GCSE WJEC Maths: Quadratic Functions – Key Points Revision | GCSE WJEC 数学：二次函数考点精讲

📚 GCSE WJEC Maths: Quadratic Functions – Key Points Revision | GCSE WJEC 数学：二次函数考点精讲

Quadratic functions form a core part of the WJEC GCSE Mathematics specification. This guide covers everything you need to know: from standard form and factorising to completing the square, using the quadratic formula, sketching graphs, and interpreting the discriminant. Each section pairs an English explanation with its Chinese counterpart so you can master the concepts and perform confidently in the exam.

二次函数是WJEC GCSE数学大纲的核心内容。本指南涵盖你需要掌握的所有要点：从标准形式和因式分解，到配方法、使用求根公式、绘制图像以及解读判别式。每个部分都配有英文解释和对应的中文讲解，帮助你牢固掌握概念并在考试中自信发挥。

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

A quadratic function is any function of the form f(x) = ax² + bx + c, where a, b and c are constants and a ≠ 0. The highest power of x is 2, which gives the name ‘quadratic’. In WJEC GCSE, you will work with quadratics that can be written in different but equivalent forms.

二次函数是形如 f(x) = ax² + bx + c 的函数，其中a、b和c为常数且 a ≠ 0。x的最高次幂为2，因此得名‘二次’。在WJEC GCSE考试中，你会遇到可以写成不同但等价形式的二次函数。

In standard form: ax² + bx + c, e.g. 2x² + 3x – 5.
标准形式：ax² + bx + c，例如 2x² + 3x – 5。
In factorised form: (px + q)(rx + s), e.g. (2x – 1)(x + 5).
因式分解形式：(px + q)(rx + s)，例如 (2x – 1)(x + 5)。
In completed square (vertex) form: a(x + p)² + q, e.g. (x – 3)² + 4.
配方法（顶点）形式：a(x + p)² + q，例如 (x – 3)² + 4。

The ability to switch between these forms is essential for solving equations and analysing graphs.

在这几种形式之间进行转换的能力，对于解方程和分析图像至关重要。

2. Solving Quadratics by Factorising | 用因式分解法解二次方程

Factorising is often the quickest method when the quadratic has nice integer roots. To solve ax² + bx + c = 0, find two numbers that multiply to give ac and add to give b. Then split the middle term and factorise by grouping.

当二次方程的根为整数时，因式分解法通常是最快的方法。要解 ax² + bx + c = 0，需要找出两个数，它们的乘积为 ac，和为 b。然后拆分中间项，通过分组进行因式分解。

Example: Solve x² + 7x + 12 = 0

示例：解方程 x² + 7x + 12 = 0

1. Look for two numbers that multiply to 12 and add to 7 → 3 and 4.

1. 寻找乘积为12、和为7的两个数 → 3 和 4。

2. Write as (x + 3)(x + 4) = 0.

2. 写成 (x + 3)(x + 4) = 0。

3. Set each bracket equal to zero: x + 3 = 0 → x = -3; x + 4 = 0 → x = -4.

3. 令每个括号等于零：x + 3 = 0 → x = -3；x + 4 = 0 → x = -4。

For quadratics where a ≠ 1, you may need to try out different combinations or use the grid method.

对于 a ≠ 1 的二次方程，你可能需要尝试不同的组合，或者使用网格法。

3. Solving Quadratics by Completing the Square | 用配方法解二次方程

Completing the square transforms ax² + bx + c into the form a(x + p)² + q. This method is especially useful when the quadratic does not factorise nicely, and it is essential for finding the turning point of a parabola.

配方法将 ax² + bx + c 转化为 a(x + p)² + q 的形式。当二次式不易因式分解时，该方法尤其有用，而且它对求抛物线的顶点至关重要。

Steps for x² + bx + c:

x² + bx + c 的步骤：

1. Write (x + b/2)² – (b/2)² + c.

1. 写出 (x + b/2)² – (b/2)² + c。

2. Simplify to get (x + p)² + q.

2. 化简得到 (x + p)² + q。

Example: x² + 6x + 1 → (x + 3)² – 9 + 1 = (x + 3)² – 8.

示例：x² + 6x + 1 → (x + 3)² – 9 + 1 = (x + 3)² – 8。

To solve an equation like x² + 6x + 1 = 0, write (x + 3)² – 8 = 0, then (x + 3)² = 8, so x + 3 = ±√8, giving x = -3 ± 2√2.

要解方程如 x² + 6x + 1 = 0，写成 (x + 3)² – 8 = 0，然后 (x + 3)² = 8，于是 x + 3 = ±√8，得到 x = -3 ± 2√2。

When a ≠ 1, factor out a first from the x² and x terms before completing the square.

当 a ≠ 1 时，在进行配方之前，先把前两项中的 a 提取出来。

4. The Quadratic Formula | 二次方程求根公式

The quadratic formula provides a direct way to find the roots of any quadratic equation ax² + bx + c = 0. You must memorise it and know when to use it.

二次方程求根公式提供了求解任何 ax² + bx + c = 0 的根的直接方法。你必须熟记公式，并知道何时使用它。

x = [-b ± √(b² – 4ac)] / 2a

Insert the values of a, b and c, then calculate carefully. The formula always works, even when the quadratic does not factorise.

代入 a、b 和 c 的值，然后仔细计算。该公式总是有效，即使二次式不能因式分解。

Example: 2x² – 3x – 2 = 0. Here a=2, b=-3, c=-2.

示例：2x² – 3x – 2 = 0。这里 a=2，b=-3，c=-2。

x = [3 ± √((-3)² – 4(2)(-2))] / (2×2) = [3 ± √(9 + 16)] / 4 = [3 ± √25] / 4 = [3 ± 5] / 4, so x = 2 or x = -½.

x = [3 ± √((-3)² – 4(2)(-2))] / (2×2) = [3 ± √(9 + 16)] / 4 = [3 ± √25] / 4 = [3 ± 5] / 4，所以 x = 2 或 x = -½。

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

The expression under the square root in the quadratic formula, b² – 4ac, is called the discriminant (Δ or D). It tells you how many real solutions the quadratic equation has without fully solving it.

求根公式中平方根号内的表达式 b² – 4ac 称为判别式（Δ 或 D）。它能在完全求解之前告诉你二次方程有多少个实数解。

Discriminant value (b² – 4ac)	判别式的值 (b² – 4ac)	Number of real roots	实数根个数
b² – 4ac > 0	b² – 4ac > 0	Two distinct real roots	两个不同的实数根
b² – 4ac = 0	b² – 4ac = 0	One repeated real root (equal roots)	一个重根（两个相等实数根）
b² – 4ac < 0	b² – 4ac < 0	No real roots	没有实数根

This knowledge is often tested in problems that ask you to find values of k for which a quadratic has equal roots, or to state whether a graph cuts, touches or does not intersect the x-axis.

这方面的知识常在题目中考察，例如要求你找出使二次函数有等根的 k 值，或者判断图像是穿过、相切还是与 x 轴无交点。

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

The graph of a quadratic function is called a parabola. Its shape depends on the sign of the coefficient a. For y = ax² + bx + c:

二次函数的图像称为抛物线。其形状取决于系数 a 的符号。对于 y = ax² + bx + c：

If a > 0, the parabola opens upwards (∪-shaped).
如果 a > 0，抛物线开口向上（呈 ∪ 形）。
If a < 0, the parabola opens downwards (∩-shaped).
如果 a < 0，抛物线开口向下（呈 ∩ 形）。

Key features to identify when sketching:

绘制草图时需要标出的关键特征：

The y-intercept is given by c (the constant term).
y 轴截距由常数项 c 给出。
The x-intercepts (roots) are found by solving ax² + bx + c = 0.
x 轴截距（根）通过求解 ax² + bx + c = 0 得到。
The turning point (vertex) can be found by completing the square or using x = -b/(2a).
顶点（拐点）可以通过配方法或公式 x = -b/(2a) 求出。

Always label your axes, intercepts and turning point on the sketch. The WJEC exam board often awards marks for correct shape and key coordinates.

在草图上一定要标出坐标轴、截距和顶点。WJEC考试局的评分通常看重正确的形状和关键坐标值。

7. Turning Point and Line of Symmetry | 顶点与对称轴

Every parabola has a turning point (maximum or minimum) and a vertical line of symmetry. In the completed square form y = a(x + p)² + q, the turning point is at (-p, q).

每条抛物线都有一个顶点（最大值或最小值）和一条垂直的对称轴。在配方式 y = a(x + p)² + q 中，顶点坐标为 (-p, q)。

If the quadratic is in standard form y = ax² + bx + c, the x-coordinate of the vertex is given by x = -b/(2a). Then substitute this value to find the y-coordinate.

如果二次函数为标准形式 y = ax² + bx + c，顶点的 x 坐标由 x = -b/(2a) 给出。然后代入该值求出 y 坐标。

The line of symmetry is the vertical line through the turning point, with equation x = -b/(2a) or x = -p.

对称轴是穿过顶点的竖直线，其方程为 x = -b/(2a) 或 x = -p。

Example: y = x² – 4x + 3. Complete the square → y = (x – 2)² – 1, so vertex (2, -1). Minimum value is -1. Line of symmetry x = 2.

示例：y = x² – 4x + 3。配方 → y = (x – 2)² – 1，因此顶点 (2, -1)。最小值为 -1。对称轴 x = 2。

8. Interpreting Quadratic Graphs in Context | 结合实际情境解读二次函数图像

WJEC often embeds quadratics in real-life contexts, such as projectile motion, area problems, or profit modelling. You may be asked to interpret intercepts, maximum/minimum points, or solve f(x) = k.

WJEC常将二次函数嵌入实际问题情境中，如抛体运动、面积问题或利润模型。你可能需要解读截距、最大值/最小值点，或求解 f(x) = k。

For example: “The height h metres of a ball after t seconds is given by h = -5t² + 20t + 1. Find the maximum height and when the ball hits the ground.”

例如：“小球在 t 秒后的高度 h 米由 h = -5t² + 20t + 1 给出。求最大高度以及小球何时落地。”

Here, the maximum height is the y-coordinate of the vertex. The ball hits the ground when h = 0, i.e. solve -5t² + 20t + 1 = 0. Always check that your answers make sense in the context (e.g. time can’t be negative).

这里，最大高度就是顶点的 y 坐标。小球落地时 h = 0，即解方程 -5t² + 20t + 1 = 0。务必检验你的答案在题目情境中是否合理（例如时间不能为负）。

9. Factorising Harder Quadratics (a ≠ 1) | 较复杂二次式的因式分解（a ≠ 1）

When the coefficient of x² is not 1, you need extra care. For ax² + bx + c, find two numbers that multiply to ac and add to b. Then rewrite the x-term and factorise by grouping.

当 x² 的系数不为 1 时，你需要额外小心。对于 ax² + bx + c，寻找乘积为 ac、和为 b 的两个数。然后改写 x 项，并通过分组进行因式分解。

Example: Factorise 2x² + 7x + 3. ac = 2 × 3 = 6. Two numbers that multiply to 6 and add to 7 are 6 and 1.

示例：因式分解 2x² + 7x + 3。ac = 2 × 3 = 6。乘积为6且和为7的两个数是6和1。

Rewrite as 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

改写为 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

Alternatively, use the systematic trial method where you list possible factor pairs. This topic is regularly assessed in both calculator and non-calculator WJEC papers.

或者，使用系统试配法，列出可能的因式组合。这个考点在WJEC允许使用计算器和禁止使用计算器的试卷中都会出现。

10. Solving Quadratic Inequalities | 解二次不等式

To solve inequalities like x² – 5x + 6 > 0, first solve the corresponding quadratic equation x² – 5x + 6 = 0 to find critical values: (x – 2)(x – 3) = 0 → x = 2, x = 3.

要解形如 x² – 5x + 6 > 0 的不等式，首先解对应的二次方程 x² – 5x + 6 = 0 找出临界值：(x – 2)(x – 3) = 0 → x = 2，x = 3。

These split the number line into three intervals: x < 2, 2 < x < 3, x > 3. Test a point from each interval in the inequality. For a positive a (∪-shaped), the graph is above the x-axis outside the roots, so x < 2 or x > 3.

这些点将数轴分为三个区间：x < 2，2 < x < 3，x > 3。从每个区间选取一个测试点代入不等式。对于 a 为正（∪ 形）的图像，在两根之外的部分位于 x 轴上方，所以解为 x < 2 或 x > 3。

Always sketch a quick graph of the quadratic to visualise where it is above or below the x-axis. This helps avoid sign errors.

务必快速画出二次函数的草图，直观判断图像在 x 轴上方或下方的区间。这有助于避免符号错误。

11. Common Mistakes and Exam Tips | 常见错误与应试技巧

Forgetting to include both the ± root when solving (x + p)² = q. For example, if (x – 3)² = 16, then x – 3 = ±4 (not just +4).
在解 (x + p)² = q 时忘记包含 ± 根。例如，若 (x – 3)² = 16，则 x – 3 = ±4（而不仅是 +4）。
Losing a negative sign when substituting into the quadratic formula, especially for b when b is negative. Use brackets: x = [-(-5) ± √…].
代入求根公式时遗漏负号，尤其当 b 为负数时。要使用括号：x = [-(-5) ± √…]。
Not checking answers in the original equation after solving, particularly with factorisation that might produce extraneous solutions.
解出答案后未代入原方程验算，特别是对于因式分解可能产生增根的情况。
Mis-identifying the y-intercept: it is always (0, c). For y = ax² + bx + c, when x=0, y=c.
错误识别 y 截距：它总是 (0, c)。对于 y = ax² + bx + c，当 x=0 时，y=c。
Confusing turning point coordinates with (p, q) when the form is a(x + p)² + q: the x-coordinate is -p, not p.
在 a(x + p)² + q 形式中混淆顶点坐标与 (p, q)：x 坐标是 -p，不是 p。

In WJEC exams, showing your method is crucial for earning part marks, even if your final answer is incorrect. Write down each step clearly.

在 WJEC 考试中，即使最终答案错误，清晰地展现解题过程对于获得部分分数至关重要。请把每一步清晰地写出来。

12. Summary and Key Points to Memorise | 总结与必须记忆的要点

Quadratic functions are everywhere in the WJEC GCSE syllabus. Make sure you can:

二次函数在 WJEC GCSE 大纲中无处不在。确保你能做到以下各项：

Recognise and generate the three forms: standard, factorised, completed square.
识别并写出三种形式：标准式、因式分解式、配方式。
Solve quadratics by factorising, completing the square, and the formula.
用因式分解法、配方法和求根公式解二次方程。
Use the discriminant to determine the nature and number of roots.
用判别式判断根的性质和个数。
Sketch a parabola accurately, labelling intercepts, turning point and line of symmetry.
准确绘制抛物线草图，标出截距、顶点和对称轴。
Apply quadratics to real-world problems and interpret solutions appropriately.
将二次函数应用于实际问题，并恰当地解读答案。

Regular practice with WJEC past papers will help you become fluent in these methods and avoid costly errors under exam pressure.

定期练习WJEC历年真题，将帮助你熟练运用这些方法，避免在考试压力下出现代价高昂的错误。

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GCSE WJEC Maths: Quadratic Functions – Key Points Revision | GCSE WJEC 数学：二次函数 考点精讲