Understanding Quadratic Graphs | 理解二次函数图像

📚 Understanding Quadratic Graphs | 理解二次函数图像

A quadratic function is one of the most important topics in IGCSE Mathematics. It appears in almost every exam paper, either as a straightforward graph-sketching question or as part of a word problem involving maximum area or projectile motion. Understanding the shape, key features, and transformation of quadratic graphs will give you a strong foundation for higher-level mathematics.

二次函数是 IGCSE 数学中最重要的内容之一。几乎每份试卷都会出现相关的题目，有时是直接要求画图，有时是结合最大面积或抛体运动的应用题。理解二次函数图像的形状、关键特征和变换，将为你在更高层次的数学学习中打下坚实基础。

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1. The General Form of a Quadratic Function | 二次函数的一般形式

A quadratic function is any function that can be written in the form y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is called a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, the parabola opens downwards and has a maximum point.

二次函数是指可以写成 y = ax² + bx + c 形式的函数，其中 a、b、c 为常数，且 a ≠ 0。二次函数的图像称为抛物线。当 a > 0 时，抛物线开口向上，有最小值点；当 a < 0 时，抛物线开口向下，有最大值点。

For example, y = 2x² + 3x − 1 has a = 2, b = 3, and c = −1. Since a is positive, the parabola opens upwards. In contrast, y = −x² + 4x has a = −1, so it opens downwards.

例如，y = 2x² + 3x − 1 中 a = 2，b = 3，c = −1。由于 a 为正数，抛物线开口向上。相比之下，y = −x² + 4x 中 a = −1，因此抛物线开口向下。

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2. The Roots and the Discriminant | 根与判别式

The roots of a quadratic equation are the x-coordinates where the graph crosses the x-axis, i.e. where y = 0. To find the roots, you solve ax² + bx + c = 0. This can be done by factorisation, completing the square, or using the quadratic formula:

二次方程的根是图像与 x 轴交点的 x 坐标，即 y = 0 时的 x 值。求根需要解 ax² + bx + c = 0，可以通过因式分解、配方法或求根公式来完成：

x = (−b ± √(b² − 4ac)) / 2a

The expression b² − 4ac is called the discriminant, often written as Δ. It tells us how many real roots the graph has:

表达式 b² − 4ac 称为判别式，通常记为 Δ。它告诉我们图像有多少个实数根：

• If Δ > 0: two distinct real roots, meaning the graph crosses the x-axis at two points.
• 如果 Δ > 0：有两个不同的实数根，图像与 x 轴有两个交点。
• If Δ = 0: one repeated real root, meaning the graph touches the x-axis at exactly one point (the vertex lies on the x-axis).
• 如果 Δ = 0：有一个重根，图像与 x 轴只有一个切点（顶点在 x 轴上）。
• If Δ < 0: no real roots, meaning the graph does not cross the x-axis at all.
• 如果 Δ < 0：没有实数根，图像与 x 轴没有交点。

For example, for y = x² − 5x + 6, we have Δ = (−5)² − 4(1)(6) = 25 − 24 = 1 > 0, so there are two roots. Factorising gives (x − 2)(x − 3) = 0, so the roots are x = 2 and x = 3.

例如，对于 y = x² − 5x + 6，计算得 Δ = (−5)² − 4(1)(6) = 25 − 24 = 1 > 0，因此有两个根。因式分解得 (x − 2)(x − 3) = 0，所以根为 x = 2 和 x = 3。

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3. The Vertex (Turning Point) | 顶点（转向点）

The vertex is the highest or lowest point of the parabola. For a quadratic written in the form y = ax² + 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。对于 y = ax² + bx + c 形式的二次函数，顶点的 x 坐标由下式给出：

x = −b / 2a

Once you have this x-value, substitute it back into the original equation to find the y-coordinate. For example, for y = x² − 4x + 3, we have a = 1 and b = −4, so x = −(−4) / (2 × 1) = 2. Substituting x = 2 gives y = 4 − 8 + 3 = −1. So the vertex is at (2, −1).

求出 x 值后，将其代回原方程即可得到 y 坐标。例如，对于 y = x² − 4x + 3，a = 1，b = −4，所以 x = −(−4) / (2 × 1) = 2。将 x = 2 代入得 y = 4 − 8 + 3 = −1。因此顶点为 (2, −1)。

Alternatively, if the quadratic is written in completed square form y = a(x − h)² + k, then the vertex is simply (h, k). This is often the fastest way to identify the turning point.

另外，如果二次函数写成配方法形式 y = a(x − h)² + k，那么顶点直接就是 (h, k)。这通常是找转向点最快的方法。

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4. The Axis of Symmetry | 对称轴

Every parabola is symmetric about a vertical line that passes through its vertex. This line is called the axis of symmetry. Its equation is simply x = −b / 2a, the same as the x-coordinate of the vertex.

每一条抛物线都关于一条穿过顶点的竖直线对称，这条线称为对称轴。它的方程就是 x = −b / 2a，与顶点的 x 坐标相同。

For y = x² − 4x + 3, the axis of symmetry is x = 2. This means that if you take any point on the curve, such as (0, 3), its mirror image across the line x = 2 will also lie on the curve. The mirror point is (4, 3), because 0 and 4 are equidistant from 2.

对于 y = x² − 4x + 3，对称轴是 x = 2。这意味着如果你取曲线上任意一点，比如 (0, 3)，它关于直线 x = 2 的镜像点也会在曲线上。镜像点为 (4, 3)，因为 0 和 4 到 2 的距离相等。

When sketching a quadratic graph, always draw the axis of symmetry as a dashed vertical line first. This helps you ensure that the two halves of the parabola are mirror images of each other.

在画二次函数草图时，应先用虚线画出对称轴。这有助于你确保抛物线的两半互为镜像。

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5. The y-Intercept | y 轴截距

The y-intercept is the point where the graph crosses the y-axis. This occurs when x = 0. For y = ax² + bx + c, substituting x = 0 gives y = c. Therefore, the y-intercept is always at (0, c). This is one of the easiest points to find on a quadratic graph.

y 轴截距是图像与 y 轴的交点，发生在 x = 0 时。对于 y = ax² + bx + c，代入 x = 0 得到 y = c。因此，y 轴截距总是 (0, c)。这是二次函数图像上最容易找到的点之一。

For example, the graph of y = 2x² + 3x − 5 crosses the y-axis at (0, −5). When sketching the graph, always mark this point clearly, along with the roots and the vertex.

例如，y = 2x² + 3x − 5 的图像在 (0, −5) 处与 y 轴相交。画图时，务必清楚标出这个点，同时标出根和顶点。

Sometimes the y-intercept can give you a quick check on whether your vertex calculation is correct. If your sketch places the vertex above the y-intercept but the parabola opens upwards, something is wrong.

有时 y 轴截距可以帮助你快速检查顶点计算是否正确。如果草图显示顶点在 y 轴截距上方，但抛物线开口向上，那么一定有地方出错了。

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6. Completing the Square | 配方法

Completing the square is a powerful technique for rewriting a quadratic expression in the form a(x − h)² + k. This form directly reveals the vertex and the line of symmetry. The process works as follows:

配方法是一种将二次表达式改写为 a(x − h)² + k 形式的重要技巧。这种形式可以直接显示顶点和对称轴。步骤如下：

x² + bx = (x + b/2)² − (b/2)²

For example, to complete the square for x² + 6x + 5, first write x² + 6x as (x + 3)² − 9. Then add the constant term: x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4. Therefore the vertex is (−3, −4) and the axis of symmetry is x = −3.

例如，对 x² + 6x + 5 配方：先将 x² + 6x 写成 (x + 3)² − 9，再加上常数项：x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4。因此顶点为 (−3, −4)，对称轴为 x = −3。

If the coefficient of x² is not 1, factor it out first before completing the square. For example, 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5.

如果 x² 的系数不是 1，先提取该系数再配方。例如，2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5。

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7. Sketching a Quadratic Graph | 画二次函数草图

To sketch a quadratic graph accurately, you should follow a systematic sequence. First, determine whether the parabola opens upwards or downwards by looking at the sign of a. Second, find the y-intercept by setting x = 0. Third, find the roots by solving ax² + bx + c = 0. Fourth, find the vertex using x = −b / 2a. Finally, plot all these points and draw a smooth curve through them.

要准确地画出二次函数草图，你应该按照系统化的步骤进行。首先，通过 a 的符号判断抛物线开口方向。其次，令 x = 0 求 y 轴截距。第三，解 ax² + bx + c = 0 求根。第四，用 x = −b / 2a 求顶点。最后，标出所有点并画一条平滑曲线穿过它们。

Let us sketch y = x² − 2x − 3. Here a = 1 > 0, so the parabola opens upwards. The y-intercept is (0, −3). Factoring gives (x − 3)(x + 1) = 0, so the roots are x = 3 and x = −1. The vertex has x = −(−2) / 2 = 1, and y = 1 − 2 − 3 = −4, so the vertex is (1, −4). The graph crosses the x-axis at −1 and 3, and the axis of symmetry is x = 1.

我们来画 y = x² − 2x − 3 的草图。这里 a = 1 > 0，所以抛物线开口向上。y 轴截距为 (0, −3)。因式分解得 (x − 3)(x + 1) = 0，所以根为 x = 3 和 x = −1。顶点 x 坐标为 −(−2) / 2 = 1，y = 1 − 2 − 3 = −4，所以顶点为 (1, −4)。图像在 −1 和 3 处穿过 x 轴，对称轴为 x = 1。

Remember that a sketch does not need to be drawn to scale like a graph plot, but it must show the correct shape and the correct coordinates of all key points. Labels are essential.

记住，草图不需要像坐标图那样严格按比例绘制，但必须显示正确的形状以及所有关键点的正确坐标。标注非常重要。

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

Quadratic graphs can be transformed by translating, reflecting, or stretching them. These transformations are summarised by changes to the equation y = a(x − h)² + k:

二次函数图像可以通过平移、反射或伸缩进行变换。这些变换可以通过方程 y = a(x − h)² + k 的变化来总结：

Transformation | 变换	Change to equation | 方程变化
Translation upwards by k | 向上平移 k 个单位	Add k to the whole expression
Translation to the right by h | 向右平移 h 个单位	Replace x by (x − h)
Reflection in the x-axis | 关于 x 轴反射	Multiply the whole expression by −1
Stretch vertically | 纵向伸缩	Increase the magnitude of a

For example, the graph of y = x² has its vertex at (0, 0). The graph of y = (x − 3)² + 2 is the same parabola shifted 3 units to the right and 2 units upwards, so its vertex is at (3, 2). The graph of y = −(x − 3)² + 2 is then reflected in the x-axis, opening downwards instead of upwards.

例如，y = x² 的图像顶点在 (0, 0)。y = (x − 3)² + 2 的图像是将同一条抛物线向右平移 3 个单位、向上平移 2 个单位，因此顶点在 (3, 2)。y = −(x − 3)² + 2 的图像则是关于 x 轴反射后的结果，开口向下而非向上。

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9. Solving Quadratic Equations Graphically | 用图像法解二次方程

One common exam question asks you to solve a quadratic equation by drawing its graph. The solutions are simply the x-coordinates where the curve crosses the x-axis. For example, if the graph of y = x² − 4x + 3 is drawn, the roots x = 1 and x = 3 can be read directly from the intersections with the x-axis.

一种常见的考试题型要求你通过画图来解二次方程。解就是曲线与 x 轴交点的 x 坐标。例如，如果画出了 y = x² − 4x + 3 的图像，根 x = 1 和 x = 3 可以直接从与 x 轴的交点读出。

Sometimes the question gives you the graph of y = x² − 4x + 3 and asks you to solve a different equation, such as x² − 4x + 1 = 0. In this case, rewrite the target equation as x² − 4x + 3 = 2. The solutions are the x-coordinates of the points where the given curve intersects the horizontal line y = 2.

有时题目给出了 y = x² − 4x + 3 的图像，却要求你解另一个方程，例如 x² − 4x + 1 = 0。此时，将目标方程改写为 x² − 4x + 3 = 2。解就是给定曲线与水平线 y = 2 的交点的 x 坐标。

This method works because the graph of a quadratic function is a visual representation of all pairs (x, y) that satisfy the equation. Intersecting it with another line isolates the points that satisfy both equations simultaneously.

这个方法之所以有效，是因为二次函数图像是满足该方程的所有 (x, y) 数对的视觉表示。将其与另一条线相交，就能找出同时满足两个方程的点。

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10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often make a few predictable mistakes when dealing with quadratic graphs. One common error is forgetting that the roots are solutions to ax² + bx + c = 0, not points on the y-axis. Another is mixing up the direction of the parabola when a is negative. A third is losing the negative sign when using the quadratic formula.

学生在处理二次函数图像时常常犯一些可预测的错误。一个常见错误是忘记根是 ax² + bx + c = 0 的解，而不是 y 轴上的点。另一个错误是在 a 为负数时搞混抛物线的开口方向。第三个错误是在使用求根公式时丢失负号。

Here are some practical exam tips. Always write down the values of a, b, and c before substituting into the formula. Check the discriminant before attempting to factorise, so that you know whether roots exist. When sketching, label every significant point: roots, y-intercept, vertex, and axis of symmetry.

这里有一些实用的考试技巧。代入公式前，先写下 a、b、c 的值。在尝试因式分解前先检查判别式，以确定根是否存在。画草图时，标出每个关键点：根、y 轴截距、顶点和对称轴。

Finally, always check whether your graph makes sense. If a > 0, the vertex must be the lowest point. If c > 0, the y-intercept must be above the x-axis. A quick sanity check can catch more than half of all algebraic mistakes.

最后，始终检查你的图像是否合理。如果 a > 0，顶点必须是最低点。如果 c > 0，y 轴截距必须在 x 轴上方。快速合理性检查可以发现一半以上的代数错误。

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11. Worked Example | 例题详解

Let us work through a full example from start to finish. Consider the quadratic function y = 2x² − 8x + 6.

让我们从头到尾完整地做一道例题。考虑二次函数 y = 2x² − 8x + 6。

Step 1: Identify a, b, and c. Here a = 2, b = −8, c = 6. Since a = 2 > 0, the parabola opens upwards.

第一步：确定 a、b、c 的值。这里 a = 2，b = −8，c = 6。因为 a = 2 > 0，抛物线开口向上。

Step 2: Find the y-intercept. Setting x = 0 gives y = 6, so the y-intercept is (0, 6).

第二步：求 y 轴截距。令 x = 0 得 y = 6，所以 y 轴截距为 (0, 6)。

Step 3: Find the roots. Solve 2x² − 8x + 6 = 0. Factor out 2 first: 2(x² − 4x + 3) = 0. Then factorise: 2(x − 1)(x − 3) = 0. Therefore x = 1 or x = 3.

第三步：求根。解 2x² − 8x + 6 = 0。先提取 2：2(x² − 4x + 3) = 0。再因式分解：2(x − 1)(x − 3) = 0。因此 x = 1 或 x = 3。

Step 4: Find the vertex. Using x = −b / 2a, we get x = −(−8) / (2 × 2) = 8 / 4 = 2. Substituting x = 2 gives y = 2(4) − 16 + 6 = 8 − 16 + 6 = −2. So the vertex is (2, −2).

第四步：求顶点。使用公式 x = −b / 2a，得 x = −(−8) / (2 × 2) = 8 / 4 = 2。将 x = 2 代入得 y = 2(4) − 16 + 6 = 8 − 16 + 6 = −2。所以顶点为 (2, −2)。

Step 5: Sketch the graph. The parabola opens upwards, crosses the y-axis at (0, 6), crosses the x-axis at (1, 0) and (3, 0), and has its vertex at (2, −2). The axis of symmetry is x = 2.

第五步：画图。抛物线开口向上，在 (0, 6) 处与 y 轴相交，在 (1, 0) 和 (3, 0) 处与 x 轴相交，顶点在 (2, −2)。对称轴为 x = 2。

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12. Conclusion | 总结

Quadratic graphs are central to IGCSE Mathematics. Once you master the general form, the discriminant, the vertex, the axis of symmetry, and the method of sketching, you will be able to solve a wide range of exam questions confidently. Remember the key formulas and always practise by drawing graphs by hand before using a calculator.

二次函数图像是 IGCSE 数学的核心内容。一旦你掌握了一般形式、判别式、顶点、对称轴以及画草图的方法，你就能自信地解决各种考试题目。记住关键公式，并且在使用计算器之前勤于动手画图练习。

Keep practising with past paper questions, especially those that ask you to interpret graphs or find unknown coefficients. The more you work with quadratic graphs, the more natural the process becomes.

坚持用历年真题练习，尤其是那些要求你解读图像或求未知系数的题目。你对二次函数图像练习得越多，这个过程就会变得更加自然。

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