A-Level Maths: Translations of Function Graphs | A-Level数学：函数图像平移变换

📚 A-Level Maths: Translations of Function Graphs | A-Level数学：函数图像平移变换

In A-Level Mathematics, understanding how algebraic changes to a function affect its graph is essential. Among all graph transformations, translations are the most intuitive yet the most frequently tested in Edexcel papers. This article will guide you through the rules, notation, and common pitfalls of function graph translations.

在A-Level数学中，理解函数代数形式的变化如何影响其图像是至关重要的。在所有图像变换中，平移是最直观的，但也是Edexcel考试中最常考查的内容。本文将系统讲解函数图像平移变换的规则、记号与常见易错点。

1. What Is a Translation? | 什么是平移变换？

A translation slides every point of a graph by the same vector without changing its shape or orientation. If the graph of y = f(x) is translated, the new curve is congruent to the original, meaning its size, shape, and rotational sense remain unchanged.

平移变换是将图像上的每一个点沿同一向量滑动，不改变图像的形状与方向。若 y = f(x) 的图像发生平移，新曲线与原曲线全等，即大小、形状和旋转方向都不改变。

In Edexcel, translations are usually described in two ways: using function notation such as y = f(x) + a or y = f(x – a), and using column vector notation. You must be able to switch fluently between these representations.

在Edexcel考试中，平移通常有两种描述方式：函数记号，例如 y = f(x) + a 或 y = f(x − a)；以及列向量记号。你必须能够熟练地在两种表示之间转换。

2. Vertical Translation: y = f(x) + a | 纵向平移：y = f(x) + a

The graph of y = f(x) + a is obtained by moving every point on y = f(x) vertically by a units. If a > 0, the graph moves upward; if a < 0, it moves downward.

y = f(x) + a 的图像是将 y = f(x) 上每一个点纵向移动 a 个单位得到的。当 a > 0 时，图像向上移动；当 a < 0 时，图像向下移动。

For example, if f(x) = x², then y = x² + 3 shifts the parabola up by 3 units, and the minimum point moves from (0, 0) to (0, 3).

例如，若 f(x) = x²，则 y = x² + 3 将抛物线向上平移3个单位，最低点从 (0, 0) 移到 (0, 3)。

This transformation does not affect the domain of f(x), but it does affect the range. If the range of f(x) is R, then the range of f(x) + a is { y + a : y ∈ R }.

该变换不影响函数 f(x) 的定义域，但会影响值域。若 f(x) 的值域为 R，则 f(x) + a 的值域为 { y + a : y ∈ R }。

Key rule: y = f(x) + a shifts up by a units.

关键规则：y = f(x) + a 向上平移 a 个单位。

3. Horizontal Translation: y = f(x − a) | 横向平移：y = f(x − a)

The graph of y = f(x − a) is obtained by shifting the graph of y = f(x) horizontally. Here the direction is opposite to the sign: y = f(x − a) moves the graph to the right by a units, while y = f(x + a) moves it to the left by a units.

y = f(x − a) 的图像是将 y = f(x) 的图像进行水平平移得到的。这里的移动方向与正负号相反：y = f(x − a) 将图像向右移动 a 个单位，而 y = f(x + a) 将图像向左移动 a 个单位。

For example, if f(x) = x², then y = (x − 2)² shifts the parabola right by 2 units, so its vertex moves from (0, 0) to (2, 0).

例如，若 f(x) = x²，则 y = (x − 2)² 将抛物线向右平移2个单位，顶点从 (0, 0) 移到 (2, 0)。

Why does the sign seem reversed? To achieve the same y-value as f(0), you now need x − 2 = 0, so x = 2. Thus the whole graph shifts right.

为什么符号看起来相反？为了得到与 f(0) 相同的 y 值，现在需要 x − 2 = 0，即 x = 2。因此整个图像向右移动。

Key rule: y = f(x − a) shifts right by a units; y = f(x + a) shifts left by a units.

关键规则：y = f(x − a) 向右平移 a 个单位；y = f(x + a) 向左平移 a 个单位。

4. Translation Vector Notation | 平移向量记号

In Edexcel, a translation may be described by a column vector. A vector written as

在Edexcel考试中，平移有时用列向量描述。向量写成

\(\begin{pmatrix} p \\ q \end{pmatrix}\)

( p 在上，q 在下 )

means the graph moves p units horizontally and q units vertically. A positive p means right, a negative p means left; a positive q means up, a negative q means down.

表示图像在水平方向上移动 p 个单位，在垂直方向上移动 q 个单位。p 为正表示向右，p 为负表示向左；q 为正表示向上，q 为负表示向下。

If the original function is y = f(x), then after translation by the vector (p, q), the new equation is y = f(x − p) + q. Note that the vector component q appears as an ordinary addition outside f, while p appears inside f with a minus sign.

若原函数为 y = f(x)，则沿向量 (p, q) 平移后的新方程为 y = f(x − p) + q。注意，向量的第二个分量 q 作为常数直接加在 f 外面，而第一个分量 p 出现在 f 内部且带负号。

For example, translating y = √x by the vector (3, −2) gives y = √(x − 3) − 2. The original start point (0, 0) moves to (3, −2).

例如，将 y = √x 沿向量 (3, −2) 平移，得到 y = √(x − 3) − 2。原起点 (0, 0) 移动到 (3, −2)。

5. Combined Translations | 组合平移

A graph can be translated both horizontally and vertically at the same time. The general form is y = f(x − p) + q, which corresponds to a translation by the vector (p, q).

图像可以同时进行水平平移和纵向平移。一般形式为 y = f(x − p) + q，对应沿向量 (p, q) 的平移。

It is important to apply the horizontal and vertical changes separately. The horizontal shift changes the input x, while the vertical shift changes the output f(x). These two effects do not interfere with each other.

重要的是分别处理水平变化和纵向变化。水平平移改变输入 x，纵向平移改变输出 f(x)。两种效果互不干扰。

For example, translating y = x³ by the vector (2, −5) gives y = (x − 2)³ − 5. Every point (a, a³) on the original curve moves to (a + 2, a³ − 5).

例如，将 y = x³ 沿向量 (2, −5) 平移，得到 y = (x − 2)³ − 5。原曲线上的每一点 (a, a³) 移动到 (a + 2, a³ − 5)。

Because horizontal and vertical translations commute, you may apply them in any order; the final graph is identical. However, be careful when other transformations such as stretches are also involved, because order then matters.

因为水平平移和纵向平移可以交换顺序，所以先做哪个后做哪个结果相同。但如果同时还涉及伸缩等其他变换，变换顺序就非常重要了。

6. Translating Key Points | 平移关键点

To sketch a translated graph, it is often enough to translate a few key points. For example, the vertex of a parabola, the centre of a circle, the intercepts, and the turning points of a cubic should all be moved by the same vector.

要画出平移后的图像，通常只需平移几个关键点即可。例如抛物线的顶点、圆心、截距以及三次函数的极值点，都应沿同一向量移动。

Suppose the original graph has a local maximum at A(2, 5). After a translation by the vector (−3, 4), the new local maximum is at (−1, 9). You can then redraw the curve with the same shape through the translated point.

假设原图像在 A(2, 5) 处有局部极大值。沿向量 (−3, 4) 平移后，新的局部极大值位于 (−1, 9)。然后可以保持图像形状不变，通过平移后的点重新画出曲线。

For intercepts, remember that the y-intercept moves with the vertical shift, but the x-intercept moves with the horizontal shift. In many questions, you must first calculate the original intercepts, then apply the vector to each point.

对于截距，y 截距随纵向平移移动，x 截距随水平平移移动。在很多题目中，需要先计算出原截距，再对每个点应用平移向量。

Vertex / minimum / maximum point: translate directly.

顶点 / 最小值点 / 最大值点：直接平移。
x-intercepts: shift by p only (their y-coordinate becomes 0 + q).

x 截距：仅水平移动 p（其 y 坐标变为 0 + q）。
y-intercept: shift by q only (its x-coordinate becomes 0 + p).

y 截距：仅纵向移动 q（其 x 坐标变为 0 + p）。

7. Effect on Asymptotes, Domain and Range | 对渐近线、定义域和值域的影响

Translations change the position of asymptotes in a predictable way. A vertical asymptote x = c is affected only by the horizontal translation: under y = f(x − p), the asymptote becomes x = c + p.

平移会以可预测的方式改变渐近线的位置。垂直渐近线 x = c 只受水平平移影响：在 y = f(x − p) 下，渐近线变为 x = c + p。

A horizontal asymptote y = d is affected only by the vertical translation: under y = f(x) + q, the asymptote becomes y = d + q. The domain of f(x) may shift if there is a vertical asymptote, while the range typically shifts with the vertical translation.

水平渐近线 y = d 只受纵向平移影响：在 y = f(x) + q 下，渐近线变为 y = d + q。如果存在垂直渐近线，函数的定义域会随之移动，而值域通常随纵向平移而整体移动。

For example, the function f(x) = 1/x has a vertical asymptote x = 0 and a horizontal asymptote y = 0. After translation by the vector (2, 3), the new function is y = 1/(x − 2) + 3, with asymptotes x = 2 and y = 3.

例如，函数 f(x) = 1/x 有垂直渐近线 x = 0 和水平渐近线 y = 0。沿向量 (2, 3) 平移后，新函数为 y = 1/(x − 2) + 3，渐近线变为 x = 2 和 y = 3。

Vertical shift affects the range; horizontal shift affects the domain.

纵向平移影响值域；水平平移影响定义域。

8. Identifying the Translation from an Equation | 从方程判断平移

Given an equation in the form y = f(x − p) + q, you can immediately read the translation vector (p, q). Always compare the new equation with the original function, not with a different base function.

给定一个形如 y = f(x − p) + q 的方程，你可以直接读出平移向量 (p, q)。一定要将新方程与原函数进行比较，而不是与其他基准函数比较。

For example, if the original function is f(x) = x² and the new equation is y = x² − 4x + 1, you must first rewrite it by completing the square: y = (x − 2)² − 3. Therefore the translation vector is (2, −3).

例如，若原函数为 f(x) = x²，新方程为 y = x² − 4x + 1，则必须先通过配方改写：y = (x − 2)² − 3。因此平移向量为 (2, −3)。

In exam questions, you may be asked to state the single transformation that maps y = f(x) onto y = f(x + 5) − 2. The correct answer is: a translation by the vector (−5, −2).

在考试中，你可能被要求说出将 y = f(x) 映射到 y = f(x + 5) − 2 的单一变换。正确答案是：沿向量 (−5, −2) 平移。

Notice that x + 5 is treated as x − (−5), so the horizontal component is −5. Many students mistakenly write +5 because they do not reverse the sign.

注意，x + 5 应看作 x − (−5)，因此水平分量为 −5。许多学生因为忘记符号反转而误写成 +5。

9. Common Pitfalls | 常见易错点

The horizontal translation sign is the most common source of error in this topic. Always remember: inside the function, minus means right, plus means left.

水平平移的符号是该知识点中最常见的错误来源。务必记住：在函数内部，减号代表向右，加号代表向左。

Another common mistake is confusing y = f(x) + 2 with y = f(x + 2). The first is a vertical translation by 2 units up; the second is a horizontal translation by 2 units left. They produce very different graphs.

另一个常见错误是混淆 y = f(x) + 2 与 y = f(x + 2)。前者是向上平移2个单位；后者是向左平移2个单位。它们产生的图像完全不同。

When translating a graph that has already been transformed, you must apply the translation to the current graph, not to the original equation. If y = f(2x) is translated by 1 unit right, the result is y = f(2(x − 1)), not y = f(2x − 1).

当对已经经过变换的图像进行平移时，你必须对当前图像进行平移，而不是对原方程直接修改。若 y = f(2x) 向右平移1个单位，结果应为 y = f(2(x − 1))，而不是 y = f(2x − 1)。

Do not swap the direction of vertical translations.

不要把纵向平移的方向弄反。
Do not forget that horizontal translations affect x-intercepts but not y-intercepts directly.

不要忘记水平平移影响 x 截距，但不直接改变 y 截距。
Read the question carefully: “translation by vector” and “translation of 3 units” require the same method but different answer formats.

仔细审题：“沿向量平移”和“平移3个单位”方法相同，但答案格式不同。

10. Worked Example 1: Quadratic | 例题1：二次函数

The curve y = x² − 2x is translated by the vector (3, 1). Find the equation of the new curve.

曲线 y = x² − 2x 沿向量 (3, 1) 平移。求新曲线的方程。

Step 1: Complete the square on the original function:

第一步：对原函数进行配方加工：

y = (x − 1)² − 1

Step 2: Apply the translation (3, 1) to get y = f(x − 3) + 1:

第二步：应用平移 (3, 1)，得到 y = f(x − 3) + 1：

y = ((x − 3) − 1)² − 1 + 1 = (x − 4)²

Therefore the new curve is y = (x − 4)², with its vertex at (4, 0). The original vertex (1, −1) has moved by (3, 1) to (4, 0).

因此新曲线为 y = (x − 4)²，其顶点在 (4, 0)。原顶点 (1, −1) 沿 (3, 1) 移动到 (4, 0)。

11. Worked Example 2: Rational Function | 例题2：有理函数

Sketch the graph of y = 2/(x − 3) + 1, stating the equations of the asymptotes and the coordinates of the y-intercept.

画出 y = 2/(x − 3) + 1 的草图，写出渐近线方程和 y 截距坐标。

The base function is y = 2/x, which has vertical asymptote x = 0 and horizontal asymptote y = 0.

基准函数为 y = 2/x，其垂直渐近线为 x = 0，水平渐近线为 y = 0。

The given function is a translation of y = 2/x by the vector (3, 1). Therefore the vertical asymptote becomes x = 3, and the horizontal asymptote becomes y = 1.

给定函数是 y = 2/x 沿向量 (3, 1) 平移的结果。因此垂直渐近线变为 x = 3，水平渐近线变为 y = 1。

To find the y-intercept, set x = 0:

求 y 截距，令 x = 0：

y = 2/(0 − 3) + 1 = −2/3 + 1 = 1/3

So the y-intercept is (0, 1/3). The domain is all real numbers except x = 3, and the range is all real numbers except y = 1.

所以 y 截距为 (0, 1/3)。定义域为除 x = 3 外的所有实数，值域为除 y = 1 外的所有实数。

12. Exam Tips and Summary | 考试技巧与总结

In Edexcel A-Level Mathematics, function translation questions may appear in pure maths, applied maths, or even in the context of solving equations graphically. Always write the translation vector clearly when the question asks for it.

在Edexcel A-Level数学中，函数平移题可能出现在纯数、应用数学，甚至以图像法解方程的情境中。当题目要求时，务必清晰地写出平移向量。

When checking your answer, test one point. Pick a convenient point on the original graph, apply the vector manually, and see whether the point satisfies the new equation. This quick check catches most arithmetic errors.

检查答案时，测试一个点。在原图像上选择一个方便的点，手动应用平移向量，然后看该点是否满足新方程。这个快速检查能发现大多数计算错误。

Finally, remember the golden rule for horizontal translations: the graph of y = f(x − a) moves in the positive x direction by a. Do not let the minus sign confuse you.

最后，记住平移变换的黄金法则：y = f(x − a) 的图像沿 x 轴正方向移动 a 个单位。不要被减号搞混。

y = f(x) + q: vertical translation by q.

y = f(x) + q：纵向平移 q 个单位。
y = f(x − p): horizontal translation by p.

y = f(x − p)：水平平移 p 个单位。
General form: y = f(x − p) + q.

一般形式：y = f(x − p) + q。

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A-Level Maths: Translations of Function Graphs | A-Level数学：函数图像平移变换