📚 Investigation 4 - Reflections | 探究4 – 反射
Reflections are fundamental geometric transformations that appear everywhere in mathematics and the real world. In this investigation we will explore the properties of reflections, their coordinate rules, matrix representations, and how multiple reflections combine to produce other transformations. Whether you are reflecting a point across a line, a function graph, or a shape, understanding reflections deepens your grasp of symmetry and linear transformations.
反射是基本的几何变换，在数学和现实世界中无处不在。在本探究中，我们将探索反射的性质、坐标规则、矩阵表示，以及多次反射如何组合产生其他变换。无论你是将点、函数图像还是图形进行反射，理解反射都能加深你对对称性和线性变换的掌握。
1. What is a Reflection? | 什么是反射？
A reflection is a transformation that flips a geometric figure over a line, called the line of reflection or mirror line. The reflected image is the same size and shape as the original, but the orientation may be reversed. Each point on the original has a corresponding point on the image such that the line of reflection is the perpendicular bisector of the segment joining these two points.
反射是一种将几何图形沿一条直线（称为反射线或镜线）翻转的变换。反射后的图像与原图形大小和形状相同，但方向可能相反。原图形上的每一点在像上都有一个对应点，使得反射线是连接这两点线段的垂直平分线。
Reflections preserve distances (isometry) and angles, but reverse orientation (e.g., clockwise becomes counterclockwise). In other words, a reflection is an improper isometry because it changes the handedness of the figure.
反射保持距离（等距变换）和角度，但反转方向（例如，顺时针变为逆时针）。换句话说，反射是一种非全等的等距变换，因为它改变了图形的左右取向。
2. The Role of the Mirror Line | 镜线的作用
The mirror line is the fixed point set under a reflection: every point on the line is invariant (maps to itself). For a given pre-image point P, its image P' is located so that the mirror line is the perpendicular bisector of PP'. This geometric property is key to constructing reflections manually and to deriving algebraic rules.
镜线是反射下的不动点集：线上每个点都保持不变（映射到自身）。对于给定的原像点 P，其像 P' 的位置使得镜线是 PP' 的垂直平分线。这一几何性质是手工画反射和推导代数规则的关键。
3. Reflection over the x-axis (y=0) | 关于x轴反射 (y=0)
) → (x, -y). The mirror line is y=0, and distances above and below the axis are preserved.
将点 (x, y) 关于x轴反射会改变y坐标的符号，而x坐标不变。映射规则是 (x, y) → (x, -y)。镜线为y=0，x轴上下方的距离保持不变。
cting across the x-axis transforms y = f(x) into y = -f(x). This is a vertical reflection.
对于函数图像，关于x轴反射将 y = f(x) 变为 y = -f(x)。这是垂直反射。
The corresponding transformation matrix is [[1, 0], [0, -1]], since (x', y') = (1*x + 0*y, 0*x - 1*y).
对应的变换矩阵是 [[1, 0], [0, -1]]，因为 (x', y') = (1*x + 0*y, 0*x - 1*y)。
4. Reflection over the y-axis (x=0) | 关于y轴反射 (x=0)
, while y stays the same. The mirror line x=0 implies symmetry with respect to the y-axis.
关于y轴反射的点 (x, y) 映射为 (-x, y)。x坐标变号，y坐标不变。镜线 x=0 意味着关于y轴对称。
tal reflection. For even functions, f(-x) = f(x), the graph is symmetric about the y-axis.
在函数中，y = f(x) 变为 y = f(-x)，是水平反射。对于偶函数，f(-x) = f(x)，其图像关于y轴对称。
The matrix for this reflection is [[-1, 0], [0, 1]].
该反射的矩阵是 [[-1, 0], [0, 1]]。
5. Reflection over the line y = x | 关于直线 y = x 反射
coordinates are equal, and it acts as a perfect diagonal mirror. This reflection yields the inverse relation of a function.
关于直线 y = x 反射会交换x和y坐标：(x, y) → (y, x)。直线 y = x 是所有坐标相等的点构成的直线，它充当完美的对角线镜。该反射得出函数的逆关系。
n f has an inverse f⁻¹, the graph of f⁻¹ is the reflection of the graph of f across y = x.
如果函数 f 有反函数 f⁻¹，那么 f⁻¹ 的图像就是 f 的图像关于 y = x 的反射。
The transformation matrix is [[0, 1], [1, 0]].
变换矩阵为 [[0, 1], [1, 0]]。
6. Reflection over the line y = -x | 关于直线 y = -x 反射
ollowed by point reflection through the origin, though a direct mirror line works as well.
关于 y = -x 反射将 (x, y) 映射为 (-y, -x)。两个坐标都变号并交换位置。这相当于组合：先关于 y = x 反射，再关于原点作点反射，当然直接用镜线也一样。
Matrix: [[0, -1], [-1, 0]].
矩阵：[[0, -1], [-1, 0]]。
This reflection swaps quadrants: a point in quadrant I maps to quadrant III, and quadrant II maps to quadrant IV.
该反射交换象限：第一象限的点映射到第三象限，第二象限的点映射到第四象限。
7. Reflection through the origin (point reflection) | 关于原点的点反射
Although not a line reflection, point reflection through the origin maps (x, y) to (-x, -y). This is equivalent to a 180° rotation about the origin. It is often called a central inversion and its transformation matrix is [[-1, 0], [0, -1]].
虽然不是线反射，但关于原点的点反射将 (x, y) 映射到 (-x, -y)。这相当于绕原点旋转180°。通常称为中心反演，变换矩阵为 [[-1, 0], [0, -1]]。
This transformation can be obtained by two successive reflections across two perpendicular lines intersecting at the origin, for example the x-axis and y-axis.
该变换可以通过关于两条相交于原点的垂直线（例如x轴和y轴）的两次连续反射得到。
8. General Reflection across any line through the origin | 关于任意过原点直线的反射
For a line through the origin making an angle θ with the positive x-axis, the reflection of a point (x,y) can be expressed in matrix form. The transformation matrix is:
[[cos(2θ), sin(2θ)], [sin(2θ), -cos(2θ)]]
对于经过原点且
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