5: Translating Graphs | 5：图形的平移

📚 5: Translating Graphs | 5：图形的平移

In A-Level mathematics, understanding how to translate graphs is an essential skill for sketching and interpreting functions. A translation moves every point of a graph by the same vector, without changing the shape or orientation of the curve. This transformation appears frequently in pure mathematics, mechanics, and even statistics. In this article, we will explore the rules for horizontal and vertical translations, learn how to apply them to equations, and practise with exam-style examples.

在 A-Level 数学中，理解如何平移图像是绘制和解读函数的重要技能。平移会按照相同的向量移动图像上的每一个点，而不改变曲线的形状或方向。这种变换在纯数学、力学甚至统计学中经常出现。在本文中，我们将探索水平和垂直平移的规则，学习如何将其应用于方程，并通过考试风格的例题进行练习。

1. What Does It Mean to Translate a Graph? | 平移图像的含义

A translation is a transformation that shifts a graph left, right, up, or down without rotating or reflecting it. In functional notation, if the original curve is y = f(x), then a translation by a horizontal vector component a and a vertical component b produces a new curve. The new curve has the same shape as the original but is located in a different part of the coordinate plane.

平移是一种将图像向左、右、上或下移动而不旋转或反射的变换。在函数表示法中，如果原始曲线为 y = f(x)，那么按水平向量分量 a 和垂直分量 b 进行平移会产生一条新曲线。新曲线与原始曲线形状相同，但位于坐标平面的不同位置。

We distinguish between two basic types of translations: vertical translations, which move the graph up or down, and horizontal translations, which move the graph left or right. The algebraic rules for these transformations are simple but often cause confusion, especially with the sign of the horizontal shift.

我们区分两种基本的平移类型：垂直平移（将图像向上或向下移动）和水平平移（将图像向左或向右移动）。这些变换的代数规则很简单，但常常造成混淆，尤其是水平移动的符号。

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

When we add a constant a to a function, the graph moves vertically. If a is positive, the graph shifts upwards by a units; if a is negative, it shifts downwards by the absolute value of a. In other words, the transformation y = f(x) + a maps the original point (x, y) to (x, y + a).

当我们给函数加上常数 a 时，图像会垂直移动。如果 a 为正，图像向上移动 a 个单位；如果 a 为负，则向下移动 |a| 个单位。换句话说，变换 y = f(x) + a 将原图像上的点 (x, y) 映射到 (x, y + a)。

For example, consider the quadratic function f(x) = x². The graph of y = x² + 3 is the same parabola shifted upward by 3 units. Its vertex moves from (0, 0) to (0, 3). The axis of symmetry remains unchanged at x = 0.

例如，考虑二次函数 f(x) = x²。图像 y = x² + 3 是同一抛物线向上平移 3 个单位。其顶点从 (0, 0) 移动到 (0, 3)，对称轴仍为 x = 0。

Vertical translation: (x, y) → (x, y + a)

垂直平移：(x, y) → (x, y + a)

A vertical translation does not affect the x-coordinates of any point on the graph. It only changes the y-coordinates by adding the constant a. This is why the graph moves directly up or down in a parallel fashion.

垂直平移不会影响图像上任何点的 x 坐标，它只通过加上常数 a 来改变 y 坐标。这就是为什么图像会以平行方式直接向上或向下移动。

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

Horizontal translations are encoded inside the function argument. The transformation y = f(x – a) shifts the graph horizontally. If a is positive, the graph moves to the right by a units. If a is negative, it moves to the left. Note the counter-intuitive sign: subtracting a from x moves the graph to the right.

水平平移被编码在函数的自变量内部。变换 y = f(x – a) 会水平移动图像。如果 a 为正，图像向右移动 a 个单位；如果 a 为负，则向左移动。注意符号与直觉相反：从 x 中减去 a 会使图像向右移动。

Let us use the same quadratic f(x) = x². The graph of y = (x – 2)² is the parabola shifted 2 units to the right. Its vertex moves from (0, 0) to (2, 0). To see why, set y = 0: then (x – 2)² = 0, so x = 2. The whole curve has effectively been delayed by 2 units along the x-axis.

我们仍然使用二次函数 f(x) = x²。图像 y = (x – 2)² 是抛物线向右平移 2 个单位。其顶点从 (0, 0) 移动到 (2, 0)。为了理解原因，令 y = 0，则 (x – 2)² = 0，所以 x = 2。整条曲线实际上沿 x 轴方向延迟了 2 个单位。

Horizontal translation: (x, y) → (x + a, y)

水平平移：(x, y) → (x + a, y)

Common mistake: students often think y = f(x – 2) moves the graph left because of the minus sign. However, consider a point where the original function has a feature, say a root at x = 0. For the new function, the same feature appears when x – 2 = 0, i.e. x = 2. Thus the feature has moved to the right.

常见错误：学生常常认为 y = f(x – 2) 会因减号而使图像左移。然而，考虑原函数的一个特征点，比如在 x = 0 处的根。对于新函数，相同的特征出现在 x – 2 = 0 时，即 x = 2。因此该特征向右移动了。

4. Combining Translations: y = f(x – a) + b | 组合平移：y = f(x – a) + b

Vertical and horizontal translations can be applied simultaneously. The general form y = f(x – a) + b represents a graph that has been shifted a units horizontally (right if a > 0, left if a < 0) and b units vertically (up if b > 0, down if b < 0). The order of applying the two translations does not matter because translation is commutative.

垂直和水平平移可以同时应用。一般形式 y = f(x – a) + b 表示图像水平移动 a 个单位（a > 0 时向右，a < 0 时向左），并垂直移动 b 个单位（b > 0 时向上，b < 0 时向下）。应用这两种平移的顺序无关紧要，因为平移满足交换律。

For instance, starting with f(x) = x², the function y = (x – 3)² + 4 has its vertex at (3, 4). The original parabola has been shifted 3 units to the right and 4 units up. This is sometimes called the completed-square form of a quadratic, but it is simply a translated version of y = x².

例如，从 f(x) = x² 出发，函数 y = (x – 3)² + 4 的顶点在 (3, 4)。原抛物线向右平移 3 个单位，再向上平移 4 个单位。这有时被称为二次函数的配方法形式，但它本质上只是 y = x² 的一个平移版本。

When dealing with combined translations, it is often helpful to locate a key point, such as a vertex, intercept, or asymptote, and move that point. The rest of the graph can then be sketched by maintaining the same shape.

处理组合平移时，通常先找到一个关键点，例如顶点、截距或渐近线，再移动该点。其余部分可以通过保持相同的形状来绘制。

5. Using Vector Notation for Translations | 用向量表示平移

In coordinate geometry, a translation can be described by a vector (a, b), where a is the horizontal displacement and b is the vertical displacement. In function notation, the translated graph is y = f(x – a) + b. This is a compact way to specify both components simultaneously.

在坐标几何中，平移可以用向量 (a, b) 来描述，其中 a 是水平位移，b 是垂直位移。在函数表示法中，平移后的图像为 y = f(x – a) + b。这是一种同时指定两个分量的简洁方式。

For example, a translation vector of (-2, 5) means the graph moves 2 units left and 5 units up. The new equation would be y = f(x + 2) + 5. Notice that the horizontal component appears with the opposite sign inside the function argument, while the vertical component keeps its sign outside.

例如，平移向量 (-2, 5) 表示图像向左移动 2 个单位并向上移动 5 个单位。新方程为 y = f(x + 2) + 5。注意水平分量在函数参数内出现时符号相反，而垂直分量在函数外保持原符号。

Translation vector (a, b): y = f(x – a) + b

平移向量 (a, b)：y = f(x – a) + b

This vector notation is particularly useful in mechanics and kinematics, where displacement vectors are common. It also helps when you need to describe a transformation precisely in exam answers.

这种向量表示法在力学和运动学中特别有用，因为那里经常使用位移向量。在考试答案中需要精确描述变换时，它也非常有帮助。

6. Translating Specific Points | 平移图像上的特定点

When translating a graph, every point moves by the same vector. Therefore, if a point (p, q) lies on the original curve y = f(x), the corresponding point on the translated curve y = f(x – a) + b is (p + a, q + b). This fact is extremely useful for sketching graphs without replotting every coordinate.

平移图像时，每个点都按相同的向量移动。因此，如果点 (p, q) 位于原曲线 y = f(x) 上，那么平移后的曲线 y = f(x – a) + b 上对应的点为 (p + a, q + b)。这一事实对于绘制图像非常有用，无需重新计算每个坐标。

For example, suppose the graph of f(x) = x³ passes through the point (1, 1). Under the translation y = (x – 2)³ + 3, this point becomes (1 + 2, 1 + 3) = (3, 4). The inflection point at (0, 0) moves to (2, 3).

例如，假设 f(x) = x³ 的图像通过点 (1, 1)。在平移 y = (x – 2)³ + 3 下，该点变为 (1 + 2, 1 + 3) = (3, 4)。拐点 (0, 0) 移动到 (2, 3)。

Roots: if x = r is a root of f(x), then x = r + a is a root of f(x – a).
Roots（根）：如果 x = r 是 f(x) 的根，那么 x = r + a 是 f(x – a) 的根。
Intercepts: the y-intercept of y = f(x) is (0, f(0)); after the translation it becomes (a, f(0) + b).
截距：y = f(x) 的 y 截距为 (0, f(0))；平移后变为 (a, f(0) + b)。
Asymptotes: vertical asymptotes shift with the same horizontal component; horizontal asymptotes shift with the vertical component.
渐近线：垂直渐近线随水平分量平移；水平渐近线随垂直分量平移。

These quick rules allow you to transform an entire graph by moving only a few key features.

这些快速规则允许你只移动少数关键特征就能变换整个图像。

7. Translating Quadratic and Cubic Graphs | 平移二次和三次函数图像

Quadratic graphs are parabolas. A quadratic in vertex form y = (x – h)² + k is simply y = x² translated by the vector (h, k). The vertex is at (h, k) and the axis of symmetry is x = h. Recognising this form makes sketching quick and easy.

二次函数图像是抛物线。顶点形式的二次函数 y = (x – h)² + k 就是 y = x² 按向量 (h, k) 平移的结果。顶点位于 (h, k)，对称轴为 x = h。识别这种形式可以快速绘制图像。

Cubic graphs, such as y = x³, have a point of inflection at the origin. Translating by (a, b) gives y = (x – a)³ + b. The inflection point moves to (a, b). The cubic retains its general shape but its position changes. For example, y = (x – 1)³ is the cubic shifted 1 unit to the right.

三次函数图像，如 y = x³，在原点有一个拐点。按 (a, b) 平移得到 y = (x – a)³ + b。拐点移动到 (a, b)。三次函数保持其大致形状，但位置发生变化。例如，y = (x – 1)³ 就是三次函数向右平移 1 个单位。

It is important to remember that translations do not change the degree of the polynomial or the multiplicity of its roots. They only relocate the roots and other features.

重要的是，平移不会改变多项式的次数或其根的重数。它们只是改变了根和其他特征的位置。

8. Translating Trigonometric Graphs | 平移三角函数图像

Trigonometric functions also follow the same translation rules. The graph of y = sin(x – a) is a horizontal shift of the sine curve, while y = sin(x) + b is a vertical shift. Combining both gives y = sin(x – a) + b.

三角函数也遵循相同的平移规则。y = sin(x – a) 的图像是正弦曲线的水平平移，而 y = sin(x) + b 是垂直平移。两者结合得到 y = sin(x – a) + b。

For example, y = cos(x – 90°) is actually equivalent to y = sin(x), because the cosine curve shifted 90° to the right aligns with the sine curve. This shows that horizontal translations can reveal important identities among trigonometric functions.

例如，y = cos(x – 90°) 实际上等于 y = sin(x)，因为余弦曲线向右平移 90° 后与正弦曲线重合。这表明水平平移可以揭示三角函数之间的重要恒等式。

When sketching translated sine or cosine graphs, first draw the original curve, then shift all key points such as maxima, minima, and x-intercepts. The period and amplitude remain unchanged under translation.

绘制平移后的正弦或余弦图像时，先绘制原始曲线，然后移动所有关键点，如最大值、最小值和 x 截距。平移不会改变周期和振幅。

9. Common Pitfalls and Exam Tips | 常见错误与考试提示

One of the most common errors in graph translation is misapplying the horizontal shift. Always remember: y = f(x – a) moves the graph to the right when a > 0, even though the minus sign might suggest a leftward movement. A useful trick is to set the inside of the bracket equal to the original argument: if x – a = 0 gives x = a, then the feature originally at x = 0 has moved to x = a.

图像平移中最常见的错误之一是错误应用水平平移。始终记住：y = f(x – a) 在 a > 0 时向右移动图像，尽管减号可能暗示向左移动。一个有用的技巧是令括号内部等于原来的自变量：如果 x – a = 0 得 x = a，那么原来在 x = 0 处的特征已经移动到 x = a。

Another pitfall is confusing translation with scaling. Stretching or compressing a graph (e.g., y = 2f(x) or y = f(2x)) changes the size or spacing of features, while translation only moves the whole graph. Do not mix the two.

另一个陷阱是混淆平移与伸缩。拉伸或压缩图像（例如 y = 2f(x) 或 y = f(2x)）会改变特征的大小或间距，而平移只是移动整个图像。不要将两者混为一谈。

Always compare the new equation to the original y = f(x) form.
总是将新方程与原始 y = f(x) 形式进行比较。
Use a known point to check the direction of the shift.
使用一个已知点来检查平移的方向。
Write your answer in the form y = f(x – a) + b when asked to describe a translation.
当要求描述平移时，用 y = f(x – a) + b 的形式写出答案。
If a question gives a translation vector, write the equation immediately using the rule above.
如果题目给出平移向量，立即用上述规则写出方程。

In Edexcel exams, a single mark is often awarded for correctly identifying both the magnitude and direction of each component, so state the full vector to avoid losing marks.

在 Edexcel 考试中，通常有一分是用于正确识别每个分量的大小和方向，因此请写出完整的向量，以免丢分。

10. Worked Example | 典型例题

Problem: The curve y = x² is translated by the vector (3, -2). Find the equation of the new curve and state the coordinates of its vertex.

题目：曲线 y = x² 按向量 (3, -2) 平移。求新曲线的方程，并写出其顶点坐标。

Solution: Using the translation rule y = f(x – a) + b with a = 3 and b = -2, we replace x by x – 3 and add -2 outside the function:

解答：根据平移规则 y = f(x – a) + b，其中 a = 3，b = -2，我们用 x – 3 替换 x，并在函数外加 -2：

y = (x – 3)² – 2

The original vertex (0, 0) moves by the same vector to (0 + 3, 0 – 2) = (3, -2). Therefore the new vertex is at (3, -2).

原始顶点 (0, 0) 按相同向量移动到 (0 + 3, 0 – 2) = (3, -2)。因此新顶点位于 (3, -2)。

Problem: A cubic graph has equation y = x³. It is shifted 2 units left and 5 units up. Write the new equation.

题目：三次函数图像 y = x³ 向左平移 2 个单位，向上平移 5 个单位。写出新方程。

Solution: A shift of 2 units left means a = -2; 5 units up means b = 5. Thus:

解答：向左平移 2 个单位意味着 a = -2；向上平移 5 个单位意味着 b = 5。因此：

y = (x – (-2))³ + 5 = (x + 2)³ + 5

The inflection point moves from (0, 0) to (-2, 5).

拐点从 (0, 0) 移动到 (-2, 5)。

11. Summary | 总结

Translating graphs is a fundamental transformation that preserves the shape of a curve. The key rules are simple:

平移图像是一种保持曲线形状的基本变换。关键规则很简单：

Vertical translation: y = f(x) + b moves the graph up for b > 0 and down for b < 0.
垂直平移：y = f(x) + b，当 b > 0 时图像向上移动，b < 0 时向下移动。
Horizontal translation: y = f(x – a) moves the graph right for a > 0 and left for a < 0.
水平平移：y = f(x – a)，当 a > 0 时图像向右移动，a < 0 时向左移动。
Vector form: A translation vector (a, b) gives the equation y = f(x – a) + b.
向量形式：平移向量 (a, b) 给出方程 y = f(x – a) + b。
Point mapping: Every point (x, y) on the original graph becomes (x + a, y + b).
点映射：原图上的每个点 (x, y) 变为 (x + a, y + b)。

Mastering translations gives you a powerful tool for graphing functions and solving problems across all branches of A-Level mathematics. In the next part of this series, we will examine reflections and stretches, which extend these ideas further.

掌握平移为你提供了绘制函数图像和解决 A-Level 数学各分支问题的有力工具。在本系列的下一部分中，我们将研究反射和伸缩，它们进一步扩展了这些概念。

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5: Translating Graphs | 5：图形的平移