Reflecting Graphs | 图形的反射

📚 Reflecting Graphs | 图形的反射

Reflection is one of the key graph transformations you need to master for IGCSE Mathematics. When you reflect a graph, you produce its mirror image across a given line. In the Edexcel syllabus, the two most important reflections are across the x-axis and across the y-axis.

反射是 IGCSE 数学中需要掌握的重要图像变换之一。当你反射一个图像时，你会得到它关于某条直线的镜像。在 Edexcel 考纲中，最重要的两种反射分别是关于 x 轴的反射和关于 y 轴的反射。

1. What Is a Graph Reflection? | 什么是图像反射？

A reflection is a transformation that flips a graph over a line, called the line of reflection or mirror line. The original graph and the reflected graph are always congruent: they have the same shape and size, but opposite orientation.

反射是一种将图像沿着一条直线翻转的变换，这条直线称为反射轴或镜线。原图像与反射后的图像始终全等：它们的形状和大小完全相同，但方向相反。

For IGCSE Edexcel, you only need to reflect graphs in the coordinate axes. However, understanding the effect on the equation is essential, because you will often be asked to write the equation of the reflected graph.

对于 IGCSE Edexcel，你只需要掌握关于坐标轴的图像反射。然而，理解对方程的影响至关重要，因为你经常需要写出反射后图像的方程。

2. Reflection in the x-axis | 关于 x 轴的反射

When a graph is reflected in the x-axis, every point moves to the other side of the x-axis while keeping its horizontal position. In coordinate terms, the y-coordinate changes sign, and the x-coordinate stays the same:

当一个图像关于 x 轴反射时，每个点都会移动到 x 轴的另一侧，同时保持其水平位置不变。用坐标来表达，就是 y 坐标改变符号，而 x 坐标保持不变：

(x, y) → (x, -y)

If the original function is y = f(x), then the reflected function is y = -f(x). This is because every y-value is multiplied by -1.

如果原函数是 y = f(x)，那么反射后的函数就是 y = -f(x)。这是因为每一个 y 值都被乘以 -1。

For example, the graph of y = x² becomes y = -x² when reflected in the x-axis. The parabola opens upwards, but after reflection it opens downwards.

例如，y = x² 的图像在关于 x 轴反射后变成 y = -x²。原来的抛物线开口向上，反射后开口向下。

3. Reflection in the y-axis | 关于 y 轴的反射

When a graph is reflected in the y-axis, every point moves to the other side of the y-axis while keeping its vertical position. This time the x-coordinate changes sign, and the y-coordinate stays the same:

当一个图像关于 y 轴反射时，每个点都会移动到 y 轴的另一侧，同时保持其垂直位置不变。这次是 x 坐标改变符号，而 y 坐标保持不变：

(x, y) → (-x, y)

If the original function is y = f(x), then the reflected function is y = f(-x). You replace every x in the original formula with -x.

如果原函数是 y = f(x)，那么反射后的函数就是 y = f(-x)。你需要用 -x 替换原公式中的每一个 x。

For example, the graph of y = 2x + 3 becomes y = 2(-x) + 3 = -2x + 3 when reflected in the y-axis.

例如，y = 2x + 3 的图像在关于 y 轴反射后变成 y = 2(-x) + 3 = -2x + 3。

4. Coordinate Rules Summary | 坐标规则总结

The table below summarises the coordinate changes for both reflections. You should memorise these rules because they appear in many questions.

下表总结了两种反射的坐标变化。你应该牢记这些规则，因为它们会出现在许多题目中。

Reflection	Coordinate change	Function change
In the x-axis	(x, y) → (x, -y)	y = f(x) → y = -f(x)
In the y-axis	(x, y) → (-x, y)	y = f(x) → y = f(-x)

A common mistake is confusing the two rules. Always ask yourself: which coordinate is changing sign? For the x-axis reflection, the y-coordinate changes. For the y-axis reflection, the x-coordinate changes.

一个常见的错误是混淆这两条规则。记得问自己：哪个坐标在改变符号？对于 x 轴反射，y 坐标改变。对于 y 轴反射，x 坐标改变。

5. Reflecting Linear Graphs | 反射一次函数图像

Linear graphs are straight lines, so reflecting them produces another straight line with a different gradient or intercept.

一次函数图像是直线，所以反射后得到的仍然是直线，只是斜率或截距发生了变化。

Consider y = 2x + 1. Reflecting in the x-axis gives y = -(2x + 1) = -2x – 1. The gradient changes from 2 to -2, and the y-intercept changes from 1 to -1.

考虑 y = 2x + 1。关于 x 轴反射得到 y = -(2x + 1) = -2x – 1。斜率从 2 变为 -2，y 截距从 1 变为 -1。

Reflecting the same line in the y-axis gives y = 2(-x) + 1 = -2x + 1. The gradient changes sign, but the y-intercept remains 1.

将同一条直线关于 y 轴反射得到 y = 2(-x) + 1 = -2x + 1。斜率改变符号，但 y 截距仍为 1。

Notice how reflection in the x-axis flips the line over the horizontal axis, while reflection in the y-axis flips it over the vertical axis.

注意，关于 x 轴的反射是将直线绕水平轴翻转，而关于 y 轴的反射是将直线绕垂直轴翻转。

6. Reflecting Quadratic Graphs | 反射二次函数图像

Quadratic graphs are parabolas. Reflecting a quadratic function often produces a parabola that opens in the opposite direction, or has a different line of symmetry.

二次函数图像是抛物线。反射二次函数通常会产生一条开口方向相反或对称轴不同的抛物线。

Take y = (x – 2)² – 3. Its vertex is at (2, -3). Reflecting in the x-axis:

以 y = (x – 2)² – 3 为例。它的顶点为 (2, -3)。关于 x 轴反射：

y = -[(x – 2)² – 3] = -(x – 2)² + 3

The new vertex is (2, 3), and the parabola now opens downwards. The axis of symmetry x = 2 is unchanged because it lies vertically through the vertex.

新顶点为 (2, 3)，抛物线现在开口向下。对称轴 x = 2 不变，因为它垂直经过顶点。

Now reflect y = (x – 2)² – 3 in the y-axis:

现在将 y = (x – 2)² – 3 关于 y 轴反射：

y = ((-x) – 2)² – 3 = (-x – 2)² – 3 = (x + 2)² – 3

The vertex moves from (2, -3) to (-2, -3). The parabola opens upwards, but its axis of symmetry has changed from x = 2 to x = -2.

顶点从 (2, -3) 移动到 (-2, -3)。抛物线开口向上，但对称轴从 x = 2 变为 x = -2。

7. Reflecting Cubic and Other Graphs | 反射三次函数及其他图像

For more complex functions, apply the same rules: replace f(x) with -f(x) for x-axis reflection, or replace x with -x for y-axis reflection.

对于更复杂的函数，应用同样的规则：关于 x 轴反射用 -f(x) 替换 f(x)，关于 y 轴反射用 -x 替换 x。

Consider y = x³ – x. Reflecting in the x-axis:

考虑 y = x³ – x。关于 x 轴反射：

y = -(x³ – x) = -x³ + x

Reflecting in the y-axis:

关于 y 轴反射：

y = (-x)³ – (-x) = -x³ + x

Interestingly, for this particular function, both reflections give the same equation. That is because the original graph has rotational symmetry about the origin. Do not assume this is always true; it only happens for odd functions.

有趣的是，对于这个特定函数，两种反射得到相同的方程。这是因为原图像关于原点具有旋转对称性。不要假设这种情况总是成立；它只对奇函数才成立。

8. Invariant Points | 不动点

An invariant point is a point that does not move when a transformation is applied. Under a reflection, every point that lies on the mirror line stays exactly where it is.

不动点是在进行变换后保持不动的点。在反射变换下，所有位于镜线上的点都保持在原处。

For reflection in the x-axis, invariant points have y-coordinate 0. That is, every point where the curve meets the x-axis remains fixed.

对于关于 x 轴的反射，不动点的 y 坐标为 0。也就是说，曲线与 x 轴相交的每个点都保持不动。

For reflection in the y-axis, invariant points have x-coordinate 0. These are the points where the curve meets the y-axis.

对于关于 y 轴的反射，不动点的 x 坐标为 0。这些是曲线与 y 轴相交的点。

Knowing invariant points can help you check whether your reflected graph is correct. If the original graph crosses the x-axis at (a, 0), that point must also appear in the x-axis reflection.

了解不动点可以帮助你检查反射后的图像是否正确。如果原图像在 (a, 0) 处穿过 x 轴，那么该点也必须出现在关于 x 轴反射后的图像中。

9. Combining Reflections with Other Transformations | 将反射与其他变换组合

In IGCSE questions, reflections are sometimes combined with translations or stretches. When dealing with combinations, apply the transformations one at a time in the order specified.

在 IGCSE 题目中，反射有时会与平移或伸缩组合。处理组合时，需要按指定顺序逐个应用变换。

For example, reflect the curve y = x² in the x-axis and then translate it 3 units upwards. First reflect to get y = -x², then add 3 to get y = -x² + 3.

例如，将曲线 y = x² 关于 x 轴反射，然后向上平移 3 个单位。先反射得到 y = -x²，再加 3 得到 y = -x² + 3。

If you reflect in the x-axis and then in the y-axis, the net effect is equivalent to a rotation of 180° about the origin. The point (x, y) becomes (-x, -y). This is a useful shortcut in some examination questions.

如果你先关于 x 轴反射，再关于 y 轴反射，净效果等价于绕原点旋转 180°。点 (x, y) 变为 (-x, -y)。在某些考试题目中，这是一个有用的捷径。

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

Students often lose marks because of small sign errors. Here are the most important points to remember.

学生经常因为小的符号错误而丢分。以下是最需要记住的要点。

For reflection in the x-axis, use y = -f(x). Do not forget to distribute the negative sign across all terms.

关于 x 轴反射时，使用 y = -f(x)。不要忘记将负号分配到所有项。
For reflection in the y-axis, use y = f(-x). Place parentheses around -x, especially when dealing with exponents.

关于 y 轴反射时，使用 y = f(-x)。在处理指数时，一定给 -x 加上括号。
Check the position of invariant points to verify your answer.

检查不动点的位置来验证你的答案。
When sketching, always label the reflected graph clearly, for example with its equation or the coordinates of key points.

画草图时，始终清楚标注反射后的图像，例如写出它的方程或关键点的坐标。

One classic error is writing y = -f(-x) when a reflection in both axes is required. That is correct only for a rotation of 180°, not for a single-axis reflection.

一个经典错误是在需要关于两个轴反射时写成 y = -f(-x)。这只有对于 180° 旋转才是正确的，并不适用于单轴反射。

11. Worked Examples and Practice | 例题与练习

Try this worked example: The graph of y = 2x² – 4x + 1 is reflected in the x-axis. Find the equation of the new graph.

试试这个例题：y = 2x² – 4x + 1 的图像关于 x 轴反射。求新图像的方程。

Solution: Reflect in the x-axis means multiply the whole function by -1.

解答：关于 x 轴反射意味着将整个函数乘以 -1。

y = -(2x² – 4x + 1) = -2x² + 4x – 1

Now reflect the same function in the y-axis. Replace x with -x:

现在将同一个函数关于 y 轴反射。用 -x 替换 x：

y = 2(-x)² – 4(-x) + 1 = 2x² + 4x + 1

Notice that the x² term is unchanged because (-x)² = x². Only the linear term changed sign.

注意 x² 项没有改变，因为 (-x)² = x²。只有一次项改变了符号。

For extra practice, reflect y = (x + 1)³ in the y-axis. The answer should be y = (1 – x)³.

作为额外练习，将 y = (x + 1)³ 关于 y 轴反射。答案应该是 y = (1 – x)³。

12. Summary | 总结

Reflecting graphs is a straightforward topic if you remember two simple rules: across the x-axis changes the sign of y, and across the y-axis changes the sign of x. In function notation, these become y = -f(x) and y = f(-x).

只要记住两条简单规则，图像反射就是一个直接了当的考点：关于 x 轴反射改变 y 的符号，关于 y 轴反射改变 x 的符号。在函数记号中，它们分别写成 y = -f(x) 和 y = f(-x)。

Always draw a quick sketch to confirm your result. Check where the original graph crosses the axes, and make sure those points behave correctly under reflection. With regular practice, reflecting graphs will become one of your most reliable skills in the examination.

永远画出草图来确认你的结果。检查原图像在坐标轴上的交点，并确保这些点在反射后位置正确。通过定期练习，反射图像将成为你考试中最可靠的技能之一。

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Reflecting Graphs | 图形的反射