IGCSE CCEA Maths: Quadratic Functions Key Points | IGCSE CCEA 数学：二次函数考点精讲

📚 IGCSE CCEA Maths: Quadratic Functions Key Points | IGCSE CCEA 数学：二次函数考点精讲

A quadratic function is one of the most fundamental concepts in the IGCSE CCEA Mathematics syllabus. Understanding its graph, algebraic manipulation, and real-world applications is essential for success in both the Core and Extended tiers. This article breaks down every key topic, from the standard form f(x) = ax² + bx + c to solving inequalities, with clear bilingual explanations and exam-ready tips.

二次函数是 IGCSE CCEA 数学大纲中最基础的概念之一。理解它的图像、代数变形以及实际应用，对于在核心与拓展层级考试中取得成功至关重要。本文将 f(x) = ax² + bx + c 的标准形式到解二次不等式等每一个关键主题进行拆解，提供清晰的双语讲解与应试技巧。

1. Introduction to Quadratic Functions | 二次函数简介

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 the variable x is 2, which gives the function its characteristic ‘U-shaped’ graph called a parabola. In CCEA IGCSE Maths, you will learn to recognise, manipulate and graph these functions.

二次函数是指可以写成 f(x) = ax² + bx + c 形式的任何函数，其中 a、b 和 c 为常数且 a ≠ 0。变量 x 的最高次幂为 2，这使得函数图像呈现特有的 “U” 形，称为抛物线。在 CCEA IGCSE 数学中，你将学习识别、处理并绘制这类函数图像。

2. The Standard Form f(x) = ax² + bx + c | 标准形式 f(x) = ax² + bx + c

The standard form is f(x) = ax² + bx + c. The coefficient ‘a’ determines the parabola’s width and direction of opening, ‘b’ influences the position of the vertex horizontally, and ‘c’ gives the y-intercept. Always ensure the expression is written with descending powers of x before identifying these coefficients.

标准形式为 f(x) = ax² + bx + c。系数 a 决定了抛物线的宽度和开口方向，b 影响顶点在水平方向上的位置，而 c 给出了 y 轴截距。在确定这些系数之前，务必确保表达式按 x 的降幂排列。

3. Shape of the Graph: Parabola | 图像形状：抛物线

The graph of a quadratic function is a smooth, symmetric curve called a parabola. If a > 0, the parabola opens upwards and has a minimum point (lowest y-value). If a < 0, it opens downwards and has a maximum point. The magnitude of a affects the steepness: larger |a| produces a narrower parabola.

二次函数的图像是一条平滑的对称曲线，称为抛物线。若 a > 0，抛物线开口向上，存在一个最低点（y 的最小值）。若 a < 0，开口向下，存在一个最高点。a 的大小影响陡峭程度：|a| 越大，抛物线越窄。

You can quickly sketch the basic shape by checking the sign of a and plotting a few points. For CCEA exams, you must be able to identify the turning point (vertex) and the axis of symmetry from the equation.

你可以通过检查 a 的符号并描出几个点，快速绘制大致形状。对于 CCEA 考试，你必须能够从方程中找出转折点（顶点）和对称轴。

4. Key Features: Vertex and Axis of Symmetry | 关键特征：顶点与对称轴

Every parabola has a vertex (turning point) and an axis of symmetry. The axis of symmetry is a vertical line passing through the vertex. For the standard form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by:

x = -b / (2a)

每条抛物线都有一个顶点（转折点）和一条对称轴。对称轴是一条经过顶点的竖直线。对于标准形式 f(x) = ax² + bx + c，顶点的 x 坐标由下式给出：

x = -b / (2a)

Substitute this x-value back into the function to find the y-coordinate of the vertex. The axis of symmetry is then the line x = -b/(2a). This formula is provided on the CCEA exam formula sheet, but knowing how to apply it is key.

将这个 x 值代回原函数即可求得顶点的 y 坐标。此时对称轴就是直线 x = -b/(2a)。该公式在 CCEA 考试公式表中给出，但知道如何应用它是关键。

5. Finding the Vertex: Completing the Square | 求顶点：配方法

Completing the square rewrites f(x) = ax² + bx + c into the form a(x + p)² + q. This directly reveals the vertex at (-p, q). For example, to complete the square for x² + 6x + 5: take half of 6, square it to get 9, and write (x + 3)² – 9 + 5 = (x + 3)² – 4. Hence the vertex is (-3, -4). When a ≠ 1, factor out a first from the x² and x terms.

配方法将 f(x) = ax² + bx + c 改写为 a(x + p)² + q 的形式。这直接显示出顶点位于 (-p, q)。例如，对 x² + 6x + 5 进行配方：取 6 的一半，平方得 9，写成 (x + 3)² – 9 + 5 = (x + 3)² – 4。因此顶点为 (-3, -4)。当 a ≠ 1 时，需要先从 x² 和 x 的项中提取公因子 a。

CCEA exam questions often ask for the minimum or maximum value of a quadratic expression. Completing the square makes this straightforward: q is the minimum value when a > 0, and the maximum when a < 0.

CCEA 考题经常要求找出二次表达式的最小值或最大值。配方法使其一目了然：当 a > 0 时，q 为最小值；当 a < 0 时，q 为最大值。

6. Intercepts: x-intercepts and y-intercept | 截距：x 轴截距与 y 轴截距

To find the y-intercept, substitute x = 0 into the function. This always gives the point (0, c). To find the x-intercepts (roots), set f(x) = 0 and solve the quadratic equation ax² + bx + c = 0. The number of x-intercepts depends on the discriminant (see section 9). Always show the points as coordinates.

要求 y 轴截距，将 x = 0 代入函数。这总是给出点 (0, c)。要求 x 轴截距（根），令 f(x) = 0 并解二次方程 ax² + bx + c = 0。x 轴截距的个数取决于判别式（见第 9 节）。始终以坐标形式标出这些点。

7. Solving Quadratic Equations: Factorising | 解二次方程：因式分解

Factorising is the quickest method when the quadratic can be expressed as a product of two linear expressions. For x² + 5x + 6, find two numbers that multiply to +6 and add to +5: these are +2 and +3, so (x + 2)(x + 3) = 0, giving solutions x = -2 and x = -3. Always set the equation to zero first and then factorise.

当二次式可以表示为两个一次因式的乘积时，因式分解是最快捷的方法。对于 x² + 5x + 6，找到两个数，其积为 +6，和为 +5：这两个数是 +2 和 +3，因此 (x + 2)(x + 3) = 0，得解 x = -2 和 x = -3。务必先将方程设为零，然后再进行因式分解。

For equations with a ≠ 1, such as 3x² + 10x – 8 = 0, use the method of splitting the middle term or trial and error. CCEA often tests this skill in both calculator and non-calculator papers.

对于 a ≠ 1 的方程，如 3x² + 10x – 8 = 0，可以使用拆中项法或试错法。CCEA 经常在可使用计算器与不可使用计算器的试卷中都考查这一技能。

8. Solving Quadratic Equations: Quadratic Formula | 解二次方程：求根公式

When factorising is difficult or impossible, use the quadratic formula. For ax² + bx + c = 0, the solutions are:

x = [ -b ± √(b² – 4ac) ] / (2a)

当因式分解困难或不可能时，使用求根公式。对于 ax² + bx + c = 0，解为：

x = [ -b ± √(b² – 4ac) ] / (2a)

Be careful when substituting negative values. Always calculate the discriminant (b² – 4ac) first to determine whether roots are real and distinct, repeated, or not real. This formula is listed in the CCEA formula booklet; memorise it so you can use it confidently.

代入负数值时要格外小心。始终先计算判别式（b² – 4ac），以确定根是实数且不相等、相等重根还是非实数根。该公式列在 CCEA 公式手册中；要熟记以便自信地使用。

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

The discriminant (Δ) = b² – 4ac tells us the nature of the roots without solving the equation. The following table summarises the possibilities:

判别式 Δ = b² – 4ac 无需解方程即可告诉我们根的性质。下表概括了各种可能情况：

Discriminant (Δ)	Nature of roots / 根的性质	Graph / 图像
Δ > 0	Two distinct real roots / 两个不相等的实根	Cuts x-axis at two points / 与 x 轴交于两点
Δ = 0	One repeated real root / 一个重根（两个相等实根）	Touches x-axis at vertex / 在顶点处与 x 轴相切
Δ < 0	No real roots / 无实根	Does not cross x-axis / 不与 x 轴相交

Questions may ask you to find the value of k that gives equal roots, so you would set Δ = 0 and solve for k. This is a very common CCEA problem type.

考题可能会要求找出使方程具有相等实根的 k 值，这时你需要令 Δ = 0 并解出 k。这是一种非常常见的 CCEA 题型。

10. Sketching Quadratic Graphs | 绘制二次图像

To sketch a quadratic graph in the exam, follow these steps: (1) Determine whether the parabola opens up or down from the sign of a. (2) Find the y-intercept (0, c). (3) Calculate the vertex using x = -b/(2a) and then find its y-coordinate. (4) Find the x-intercepts if possible by solving f(x)=0. (5) Draw a symmetrical curve through these points, labelling the vertex and intercepts. A well-labelled sketch does not need perfect scale but must show all key features.

在考试中绘制二次函数图像时，遵循下列步骤：(1) 从 a 的符号判断抛物线开口向上还是向下。(2) 找出 y 轴截距 (0, c)。(3) 使用 x = -b/(2a) 计算顶点，然后求 y 坐标。(4) 若可能，解 f(x)=0 求出 x 轴截距。(5) 通过这些点画出一条对称的曲线，标出顶点和截距。一幅标注清晰的草图不需要完美的比例，但必须显示所有关键特征。

11. Quadratic Inequalities | 二次不等式

Solving quadratic inequalities often appears in the Extended tier. To solve x² – 4x + 3 < 0, first solve x² - 4x + 3 = 0 to get roots x = 1 and x = 3. Sketch the parabola y = x² - 4x + 3 (a = 1 > 0, so U-shaped). The graph is below the x-axis between the roots. Hence the solution is 1 < x < 3. Always show the solution set on a number line or using interval notation as required. For inequalities like ax² + bx + c > 0, the solution lies outside the roots for a > 0. Remember to flip the inequality sign when multiplying or dividing by a negative number.

解二次不等式常见于拓展层级考试。要求解 x² – 4x + 3 < 0，先解方程 x² - 4x + 3 = 0，得根 x = 1 和 x = 3。画出抛物线 y = x² - 4x + 3 的草图（a = 1 > 0，开口向上）。图像在两根之间位于 x 轴下方。因此解为 1 < x < 3。始终按要求用数轴或区间记号表示解集。对于形如 ax² + bx + c > 0 的不等式，当 a > 0 时，解在两根之外。切记，乘以或除以负数时要翻转不等号方向。

12. Applications and Exam Tips | 应用与考试技巧

Quadratic functions are often embedded in real-life contexts such as projectile motion, area problems, or maximising profit. In these word problems, define the unknown variable, form the quadratic equation or function, and then use the appropriate solving method. CCEA examiners love to ask for the maximum/minimum value in context, which requires completing the square.

二次函数常被嵌入现实情境中，如抛体运动、面积问题或利润最大化。在这些文字题中，要设出未知变量，列出二次方程或函数，然后选择合适的求解方法。CCEA 考官喜欢结合情境考查最大/最小值，这需要用配方法解答。

Check your signs when substituting negative numbers into formulas. / 将负数代入公式时，务必检查符号。
Show all working – method marks are generous in CCEA exams. / 展示所有步骤 – CCEA 考试中对方法分给得很大方。
Use the discriminant to verify your roots make sense. / 运用判别式 来验证你的根是否合理。
Always label the vertex and intercepts when sketching. / 画草图时务必标出顶点和截距。

Practise past paper questions from both the Core and Extended tiers to solidify these concepts. Consistent revision on transformaions such as y = f(x) + a and y = f(x + a) also helps, as they often appear together with quadratic curves.

练习过往试卷中的核心和拓展层级题目，以巩固这些概念。对诸如 y = f(x) + a 和 y = f(x + a) 等图像变换持续复习也很有帮助，因为它们常与二次曲线一同出现。

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IGCSE CCEA Maths: Quadratic Functions Key Points | IGCSE CCEA 数学：二次函数 考点精讲