Reflection Transformations of Function Graphs | IB数学：函数图像的反射变换

📚 Reflection Transformations of Function Graphs | IB数学：函数图像的反射变换

In IB Mathematics, understanding how a function’s graph changes under reflection is essential for analysing transformations. Reflection is a type of transformation that produces a mirror image of the original graph across a given line, such as the x-axis, y-axis, or the line y = x.

在IB数学中，理解函数图像在反射变换下如何变化，是分析变换的核心内容。反射是一种变换，它沿某条给定直线（如x轴、y轴或直线 y = x）产生原始图形的镜像。

1. Reflection in the x-axis: y = −f(x) | 关于x轴的反射：y = −f(x)

When every point (x, y) on the graph of y = f(x) is mapped to (x, −y), the graph is reflected in the x-axis. The new function is written as y = −f(x).

当函数 y = f(x) 图像上的每一个点 (x, y) 被映射到 (x, −y) 时，图像就关于x轴发生了反射。新的函数记作 y = −f(x)。

Coordinates: (x, y) → (x, −y).

坐标变换：(x, y) → (x, −y)。
The x-intercepts remain unchanged; the sign of every y-value is reversed.

x轴截距保持不变；所有y值的符号被取反。
This is equivalent to multiplying the output of the function by −1.

这等价于将函数的输出值乘以−1。

If y = f(x), then its reflection in the x-axis is y = −f(x).

若 y = f(x)，则其关于x轴的反射为 y = −f(x)。

2. Reflection in the y-axis: y = f(−x) | 关于y轴的反射：y = f(−x)

Reflecting the graph of y = f(x) in the y-axis maps each point (x, y) to (−x, y). The resulting function is y = f(−x).

将 y = f(x) 的图像关于y轴反射，每个点 (x, y) 被映射到 (−x, y)。得到的函数是 y = f(−x)。

Coordinates: (x, y) → (−x, y).

坐标变换：(x, y) → (−x, y)。
The y-intercept remains unchanged; the domain is effectively mirrored.

y轴截距保持不变；定义域被镜像。
For even functions, f(−x) = f(x), so the graph is unchanged under this reflection.

对于偶函数，f(−x) = f(x)，因此图像在该反射下保持不变。

If y = f(x), then its reflection in the y-axis is y = f(−x).

若 y = f(x)，则其关于y轴的反射为 y = f(−x)。

3. Reflection in the origin: y = −f(−x) | 关于原点的反射：y = −f(−x)

A reflection in the origin, also called a central reflection, maps (x, y) to (−x, −y). Combining reflections in both axes gives y = −f(−x).

关于原点的反射，也称为中心反射，将 (x, y) 映射到 (−x, −y)。将关于两个坐标轴的反射结合，就得到 y = −f(−x)。

Coordinates: (x, y) → (−x, −y).

坐标变换：(x, y) → (−x, −y)。
This transformation is equivalent to a 180° rotation about the origin.

这个变换等价于绕原点旋转180°。
If f is an odd function, then −f(−x) = f(x), meaning the graph is invariant.

如果f是奇函数，则 −f(−x) = f(x)，意味着图像不变。

Reflection in the origin: y = −f(−x).

关于原点的反射：y = −f(−x)。

4. Reflection in the line y = x: inverse relation | 关于直线 y = x 的反射：反函数关系

Reflecting the graph of y = f(x) across the line y = x swaps the x- and y-coordinates of every point. This produces the graph of the inverse relation.

将 y = f(x) 的图像沿直线 y = x 反射，会交换每一个点的x坐标和y坐标。这产生的是反关系的图像。

Coordinates: (x, y) → (y, x).

坐标变换：(x, y) → (y, x)。
If f is a one-to-one function, the reflected graph represents its inverse function y = f⁻¹(x).

如果f是一一对应函数，反射后的图像就表示其反函数 y = f⁻¹(x)。
The domain of the original function becomes the range of the reflected relation, and vice versa.

原函数的定义域变为反射后关系的值域，反之亦然。

A function and its inverse are reflections of each other across y = x.

函数与其反函数关于 y = x 互为反射。

5. Reflection in the line y = −x | 关于直线 y = −x 的反射

Reflecting across the line y = −x maps (x, y) to (−y, −x). This transformation is less common but appears in advanced graph analysis.

关于直线 y = −x 的反射将 (x, y) 映射到 (−y, −x)。这个变换不太常见，但在高级图像分析中会出现。

Coordinates: (x, y) → (−y, −x).

坐标变换：(x, y) → (−y, −x)。
The slope of any line segment is negated and inverted in a specific way.

任何线段的斜率都会以特定方式被取负并求倒数。
This reflection is equivalent to a reflection in y = x followed by a reflection in the x-axis (or y-axis).

该反射等价于先关于 y = x 反射，再关于x轴（或y轴）反射。

6. Vertical vs. Horizontal Reflection: key differences | 垂直反射与水平反射：关键区别

Reflections in the x-axis and y-axis affect the graph in fundamentally different ways. It is important to distinguish between them.

关于x轴和y轴的反射在根本上是不同的，重要的是要区分它们。

Reflection axis	Transformation	Change to graph
x-axis	y = −f(x)	Vertical flip; y-values change sign
y-axis	y = f(−x)	Horizontal flip; x-values change sign
origin	y = −f(−x)	Both axes flipped

关键区别：关于x轴的反射改变y值的符号，关于y轴的反射改变x值的符号，而关于原点的反射两者都改变。

The key difference: reflection in the x-axis changes the sign of y-values, reflection in the y-axis changes the sign of x-values, and reflection in the origin changes both.

7. Worked example: quadratic function | 例题：二次函数

Consider f(x) = x² − 2x. Reflect it in the x-axis and in the y-axis, and describe each resulting graph.

考虑 f(x) = x² − 2x。将其分别关于x轴和y轴反射，并描述每个结果图像。

Reflection in x-axis: g(x) = −f(x) = −x² + 2x. This is a downward-opening parabola with the same x-intercepts as f.

关于x轴反射：g(x) = −f(x) = −x² + 2x。这是一个开口向下的抛物线，与f有相同的x轴截距。
Reflection in y-axis: h(x) = f(−x) = (−x)² − 2(−x) = x² + 2x. This shifts the vertex horizontally while preserving the upward opening.

关于y轴反射：h(x) = f(−x) = (−x)² − 2(−x) = x² + 2x。这使顶点水平移动，但保持开口向上。
Reflection in origin: k(x) = −f(−x) = −(x² + 2x) = −x² − 2x. This flips the parabola both vertically and horizontally.

关于原点反射：k(x) = −f(−x) = −(x² + 2x) = −x² − 2x。这使抛物线在垂直和水平方向都被翻转。

8. Worked example: exponential function | 例题：指数函数

Let f(x) = 2ˣ. Analyse the reflections of this function.

设 f(x) = 2ˣ。分析该函数的反射。

y = −f(x) = −2ˣ: the graph is flipped below the x-axis; the horizontal asymptote y = 0 remains, but values are negative.

y = −f(x) = −2ˣ：图像被翻转到x轴下方；水平渐近线 y = 0 不变，但函数值为负。
y = f(−x) = 2⁻ˣ = (1/2)ˣ: this reflects exponential growth into exponential decay.

y = f(−x) = 2⁻ˣ = (1/2)ˣ：这使指数增长反射为指数衰减。
y = −f(−x) = −2⁻ˣ: the graph is a decreasing curve below the x-axis, approaching y = 0 from below as x → ∞.

y = −f(−x) = −2⁻ˣ：图像是x轴下方的递减曲线，当 x → ∞ 时从下方趋近 y = 0。

9. Reflection and invariant points | 反射与不动点

Invariant points are points that stay fixed under a reflection. Identifying them helps to verify whether a transformation has been applied correctly.

不动点是在反射下保持不动的点。识别它们有助于验证变换是否被正确应用。

For y = −f(x), points on the x-axis (y = 0) are invariant.

对于 y = −f(x)，x轴上的点（y = 0）是不动点。
For y = f(−x), points on the y-axis (x = 0) are invariant.

对于 y = f(−x)，y轴上的点（x = 0）是不动点。
For y = −f(−x), the only invariant point is the origin (0, 0), provided it lies on the graph.

对于 y = −f(−x)，唯一的不动点是原点 (0, 0)，前提是原点在图像上。

10. Order of transformations with reflection | 反射与其他变换的顺序

When a graph involves multiple transformations, the order matters. Reflection is not always commutative with translations and scalings.

当图像涉及多种变换时，顺序很重要。反射并不总是与平移和伸缩可交换。

Example: starting from y = f(x), reflect in x-axis then shift up by 2 gives y = −f(x) + 2.

示例：从 y = f(x) 出发，先关于x轴反射，再向上平移2个单位，得到 y = −f(x) + 2。
If you shift first then reflect: y = −(f(x) + 2) = −f(x) − 2, which is different.

如果先平移再反射：y = −(f(x) + 2) = −f(x) − 2，结果是不同的。
Always follow the standard order: horizontal stretches/reflections affect x first, then horizontal shifts; vertical stretches/reflections affect y first, then vertical shifts.

始终遵循标准顺序：水平伸缩/反射先影响x，再进行水平平移；垂直伸缩/反射先影响y，再进行垂直平移。

11. Reflection in graph sketching: practical tips | 徒手画图中的反射技巧

When sketching reflected graphs by hand, use key points and asymptotes as guides.

徒手绘制反射图时，应以关键点和渐近线作为参照。

Identify the key points such as intercepts, turning points, and endpoints.

确定关键点，例如截距、转折点和端点。
Reflect each key point across the chosen line, then join them smoothly.

将每个关键点沿选定直线反射，然后平滑连接。
Check the behaviour near asymptotes: a horizontal asymptote y = c becomes y = −c under x-axis reflection.

检查渐近线附近的行为：水平渐近线 y = c 在关于x轴反射后变为 y = −c。

12. Summary: reflection rules for IB exams | 总结：IB考试中的反射规则

These rules are essential for examinations, both for Paper 2 (graphical display calculator) and Paper 3 (analysis).

这些规则在考试中至关重要，无论是Paper 2（图形计算器）还是Paper 3（分析）。

Transformation	Function	Effect on graph
Reflection in x-axis	−f(x)	Vertical flip
Reflection in y-axis	f(−x)	Horizontal flip
Reflection in origin	−f(−x)	Both flips
Reflection in y = x	f⁻¹(x) (if exists)	Inverse graph

Practice sketching these transformations on the same coordinate axes until the visual patterns become automatic.

在同一坐标系中反复练习绘制这些变换图，直到视觉模式变得自动熟练。

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Reflection Transformations of Function Graphs | IB数学：函数图像的反射变换