Quadratic Functions Exam Focus | GCSE OCR 数学：二次函数考点精讲

📚 Quadratic Functions Exam Focus | GCSE OCR 数学：二次函数考点精讲

A quadratic function is one of the most important topics in the OCR GCSE Mathematics syllabus. Mastering its forms, graphs, and solution methods will give you confidence across algebra and problem-solving questions, where it frequently appears in both calculator and non-calculator papers.

二次函数是 OCR GCSE 数学大纲中最重要的主题之一。掌握它的各种形式、图像和求解方法，能让你在代数和应用问题中充满信心，而它常出现在计算器与非计算器试卷中。

1. What is a Quadratic Function? | 什么是二次函数？

A quadratic function is any function that can be written in the general form f(x) = ax² + bx + c, where a, b and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, which either opens upwards (when a > 0) or downwards (when a < 0).

二次函数是可以写成一般形式 f(x) = ax² + bx + c 的任何函数，其中 a、b 和 c 是常数且 a ≠ 0。二次函数的图像是一条抛物线，当 a > 0 时开口向上，当 a < 0 时开口向下。

In OCR exams, you will be expected to recognise quadratics in standard form, factorised form, and completed square form. Each form reveals different key features of the parabola.

在 OCR 考试中，你需要识别二次函数的标准形式、因式分解形式和配方法形式。每种形式都能揭示抛物线的不同关键特征。

2. Standard Form and Coefficients | 标准形式与系数

The standard form is y = ax² + bx + c. The coefficient a controls the width and direction of the parabola; b affects the position of the vertex and axis of symmetry; c is the y-intercept, the point where the curve crosses the y-axis.

标准形式为 y = ax² + bx + c。系数 a 控制抛物线的宽度和开口方向；b 影响顶点和对称轴的位置；c 是 y 轴截距，即曲线与 y 轴的交点。

Always check the sign of a first: positive a gives a ∪-shaped minimum turning point, negative a gives an ∩-shaped maximum. OCR questions often ask you to identify which of several equations matches a given sketch.

一定要首先检查 a 的符号：a 为正时，图像呈 ∪ 形且存在最小值转点；a 为负时，图像呈 ∩ 形且存在最大值。OCR 题目常会让你根据草图匹配对应的方程。

3. Factorising Quadratics | 因式分解二次式

Factorising a quadratic is the process of writing ax² + bx + c as a product of two linear brackets. For simple quadratics where a = 1, look for two numbers that multiply to give c and add to give b. For example, x² + 7x + 10 = (x + 2)(x + 5).

因式分解二次式是将 ax² + bx + c 写成两个一次括号的乘积的过程。对于 a = 1 的简单二次式，需寻找两个数，它们的乘积等于 c，和等于 b。例如，x² + 7x + 10 = (x + 2)(x + 5)。

When a ≠ 1, you must consider the product of a and c. Split the middle term accordingly and factor by grouping. Always expand your brackets to check your answer.

当 a ≠ 1 时，必须考虑 a 与 c 的积，据此拆分中间项，然后通过分组进行分解。务必展开括号来检验答案。

Factorising is often the quickest way to solve a quadratic equation. Once written as a product equal to zero, apply the null factor law: if A × B = 0, then A = 0 or B = 0.

因式分解通常是解二次方程最快的方法。一旦写成乘积等于零的形式，就应用零因子法则：若 A × B = 0，则 A = 0 或 B = 0。

4. Completing the Square | 配方法

Completing the square allows you to rewrite y = ax² + bx + c in the form y = a(x + p)² + q. Start by halving the coefficient of x when a = 1, then adjust the constant term. For instance, x² + 6x + 1 = (x + 3)² – 8.

配方法能将 y = ax² + bx + c 重新写成 y = a(x + p)² + q 的形式。先在 a = 1 时将 x 项系数折半，再调整常数项。例如，x² + 6x + 1 = (x + 3)² – 8。

When a ≠ 1, first factor a out of the x² and x terms, perform the process on the bracket, then multiply out the constant correction. This form immediately gives the coordinates of the turning point (–p, q).

如果 a ≠ 1，先提取 a 从 x² 和 x 项，括号内完成配方，再乘出常数修正项。这种形式可直接给出转点坐标 (–p, q)。

OCR frequently asks for the turning point and the line of symmetry using this method. The resulting form also makes it easier to solve equations when factorisation is not obvious.

OCR 常要求用这种方法求转点和对称轴。所得的形式在因式分解不明显时，也能使方程求解变得更容易。

5. The Quadratic Formula | 二次公式

The quadratic formula provides the solutions to ax² + bx + c = 0 and is given in the OCR formula sheet:

二次公式给出了 ax² + bx + c = 0 的解，并出现在 OCR 公式表中：

x = ⁻b ± √(b² – 4ac) / 2a

Make sure you can substitute negative values correctly, especially when b is negative. Use brackets around negative numbers when squaring to avoid sign errors.

确保你能正确代入负值，尤其是 b 为负时。对负数平方时请使用括号，以避免符号错误。

The formula works for any quadratic, even those that do not factorise. Always simplify the surd where possible, and leave your answer in exact form unless told otherwise.

该公式适用于任何二次方程，即使无法因式分解的也可以。尽可能化简根式，除非另有说明，否则答案保留准确形式。

6. The Discriminant | 判别式

The discriminant D = b² – 4ac tells you about the number and nature of the roots without solving the equation. If D > 0, there are two distinct real roots; if D = 0, there is exactly one real root (a repeated root); if D < 0, there are no real roots.

判别式 D = b² – 4ac 能在不求解方程的情况下，告诉你根的数量和性质。若 D > 0，有两个不等实根；若 D = 0，恰好有一个实根（重根）；若 D < 0，没有实根。

OCR exam questions often ask: “Find the value of k for which the equation has equal roots.” Set D = 0 and solve for k. Understanding the discriminant also helps when interpreting graphs and completing the square.

OCR 试题常会问：“求出使方程具有相等根的 k 值。”此时令 D = 0 并解出 k。理解判别式也有助于解读图像和配方法。

7. Graphs of Quadratics – Parabolas | 二次函数图像 – 抛物线

The graph of y = ax² + bx + c is a smooth, symmetrical curve called a parabola. Its most important features are the y-intercept (0, c), the x-intercepts (roots), and the turning point (vertex).

y = ax² + bx + c 的图像是一条光滑、对称的曲线，称为抛物线。它最重要的特征是 y 轴截距 (0, c)、x 轴截距（根）以及转点（顶点）。

The axis of symmetry is the vertical line that passes through the turning point. For a quadratic in the form y = a(x + p)² + q, the axis of symmetry is x = –p.

对称轴是通过转点的垂直线。对于形如 y = a(x + p)² + q 的二次函数，对称轴为 x = –p。

In OCR, you may need to plot graphs accurately by creating a table of values, or sketch them by identifying key points only. Always label intercepts and the turning point on sketches.

在 OCR 中，你可能需要通过数值表准确描点画图，或仅通过确定关键点来画草图。务必在草图上标注截距和转点。

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

The vertex is the maximum or minimum point of the parabola. Using completing the square, y = a(x + p)² + q, the vertex is at (–p, q). If the quadratic is in standard form, you can also find the vertex by first calculating x = –b/(2a), then substituting to find y.

顶点是抛物线的最大值或最小值点。利用配方法得到的 y = a(x + p)² + q，顶点坐标为 (–p, q)。若二次函数为标准形式，也可以先计算 x = –b/(2a)，再代入求 y。

The axis of symmetry is the line x = –b/(2a). This appears in many symmetry-based problems, such as finding the other x-intercept when one is given.

对称轴是直线 x = –b/(2a)。这出现在许多基于对称性的问题中，例如已知一个 x 轴截距时求出另一个。

9. Finding Roots (x-intercepts) | 求根（x 轴截距）

Roots are the x-values where y = 0, i.e. the solutions of ax² + bx + c = 0. They can be found by factorising, using the quadratic formula, or completing the square. The roots represent the intersections of the graph with the x-axis.

根是 y = 0 时的 x 值，即 ax² + bx + c = 0 的解。可以通过因式分解、二次公式或配方法求得。根表示图像与 x 轴的交点。

If the discriminant is negative, the parabola does not cross the x‑axis, so there are no real roots. When D = 0, the parabola touches the x‑axis at exactly one point, giving a repeated root.

如果判别式为负，抛物线不与 x 轴相交，因此没有实根。当 D = 0 时，抛物线在恰好一点接触 x 轴，产生重根。

In many OCR questions, you will be asked to find the roots and then use them to write the quadratic in factorised form, or the other way round.

在许多 OCR 题目中，会要求你求出根，然后用它们写出二次式的因式分解形式，反之亦然。

10. Sketching Quadratic Graphs | 绘制二次函数草图

To sketch a quadratic, determine: whether the parabola opens upwards or downwards; the y-intercept; the roots (if any); and the coordinates of the vertex. Plot these points and draw a smooth, symmetrical curve.

要画二次函数的草图，需确定：抛物线开口向上还是向下；y 轴截距；根（如果有）；以及顶点坐标。标出这些点，然后画一条光滑对称的曲线。

Even if you do not have exact roots, you can still sketch the shape and position using the vertex and y-intercept. OCR often awards marks for the general shape and key points labelled.

即使没有精确的根，你仍然可以用顶点和 y 轴截距画出形状和位置。OCR 常会对整体形状和标注的关键点给分。

Watch out for the scale: a narrow parabola has |a| > 1, a wide one has |a| < 1. Use symmetry to find additional points quickly.

注意比例：狭窄的抛物线 |a| > 1，宽缓的抛物线 |a| < 1。利用对称性可以快速找到其他点。

11. Solving Quadratic Equations – Mixed Methods | 解二次方程 – 综合方法

OCR expects you to choose the most efficient method for each equation. If the quadratic factorises easily, factorising is fastest. When coefficients are large or awkward, the formula is reliable. Completing the square works well when you need the turning point.

OCR 期望你为每个方程选择最有效的方法。如果可以轻松因式分解，分解法最快。当系数较大或棘手时，二次公式很可靠。当你需要转点时，配方法很适用。

Always set the equation to zero first. Remember that some equations may first need rearranging or expanding. Practice deciding the method based on the form of the quadratic.

务必先将方程化为等于零。记住，有些方程可能需要先移项或展开。根据二次式的形式，练习决定使用哪种方法。

In exam contexts, you might need to solve geometrically, such as finding the intersection of a quadratic and a straight line. Set the expressions equal and solve the resulting quadratic.

在考试情境中，你可能需要通过几何方式求解，例如求二次函数与直线的交点。令两者的表达式相等，并解所得的二次方程。

12. Applications and Word Problems | 应用与文字题

Quadratic functions model projectile motion, area problems, and optimisation tasks. OCR frequently presents a scenario and asks for the maximum height of a thrown object or the dimensions giving maximum area.

二次函数可以模拟抛体运动、面积问题和优化任务。OCR 经常给出一个情景，并问抛出物体的最大高度或给出最大面积的尺寸。

Identify the quadratic expression, decide what variable represents, and form an equation. The maximum or minimum value corresponds to the turning point. Translate the context into inequalities if required, such as time > 0 in flight problems.

识别二次表达式，确定变量所代表的含义，并建立方程。最大值或最小值对应转点。如有需要，将上下文转化为不等式，例如飞行问题中时间 > 0。

When solving word problems, always check whether your solution makes sense in the real‑world situation. Negative lengths or times, for instance, should normally be rejected.

在解文字题时，始终检查你的解在现实情境中是否有意义。例如，负的长度或时间通常应舍去。

Regular practice with contextual problems will improve your confidence, as the underlying mathematics is the same but your interpretation of the question becomes crucial.

定期练习情境题会增强你的信心，因为底层的数学是相同的，但对题目的理解变得尤为关键。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Quadratic Functions Exam Focus | GCSE OCR 数学：二次函数 考点精讲