Quadratic Functions for IGCSE WJEC Mathematics | IGCSE WJEC 数学：二次函数考点精讲

📚 Quadratic Functions for IGCSE WJEC Mathematics | IGCSE WJEC 数学：二次函数考点精讲

Quadratic functions are a cornerstone of the IGCSE WJEC Mathematics syllabus. Whether you are tackling equations, sketching graphs, or applying them to real‑world problems, a solid grasp of quadratics is essential for top marks. This article walks you through every key learning point, from standard form to the discriminant, with clear explanations and WJEC‑style tips.

二次函数是 IGCSE WJEC 数学课程的核心内容。无论是解方程、绘制图像，还是将二次函数应用到实际问题中，牢固掌握相关知识对于取得高分至关重要。本文将带你逐一梳理每个重要考点，从标准形式到判别式，并提供清晰的解释和符合 WJEC 风格的考试技巧。

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 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 function its characteristic U‑shaped curve called a parabola. On the WJEC papers, you may see it expressed as y = ax² + bx + c or using function notation.

二次函数是任何可以写成 f(x) = ax² + bx + c 形式的函数，其中 a、b、c 为常数且 a ≠ 0。x 的最高次数为 2，这使得函数图像呈现特有的 U 形曲线，称为抛物线。在 WJEC 试卷中，你可能会看到它被表示为 y = ax² + bx + c 或使用函数记号。

2. Standard Form and Its Features | 标准形式及其特征

In the standard form y = ax² + bx + c, the coefficient a controls the direction and width of the parabola. If a > 0, the parabola opens upwards (a ‘happy’ curve); if a < 0, it opens downwards (a 'sad' curve). The constant c gives the y‑intercept, because when x = 0, y = c. The x‑intercepts (roots) are found by solving ax² + bx + c = 0.

在标准形式 y = ax² + bx + c 中，系数 a 控制抛物线的开口方向和宽度。若 a > 0，抛物线开口向上（“微笑”曲线）；若 a < 0，则开口向下（“沮丧”曲线）。常数 c 表示 y 轴截距，因为当 x = 0 时，y = c。x 轴截距（根）通过求解 ax² + bx + c = 0 得到。

WJEC often asks candidates to identify these features directly from an equation, so practice reading off the y‑intercept and the sign of a at a glance.

WJEC 常要求考生直接从方程中识别这些特征，因此请练习一眼看出 y 轴截距和 a 的符号。

3. Graphing Quadratic Functions: The Parabola | 二次函数图像：抛物线

When sketching a quadratic graph, WJEC examiners expect you to label the turning point, the axis of symmetry, and the intercepts. The turning point is the maximum or minimum of the parabola. The axis of symmetry is a vertical line passing through the turning point, given by x = –b/(2a). To find the coordinates of the turning point, substitute this x‑value back into the function.

在绘制二次函数图像时，WJEC 考官期望你标出转折点、对称轴和截距。转折点是抛物线的最大值或最小值。对称轴是一条穿过转折点的垂直线，其方程为 x = –b/(2a)。要求转折点坐标，只需将该 x 值代回原函数即可。

You can also construct a table of values, a favourite task on Paper 1 without a calculator. Always plot points smoothly – a parabola has no sharp corners.

你还可以通过列表取值来作图，这在不使用计算器的 Paper 1 中是常见题型。描点时要光滑——抛物线没有尖角。

4. Vertex Form and Completing the Square | 顶点形式与配方法

Completing the square rewrites ax² + bx + c into the vertex form a(x + p)² + q, where (–p, q) is the turning point. The steps are: factor a from the first two terms, then add and subtract (b/(2a))² inside the bracket. For example, x² + 6x + 5 becomes (x + 3)² – 4.

配方法将 ax² + bx + c 改写为顶点形式 a(x + p)² + q，其中 (–p, q) 即为转折点。步骤是：从前两项中提取 a，然后在括号内加上并减去 (b/(2a))²。例如，x² + 6x + 5 可写成 (x + 3)² – 4。

WJEC frequently uses completing the square to ask for the minimum or maximum value of a quadratic, or to solve equations when factorisation is awkward. Remember: if a is negative, the turning point is a maximum.

WJEC 经常用配方法来求二次函数的最小值或最大值，或在因式分解困难时求解方程。记住：若 a 为负，则转折点为最大值。

5. Finding Roots by Factorising | 通过因式分解求根

Factorising a quadratic into two linear brackets (mx + n)(px + q) = 0 gives the solutions x = –n/m and x = –q/p. The method works when the quadratic can be expressed as a product of two binomials with integer coefficients. Always check your factorisation by expanding back.

将二次式因式分解为两个一次因式 (mx + n)(px + q) = 0，即可得到解 x = –n/m 和 x = –q/p。当二次式可以表示为两个整系数二项式的乘积时，这种方法行之有效。务必通过展开来验算因式分解。

In WJEC exams, factorising is often the fastest way to solve a quadratic, especially on non‑calculator papers. Look out for common factors first – they can simplify the process.

在 WJEC 考试中，因式分解通常是解二次方程最快的方法，尤其是在没有计算器的试卷中。注意先寻找公因式——它可以简化过程。

6. The Quadratic Formula | 二次公式

When factorising is not possible or too time‑consuming, the quadratic formula provides a universal method:

x = [ –b ± √(b² – 4ac) ] / (2a)

It comes directly from completing the square on the general quadratic. Make sure you set the equation equal to zero before identifying a, b and c. Plug the values carefully, particularly watching signs.

当无法因式分解或因式分解过于耗时时，二次公式提供了通用的求解方法：

x = [ –b ± √(b² – 4ac) ] / (2a)

它直接源于对一般二次式进行配方。在确定 a、b、c 之前，务必先将方程设为零。代入数值时要仔细，尤其要注意符号。

WJEC will often require you to give answers in surd form or to a specified number of decimal places. Leave your answer in exact form unless told otherwise.

WJEC 通常会要求答案以根式形式或指定的小数位数给出。除非另有说明，否则保留准确值。

7. The Discriminant | 判别式

The discriminant D = b² – 4ac reveals the 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 (the parabola does not cross the x‑axis).

判别式 D = b² – 4ac 在不求解方程的情况下就能揭示根的性质。若 D > 0，有两个不等实根；若 D = 0，有一个实根（重根）；若 D < 0，则无实根（抛物线不与 x 轴相交）。

WJEC often asks: ‘State the number of real solutions’ or ‘Find the values of k for which the equation has equal roots’. These rely entirely on interpreting the discriminant correctly.

WJEC 经常提问：“指出实数解的个数”或“求使方程有等根的 k 值”。这些问题完全依赖于正确解读判别式。

8. Solving Quadratic Inequalities | 解二次不等式

To solve an inequality like x² – 4x – 5 < 0, first solve the corresponding equation to find critical values (here x = –1 and x = 5). Then sketch a quick parabola or use a sign table to identify the intervals where the quadratic is below or above the x‑axis. The solution is –1 < x < 5.

要解不等式如 x² – 4x – 5 < 0，首先解对应的方程以找到临界值（这里为 x = –1 和 x = 5）。然后快速绘制抛物线草图或使用符号表，确定二次函数位于 x 轴下方或上方的区间。解为 –1 < x < 5。

Be careful with strict versus non‑strict inequalities and when multiplying or dividing by a negative a. WJEC often includes a quadratic inequality as part of a larger problem, such as finding the domain for which a shape’s area is below a certain value.

注意严格不等号与非严格不等号的区别，以及乘除以负数 a 的情况。WJEC 常将二次不等式作为较大问题的一部分，例如求使某个图形面积小于某值的定义域。

9. Applications and Word Problems | 应用题与实际问题

Quadratic functions appear in projectile motion, area problems, and economics. A typical WJEC question might describe a ball thrown upwards and ask for the maximum height (turning point) or the time it hits the ground (roots). Translate the words into an equation, identify the unknown, and decide which part of the quadratic you need.

二次函数出现在抛体运动、面积问题以及经济学中。一道典型的 WJEC 题目可能描述一个向上抛出的球，要求最大高度（转折点）或落地时间（根）。将文字转化为方程，确定未知量，并判断你需要使用二次函数的哪部分。

Always re‑read the question to check that your answers make sense in context. Negative times or heights, for instance, should usually be discarded.

始终重新审题，确保答案在情境中合理。例如，负的时间或高度通常应舍去。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Don’t forget a ≠ 0: if a = 0 the function becomes linear.
Sign errors: when using the formula, bracket negative numbers. –b means the opposite sign of b.
Completing the square: after writing (x + p)², don’t forget to subtract p² or adjust for the coefficient a.
Graph sketching: always label the axes, turning point, and intercepts. If the question gives a domain, only plot within it.
Discriminant: remember it’s b² – 4ac, not the square root of it. The roots formula uses the square root of the discriminant.
Units: in word problems, include units in your final answer.

不要忘记 a ≠ 0：若 a = 0，函数变为一次函数。
符号错误：使用公式时，给负数加上括号。–b 表示与 b 相反的符号。
配方法：在写出 (x + p)² 后，不要忘记减去 p² 或根据系数 a 进行调整。
绘制图像：始终标注坐标轴、转折点和截距。如果题目给出了定义域，只在该范围内作图。
判别式：记住是 b² – 4ac，而不是它的平方根。求根公式中使用的是判别式的平方根。
单位：在应用题中，最终答案要包含单位。

Practice with past WJEC papers; the style is consistent and you will learn to spot the common twists. Take your time with setting out working clearly – method marks are generous.

多练习 WJEC 历年真题；题型风格一致，你很快就能学会识别常见的出题套路。花时间把解题步骤写清楚——过程分给得很大方。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Quadratic Functions for IGCSE WJEC Mathematics | IGCSE WJEC 数学：二次函数 考点精讲