Translating Graphs | 函数图像的平移

📚 Translating Graphs | 函数图像的平移

Graph transformations are an important topic in IGCSE Edexcel Mathematics. Among these, translation is the simplest because it changes the position of a graph without changing its shape, orientation, or size. In this revision guide, we will explain how to translate graphs using vector notation, how to write down the equation of a translated graph, and how to avoid the most common mistakes.

图像变换是爱德思 IGCSE 数学中的一个重要章节。其中，平移是最简单的一种，因为它只改变图像的位置，而不改变其形状、方向或大小。在本复习指南中，我们将讲解如何使用向量记号平移图像、如何写出平移后图像的方程，以及如何避免最常见的错误。

1. What is a Translation? | 什么是平移？

A translation moves every point on a graph by the same horizontal and vertical component. If a point (x, y) is translated by the vector (a, b), its new position is (x + a, y + b). A positive a moves the point to the right; a negative a moves it to the left. A positive b moves the point upwards; a negative b moves it downwards.

平移按相同的水平和垂直分量移动图形上的每一点。若点 (x, y) 沿向量 (a, b) 平移，则新位置为 (x + a, y + b)。a 为正时点向右移动，a 为负时向左移动；b 为正时点向上移动，b 为负时向下移动。

For example, the point (2, 3) translated by the vector (-1, 4) becomes (1, 7). The whole graph is made up of infinitely many points, and every one of those points moves using the same rule.

例如，点 (2, 3) 沿向量 (-1, 4) 平移后变为 (1, 7)。整幅图像由无数个点组成，其中每个点都按照相同的规则移动。

2. Vertical Translations | 垂直平移

Adding a constant outside a function moves the graph vertically. If k > 0, the graph of y = f(x) + k is obtained by shifting y = f(x) upwards by k units. If k < 0, the graph of y = f(x) + k is shifted downwards by |k| units.

在函数外侧加上常数会使图像垂直移动。当 k > 0 时，y = f(x) + k 的图像是 y = f(x) 向上平移 k 个单位；当 k < 0 时，y = f(x) + k 的图像是向下平移 |k| 个单位。

Vertical translations change only the y-coordinates of every point. The x-coordinates stay the same, so the shape and horizontal spacing of the graph are completely preserved.

垂直平移只改变每个点的 y 坐标，x 坐标保持不变，因此图像的形状和水平间距完全不会改变。

Example: the graph of y = x² is shifted up by 3 units. The new equation is y = x² + 3. The vertex moves from (0, 0) to (0, 3), and the point (1, 1) moves to (1, 4).

例如：y = x² 的图像向上平移 3 个单位，新方程为 y = x² + 3。顶点从 (0, 0) 移动至 (0, 3)，点 (1, 1) 移动至 (1, 4)。

3. Horizontal Translations | 水平平移

To translate a graph horizontally, you replace x in f(x) by (x – a). For a > 0, y = f(x – a) is the graph of y = f(x) shifted right by a units. For a > 0, y = f(x + a) is shifted left by a units. Notice that the sign inside the bracket appears to be the opposite of the direction of movement: subtract for a shift to the right, add for a shift to the left.

要水平平移图像，需要将 f(x) 中的 x 替换为 (x – a)。当 a > 0 时，y = f(x – a) 是 y = f(x) 向右平移 a 个单位；y = f(x + a) 是向左平移 a 个单位。注意，括号内的符号看起来与移动方向相反：向右移用减，向左移用加。

Why does this happen? Suppose the original graph has a point at (x₀, y₀). After a shift to the right by a, that point becomes (x₀ + a, y₀). To get the new y-coordinate from the new x-coordinate, we must input (x – a). Therefore the new function is y = f(x – a).

这是为什么呢？假设原图像上有一点 (x₀, y₀)。向右平移 a 后，该点变为 (x₀ + a, y₀)。为了从新的 x 坐标得到新的 y 坐标，我们必须计算 f(x – a)。因此新函数为 y = f(x – a)。

Example: the graph of y = x² shifted right by 2 units becomes y = (x – 2)². Shifted left by 2 units, it becomes y = (x + 2)².

例如：y = x² 的图像向右平移 2 个单位后变为 y = (x – 2)²；向左平移 2 个单位后变为 y = (x + 2)²。

4. General Translation Vector | 一般平移向量

The most compact way to describe a translation is with a vector. The transformation that maps y = f(x) to y = f(x – p) + q is a translation by the vector (p, q). Here p is the horizontal shift and q is the vertical shift. A positive p means the graph moves to the right; a positive q means the graph moves upwards.

描述平移最简洁的方式是使用向量。将 y = f(x) 变为 y = f(x – p) + q 的变换是沿向量 (p, q) 的平移。其中 p 是水平位移，q 是垂直位移。p 为正表示图像向右移动，q 为正表示图像向上移动。

y = f(x) → y = f(x – p) + q

For example, a translation by vector (1, 2) applied to y = 1/x gives the new equation y = 1/(x – 1) + 2. The vertical asymptote x = 0 moves to x = 1, and the horizontal asymptote y = 0 moves to y = 2.

例如，将平移向量 (1, 2) 作用于 y = 1/x，得到新方程 y = 1/(x – 1) + 2。垂直渐近线 x = 0 移动至 x = 1，水平渐近线 y = 0 移动至 y = 2。

5. Worked Example 1 | 例题一

Question: The graph of y = f(x) is translated by the vector (3, -2). Write down the equation of the translated graph. If f(x) = x² – 4x, find the new equation in simplified form.

题目：函数 y = f(x) 的图像沿向量 (3, -2) 平移，写出平移后图像的方程。若 f(x) = x² – 4x，求化简后的新方程。

The general rule is y = f(x – p) + q for a translation by vector (p, q). Here p = 3 and q = -2, so the new equation is y = f(x – 3) – 2.

通用规则是：沿向量 (p, q) 平移后，新方程为 y = f(x – p) + q。这里 p = 3，q = -2，所以新方程为 y = f(x – 3) – 2。

Now substitute f(x) = x² – 4x. We need f(x – 3): f(x – 3) = (x – 3)² – 4(x – 3). Expand: (x – 3)² = x² – 6x + 9, and -4(x – 3) = -4x + 12. Therefore f(x – 3) = x² – 10x + 21. Then subtract 2: y = x² – 10x + 19.

接着代入 f(x) = x² – 4x。先求 f(x – 3)：f(x – 3) = (x – 3)² – 4(x – 3)。展开得 (x – 3)² = x² – 6x + 9，-4(x – 3) = -4x + 12，所以 f(x – 3) = x² – 10x + 21。最后再减 2：y = x² – 10x + 19。

Check: the original quadratic y = x² – 4x has its turning point at (2, -4). After translation by (3, -2), the new turning point should be (5, -6). The new equation y = (x – 5)² – 6 does indeed give a turning point at (5, -6), which matches.

检验：原二次函数 y = x² – 4x 的顶点在 (2, -4)。沿向量 (3, -2) 平移后，新顶点应为 (5, -6)。新方程 y = (x – 5)² – 6 确实给出顶点 (5, -6)，完全一致。

6. Worked Example 2 | 例题二

Question: The graph of y = x³ is translated 2 units left and 5 units up. Find the equation of the resulting graph.

题目：y = x³ 的图像先向左平移 2 个单位，再向上平移 5 个单位，求所得图像的方程。

A shift 2 units left is a horizontal translation with p = -2. A shift 5 units up gives q = 5. Therefore y = f(x + 2) + 5. Since f(x) = x³, the new equation is y = (x + 2)³ + 5.

向左平移 2 个单位相当于水平

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Translating Graphs | 函数图像的平移