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9c Homework Answers: Reflections from the Title Verso | 9c作业答案：从扉页背面看反射

2026-08-21 14:11 · 本文部分内容由AI辅助创作

📚 9c Homework Answers: Reflections from the Title Verso | 9c作业答案：从扉页背面看反射

Have you ever opened a textbook and noticed that the back of the title page - the title verso - often shows a mirror-image layout or reversed text in design? In mathematics, the idea of "reversing" or "flipping" an object so that it becomes its mirror image is called a reflection. This article walks through the key points tested under reflection transformations, using the title verso as a visual reminder of what it means to flip across a line.

你是否曾翻开课本，注意到书名页背面——也就是扉页的反面——常常呈现出镜像排版或反向文字？在数学中，将一个对象“翻转”或“镜像”变成它的镜像图形的操作就叫反射变换。本文将以扉页背面为直观提醒，梳理反射变换的主要考点。

1. What Is a Reflection? | 什么是反射？

A reflection is a transformation that flips a shape over a fixed line called the mirror line. Every point of the original shape is mapped to a corresponding point on the opposite side of the mirror line, so that the mirror line is the perpendicular bisector of the segment joining each point and its image.

反射是一种将图形沿一条固定直线（称为镜面线）翻转的变换。原图形的每个点都映射到镜面线另一侧的对应点，使得镜面线是连接该点与其像点的线段的垂直平分线。

P → P′ with line ℓ as perpendicular bisector of PP′

We often write P′ (read "P prime") for the image of P under a reflection. The notation makes it clear which point corresponds to which after the flip.

我们常用 P′（读作“P 撇”）表示点 P 在反射下的像。这种记法可以清楚地表明翻转后点的对应关系。

2. The Mirror Line | 镜面线

and y = −x. Identifying the mirror line is the first step in solving a reflection problem.

镜面线可以是平面内的任意直线，但考试中常用坐标轴或对角线 y = x 和 y = −x。确定镜面线是解决反射问题的第一步。

对应的中文为：x 轴：y = 0；y 轴：x = 0；上升对角线：y = x；下降对角线：y = −x。

Once you have identified the mirror line, you can apply the correct coordinate rule or draw the perpendicular distance accurately. Many marks are lost simply by using the wrong mirror line.

一旦确定了镜面线，你就可以应用正确的坐标规则或准确绘制垂直距离。很多失分都是因为选错了镜面线。

3. Reflecting Points Across the x-axis and y-axis | 关于 x 轴和 y 轴的反射

When a point is reflected across the x-axis, its x-coordinate stays the same while its y-coordinate changes sign. This is because the point moves vertically across the horizontal axis.

当点关于 x 轴反射时，其 x 坐标保持不变，而 y 坐标变号。这是因为点沿竖直方向穿过水平轴移动。

Mₓ: (x, y) → (x, −y)

When a point is reflected across the y-axis, its y-coordinate stays the same while its x-coordinate changes sign. Here the point moves horizontally across the vertical axis.

当点关于 y 轴反射时，其 y 坐标保持不变，而 x 坐标变号。此时点沿水平方向穿过竖直轴移动。

My: (x, y) → (−x, y)

Original point P(3, 5) Reflection in x-axis: P′(3, −5) Reflection in y-axis: P′(−3, 5)
原有点 P(3, 5) 关于 x 轴反射：P′(3, −5) 关于 y 轴反射：P′(−3, 5)

Notice that only one coordinate changes sign for these simple reflections. Keeping a small table like this in your notes can help you avoid sign errors under time pressure.

注意这些简单反射中只有一个坐标变号。在笔记中保留这样的小表格可以帮助你在时间紧张时避免符号错误。

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4. Reflecting Across the Lines y = x and y = −x | 关于 y = x 和 y = −x 的反射

nd y-coordinates: (x, y) → (y, x). This line has gradient 1 and passes through the origin.

关于 y = x 的反射交换 x 坐标和 y 坐标：(x, y) → (y, x)。这条直线的斜率为 1，且经过原点。

M on y = x: (x, y) → (y, x)

th signs: (x, y) → (−y, −x). This line has gradient −1 and also passes through the origin.

关于 y = −x 的反射交换坐标并同时改变两个符号：(x, y) → (−y, −x)。这条直线的斜率为 −1，也经过原点。

M on y = −x: (x, y) → (−y, −x)

flection involves both swapping and sign changes. A good check is to take a test point such as (2, 1) and confirm that its image is (−1, −2).

许多学生忘记了关于 y = −x 的反射既涉及交换又涉及变号。一个有效的检验方法是取一个测试点，如 (2, 1)，并确认它的像是 (−1, −2)。

5. Matrix Form of Reflections | 反射的矩阵形式

epresented by a 2×2 matrix. For the standard lines, these matrices are written as follows.

在线性代数和 A-level 进阶数学中，关于过原点直线的反射可用 2×2 矩阵表示。对于标准直线，这些矩阵如下所示。

Reflection in x-axis [[1, 0], [0, −1]]
关于 x 轴反射 [[1, 0], [0, −1]]
Reflection in y-axis [[−1, 0], [0, 1]]
关于 y 轴反射 [[−1, 0], [0, 1]]
Reflection in y = x [[0, 1], [1, 0]]
关于 y = x 反射 [[0, 1], [1, 0]]
Reflection in y = −x [[0, −1], [−1, 0]]
关于 y = −x 反射 [[0, −1], [−1, 0]]

Applying the matrix to a column vector (x, y) gives the reflected coordinates. For example, multiplying the x-axis matrix by (3, 5) gives (3, −5), matching the rule from Section 3.

对列向量 (x, y) 应用该矩阵即可得到反射后的坐标。例如，将 x 轴反射矩阵乘以 (3, 5) 得到 (3, −5)，与第 3 节的规则一致。

6. Properties of Reflections | 反射的性质

A reflection is an isometry, meaning it preserves lengths and angles. The image is congruent to the original shape. However, it reverses orientation: a clockwise arrangement becomes counter-clockwise after reflection.

反射是一种等距变换，即保持长度和角度不变。像图形与原图形全等。但它会反转方向：顺时针排列在反射后变为逆时针排列。