Understanding Quadratic Functions and Their Graphs | 理解二次函数及其图像

📚 Understanding Quadratic Functions and Their Graphs | 理解二次函数及其图像

A quadratic function is one of the central topics in IGCSE Mathematics. It connects algebra, coordinate geometry and problem solving, and it appears in many real-world situations such as projectile motion, area optimisation and profit modelling. In this article, you will learn how to recognise a quadratic function, sketch its graph, find its vertex and intercepts, use completing the square, and apply transformations to sketch new curves.

二次函数是 IGCSE 数学的核心内容之一，它把代数、坐标几何与问题解决联系起来，并出现在许多现实情境中，例如抛体运动、面积最优化和利润建模。在本文中，你将学习如何识别二次函数、画出它的图像、求顶点与截距、使用配方法，以及运用变换来绘制新曲线。

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

A quadratic function is a polynomial of degree 2. Its general form is:

f(x) = ax² + bx + c, a ≠ 0

ion becomes linear. The graph of a quadratic function is a smooth curve called a parabola.

这里的 a、b 和 c 都是实数。系数 a 不能为零，因为如果 a = 0，函数就变成了一次函数。二次函数的图像是一条平滑的曲线，称为抛物线。

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2. Shape and Direction of a Parabola | 抛物线的形状与开口方向

The sign of a determines whether the parabola opens upward or downward, and the size of a controls how wide or narrow the curve is.

a 的正负决定了抛物线开口向上还是向下，而 a 的大小控制着曲线的宽窄。

• If a > 0, the parabola opens upward and it has a minimum point.
• 如果 a > 0，抛物线开口向上，并且有最小值点。
• If a < 0, the parabola opens downward and it has a maximum point.
• 如果 a < 0，抛物线开口向下，并且有最大值点。
• A larger value of |a| makes the curve narrower; a smaller value of |a| makes it wider.
• |a| 的值越大，曲线越狭窄；|a| 的值越小，曲线越宽阔。

a > 0	Opens upward, like a cup, and has a minimum value.
a < 0	Opens downward, like a cap, and has a maximum value.

x², the curve is symmetric about the y-axis and its minimum point is at the origin (0, 0).

对于 f(x) = x² 的图像，曲线关于 y 轴对称，其最小值点在原点 (0, 0) 处。

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

as a vertical line of symmetry. For f(x) = ax² + bx + c, the axis of symmetry is given by:

x = −b/(2a)

The vertex is the turning point of the parabola. Its x-coordinate is the same as the axis of symmetry, and its y-coordinate is found by substituting this value into the function:

顶点是抛物线的转折点。它的 x 坐标与对称轴相同，y 坐标则通过把该值代入函数求得：

Vertex = (−b/(2a), f(−b/(2a)))

If a > 0, this vertex is the minimum point; if a < 0, it is the maximum point.

如果 a > 0，该顶点就是最低点；如果 a < 0，该顶点就是最高点。

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4. Intersections with the Axes | 与坐标轴的交点

To find the y-intercept, set x = 0. Then f(0) = c, so the y-intercept is (0, c).

求 y 截距时，令 x = 0。此时 f(0) = c，所以 y 截距为 (0, c)。

e the equation ax² + bx + c = 0. The number of real solutions depends on the discriminant:

求 x 截距时，令 f(x) = 0，并解方程 ax² + bx + c = 0。实数解的个数由判别式决定：

Δ = b² − 4ac

• If Δ > 0, there are two distinct x-intercepts.
• 如果 Δ > 0，有两个不同的 x 截距。
• If Δ = 0, there is one repeated x-intercept, so the vertex lies on the x-axis.
• 如果 Δ = 0，有一个重根 x 截距，因此顶点位于 x 轴上。
• If Δ < 0, there are no real x-intercepts, and the curve does not cross the x-axis.
• 如果 Δ < 0，没有实数 x 截距，曲线不穿过 x 轴。

Δ > 0	Two real roots; the curve cuts the x-axis twice.
Δ = 0	One repeated root; the curve touches the x-axis.
Δ < 0	No real roots; the curve does not meet the x-axis.

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5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form:

配方法将二次函数改写为以下形式：

f(x) = a(x − h)² + k

function is in this form, the vertex is simply (h, k), and the axis of symmetry is x = h.

一旦函数写成这种形式，顶点就是 (h, k)，对称轴就是 x = h。

For example, take f(x) = x² − 6x + 11. Half of −6 is −3, so:

例如，取 f(x) = x² − 6x + 11。−6 的一半是 −3，所以：

f(x) = (x − 3)² − 9 + 11 = (x − 3)² + 2

Therefore the minimum point is (3, 2) and the axis of symmetry is x = 3.

因此最低点是 (3, 2)，对称轴是 x = 3。

When a ≠ 1, factor out a from the first two terms before completing the square. This is a common IGCSE skill that helps you sketch graphs and solve maximum or minimum problems quickly.

当 a ≠ 1 时，应先把前两项的公因数 a 提出来，再配方。这是 IGCSE 常见技能，可以帮助你快速画图和解决最大值或最小值问题。

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6. Transformations of Quadratic Graphs | 二次函数图像的变换

mple transformations to y = x². These transformations also apply to any function y = f(x).

你可以通过对 y = x² 进行简单变换来画出新的二次函数图像。这些变换也适用于任意函数 y = f(x)。

• y = f(x) + k: translate the graph upward by k units.

  y = f(x) + k：将图像向上平移 k 个单位。

• y = f(x) − k: translate the graph downward by k units.

  y = f(x) − k：将图像向下平移 k 个单位。

• y = f(x + k): translate the graph to the left by k units.

  y = f(x + k)：将图像向左平移 k 个单位。

• y = f(x − k): translate the graph to the right by k units.

  y = f(x − k)：将图像向右平移 k 个单位。

• y = −f(x): reflect the graph in the x-axis.

  y = −f(x)：将图像关于 x 轴反射。

• y = f(−x): reflect the graph in the y-axis.

  y = f(−x)：将图像关于 y 轴反射。

es the graph left, not right. This sign rule is a common source of errors in examinations.

注意 y = f(x + k) 使图像向左移动，而不是向右移动。这个符号规则是考试中常见的错误来源。

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7. Finding the Equation of a Quadratic from Its Graph | 由图像求二次函数的表达式

If you know the vertex of a parabola and one other point, you can find the equation in vertex form.

如果知道抛物线的顶点和另一个点的坐标，就可以用顶点式求出函数表达式。

Suppose the vertex is (2, −3) and the curve passes through (0, 5). Since the vertex is (h, k), we write:

假设顶点为 (2, −3)，并且曲线经过点 (0, 5)。因为顶点是 (h, k)，我们写成：

y = a(x − 2)² − 3

Substitute x = 0 and y = 5 into the equation:

把 x = 0 和 y = 5 代入方程：

5 = a(0 − 2)² − 3

5 = 4a − 3

4a = 8, so a = 2

he equation is y = 2(x − 2)² − 3. Expanding this gives the standard form y = 2x² − 8x + 5.

因此函数表达式为 y = 2(x − 2)² − 3。展开后得到标准形式 y = 2x² − 8x + 5。

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8. Worked Examination Example | 考试例题解析

− 8x + 5. Write f(x) in the form a(x − h)² + k, state the minimum value, and find the y-intercept.

已知函数 f(x) = 2x² − 8x + 5。请将 f(x) 写成 a(x − h)² + k 的形式，说明最小值，并求 y 截距。

First, factor out 2 from the first two terms:

首先，把 2 从前两项中提出来：

f(x) = 2(x² − 4x) + 5

Complete the square inside the brackets. Half of −4 is −2, so:

在括号内配方。−4 的一半是 −2，所以：

x² − 4x = (x − 2)² − 4

Then continue:

然后继续计算：

f(x) = 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 8 + 5 = 2(x − 2)² − 3

2)² − 3. Since a = 2 > 0, the graph opens upward and has a minimum value of −3 when x = 2.

现在顶点式为 f(x) = 2(x − 2)² − 3。因为 a = 2 > 0，图像开口向上，当 x = 2 时取得最小值 −3。

The y-intercept is found by setting x = 0:

求 y 截距时令 x = 0：

f(0) = 0 − 0 + 5 = 5

So the curve crosses the y-axis at (0, 5).

所以曲线与 y 轴交于点 (0, 5)。

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9. Common Mistakes and Tips | 常见错误与提示

Many students lose marks on quadratic questions because of small sign errors or missing steps. Here are the most common pitfalls.

许多学生在二次函数问题上失分，原因往往是小符号错误或漏写步骤。以下是最常见的陷阱。

• gn in the axis of symmetry formula. Remember that the axis is x = −b/(2a), not x = b/(2a).

  对称轴公式的符号用错。记住对称轴是 x = −b/(2a)，而不是 x = b/(2a)。

• Forgetting to factor out a before completing the square. This leads to an incorrect vertex.

  配方法前忘记提出 a。这会导致顶点错误。

• Confusing horizontal translations. y = f(x + k) moves the graph left, not right.

  混淆水平平移。y = f(x + k) 使图像向左移动，而不是向右移动。

• Assuming every quadratic has two x-intercepts. Always check the discriminant Δ = b² − 4ac.

  假设每个二次函数都有两个 x 截距。一定要检查判别式 Δ = b² − 4ac。

• dratic formula, write the equation in the form ax² + bx + c = 0 before reading a, b and c.

  使用求根公式前，先把方程写成 ax² + bx + c = 0 的形式，再确定 a、b、c。

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10. Summary and Practice | 总结与练习

unction f(x) = ax² + bx + c is one of the most heavily tested topics in IGCSE Mathematics.

二次函数 f(x) = ax² + bx + c 是 IGCSE 数学中考查最频繁的内容之一。

• The graph is a parabola. It opens upward when a > 0 and downward when a < 0.

  图像是抛物线。当 a > 0 时开口向上，当 a < 0 时开口向下。

• The axis of symmetry is x = −b/(2a), and the vertex lies on this line.

  对称轴是 x = −b/(2a)，顶点位于这条直线上。

• Completing the square gives the minimum or maximum point directly.

  配方法可以直接给出最小值点或最大值点。

• The discriminant Δ = b² − 4ac tells you how many x-intercepts the graph has.

  判别式 Δ = b² − 4ac 可以告诉你图像与 x 轴有多少个交点。

), and the x-intercepts are 1 and 3; the graph is translated 1 unit left and 2 units down.

请快速完成以下练习。第一题：对 f(x) = x² − 4x + 3，求顶点和 x 截距。第二题：描述从 y = x² 到 y = (x + 1)² − 2 的变换。第一题答案是顶点为 (2, −1)，x 截距为 1 和 3；第二题答案是图像向左平移 1 个单位，再向下平移 2 个单位。

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