Translating Graphs | 函数图像平移

📚 Translating Graphs | 函数图像平移

In Edexcel A-Level Mathematics, graph translations are one of the most important transformation skills. A translation slides a graph to a new position without changing its shape, size or orientation. This article covers vertical and horizontal translations, vector notation, combined translations, sketching, and common exam pitfalls.

在 Edexcel A-Level 数学中，函数图像平移是最重要的图像变换技能之一。平移将图像滑动到新位置，不改变其形状、大小或方向。本文涵盖垂直平移、水平平移、向量表示、复合平移、绘图以及常见考试易错点。

1. What is a Graph Translation? | 什么是图像平移

A graph translation moves every point of a curve by the same vector. If a graph y = f(x) is translated by the vector (a, b), the image is y = f(x − a) + b. The original curve and its image are congruent: no stretching, rotation or reflection occurs.

图像平移将曲线上每个点按同一向量移动。若 y = f(x) 的图像按向量 (a, b) 平移，所得图像为 y = f(x − a) + b。原曲线与其像是全等的：没有拉伸、旋转或反射。

y = f(x) → y = f(x − a) + b is a translation by vector (a, b)

This single form is the key to nearly all translation problems in the Edexcel specification. The letter a represents the horizontal shift and b represents the vertical shift.

这个统一形式是 Edexcel 考试大纲中几乎所有平移问题的关键。字母 a 表示水平位移，b 表示垂直位移。

2. Vertical Translations: y = f(x) + a | 垂直平移：y = f(x) + a

For a vertical translation, the x-coordinate stays the same while the y-coordinate changes. The curve y = f(x) + a is the graph of y = f(x) moved up by a units when a > 0, and down by |a| units when a < 0.

垂直平移时，x 坐标保持不变，y 坐标发生变化。若 a > 0，曲线 y = f(x) + a 是 y = f(x) 向上移动 a 个单位；若 a < 0，则向下移动 |a| 个单位。

y = f(x) + a → translation vector (0, a)

Example: y = x² + 3 is y = x² translated up by 3 units. The vertex moves from (0,0) to (0,3).

例如：y = x² + 3 是 y = x² 向上平移 3 个单位。顶点从 (0,0) 移到 (0,3)。

The constant added outside the function directly changes every output value. This is why the vertical translation is the easier of the two directions to interpret.

函数外部添加的常数会直接改变每一个输出值。这就是垂直平移比水平平移更容易理解的原因。

3. Horizontal Translations: y = f(x − a) | 水平平移：y = f(x − a)

Horizontal translations affect the x-coordinate, and the sign inside the bracket is the opposite of the direction of movement. y = f(x − a) moves the graph right by a units; y = f(x + a) moves it left by a units.

水平平移影响 x 坐标，括号内的符号与移动方向相反。y = f(x − a) 将图像向右平移 a 个单位；y = f(x + a) 将图像向左平移 a 个单位。

y = f(x − a) → translation vector (a, 0)
y = f(x + a) → translation vector (−a, 0)

Example: y = (x − 2)² is y = x² translated right by 2 units. The vertex becomes (2,0).

例如：y = (x − 2)² 是 y = x² 向右平移 2 个单位。顶点变为 (2,0)。

This opposite sign rule is a frequent source of errors. Always ask: what value of x makes the inside of the bracket equal to zero? That value is the new x-coordinate of the corresponding key point.

这种相反符号规则是常见的错误来源。始终要问：x 取什么值时括号内部等于零？这个值就是对应关键点的新 x 坐标。

4. Reading Translations from Equations | 从解析式读取平移

You can read off the translation vector directly from the form y = f(x − a) + b. The number inside the bracket gives the horizontal displacement; the number outside gives the vertical displacement.

你可以直接从 y = f(x − a) + b 的形式读出平移向量。括号内的数给出水平位移，括号外的数给出垂直位移。

In the first row, the graph has moved 3 units to the right and 2 units up. In the second row, it has moved 4 units to the left and 1 unit down.

第一行中，图像向右移动 3 个单位、向上移动 2 个单位。第二行中，图像向左移动 4 个单位、向下移动 1 个单位。

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