📚 Graphs of Quadratic Functions | 二次函数图像
Quadratic graphs are one of the most important topics in IGCSE Mathematics. Understanding how to sketch and interpret these curves is essential for solving quadratic equations, analysing real-world motion, and preparing for calculus at A-Level.
二次函数图像是 IGCSE 数学中最重要的内容之一。理解如何绘制和解读这些曲线，对于解二次方程、分析实际运动问题，以及为 A-Level 微积分学习做准备，都非常关键。
1. What Is a Quadratic Function? | 什么是二次函数
A quadratic function is a relationship that can be written in the form:
二次函数是一种可以写成如下形式的关系式：
y = ax² + bx + c, where a ≠ 0
Here a, b and c are constants. The value of a cannot be zero because zero would make the expression linear rather than quadratic. The graph of every quadratic function is a smooth curve called a parabola.
这里 a、b、c 是常数。a 不能等于零，因为如果 a = 0，表达式就会变成一次函数而不是二次函数。每一个二次函数的图像都是一条平滑的曲线，称为抛物线。
- If a > 0, the parabola opens upward and looks like a cup, with a minimum point at the bottom.
- 如果 a > 0，抛物线开口向上，形状像一个杯子，底部有最低点。
- If a < 0, the parabola opens downward and looks like a cap, with a maximum point at the top.
- 如果 a < 0，抛物线开口向下，形状像一个帽子，顶部有最高点。
| Sign of a | a 的符号 | Graph shape | 图像形状 |
|---|---|
| a > 0 | U-shaped parabola, opens upwards | U 形抛物线，开口向上 |
| a < 0 | n-shaped parabola, opens downwards | 倒 U 形抛物线，开口向下 |
2. The Standard Form and Its Coefficients | 标准形式与系数
ost common way to write a quadratic function. Each coefficient carries special information about the graph.
标准形式 y = ax² + bx + c 是书写二次函数最常用的方式。每个系数都包含图像的重要信息。
- The coefficient a controls whether the parabola opens up or down and how steep or wide it is.
- 系数 a 控制抛物线的开口方向，也影响抛物线的陡峭程度和宽窄。
- The coefficient b affects the horizontal position of the parabola and helps locate the axis of symmetry.
- 系数 b 影响抛物线的水平位置，并帮助确定对称轴的位置。
- The constant c gives the y-intercept, the point where the graph crosses the y-axis.
- 常数 c 给出 y 轴截距，即图像与 y 轴相交的位置。
2x² - 4x + 5, the y-intercept is (0, 5). The graph opens upward because a = 2 is positive.
例如，在 y = 2x² - 4x + 5 中，y 轴截距是 (0, 5)。因为 a = 2 是正数，所以图像开口向上。
3. Plotting y = x² | 绘制 y = x²
. To plot it accurately, choose a set of x-values and calculate the corresponding y-values.
最简单的二次函数图像是 y = x²。要准确作图，可以取一组 x 值并计算对应的 y 值。
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y = x² | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
. The point (0, 0) is the vertex of the graph, and the line x = 0 is the axis of symmetry.
这些点位于一条对称的 U 形曲线上。点 (0, 0) 是图像的顶点，直线 x = 0 是对称轴。
4. Key Features: y-Intercept, Axis and Vertex | 关键特征：y 轴截距、对称轴与顶点
Every quadratic graph has three important features: the y-intercept, the axis of symmetry and the vertex.
每个二次函数图像都有三个重要特征：y 轴截距、对称轴和顶点。
For the function y = ax² + bx + c:
对于函数 y = ax² + bx + c：
- The y-intercept is always (0, c), because substituting x = 0 leaves y = c.
- y 轴截距始终是 (0, c)，因为代入 x = 0 后只剩 y = c。
- The axis of symmetry is the vertical line x = -b ⁄ (2a).
- 对称轴是竖直线 x = -b ⁄ (2a)。
- The vertex lies on the axis of symmetry, so its x-coordinate is -b ⁄ (2a).
- 顶点位于对称轴上，所以它的 x 坐标是 -b ⁄ (2a)。
Axis of symmetry: x = -b ⁄ (2a)
Vertex: ( -b ⁄ (2a), y after substitution )
y = x² - 4x + 3 has axis x = 2. Substituting x = 2 gives y = -1, so the vertex is (2, -1).
例如，y = x² - 4x + 3 的对称轴是 x = 2。代入 x = 2 得到 y = -1，所以顶点是 (2, -1)。
5. Roots and the Discriminant | 根与判别式
the graph crosses the x-axis, so they are the solutions of the equation ax² + bx + c = 0.
二次函数的根是图像与 x 轴相交处的 x 值，也就是方程 ax² + bx + c = 0 的解。
Roots can be found by factorising, by completing the square, or by using the quadratic formula:
根可以通过因式分解、配方法或求根公式求得：
x = ( -b ± √b² - 4ac ) ⁄ (2a)
The expression b² - 4ac is called the discriminant, often written as Δ.
表达式 b² - 4ac 称为判别式，通常记作 Δ。
| Discriminant | 判别式 | Number of real roots | 实数根个数 | Graph meaning | 图像意义 |
|---|---|---|
| b² - 4ac > 0 | Two real roots | The graph cuts the x-axis at two points | 图像与 x 轴相交于两点 |
| b² - 4ac = 0 | One repeated real root | The graph touches the x-axis at the vertex | 图像在顶点处与 x 轴相切 |
| b² - 4ac < 0 | No real roots | The graph never crosses the x-axis | 图像不与 x 轴相交 |
This discriminant test is extremely useful in IGCSE questions that ask whether a quadratic graph has 0, 1 or 2 x-intercepts.
在 IGCSE 考试中，判断二次函数图像与 x 轴交点是 0 个、1 个还是 2 个时，判别式非常有用。
6. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form:
配方法可以将二次函数改写成如下形式：
y = a(x - h)² + k
In this form, the vertex is simply (h, k), and the axis of symmetry is x = h.
在这种形式下，顶点直接是 (h, k)，对称轴是 x = h。
For example, take y = x² - 6x + 5. Complete the square by taking half of -6, which is -3:
例如，考虑 y = x² - 6x + 5。取 -6 的一半，即 -3：
y = (x - 3)² - 9 + 5 = (x - 3)² - 4
and the axis of symmetry is x = 3. This form also shows that the minimum value of y is -4.
因此顶点是 (3, -4)，对称轴是 x = 3。这种形式还显示出 y 的最小值是 -4。
7. Sketching a Quadratic Graph | 画二次函数草图
In an IGCSE exam, you may be asked to sketch a quadratic graph without plotting many points. The quickest method uses four main features.
在 IGCSE 考试中，可能会要求不逐点描图而直接画出二次函数的草图。最快的方法是利用四个主要特征。
- Determine the direction of the parabola using the sign of a.
- 根据 a 的符号确定抛物线的开口方向。
- Write down the y-intercept (0, c).
- 写出 y 轴截距 (0, c)。
- Find the vertex by completing the square or using x = -b ⁄ (2a).
- 通过配方或使用 x = -b ⁄ (2a) 求出顶点。
- If possible, find the roots and mark them on the x-axis.
- 如果可能，求出根并在 x 轴上标出。
Then draw a smooth symmetric parabola through these key points. Always label the intersections with the axes and the vertex.
然后通过这些关键点画一条平滑对称的抛物线。一定要标出与坐标轴的交点和顶点。
8. Solving Equations Graphically | 用图像解方程
Quadratic graphs can be used to solve equations. The x-coordinates of any points where the graph crosses a horizontal line give the solutions.
二次函数图像可以用来解方程。图像与水平线交点的 x 坐标就是方程的解。
- 3 crosses the x-axis at x = -1 and x = 3, so these are the solutions of x² - 2x - 3 = 0.
例如，y = x² - 2x - 3 的图像与 x 轴相交于 x = -1 和 x = 3，所以它们是 x² - 2x - 3 = 0 的解。
2 on the same diagram. The x-coordinates of the two intersection points are the solutions.
要解 x² - 2x - 3 = 2，可以在同一坐标系中画水平线 y = 2。两个交点的 x 坐标就是方程的解。
This method is especially helpful when the equation does not factorise neatly and you only need an approximate answer.
当方程不易因式分解，而且只需要近似答案时，这种方法尤其有用。
9. Transformations of Quadratic Graphs | 二次函数图像的变换
Starting from y = x², simple transformations can produce a family of quadratic graphs.
从 y = x² 出发，通过简单的变换可以产生一系列二次函数图像。
| Function | 函数 | Transformation | 变换 |
|---|---|
| y = x² + k | Translates the graph vertically by k units | 图像上下平移 k 个单位 |
| y = (x - h)² | Translates the graph horizontally by h units | 图像左右平移 h 个单位 |
| y = -x² | Reflects the graph in the x-axis | 图像关于 x 轴翻折 |
| y = ax² | Stretches or compresses the graph vertically | 图像纵向拉伸或压缩 |
right when h is positive and to the left when h is negative. Students often confuse this direction.
对于 y = (x - h)²，当 h 为正时图像向右平移，当 h 为负时向左平移。学生经常在这个方向上混淆。
10. Common Mistakes and Exam Tips | 常见错误与考试提示
Many marks are lost in quadratic graph questions because of small but repeated mistakes. Here are the most important points to remember.
很多学生在二次函数图像题目中因为微小但反复出现的错误而失分。以下是需要记住的最重要几点。
- The y-intercept is (0, c), not (c, 0).
- y 轴截距是 (0, c)，不是 (c, 0)。
- The axis of symmetry is x = -b ⁄ (2a). Do not forget the negative sign.
- 对称轴是 x = -b ⁄ (2a)，不要忘记负号。
- Use the discriminant before writing down roots. If b² - 4ac < 0, the graph has no x-intercepts.
- 写出根之前先使用判别式。如果 b² - 4ac < 0，图像与 x 轴没有交点。
- ompleting the square, check the vertex by substituting x = h into the original expression.
- 配方时，把 x = h 代入原式来检查顶点是否正确。
- When sketching, always draw a smooth curve, not a set of straight line segments.
- 画草图时，一定要画平滑曲线，不要画成折线段。
With regular practice, quadratic graphs become a reliable source of marks. Focus on the vertex, roots, discriminant and transformations, and you will be well prepared.
通过有规律的练习，二次函数图像会成为你的可靠得分点。重点关注顶点、根、判别式和图像变换，你一定可以做好准备。
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