Completing the Square & Quadratic Graphs | 配方法与二次函数图像

📚 Completing the Square & Quadratic Graphs | 配方法与二次函数图像

Completing the square is a core algebraic skill in the IGCSE Mathematics syllabus. It allows you to rewrite any quadratic expression in a form that reveals the turning point of its graph, making it indispensable for sketching parabolas and solving optimisation problems. In this revision guide, we will break down the method step by step, illustrate it with worked examples, and highlight the most common exam errors to avoid.

配方法是 IGCSE 数学课程中的核心代数技能。它使你能够将任何二次表达式改写为一种能直接揭示图像转折点的形式，因此在绘制抛物线和解决最优化问题时必不可少。在这份复习指南中，我们将逐步拆解这一方法，用完整例题进行说明，并指出考试中最常见的错误，帮助你避开这些失分陷阱。

1. The Standard Form of a Quadratic | 二次函数的标准形式

A quadratic expression in x is conventionally written in the standard form:

ax² + bx + c

Here, a, b and c are constants, and a ≠ 0. The sign of a determines the direction in which the parabola opens: if a > 0, the curve opens upward and has a minimum point; if a < 0, the curve opens downward and has a maximum point.

其中 a、b、c 是常数，且 a ≠ 0。a 的符号决定抛物线的开口方向：如果 a > 0，曲线开口向上，存在最小值点；如果 a < 0，曲线开口向下，存在最大值点。

The standard form is useful for quickly finding the y-intercept, because setting x = 0 gives y = c. However, it does not directly reveal the vertex or the axis of symmetry. To locate those, we use the technique of completing the square.

标准形式便于快速求出 y 截距，因为令 x = 0 可得 y = c。但它不能直接显示顶点或对称轴。要找到这些信息，我们需要使用配方法。

2. The Method of Completing the Square | 配方法的基本步骤

Consider the quadratic expression x² + 6x + 5. Take the coefficient of x, which is 6, halve it to get 3, and square it to obtain 9. We then rewrite the first two terms as a perfect square:

考虑二次表达式 x² + 6x + 5。取 x 的系数 6，将其减半得到 3，再平方得到 9。然后我们将前两项改写为一个完全平方式：

x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4

For the general monic quadratic x² + bx + c, the completed-square form is:

对于一般首项系数为 1 的二次式 x² + bx + c，配方法形式为：

(x + b/2)² – (b/2)² + c

If the coefficient of x² is not 1, factor it out first. For example:

如果 x² 的系数不是 1，需要先提取该系数。例如：

2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² – 4] + 3 = 2(x + 2)² – 5

Always check your result by expanding the completed-square form: it must return exactly to the original expression.

务必通过展开来检验你的结果：展开配方法形式后必须完全回到原表达式。

3. Vertex Form: y = a(x − h)² + k | 顶点形式：y = a(x − h)² + k

The completed-square form is also called vertex form. In y = a(x – h)² + k, the vertex is the point (h, k), and the axis of symmetry is the vertical line x = h. If a > 0, the vertex is the lowest point; if a < 0, the vertex is the highest point.

配方法形式也称为顶点形式。在 y = a(x – h)² + k 中，顶点是点 (h, k)，对称轴是直线 x = h。如果 a > 0，顶点是最低点；如果 a < 0，顶点是最高点。

Take the expression from the previous section, y = 2(x + 2)² – 5. Its vertex is (-2, -5) and its axis of symmetry is x = -2. Notice the sign change: x + 2 corresponds to h = -2.

以本节前面的表达式 y = 2(x + 2)² – 5 为例，其顶点为 (-2, -5)，对称轴为 x = -2。注意符号变化：x + 2 对应 h = -2。

Form	Immediate information	中文说明
Standard ax² + bx + c	y-intercept is c	y 截距为 c
Vertex a(x – h)² + k	Vertex (h, k), axis x = h	顶点 (h, k)，对称轴 x = h
Factorised a(x – p)(x – q)	x-intercepts at p, q	x 截距为 p、q

Switching confidently between these forms is a key examination skill.

在这几种形式之间熟练转换是考试中非常关键的技能。

4. Solving Quadratic Equations by Completing the Square | 用配方法解二次方程

Completing the square also provides a systematic way to solve quadratic equations. For example, to solve x² + 6x + 5 = 0, rewrite the left side as (x + 3)² – 4 = 0. Then add 4 to both sides and take square roots:

配方法也提供了一种解二次方程的系统化方法。例如，解方程 x² + 6x + 5 = 0 时，先将左边改写为 (x + 3)² – 4 = 0，然后在两边加 4，再开平方：

(x + 3)² = 4 → x + 3 = ±2 → x = -1 or x = -5

In general, if the completed-square form is (x + h)² = k with k ≥ 0, then:

一般来说，如果配方法形式为 (x + h)² = k，且 k ≥ 0，那么：

x = -h ± √k

If k < 0, the quadratic equation has no real solutions, since the square of any real number is non-negative.

如果 k < 0，则该二次方程没有实数解，因为任何实数的平方都是非负的。

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

To sketch the graph of a quadratic function without plotting many points, collect four pieces of information.

要画出二次函数的草图，无需逐点描画，只需收集四项信息。

Direction of opening: determined by the sign of a.

开口方向：由 a 的符号决定。
Turning point: read directly from vertex form as (h, k).

转折点：直接从顶点形式中读出，为 (h, k)。
y-intercept: set x = 0 and evaluate the expression.

y 截距：令 x = 0 并计算表达式的值。
x-intercepts (if they exist): set y = 0 and solve the quadratic equation.

x 截距（如果存在）：令 y = 0 并解二次方程。

For example, y = x² – 4x + 3 has vertex (2, -1), y-intercept 3, and x-intercepts 1 and 3. With these four facts, the sketch is straightforward.

例如，y = x² – 4x + 3 的顶点为 (2, -1)，y 截距为 3，x 截距为 1 和 3。有了这四个事实，绘制草图就非常简单。

6. Maximum and Minimum Values | 最大值和最小值

Many IGCSE problems ask for the greatest or least value of a quadratic function. Once the expression is written in vertex form y = a(x – h)² + k, the extremum is simply k. If a > 0, k is the minimum; if a < 0, k is the maximum.

许多 IGCSE 题目要求二次函数的最大值

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Completing the Square & Quadratic Graphs | 配方法与二次函数图像