A-Level OCR Mathematics: Quadratic Functions Exam Focus | A-Level OCR 数学：二次函数考点精讲

📚 A-Level OCR Mathematics: Quadratic Functions Exam Focus | A-Level OCR 数学：二次函数考点精讲

Quadratic functions lie at the heart of A-Level OCR Mathematics. From solving simple equations to analysing real-world projectile paths, mastering quadratics is essential for Paper 1 and beyond. This revision guide breaks down every key concept, formula, and exam technique you need, with paired explanations in both English and Chinese to strengthen your understanding and bilingual mathematical vocabulary.

二次函数是 A-Level OCR 数学的核心内容。从求解简单方程到分析真实的抛体轨迹，掌握二次函数对卷一和后续学习至关重要。这份考点精讲梳理了每一个关键概念、公式和应试技巧，提供中英对照讲解，帮助你加深理解并强化双语数学词汇。

1. Definition and Standard Form | 定义与标准形式

A quadratic function is a polynomial function of degree 2. In OCR exams, it is almost always written in standard form as y = ax² + bx + c, where a, b and c are real constants and a cannot be zero. If a were zero, the function would become linear, losing its quadratic nature.

二次函数是最高次为 2 的多项式函数。在 OCR 考试中，它通常写成标准形式 y = ax² + bx + c，其中 a、b 和 c 为实数常数，且 a 不能为零。若 a 为零，函数就退化成一次函数，不再是二次。

y = ax² + bx + c (a ≠ 0)

The coefficient a determines the direction and ‘width’ of the parabola: a > 0 gives a U-shaped (happy) curve, while a < 0 gives an ∩-shaped (sad) curve. The constant term c is the y-intercept. The linear coefficient b influences the position of the vertex and the axis of symmetry.

系数 a 决定了抛物线的开口方向和“宽窄”：a > 0 时开口向上（笑脸型），a < 0 时开口向下（哭脸型）。常数项 c 就是 y 轴截距。一次项系数 b 影响着顶点和对称轴的位置。

2. Completing the Square | 配方法

Completing the square rewrites a quadratic from standard form into vertex form: y = a(x − h)² + k. This technique is not only tested directly but is also used to find vertices, solve equations, and integrate rational functions later. The basic idea is to make a perfect square trinomial from the x² and x terms.

配方法能将二次函数从标准形式转化为顶点式：y = a(x − h)² + k。这一技巧不仅直接出现在考题中，还用于求顶点、解方程以及日后有理函数的积分。基本思路就是把 x² 项和 x 项配成完全平方式。

x² + bx = (x + b/2)² − (b/2)²

For example, to complete the square for x² + 6x + 5: write x² + 6x as (x + 3)² − 9, then add the constant: (x + 3)² − 9 + 5 = (x + 3)² − 4. If the coefficient a is not 1, factor it out of the x² and x terms first. Practice this until it becomes automatic — many OCR questions rely on it.

例如，对 x² + 6x + 5 进行配方：把 x² + 6x 写成 (x + 3)² − 9，再加上常数项，得到 (x + 3)² − 9 + 5 = (x + 3)² − 4。如果 a 不是 1，则需要先从 x² 和 x 项中提取 a。要反复练习直到熟练——很多 OCR 试题都依赖这种方法。

3. Discriminant and Nature of Roots | 判别式与根的性质

The discriminant, denoted by the Greek letter delta Δ or simply D, is the expression under the square root in the quadratic formula: Δ = b² − 4ac. It tells us the number and type of real roots without solving the equation.

判别式用希腊字母 Δ 或 D 表示，是求根公式中根号下的表达式：Δ = b² − 4ac。它可以在不解方程的情况下告诉我们实根的个数和类型。

Δ = b² − 4ac

If Δ > 0, the quadratic has two distinct real roots. The graph cuts the x-axis at two points.

若 Δ > 0，二次方程有两个不同的实根。图像与 x 轴交于两点。
If Δ = 0, the quadratic has exactly one real root (a repeated root). The graph touches the x-axis at the vertex.

若 Δ = 0，二次方程有一个实根（重根）。图像在顶点处与 x 轴相切。
If Δ < 0, the quadratic has no real roots. The graph does not intersect the x-axis.

若 Δ < 0，二次方程没有实根。图像与 x 轴不相交。

OCR frequently asks you to find the set of values of k for which a quadratic equation has two distinct real roots, or no real roots. Set up the discriminant inequality accordingly and solve it carefully.

OCR 常考求使得二次方程有两个不同实根或无实根的 k 的取值范围。需要根据题意列出判别式的不等式，并仔细求解。

4. Quadratic Formula and Solving Equations | 求根公式与解方程

The quadratic formula provides the roots of any quadratic equation ax² + bx + c = 0 directly. It is derived by completing the square and is given on the formula sheet, but you must memorise it and know how to apply it confidently.

求根公式可以直接给出任何二次方程 ax² + bx + c = 0 的根。它由配方法推导而得，公式表上会提供，但你仍需牢记并熟练运用。

x = (−b ± √(b² − 4ac)) / (2a)

Always get the equation into the form ax² + bx + c = 0 before substituting into the formula. Simplify surds where possible; OCR expects simplified exact answers unless a decimal approximation is specifically requested. When the discriminant is a perfect square, the quadratic will factorise nicely — but using the formula is still valid.

代入公式前，务必先将方程化为 ax² + bx + c = 0 的形式。根式要尽量化简；OCR 要求给出化简后的精确值，除非题目明确要求小数近似。当判别式为完全平方数时，二次式可以顺利因式分解——但使用公式同样正确。

5. Factorisation Techniques | 因式分解技巧

Factorising a quadratic into (px + q)(rx + s) is often the quickest way to find roots, especially when coefficients are small. For monic quadratics (a = 1), find two numbers that multiply to c and add to b. For non-monic quadratics, use the ‘ac method’ or systematic trial and error.

将二次式分解成 (px + q)(rx + s) 往往是求根最快的方法，尤其是系数较小时。对于首项系数为 1 的二次式，需要找到两个数，乘积为 c，和为 b。对于系数不为 1 的二次式，可以使用“ac 法”或系统地尝试。

Example: 2x² + 7x + 3. Multiply a and c: 2 × 3 = 6. Find two numbers that multiply to 6 and add to 7 — that is 6 and 1. Split the middle term: 2x² + 6x + x + 3, then factor by grouping: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). Practice recognising common patterns like the difference of two squares: x² − a² = (x − a)(x + a).

例如：2x² + 7x + 3。把 a 和 c 相乘：2 × 3 = 6。找到两个数，乘积为 6，和为 7——即 6 和 1。将中间项拆开：2x² + 6x + x + 3，再分组分解：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。注意识别平方差等常见模式：x² − a² = (x − a)(x + a)。

6. Graphing Quadratic Functions | 二次函数图像

The graph of a quadratic function is a parabola. In OCR exam questions, you will often need to sketch graphs, identifying key features: shape (U or ∩), y-intercept (0, c), and the coordinates of the vertex. If the quadratic is given in completed square form y = a(x − h)² + k, the vertex is simply (h, k).

二次函数的图像是一条抛物线。OCR 试题常要求你画出草图，并识别关键特征：形状（开口向上或向下）、y 轴截距 (0, c) 以及顶点坐标。如果给出的是配方后的形式 y = a(x − h)² + k，顶点就是 (h, k)。

If you have the factorised form y = (px + q)(rx + s), you can immediately read off the x-intercepts by setting each factor to zero. The axis of symmetry passes through the midpoint of the roots. The y-intercept is found by setting x = 0. Always label your axes and mark intercepts clearly.

如果给出的是因式分解形式 y = (px + q)(rx + s)，令每个因式为零就能直接读出 x 轴截距。对称轴穿过两根的中点。y 轴截距则在 x = 0 时求得。画图时务必标注坐标轴，并清楚标出截距。

7. Vertex and Axis of Symmetry | 顶点与对称轴

The vertex is the turning point of the parabola. In OCR A-Level, you must be able to find the vertex using either completing the square or the formula x = −b / (2a) for the axis of symmetry, then substitute to find y. The completed square form y = a(x − h)² + k instantly gives vertex (h, k).

顶点是抛物线的转折点。在 OCR A-Level 中，你必须能通过配方法或公式 x = −b / (2a) 求出对称轴，再代入求顶点 y 坐标。配方式 y = a(x − h)² + k 能直接给出顶点 (h, k)。

x = −b / (2a)

For example, given y = 2x² − 8x + 3, the axis of symmetry is x = −(−8) / (2×2) = 8/4 = 2. Substitute x = 2 into the equation: y = 2(4) − 16 + 3 = −5. Vertex is (2, −5). Knowing the vertex helps you find the range of the function and solve optimisation problems.

例如，已知 y = 2x² − 8x + 3，对称轴为 x = −(−8) / (2×2) = 8/4 = 2。将 x = 2 代回：y = 2(4) − 16 + 3 = −5，顶点为 (2, −5)。掌握顶点有助于求函数值域以及解决最优化问题。

8. Quadratic Inequalities | 二次不等式

Solving quadratic inequalities such as x² − 5x + 6 > 0 requires a clear method. First, solve the corresponding quadratic equation to find critical values where the expression equals zero. Then sketch a quick parabola (or use a sign table) to determine the intervals where the inequality is satisfied.

求解形如 x² − 5x + 6 > 0 的二次不等式需要清晰的方法。首先，解对应的二次方程，找出使得表达式为零的临界值。然后快速勾画抛物线（或使用符号表），确定不等式成立的区间。

Example: x² − 5x + 6 > 0 → (x − 2)(x − 3) = 0 gives critical values x = 2, 3. Since a > 0, the parabola opens upwards and is above the x-axis outside the roots. The solution is x < 2 or x > 3, often written in set notation {x: x < 2} ∪ {x: x > 3}. Be meticulous with open and closed intervals depending on whether the inequality is strict.

例题：x² − 5x + 6 > 0 → (x − 2)(x − 3) = 0，得临界值 x = 2, 3。由于 a > 0，抛物线开口向上，在两根之外函数值大于零。解为 x < 2 或 x > 3，常写作 {x: x < 2} ∪ {x: x > 3}。需要根据不等号是否带等号，仔细区分开区间和闭区间。

9. Intersections with Lines | 与直线的交点

Finding the intersection of a quadratic curve and a straight line involves solving simultaneous equations. Substitute the linear equation (often y = mx + c) into the quadratic and set up a quadratic equation in x. The discriminant of this new quadratic tells you how many intersection points exist.

求二次曲线与直线的交点涉及解联立方程组。将直线方程（通常为 y = mx + c）代入二次函数，得到一个关于 x 的二次方程。这个新二次式的判别式能告诉我们交点的个数。

If Δ > 0: line cuts the curve at two distinct points.

若 Δ > 0，直线与曲线交于两个不同的点。
If Δ = 0: line is tangent to the curve (touches at exactly one point).

若 Δ = 0，直线与曲线相切（恰好接触于一点）。
If Δ < 0: line does not meet the curve.

若 Δ < 0，直线与曲线无交点。

OCR often asks you to find the value of k for which a line is tangent to a given quadratic, or to show that a line and curve do not intersect. Set up the quadratic in x, apply the discriminant condition Δ = 0 for tangency, and solve for k.

OCR 常考求使直线与给定二次曲线相切的 k 值，或证明直线与曲线不相交。建立一个关于 x 的二次方程，利用相切条件 Δ = 0，再解出 k。

10. Modelling with Quadratics | 二次函数建模

Quadratic models appear regularly in OCR examination contexts, such as projectile motion, area optimisation, profit and cost analysis, or bridge arch shapes. You might be given a scenario where a quantity reaches a maximum or minimum value. The vertex of the parabola then represents the optimal solution.

二次模型经常出现在 OCR 考题中，例如抛体运动、面积优化、利润与成本分析，或桥拱形状。你可能遇到一个量达到最大值或最小值的情景，此时抛物线的顶点就代表了最优解。

Always define your variables clearly. Express the quantity to be optimised as a quadratic function of one variable. Use completing the square or the vertex formula to find the maximum or minimum value. Finally, interpret your mathematical answer in the context of the problem and ensure it makes practical sense (e.g., dimensions cannot be negative).

务必清晰地定义变量。将要优化的量表示为单个变量的二次函数。利用配方法或顶点公式求出最大值或最小值。最后，结合题目情境解释所得数学答案，并确保其现实合理性（例如尺寸不能为负）。

11. Hidden Quadratics and Substitution | 隐藏二次型与换元法

OCR papers often include equations that are not immediately quadratic but can be transformed into one through substitution. Common examples include x⁴ − 5x² + 4 = 0 (let t = x²) or exponential types like e²ˣ − 3eˣ + 2 = 0 (let u = eˣ). These are sometimes called ‘disguised quadratics’.

OCR 试题常出现看似不是二次的方程，但通过换元可以化为二次。常见例子有：x⁴ − 5x² + 4 = 0（令 t = x²），或指数型如 e²ˣ − 3eˣ + 2 = 0（令 u = eˣ）。这有时被称为“隐藏的二次型”。

After substitution, solve the resulting quadratic in the new variable, then substitute back to find the original variable. Be careful to check the domain of the substitution — for instance, if t = x², then t ≥ 0, so reject any negative t solutions. This checking step is essential to avoid losing marks.

换元后，解出新变量的二次方程，再回代求出原变量。记得检查换元后的定义域——例如，若 t = x²，则 t ≥ 0，所以要舍去任何负的 t 值。这个检验步骤对避免失分至关重要。

12. Exam Tips and Common Mistakes | 应试技巧与常见错误

To excel in quadratic questions on OCR A-Level Maths, adopt a strategic approach. Always write down the standard form and identify a, b, c before using the discriminant or quadratic formula. Double-check arithmetic: sign errors in b² or the root expression are extremely common.

想在 OCR A-Level 数学的二次函数题上取得高分，应策略性地解题。在使用判别式或求根公式前，先写下标准形式并识别 a、b、c。要仔细检查算术：b² 或根号内的符号错误极为常见。

When factorising, expand your brackets mentally to verify. In inequalities, never blindly divide by a variable without considering its sign. For modelling questions, remember to state your final answer with correct units. And most importantly, practice past paper questions under timed conditions — many quadratic topics are combined with coordinate geometry or calculus in Paper 1.

因式分解后，在心中展开括号进行验证。解不等式时，绝不在不考虑符号的情况下盲目除以变量。建模题要记得附带正确的单位写出最终答案。最重要的是，在限时条件下练习历年真题——许多二次函数考点在卷一常与解析几何或微积分结合考查。

Common pitfalls: forgetting that a repeated root means the discriminant equals zero, misreading < or ≤, and failing to simplify surds. Also, many students lose marks by not showing clear steps when completing the square for a ≠ 1. Keep your working logical, and when a question says 'hence', use your previous result — it is there for a reason.

常见误区：忘记重根意味着判别式等于零；误读 < 或 ≤；未能化简根式。此外，许多学生在 a ≠ 1 的配方法中没有清晰展示步骤而失分。保持解题过程逻辑清晰，当题目出现“hence”字眼时，要利用前面的结果——它一定有作用。

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A-Level OCR Mathematics: Quadratic Functions Exam Focus | A-Level OCR 数学：二次函数考点精讲