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

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

A quadratic function is one of the most important topics in IGCSE Mathematics. It appears frequently in exams and forms the foundation for calculus and higher-level algebra. In this article, we will explore its standard form, key features, graphing techniques, and common exam pitfalls.

二次函数是IGCSE数学中最重要的主题之一。它频繁出现在考试中，也是微积分和更高阶代数的基础。本文将探讨它的标准形式、关键特征、作图技巧以及常见考试误区。

1. The Standard Form | 标准形式

A quadratic function is any function that can be written as \(f(x) = ax^2 + bx + c\) where \(a \neq 0\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The graph of a quadratic function is a parabola.

二次函数是能够写成 \(f(x) = ax^2 + bx + c\) 的函数，其中 \(a \neq 0\)。这里的 \(a\)、\(b\) 和 \(c\) 是常数，\(x\) 是变量。二次函数的图像是一条抛物线。

For example, \(f(x) = 2x^2 – 4x + 1\) is a quadratic function with \(a = 2\), \(b = -4\), \(c = 1\). The sign of \(a\) determines the orientation of the parabola: if \(a > 0\), it opens upward; if \(a < 0\), it opens downward.

例如，\(f(x) = 2x^2 – 4x + 1\) 是二次函数，其中 \(a = 2\)，\(b = -4\)，\(c = 1\)。\(a\) 的正负决定了抛物线的开口方向：若 \(a > 0\)，开口向上；若 \(a < 0\)，开口向下。

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

The vertex is the highest or lowest point of the parabola. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. For a quadratic in standard form \(y = ax^2 + bx + c\), the \(x\)-coordinate of the vertex is given by:

顶点是抛物线的最高点或最低点。对称轴是穿过顶点的一条垂直线，将抛物线分成两个对称的部分。对于标准形式的二次函数 \(y = ax^2 + bx + c\)，顶点的 \(x\) 坐标由下式给出：

x = −b/(2a)

Once you have \(x\), substitute it back into the original equation to find the \(y\)-coordinate. For example, for \(y = x^2 – 6x + 5\), we have \(x = -(-6)/(2 \times 1) = 3\). Then \(y = 9 – 18 + 5 = -4\). So the vertex is \((3, -4)\).

求出 \(x\) 后，将其代回原方程即可得到 \(y\) 坐标。例如，对于 \(y = x^2 – 6x + 5\)，有 \(x = -(-6)/(2 \times 1) = 3\)，然后 \(y = 9 – 18 + 5 = -4\)。所以顶点为 \((3, -4)\)。

3. The Roots (x-intercepts) | 根（与 x 轴的交点）

The roots of a quadratic equation are the values of \(x\) where \(y = 0\). These correspond to the points where the parabola crosses the \(x\)-axis. You can find them by factoring, completing the square, or using the quadratic formula:

二次方程的根是使 \(y = 0\) 的 \(x\) 值，对应抛物线与 \(x\) 轴的交点。你可以通过因式分解、配方法或求根公式来求解：

x = [−b ± √(b² − 4ac)] / (2a)

The expression \(b^2 – 4ac\) is called the discriminant. It tells us the number of real roots:

表达式 \(b^2 – 4ac\) 称为判别式，它告诉我们实数根的个数：

If \(b^2 – 4ac > 0\), there are two distinct real roots.
If \(b^2 – 4ac = 0\), there is exactly one real root (the vertex touches the x-axis).
If \(b^2 – 4ac < 0\), there are no real roots (the parabola does not intersect the x-axis).

若 \(b^2 – 4ac > 0\)，有两个不同的实数根。
若 \(b^2 – 4ac = 0\)，只有一个实数根（顶点与 x 轴相切）。
若 \(b^2 – 4ac < 0\)，没有实数根（抛物线不与 x 轴相交）。

4. Completed Square Form | 配方法（顶点式）

Writing a quadratic in the form \(y = a(x – h)^2 + k\) is extremely useful for identifying the vertex directly: the vertex is \((h, k)\). The axis of symmetry is \(x = h\). This is called the completed square form.

将二次函数写成 \(y = a(x – h)^2 + k\) 的形式对于直接识别顶点非常有用：顶点为 \((h, k)\)，对称轴为 \(x = h\)。这被称为配方法形式（顶点式）。

For example, \(y = 2x^2 – 8x + 7\) can be rewritten by completing the square. First, factor out 2 from the first two terms: \(y = 2(x^2 – 4x) + 7\). Inside the bracket, add and subtract \((4/2)^2 = 4\): \(y = 2[(x – 2)^2 – 4] + 7 = 2(x – 2)^2 – 8 + 7 = 2(x – 2)^2 – 1\). Thus the vertex is \((2, -1)\).

例如，\(y = 2x^2 – 8x + 7\) 可以通过配方法改写。先从前两项中提取 2：\(y = 2(x^2 – 4x) + 7\)。在括号内加上并减去 \((4/2)^2 = 4\)：\(y = 2[(x – 2)^2 – 4] + 7 = 2(x – 2)^2 – 8 + 7 = 2(x – 2)^2 – 1\)。因此顶点为 \((2, -1)\)。

5. Sketching a Quadratic Graph | 画二次函数图像

To sketch a quadratic graph accurately, follow these steps:

为了准确地画出二次函数图像，请遵循以下步骤：

Determine the direction of the parabola (upward if \(a > 0\), downward if \(a < 0\)).
Find the vertex using \(x = -b/(2a)\), or from the completed square form.
Find the \(y\)-intercept by setting \(x = 0\).
Find the \(x\)-intercepts (roots), if they exist, by solving \(ax^2 + bx + c = 0\).
Plot these key points and draw a smooth curve through them.

确定抛物线开口方向（\(a > 0\) 向上，\(a < 0\) 向下）。
使用 \(x = -b/(2a)\) 或顶点式求顶点。
令 \(x = 0\) 求 \(y\) 轴截距。
解 \(ax^2 + bx + c = 0\) 求 \(x\) 轴截距（根），如果存在的话。
标出这些关键点，并用平滑曲线连接。

Remember that the graph is symmetric about the axis of symmetry. You can always reflect points across this line to get extra points.

请记住图像关于对称轴是对称的。你总可以将点关于对称轴反射来获得额外点。

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

Given the graph of \(y = x^2\), you can obtain other parabolas through translations and stretches. The general vertex form \(y = a(x – h)^2 + k\) combines these:

给定 \(y = x^2\) 的图像，你可以通过平移和伸缩得到其他抛物线。一般的顶点式 \(y = a(x – h)^2 + k\) 综合了这些变换：

\(k\) shifts the graph vertically upward if \(k > 0\), downward if \(k < 0\).
\(h\) shifts the graph horizontally to the right if \(h > 0\), to the left if \(h < 0\).
Changing \(a\) stretches or compresses the graph vertically; if \(a < 0\), it also reflects the graph in the \(x\)-axis.

\(k\) 使图像垂直移动：\(k > 0\) 向上，\(k < 0\) 向下。
\(h\) 使图像水平移动：\(h > 0\) 向右，\(h < 0\) 向左。
改变 \(a\) 会使图像垂直伸缩；若 \(a < 0\)，还会关于 \(x\) 轴反射。

7. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities are solved by first finding the roots and then testing intervals on a number line. For example, solve \(x^2 – 3x – 4 > 0\). Factor to \((x – 4)(x + 1) > 0\). The critical points are \(x = -1\) and \(x = 4\). Test intervals: \(x < -1\), \(-1 < x < 4\), and \(x > 4\).

解二次不等式需要先求根，然后在数轴上测试区间。例如，解 \(x^2 – 3x – 4 > 0\)。因式分解得 \((x – 4)(x + 1) > 0\)。临界点为 \(x = -1\) 和 \(x = 4\)。测试区间：\(x < -1\)，\(-1 < x < 4\)，\(x > 4\)。

If \(x = -2\), the product is \((-6)(-1) = 6 > 0\), so this interval works. If \(x = 0\), the product is \((-4)(1) = -4 < 0\), so it does not. If \(x = 5\), the product is \((1)(6) = 6 > 0\), so it works. The solution is \(x < -1\) or \(x > 4\).

若 \(x = -2\)，乘积为 \((-6)(-1) = 6 > 0\)，因此这个区间满足。若 \(x = 0\)，乘积为 \((-4)(1) = -4 < 0\)，因此不满足。若 \(x = 5\)，乘积为 \((1)(6) = 6 > 0\)，因此满足。解为 \(x < -1\) 或 \(x > 4\)。

8. Applications in Real Life | 实际生活中的应用

Quadratic functions appear in physics, business, and geometry. For instance, the height of a projectile thrown upward can be modeled by \(h(t) = -4.9t^2 + v_0 t + h_0\), where \(h_0\) is the initial height and \(v_0\) is the initial velocity. The maximum height occurs at the vertex.

二次函数出现在物理、商业和几何中。例如，向上抛出的物体高度可以用 \(h(t) = -4.9t^2 + v_0 t + h_0\) 建模，其中 \(h_0\) 是初始高度，\(v_0\) 是初始速度。最大高度出现在顶点处。

In business, profit functions often take a quadratic form. If a company’s profit \(P(x)\) in dollars is \(P(x) = -2x^2 + 200x – 1000\), where \(x\) is the number of items sold, then the maximum profit is found by locating the vertex: \(x = -200/(2 \times -2) = 50\). The maximum profit is \(P(50) = -2(2500) + 10000 – 1000 = 4000\) dollars.

在商业中，利润函数通常呈现二次形式。如果公司利润 \(P(x)\)（美元）为 \(P(x) = -2x^2 + 200x – 1000\)，其中 \(x\) 是售出的物品数量，那么最大利润通过求顶点找到：\(x = -200/(2 \times -2) = 50\)。最大利润为 \(P(50) = -2(2500) + 10000 – 1000 = 4000\) 美元。

9. Common Mistakes | 常见错误

Here are some frequent errors students make with quadratic functions:

以下是学生在二次函数中经常犯的一些错误：

Forgetting that \(a\) cannot be zero in the standard form.
Mixing up the sign of the vertex coordinate when using \(h\) and \(k\) in \(a(x – h)^2 + k\).
When finding roots, equating \(c\) to zero incorrectly in the quadratic formula.
Misinterpreting the discriminant as the roots themselves.
Sketching the graph without checking the direction of opening.

忘记标准形式中 \(a\) 不能为零。
在使用 \(a(x – h)^2 + k\) 时混淆顶点坐标的符号。
求根时在求根公式中错误地将 \(c\) 当作零。
将判别式误认为是根本身。
画图时不检查开口方向。

10. Practice Question | 练习题目

Let’s try an exam-style question. The curve \(y = x^2 – 4x + 3\) intersects the \(x\)-axis at points A and B. Find the coordinates of A and B, the vertex, and the \(y\)-intercept. Then sketch the graph.

我们来尝试一道考试风格的题目。曲线 \(y = x^2 – 4x + 3\) 与 \(x\) 轴交于点 A 和 B。求 A、B 的坐标、顶点和 \(y\) 轴截距，并画出草图。

Solution: Factor the quadratic: \(x^2 – 4x + 3 = (x – 1)(x – 3)\). So the roots are \(x = 1\) and \(x = 3\), meaning A = (1, 0) and B = (3, 0). The \(x\)-coordinate of the vertex is the midpoint: \(x = (1 + 3)/2 = 2\). Then \(y = 4 – 8 + 3 = -1\). So the vertex is (2, -1). The \(y\)-intercept is found by setting \(x = 0\): \(y = 3\), so (0, 3).

解答：对二次项因式分解：\(x^2 – 4x + 3 = (x – 1)(x – 3)\)。因此根为 \(x = 1\) 和 \(x = 3\)，即 A = (1, 0)，B = (3, 0)。顶点的 \(x\) 坐标是中点：\(x = (1 + 3)/2 = 2\)。然后 \(y = 4 – 8 + 3 = -1\)。所以顶点为 (2, -1)。\(y\) 轴截距通过令 \(x = 0\) 求得：\(y = 3\)，即 (0, 3)。

11. Summary | 总结

Quadratic functions are a core part of IGCSE Mathematics. You must master the standard form, the completed square form, the quadratic formula, and the discriminant. Sketching graphs requires identifying the direction, vertex, intercepts, and symmetry. With practice, these problems become straightforward.

二次函数是IGCSE数学的核心内容。你必须掌握标准形式、配方法形式、求根公式和判别式。画图需要确定开口方向、顶点、截距和对称性。通过练习，这些问题会变得简单直接。

Keep practicing with past papers and always check your graph for common mistakes. A clear understanding here will help you succeed in exams and future mathematics courses.

坚持练习历年真题，并始终检查你的图像是否存在常见错误。清晰理解这一主题将帮助你在考试和未来的数学课程中取得成功。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

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