📚 Translation of Functions | 函数平移
In A-Level mathematics, transforming graphs is a fundamental skill. Among the various transformations, translation is one of the simplest yet most frequently examined topics. A translation shifts a graph horizontally or vertically without changing its shape. Understanding how to describe and apply translations using function notation is essential for success in Edexcel exams, especially when tackling questions on curve sketching, equation solving, and modelling.
在A-Level数学中，图形变换是一项基本技能。在众多变换中，平移是最简单却也最常考察的主题之一。平移会将图形沿水平或垂直方向移动，而不改变其形状。理解如何使用函数符号描述和应用平移，对于在Edexcel考试中取得成功至关重要，尤其是在处理曲线草图、方程求解和建模问题时。
1. What is a Translation? | 什么是平移？
If a function y = f(x) is translated, the new function can be written in the form y = f(x – ly and b units vertically. The shape, orientation, and size of the graph remain unchanged.
平移是一种变换，将图形上的每个点按相同的向量移动。如果函数 y = f(x) 被平移，新函数可以写成 y = f(x – a) + b 的形式，其中 a 和 b 是实数。图形会水平移动 a 个单位，垂直移动 b 个单位。图形的形状、方向和大小保持不变。
2. Vertical Translation | 垂直平移
shifted upward by k units, the new equation is y = f(x) + k. If shifted downward, y = f(x) – s (0, k). Always remember: adding outside the function affects the y-coordinates directly.
垂直平移是在函数输出上加一个常数。如果 y = f(x) 向上平移 k 个单位，新方程为 y = f(x) + k。如果向下平移，则是 y = f(x) – k。例如，将 y = x² 的图像向上移动3个单位得到 y = x² + 3。变换向量为 (0, k)。请始终记住：在函数外部加常数会直接影响 y 坐标。
3. Horizontal Translation | 水平平移
put of the function. To shift the graph of y = f(x) to the right by h units, write y = f(x – y = f(x + h). Notice the sign is opposite to the direction of the shift. For instance, y = √x – 2 represents y = √x shifted 2 units to the right. The transformation vector is (h, 0). This is a common source of confusion, so pay close attention to the sign inside the bracket.
水平平移改变函数的输入。要将 y = f(x) 的图像向右平移 h 个单位，写作 y = f(x – h)。向左平移 h 个单位，则写作 y = f(x + h)。注意符号与移动方向相反。例如，y = √x – 2 表示 y = √x 向右平移2个单位。变换向量为 (h, 0)。这是常见的混淆点，要特别留意括号内的符号。
4. General Form and Vector Notation | 一般形式和向量表示法
shifts, a general translation of the function f(x) by the vector (a, b) results in y = f(x – he transformation succinctly. In Edexcel mark schemes, using vector notation is often rewarded.
将两种平移结合起来，按向量 (a, b) 对 f(x) 进行一般平移，结果为 y = f(x – a) + b。其中，a 是水平移动量（正向右），b 是垂直移动量（正向上）。向量 (a, b) 可用来简洁地描述该变换。在Edexcel评分方案中，使用向量表示法通常会得到加分。
5. Translating Specific Curves | 平移特定曲线
s build intuition. For the quadratic y = x², a translation by vector (2, -5) yields y = (x – 2)² – 5. The vertex moves from (0,0) to (2, -5). For the exponential function y = eˣ, shifting up by 1 gives y = eˣ + 1, and shifting left by 3 gives y = eˣ⁺³. In all cases, the pattern y = f(x – a) + b holds.
将平移规则应用于熟悉的函数有助于建立直觉。对于二次函数 y = x²，按向量 (2, -5) 平移得到 y = (x – 2)² – 5。顶点从 (0,0) 移动到 (2, -5)。对于指数函数 y = eˣ，向上移动1个单位得到 y = eˣ + 1，向左移动3个单位得到 y = eˣ⁺³。所有情况下，模式 y = f(x – a) + b 都成立。
6. Combining Translations with Other Transformations | 平移与其他变换的组合
l transformations first, then vertical ones. For example, to transform y = x² into y = 2(x – 2, and finally translate up by 4. Writing the intermediate steps clearly avoids mistakes.
在Edexcel考试中，平移常与拉伸和反射组合出现。运算顺序很重要：对变换 y = a f(bx + c) + d 应逐步解读。通常我们先执行水平变换，再执行垂直变换。例如，要将 y = x² 变换为 y = 2(x – 3)² + 4，先向右平移3个单位，然后施加垂直拉伸因子2，最后向上平移4个单位。清晰地写出中间步骤可避免错误。
7. Vertex Form and Completing the Square | 顶点形式与配方法
nked to completing the square. A quadratic expression ax² + bx + c can be rewritten as a(x – l stretch by a. This is the vertex form. The vertex of the parabola is (h, k). Edexcel frequently asks students to find the vertex or line of symmetry using this technique.
平移与配方法密切相关。二次表达式 ax² + bx + c 可重写为 a(x – h)² + k，这揭示了 y = x² 经过垂直拉伸 a 倍后再按向量 (h, k) 平移。这就是顶点形式。抛物线的顶点为 (h, k)。Edexcel经常要求学生利用该技巧求顶点或对称轴。
8. Solving Equations Using Translations | 利用平移求解方程
n can simplify solving equations that involve shifted functions. For instance, to solve (x – 2)³ + 5 = 13, we can set X = x – tal shift, making the equation easier. It reflects the substitution technique commonly used in integration and differential equations.
平移可以简化涉及移位函数的方程求解。例如，解方程 (x – 2)³ + 5 = 13，我们可以设 X = x – 2，则方程变为 X³ + 5 = 13，因此 X³ = 8，得到 X = 2，从而 x = 4。这种方法本质上是平移变量以消除水平位移，使方程更易解。它反映了积分和微分方程中常用的代换技巧。
9. Common Mistakes and How to Avoid Them | 常见错误及如何避免
Many students reverse the sign in horizontal translations. Remember: f(x – dinates after a translation, apply the vector directly to the original key points. Instead of memorising, understand that a translation adds to x and y; rewriting the function as y – b = f(x – a) can clarify the effect.
许多学生会在水平平移中弄反符号。记住：f(x – a) 将图像向右移动，而非向左。另一个错误是将 y = f(x) + b 误认为水平平移，它始终是垂直的。此外，在平移后读取坐标时，应直接将向量加到原始关键点上。与其死记硬背，不如理解平移就是在 x 和 y 上相加；将函数改写为 y – b = f(x – a) 可清晰展示效果。
10. Exam-Style Question Walkthrough | 考试风格题目演练
ated by vector (-3, 1) to give curve C₂. Write the equation of C₂. Solution: using y = f(x – a) + b with a = -3 and b = 1 gives y = ln(x – ally, sketch both curves, labelling intercepts. Practice like this builds exam confidence.
考虑曲线 C₁：y = ln x。将其按向量 (-3, 1) 平移得到曲线 C₂。写出 C₂ 的方程。解答：使用 y = f(x – a) + b，其中 a = -3，b = 1，得到 y = ln(x – (-3)) + 1 = ln(x + 3) + 1。接着求渐近线：原为 x = 0，现向左移动3，所以新渐近线为 x = -3。最后，画出两条曲线并标注截距。像这样的练习能建立考试信心。
11. Key Points Summary | 要点总结
To quickly revise: Translation of y = f(x) by vector (a, b) gives y = f(x – ape never changes. When describing transformations, specify the vector or the exact phrase 'translation by vector' e . For combined transformations, follow the order inside the bracket first, then outside. Repeated practice with different functions ensures mastery.
快速复习：y = f(x) 按向量 (a, b) 平移得到 y = f(x – a) + b。水平移动 a：正向右，负向左。垂直移动 b：正向上，负向下。形状永不改变。描述变换时，要指明向量或明确说法“按向量平移”。对于组合变换，先处理括号内，再处理括号外。针对不同函数的反复练习可确保掌握。
12. Final Tips for Edexcel Success | 决胜Edexcel的最终建议
In the exam, always show the modified equation clearly. Even if you sketch roughly, label the translated key points. If a question asks for a single transformation, check whether a combination of translations can be condensed into one vector. Use parentheses correctly when substituting into f. And most importantly, don't panic—translations are straightforward once you internalise the rule: 'outside changes y, inside changes x, opposite sign for x'. Good luck!
考试中，一定要清晰地写出修改后的方程。即便只是粗略画图，也要标出平移后的关键点。如果题目要求单一变换，检查一下组合平移是否可以浓缩成一个向量。代入 f 时要正确使用括号。最重要的是不要慌张——一旦你内化了“外部改变 y，内部改变 x，x 的符号相反”这一规则，平移就变得很简单。祝好运！
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