setting up an inequality involving the discriminant and solving it carefully.

AQA 考试中常出现这样的问题：“求 k 的取值范围，使方程有两个不相等的实根”或“没有实数解”。这需要建立含判别式的不等式并仔细求解。

4. Sketching Quadratic Graphs | 绘制二次函数图像

To sketch the graph of a quadratic, you should identify: the shape (∪ if a > 0, ∩ if a < 0), the y-intercept (0, c), any x-intercepts (roots), and the vertex (turning point). The line of symmetry is the vertical line x = h passing through the vertex.

要绘制二次函数图像，你需要确定：形状（a > 0 时开口向上 ∪，a < 0 时开口向下 ∩）、y 轴截距 (0, c)、x 轴截距（根）以及顶点（转折点）。对称轴是经过顶点的竖直线 x = h。

If the quadratic does not factorise, you can find the vertex by completing the square or by using x = −b/(2a) and substituting to find the y-coordinate. Always label the key points clearly on your sketch, as AQA mark schemes award marks for correct coordinates and shape.

如果二次式不能因式分解，你可以通过配方法求出顶点，或利用 x = −b/(2a) 求出顶点的 x 坐标并代入求出 y 坐标。在草图上务必清晰标注关键点，因为 AQA 的评分标准会给正确的坐标和形状赋分。

Remember that the vertex represents the maximum or minimum value of the function. If the domain is restricted, the maximum or minimum may occur at an endpoint instead, so always consider the domain given in the question.

请记住，顶点代表函数的最大值或最小值。如果定义域受限，最大值或最小值可能出现在端点处，因此务必考虑题目给定的定义域。

5. Solving Quadratic Equations by Factorising | 因式分解法解二次方程

When a quadratic expression can be written as a product of two linear factors, solving the equation becomes straightforward. For example, x² − 5x + 6 = 0 factorises to (x − 2)(x − 3) = 0, giving x = 2 or x = 3. This method relies on the principle that if mn = 0, then m = 0 or n = 0.

当二次式可以写成两个一次因式的乘积时，解方程就变得简单直观。例如，x² − 5x + 6 = 0 因式分解为 (x − 2)(x − 3) = 0，得到 x = 2 或 x = 3。此方法依据的原理是：若 mn = 0，则 m = 0 或 n = 0。

Factorising quadratics with a ≠ 1 requires more care. For 2x² + 7x + 3, look for two numbers that multiply to ac = 2×3 = 6 and add to b = 7, namely 6 and 1. Split the middle term: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

对 a ≠ 1 的二次式进行因式分解需要更加细致。以 2x² + 7x + 3 为例，寻找两个数，使其积为 ac = 2×3 = 6，和为 b = 7，这两个数为 6 和 1。拆分中项：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

AQA may ask you to solve equations by factorising or to use factorisation as part of a larger problem, such as simplifying rational expressions or finding the intersections of a line and a curve.

AQA 可能会要求你用因式分解法解方程，或将因式分解作为更大问题的一部分，例如化简有理式或求直线与曲线的交点。

6. Quadratic Inequalities | 二次不等式

To solve a quadratic inequality such as x² − 4x + 3 > 0, first find the critical values by solving the corresponding equation x² − 4x + 3 = 0, which gives x = 1 and x = 3. These values divide the number line into three intervals.

要解二次不等式（如 x² − 4x + 3 > 0），首先通过解相应方程 x² − 4x + 3 = 0 求出临界值，得到 x = 1 和 x = 3。这些值将数轴分成三个区间。

Test a value from each interval in the original inequality. Alternatively, sketch the graph of y = x² − 4x + 3. Since a > 0, the parabola opens upward and is below the x-axis between the roots. Thus the inequality > 0 is satisfied for x < 1 or x > 3. Always present your final answer using set notation or interval notation as required.

从每个区间选取一个值代入原不等式进行检验。或者，画出 y = x² − 4x + 3 的图像。由于 a > 0，抛物线开口向上，且在两根之间位于 x 轴下方。因此不等式 > 0 的解为 x < 1 或 x > 3。作答时务必按要求使用集合符号或区间符号表示最终答案。

For inequalities involving a ≤ or ≥ sign, the critical values themselves are included in the solution set. Be mindful of strict versus non-strict inequalities when writing your final answer.

对于含 ≤ 或 ≥ 的不等式，临界值本身包含在解集中。书写最终答案时要注意区分严格不等式与非严格不等式。

7. The Relationship Between Roots and Coefficients | 根与系数的关系

For the quadratic equation ax² + bx + c = 0 with roots α and β, the sum of the roots is α + β = −b/a and the product is αβ = c/a. These relations arise from comparing coefficients in the identity a(x − α)(x − β) ≡ ax² + bx + c.

对于二次方程 ax² + bx + c = 0，若根为 α 和 β，则根的和为 α + β = −b/a，根的积为 αβ = c/a。这些关系源于将恒等式 a(x − α)(x − β) ≡ ax² + bx + c 展开后比较系数。

AQA may ask questions such as “The equation x² + px + q = 0 has roots 2α and 2β. Find p and q in terms of α and β.” You would substitute into the sum and product formulas and simplify. This topic bridges quadratics and the broader theory of polynomial equations.

AQA 可能会出这样的题目：“方程 x² + px + q = 0 的两个根为 2α 和 2β。用 α 和 β 表示 p 和 q。”你需要将它们代入求和与求积公式并进行化简。这一知识点连接了二次函数与更广泛的多项式方程理论。

8. Quadratic Modelling and Optimisation | 二次建模与优化

Many real-world situations can be modelled by a quadratic function, such as the trajectory of a projectile, the area of a rectangular enclosure, or the profit from selling a product. Typically, you will form a quadratic expression, then use completing the square or differentiation to find the maximum or minimum value.

许多现实情境都可用二次函数建模，如抛射体的轨迹、矩形围栏的面积或销售产品的利润。通常，你需要建立一个二次表达式，然后通过配方法或微分来寻找最大值或最小值。

When modelling, pay careful attention to the practical constraints on the domain. For instance, lengths must be positive, and sales cannot be negative. The global turning point might lie outside the feasible domain, in which case the optimum occurs at a boundary.

建模时，要特别注意定义域的实际限制。例如，长度必须为正，销售量不能为负。全局转折点可能位于可行域之外，此时最优值出现在边界上。

AQA exam questions often require you to “prove that the maximum area is …” or “find the value of x that maximises the profit”. Always check that your answer makes sense in the context of the problem.

AQA 真题常要求你“证明最大面积为…”或“求使利润最大化的 x 值”。务必结合实际情境检查答案是否合理。

9. Hidden Quadratics and Substitutions | 隐藏型二次方程与换元法

Some equations are not obviously quadratic but can be transformed into a quadratic by a suitable substitution. Examples include x⁴ − 5x² + 4 = 0 (let u = x²), or 3²ˣ − 12×3ˣ + 27 = 0 (let u = 3ˣ), or 2sin²θ − sinθ − 1 = 0 (let u = sinθ).

有些方程并非明显是二次方程，但通过适当的换元可转化为二次方程。例如 x⁴ − 5x² + 4 = 0（令 u = x²），或 3²ˣ − 12×3ˣ + 27 = 0（令 u = 3ˣ），或 2sin²θ − sinθ − 1 = 0（令 u = sinθ）。

After solving for u, remember to substitute back to the original variable and solve for all possible solutions within the given domain. These questions test your algebraic fluency and ability to recognise standard quadratic structures.

在解出 u 之后，记住要代回原变量，并在给定定义域内求出所有可能的解。这类题目考查你的代数熟练度以及识别标准二次结构的能力。

10. Transformations of Quadratic Graphs | 二次函数图像的变换

Understanding graph transformations is essential for AQA. For the basic parabola y = x², the effect of f(x) + a, f(x + a), a f(x) and f(ax) are tested. A combination of transformations must be applied in the correct order: generally, horizontal shifts and stretches first, then vertical shifts and stretches.

理解图像变换对 AQA 考试至关重要。对于基本抛物线 y = x²，f(x) + a、f(x + a)、a f(x) 和 f(ax) 的效果都会被考查。组合变换必须按照正确的顺序进行：通常先进行水平平移和伸缩，再进行垂直平移和伸缩。

For a quadratic written in vertex form a(x − h)² + k, you can see that the graph of y = x² has been translated horizontally by h and vertically by k, and stretched vertically by a factor of a. If a is negative, it is also reflected in the x-axis.

对于写成顶点式 a(x − h)² + k 的二次函数，可以看出 y = x² 的图像沿水平方向平移了 h，沿竖直方向平移了 k，并在竖直方向上拉伸了 a 倍。若 a 为负数，图像还会关于 x 轴反射。

The order of transformations matters. For instance, starting from y = x², the transformation to y = 2(x − 3)² + 1 involves a shift right by 3, then a vertical stretch by factor 2, then a shift up by 1. Applying them in a different order may lead to an incorrect graph.

变换的顺序至关重要。例如，从 y = x² 出发，变换到 y = 2(x − 3)² + 1 涉及先向右平移 3 个单位，再垂直拉伸 2 倍，再向上平移 1 个单位。以不同顺序应用这些变换可能会导致错误的图像。

11. Intersection of a Line and a Parabola | 直线与抛物线的交点

Finding the intersection points between a line y = mx + c and a parabola y = ax² + bx + c (or other forms) leads to a quadratic equation in x. The discriminant of this equation tells you how many times the line cuts the curve: two intersections (D > 0), one if it is a tangent (D = 0), or none (D < 0).

求直线 y = mx + c 与抛物线 y = ax² + bx + c（或其他形式）的交点，会得到一个关于 x 的二次方程。该方程的判别式告诉我们直线与曲线相交的次数：两个交点（D > 0），相切时一个交点（D = 0），或无交点（D < 0）。

Questions often ask: “Prove that the line is a tangent to the curve” or “Find the value of k for which the line touches the parabola.” The approach is to set the equations equal, rearrange to a quadratic, and set the discriminant equal to zero.

题目常会问：“证明该直线是曲线的切线”或“求使直线与抛物线相切的 k 值”。解题方法是令两式相等，重组为二次方程，再令判别式等于零。

12. Common Exam Pitfalls | 常见考试陷阱

Many marks are lost through simple algebraic slips. When completing the square, ensure you account for the a coefficient correctly when taking it out as a factor. When using the quadratic formula, double-check the signs of b and the discriminant. And when solving inequalities, never divide by x unless you are sure of its sign, as this can reverse the inequality.

许多分数因简单的代数失误而丢失。配方法时，确保在将 a 作为因子提出时正确处理 a 系数。使用求根公式时，仔细检查 b 和判别式的符号。解不等式时，除非确定 x 的符号，否则切勿除以 x，因为这可能使不等号方向反转。

Another common error is forgetting to find both x- and y-coordinates of the vertex when sketching. The vertex is a coordinate pair, not just an x-value. AQA mark schemes frequently penalise incomplete answers.

另一个常见错误是在画图时忘记求顶点的 x 和 y 坐标。顶点是一个坐标对，不仅仅是 x 值。AQA 评分标准通常会扣掉不完整答案的分数。

Finally, always read the question carefully to see what form it requires. Some questions specify ‘in the form a(x + p)² + q’, where p may be a negative number. If the question states this, you must write it with a plus sign inside the bracket, even if that makes your p negative.

最后，务必仔细读题，明确题目要求的表达形式。有些问题会指定“写成 a(x + p)² + q 的形式”，其中 p 可为负数。如果题目要求如此，你必须将括号内的符号写成加号，即便这意味着你的 p 为负值。

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Quadratic Functions for AQA A-Level | A-Level AQA 数学：二次函数 考点精讲