📚 Translating Graphs | 图形变换
Graph translation is one of the most fundamental transformations in A-Level mathematics. It involves shifting the entire graph of a function either vertically, horizontally, or both, without altering its shape. Understanding translations allows you to quickly sketch related functions and analyse how changing a function's equation affects its graphical representation.
图形平移是A-Level数学中最基本的变换之一。它是指将函数的整个图像沿垂直方向、水平方向或两个方向同时移动，而图像形状保持不变。理解平移变换可以帮助你快速绘制相关函数图像，并分析函数方程的变化如何影响其图形表现。
1. Introduction to Graph Translations | 图形平移变换导论
A translation moves every point on the graph of a function by the same fixed distance in a given direction. Unlike stretches or reflections, a translation does not change the size, orientation, or overall shape of the graph. In coordinate geometry, we can describe translations using vector notation or by modifying the function's equation.
平移变换将函数图像上的每一个点沿着给定方向移动相同的固定距离。与拉伸或反射不同，平移不会改变图形的大小、方向或整体形状。在坐标几何中，我们可以用向量表示来描述平移，或者通过修改函数方程来表示。
The two basic types of translation covered in the Edexcel specification are vertical translations and horizontal translations. When combined, they can produce diagonal shifts, which are also part of the syllabus.
Edexcel考试大纲中涵盖的两种基本平移类型是垂直平移和水平平移。当两者组合时，可以产生对角线移动，这也属于考纲范围。
2. Vertical Translations: y = f(x) + a | 垂直平移：y = f(x) + a
a is positive, the graph moves upwards by a units; if a is negative, it moves downwards by |a| tion directly adds or subtracts from the y-coordinate of each point on the original curve.
垂直平移由方程 y = f(x) + a 表示。如果 a 为正数，图像向上移动 a 个单位；如果 a 为负数，则向下移动 |a| 个单位。这种变换直接对原曲线上每个点的 y 坐标进行加减。
= sin x + 2 shifts the entire sine wave up by 2, changing the midline from y = 0 to y = 2.
例如，若 f(x) = x²，那么 y = x² + 3 使抛物线向上平移 3 个单位，其顶点从 (0,0) 移动到 (0,3)。类似地，y = sin x + 2 将整个正弦波向上移动 2 个单位，其中线从 y = 0 变为 y = 2。
It is important to note that vertical translations affect the y-intercept, maximum and minimum points, and asymptotes in exactly the vertical direction, but they do not affect the x-coordinates of any features.
需要注意，垂直平移会在垂直方向上影响 y 轴截距、最大值和最小值点以及渐近线，但不会改变任何特征的 x 坐标。
3. Horizontal Translations: y = f(x + a) | 水平平移：y = f(x + a)
sitive, the graph moves to the left by a units; if a is negative, it moves to the right by |a| ve at first because the sign inside the bracket is opposite to the direction of the shift.
水平平移写作 y = f(x + a)。其中，如果 a 为正数，图像向左移动 a 个单位；如果 a 为负数，则向右移动 |a| 个单位。初看这会显得有点反直觉，因为括号内的符号与移动方向相反。
ns that a point that was originally at x = 0 now appears at x = 2 in order to achieve the same input value for the function.
例如，y = f(x - 2) 将图像向右平移 2 个单位，而 y = f(x + 2) 则将其向左平移 2 个单位。这是因为用 (x - 2) 替代 x 意味着原本在 x = 0 的点现在需要出现在 x = 2 处，才能得到函数相同的输入值。
point (p, q) lies on y = f(x), then the corresponding point on y = f(x + a) is (p - a, q) – the y-coordinate remains unchanged.
从坐标角度看，如果原图像上的点 (p, q) 满足 y = f(x)，那么在 y = f(x + a) 上对应的点为 (p - a, q)——y 坐标保持不变。
4. Direction of Translation and Sign Confusion | 平移方向与符号混淆
: for y = f(x + a), the graph moves in the opposite direction of the sign of a. If you see 'x + a', think 'shift left'; if you see 'x - a', think 'shift right'.
许多学生因符号规则而混淆水平平移方向。一个有用的记忆方法是：对于 y = f(x + a)，图像移动的方向与 a 的符号相反。如果你看见“x + a”，就想“向左移动”；如果看见“x - a”，就想“向右移动”。
ph up when a > 0 and down when a < 0. Matching the direction with the sign avoids errors when applying transformations to unfamiliar functions.
垂直平移则更直接：y = f(x) + a 在 a > 0 时向上移动图像，在 a < 0 时向下移动。将方向与符号匹配，可以避免在对陌生函数应用变换时出错。
A typical exam question may ask: 'The graph of y = f(x + 3) is obtained by translating y = f(x). Describe the translation.' The answer is a translation of 3 units in the negative x-direction.
一个典型的考题可能会问：“y = f(x + 3) 的图像是通过平移 y = f(x) 得到的。请描述该平移。”答案是沿负 x 方向平移 3 个单位。
5. Using Vector Notation for Translations | 用向量表示平移
ection and q units in the y-direction. This can be applied to the graph of y = f(x) to give y - q = f(x - p), which is equivalent to y = f(x - p) + q.
在 Edexcel A-Level 数学中，平移常用列向量表示。向量 (p, q) 的平移意味着在 x 方向上移动 p 个单位，在 y 方向上移动 q 个单位。将其应用于 y = f(x) 的图像可以得到 y - q = f(x - p)，等价于 y = f(x - p) + q。
r notation is a clear way to describe combined horizontal and vertical shifts in a single instruction.
例如，向量 (2, -3) 将图像向右平移 2 个单位、向下平移 3 个单位。变换后的方程变为 y = f(x - 2) - 3。使用向量表示法可以清晰地用一条指令描述同时进行的水平和垂直平移。
In exam answers, you might write: 'Translation by vector (a, b)' or 'Shift (a, b)'. Both are acceptable as long as the direction and magnitude are unambiguous.
在考试作答中，你可以写：“平移向量 (a, b)”或“移动 (a, b)”。只要方向和大小明确，两者均可接受。
6. Translating Specific Points | 特定点的平移
then after a translation by vector (h, k), its new position is (x₀ + h, y₀ + k). This applies to key features such as intercepts, turning points, and roots.
要确定给定点在平移下的像，只需将平移向量与其坐标相加。若点 (x₀, y₀) 在 y = f(x) 上，那么经过向量 (h, k) 的平移后，它的新位置为 (x₀ + h, y₀ + k)。这适用于截距、转折点以及根等关键特征。
minimum becomes (1 - 2, 0 + 5) = (-1, 5). The translated graph would be y = (x + 1)² + 5.
例如，y = (x - 1)² 的最小值点位于 (1, 0)。如果施加平移向量 (-2, 5)，新的最小值点变为 (1 - 2, 0 + 5) = (-1, 5)。平移后的图像将是 y = (x + 1)² + 5。
Being able to translate points without fully rewriting the equation is extremely useful when sketching graphs or verifying that a transformed graph matches the required vector shift.
能够不完整重写方程就完成点的平移，在绘制图像草图或验证变换后的图像是否符合所要求的向量移动时极为有用。
7. Finding the Equation After Translation | 求平移后的方程
e function. So the translated graph has equation y = f(x - h) + k. This formula works for all functions and ensures the correct relationship between the variables.
给定平移向量 (h, k) 和原方程 y = f(x)，你可以通过将 x 替换为 (x - h) 然后整体加 k 来求得新方程。因此平移后的图像方程为 y = f(x - h) + k。该公式适用于所有函数，并能确保变量间的正确关系。
al asymptote moves from x = 0 to x = 3, and the horizontal asymptote from y = 0 to y = -4.
例如，若 f(x) = 1/x，平移向量为 (3, -4)，则新方程为 y = 1/(x - 3) - 4。垂直渐近线从 x = 0 移动到 x = 3，水平渐近线从 y = 0 移动到 y = -4。
tify the translation by rewriting it in the form y - k = f(x - h) and reading off h and k.
此替换法可逆向使用：给定一个平移后的方程，你可以通过将其改写为 y - k = f(x - h) 的形式并读出 h 和 k 来识别平移。
8. Translating Quadratic and Trigonometric Graphs | 二次函数与三角函数的平移
the vertex (a, b) is a key indicator of the translation applied to the standard parabola.
平移在二次函数中频繁出现。配平方形式 y = (x - a)² + b 直接表明 y = x² 经过了向量 (a, b) 的平移。因此顶点 (a, b) 就是对标准抛物线所施平移的关键指示。
, and y = cos x + 1 shifts the cosine graph up by 1. Combining these can model alternating currents and wave phenomena.
对于三角函数图像，水平平移对应相位移动。例如，y = sin(x - 30°) 将正弦曲线向右移动 30°，而 y = cos x + 1 将余弦图像向上移动 1 个单位。组合这些变换可以对交变电流和波动现象进行建模。
It is essential to work with degrees in Edexcel unless otherwise specified, and to remember that the period of the graph remains unchanged after translation, because only the position, not the shape, is altered.
除非另有说明，Edexcel 要求使用度数，并且要记住平移后图像的周期保持不变，因为变化的只是位置而非形状。
9. Reversing a Translation | 逆向平移
quations, if the translated graph is y = g(x), then the original graph is y = g(x + h) - k.
如果一个图像经过了向量 (h, k) 的平移，那么施加相反的平移向量 (-h, -k) 就会将其映射回原位置。就方程而言，若平移后的图像为 y = g(x)，则原图像为 y = g(x + h) - k。
Reverse translation is often used in problems where you are given the graph after a transformation and must recover the original function. By applying the inverse vector shift, you can eliminate the translation and identify the base graph.
逆向平移常用于给出变换后图像、要求恢复原函数的问题。通过施加逆向向量移动，你可以消除平移并识别基本图像。
This concept reinforces the idea that translations are invertible mappings in the coordinate plane, and the order of applying forward and backward translations cancels out exactly.
这一概念强调了平移是坐标平面上的可逆映射，且正向与反向平移的叠加恰好抵消。
10. Combining Horizontal and Vertical Translations | 组合水平和垂直平移
When both horizontal and vertical shifts are present, the overall transformation is a diagonal translation. The order in which you apply them does not matter: shifting right by 2 and up by 3 yields the same final graph whether you move horizontally first or vertically first.
当水平和垂直移动同时存在时，整体变换为对角平移。其应用顺序无关紧要：无论先水平移动还是先垂直移动，向右 2 个单位、向上 3 个单位的最终图像都是相同的。
you will most often use. You can think of it as first replacing x with (x - h) to handle the horizontal shift, and then adding k for the vertical shift. However, commutativity means the order is not a restriction.
最常用的组合形式是 y = f(x - h) + k。你可以将其视为首先用 (x - h) 替代 x 以处理水平移动，然后加上 k 进行垂直移动。不过，交换性意味着顺序并不是限制。
In Edexcel questions, you may be asked to describe two successive translations as a single translation vector, which is simply the sum of the individual vectors. This makes complex transformations easier to communicate.
在 Edexcel 考题中，可能会要求你将两个相继的平移描述为一个平移向量，这其实就是各向量之和。这使得复杂变换更易于表达。
11. Effect on Asymptotes, Intercepts and Roots | 对渐近线、截距和根的影响
nt as the horizontal shift. Horizontal asymptotes, y = constant, move vertically by the vertical shift. Roots (x-intercepts) are affected by both horizontal and vertical translations, but their new positions can be found by solving the translated equation.
平移图像会移动其所有重要特征。垂直渐近线（即垂直线 x = 常数）会水平移动，移动量与水平平移相同。水平渐近线（即 y = 常数）会垂直移动，移动量与垂直平移相同。根（x 轴截距）同时受水平与垂直平移影响，但它们的新位置可通过解平移后的方程求得。
(x - 3) + 3; the vertical asymptote moves to x = 3, and the horizontal asymptote to y = 3.
例如，考虑 f(x) = 1/(x - 1)，其垂直渐近线为 x = 1，水平渐近线为 y = 0。平移向量 (2, 3) 得到 y = 1/(x - 3) + 3；垂直渐近线移至 x = 3，水平渐近线移至 y = 3。
Understanding how these features shift helps in sketching transformed graphs accurately without plotting many individual points, a skill regularly tested in the Pure Mathematics papers.
理解这些特征如何移动，有助于在不描绘大量个别点的情况下准确绘制变换后的图像，这是纯数学试卷中经常考查的一项技能。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
ves f(0), confirming a leftward shift. Another error is forgetting that vertical translation is outside the function, while horizontal translation is inside the argument.
最常见的错误之一是混淆水平平移的方向。应始终用一个简单点来检验：对于 y = f(x + 2)，检查 x = -2 会得到 f(0)，这就确认了左移。另一个错误是忘记垂直平移在函数外部，而水平平移在函数内部的自变量上。
When writing vector translations, ensure you state the vector in column form with the x-component first. Also, be precise with language: 'Translate the graph by the vector (3, -2)' is better than 'move right 3 down 2'. Both may be accepted, but the former is more mathematically rigorous.
在书写向量平移时，务必以列向量形式并先写 x 分量。同时，语言要精确：“将图像按向量 (3, -2) 平移”好于“向右 3 向下 2”。两者可能都算对，但前者在数学上更严谨。
, the original is y = f(x + h) - k. Double-check with a known coordinate to avoid sign errors.
最后，若给出变换后的图像并要求原函数，记得施加逆向平移。对于平移 (h, k)，原函数为 y = f(x + h) - k。用一个已知坐标进行双重检查，以避免符号错误。
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