📚 Combined Graph Transformations | 图像变换的组合
Graph transformations are fundamental in A‑Level Mathematics, allowing you to sketch complicated functions quickly by applying a sequence of shifts, stretches and reflections to a known base graph. When more than one transformation is performed on a function, the order in which they are applied can alter the final result. This tutorial explains each type of transformation, shows how to combine them correctly, and gives plenty of exam‑style practice to build your confidence.
图像变换是 A‑Level 数学的重要基础，通过对已知基本图像进行平移、伸缩和反射的序列变换，可以快速绘制出复杂函数的草图。当对函数连续施加多个变换时，执行的顺序可能会改变最终结果。本教程逐一解释每种变换类型，演示如何正确地进行组合，并提供大量考试风格的练习，帮助你建立信心。
1. Why Graph Transformations Matter | 为什么图像变换很重要
m graphs allows you to move from a simple parent function such as y = x², y = sin x, y = eˣ ng points one by one, you can apply a few geometric rules to the basic shape. In Edexcel A‑ f asymptotes, or describe the order of transformations that map one function onto another.
掌握图像变换能让你从简单的母函数（如 y = x²、y = sin x、y = eˣ、y = ln x 或 y = 1/x）出发，轻松得到更广泛的相关曲线族。无需逐点描图，你只需对基本形状应用几个几何规则即可。在 Edexcel A‑Level 考试中，题目常常要求你绘制变换后的图像、写出渐近线方程，或描述将一个函数映射为另一个函数时的变换顺序。
2. Types of Basic Transformations | 基本变换的类型
ranslations (shifts), stretches (scalings) and reflections. Each can act in the vertical direction (affecting the y‑coordinates) or in the horizontal direction (affecting the x‑coordinates). The table below summarises the standard notation.
对于 y = f(x) 的图像，主要有三种变换类型：平移、伸缩和反射。每种变换既可以在垂直方向进行（影响 y 坐标），也可以在水平方向进行（影响 x 坐标）。下表总结了标准记法。
| Transformation / 变换 | Vertical / 垂直 | Horizontal / 水平 |
|---|---|---|
| Translation / 平移 | y = f(x) + a a > 0: up / 向上 a < 0: down / 向下 | y = f(x + a) a > 0: left / 向左 a < 0: right / 向右 |
| Stretch / 伸缩 | y = a f(x) a > 1: vertical stretch 0 < a < 1: vertical compression 垂直拉伸 / 垂直压缩 | y = f(ax) a > 1: horizontal compression 0 < a < 1: horizontal stretch 水平压缩 / 水平拉伸 |
| Reflection / 反射 | y = – f(x) reflect in x‑axis / 关于 x 轴反射 | y = f(– x) reflect in y‑axis / 关于 y 轴反射 |
3. Translation in Detail | 平移详解
le curve upward, while subtracting b lowers it. For example, y = x² + 3 is the parabola y = x² moved 3 units up.
平移将图像上的每一点按同一向量移动。方程 y = f(x) + b 实现垂直平移：加上正数 b 使整条曲线向上移动，减去 b 则向下移动。例如，y = x² + 3 就是将抛物线 y = x² 向上平移 3 个单位。
ction to the sign of c. y = f(x + 2) shifts the graph 2 units to the left, whereas y = f(x – 2) shifts it 2 units to the right. This ‘inside‑the‑bracket’ effect frequently causes mistakes, so always pause and check the direction.
水平平移的形式为 y = f(x + c)，其移动方向与 c 的符号相反。y = f(x + 2) 将图像向左平移 2 个单位，而 y = f(x – 2) 则向右平移 2 个单位。这种“括号内取反”的效应经常引起错误，因此一定要停下来核对方向。
4. Stretch in Detail | 伸缩详解
Vertical stretches multiply the whole output of the function. y = 2 f(x) doubles every y‑coordinate, pulling points away from the x‑ plier is between 0 and 1, say y = ½ f(x), the graph is vertically compressed towards the x‑axis. The x‑intercepts remain unchanged because f(x) = 0 still gives 0.
垂直伸缩会将函数的输出全部乘以一个倍数。y = 2 f(x) 让每个 y 坐标翻倍，把各点拉离 x 轴。若乘数介于 0 与 1 之间，例如 y = ½ f(x)，图像则被垂直压缩并贴近 x 轴。x 轴截距保持不变，因为 f(x) = 0 时结果仍然为 0。
stretches alter the input. y = f(2x) compresses the graph horizontally by a factor of 1/2 – the curve is squeezed towards the y‑axis. y = f(x/3) – equivalently y = f(⅓ x) – stretches the graph horizontally by a factor of 3, pulling it away from the y‑ and the stretch factor is essential: f(ax) gives a horizontal stretch of scale factor 1/a.
水平伸缩改变的是输入。y = f(2x) 将图像沿水平方向压缩为原来的 1/2，曲线被挤向 y 轴。y = f(x/3)（即 y = f(⅓ x)）则将图像水平拉伸至原来的 3 倍，使其远离 y 轴。系数与伸缩因子之间的倒数关系非常重要：f(ax) 表示水平伸缩的比例因子为 1/a。
5. Reflection in Detail | 反射详解
Reflecting the graph of y = f(x) in the x‑axis changes the sign of all y‑values, giving y = – f(x). Peaks become troughs and vice versa, while the x‑coordinates of points stay fixed. This is often used when the function is multiplied by –1.
将 y = f(x) 的图像关于 x 轴反射会改变所有 y 值的符号，得到 y = – f(x)。峰顶变为谷底，反之亦然，而各点的 x 坐标保持不变。这通常发生在函数被乘以 –1 时。
Reflection in the y‑axis changes the sign of the x‑variable inside the function: y = f(– x). The graph is mirrored left‑to‑right. Even functions, where f(– x) = f(x), show symmetry about the y‑axis and therefore appear unchanged under this reflection.
关于 y 轴反射会改变函数内部 x 变量的符号：y = f(– x)。图像左右镜像翻转。对于满足 f(– x) = f(x) 的偶函数，图像本身关于 y 轴对称，因此在这种反射下看起来不变。
6. Combining Transformations: Why Order Matters | 组合变换：顺序为何重要
e mixed. Consider performing a vertical stretch followed by a vertical translation: y = 2 f(x) + 3 is clearly different from y = 2[f(x) + 3] = 2 f(x) + 6. The latter stretches the translation as well, shifting the graph twice as far vertically.
当对 y = f(x) 施加两个或更多变换时，执行顺序会影响最终结果，尤其当水平与垂直变化混合时。考虑先垂直拉伸再垂直平移：y = 2 f(x) + 3 显然不同于 y = 2[f(x) + 3] = 2 f(x) + 6，后者对平移也进行了拉伸，将图像垂直移动了两倍的距离。
y = f(2x + 1) = f(2(x + 0.5)) represents a left shift of 0.5 followed by the same compression. A robust habit is to factorise the coefficient of x before describing horizontal transformations.
水平变换同样表现出这种敏感性。y = f(2(x + 1)) 是先向左平移 1 再水平压缩为原来的 1/2，而 y = f(2x + 1) = f(2(x + 0.5)) 则是先向左平移 0.5 再进行同样的压缩。一个可靠的习惯是：在描述水平变换之前，先将 x 的系数提取出来。
7. Horizontal vs Vertical: The ‘Inside‑Outside’ Rule | 水平与垂直：“先内后外”规则
A helpful guideline when combining shifts, stretches and reflections is to apply transformations that act ‘inside’ the function (on x) first, then those that act ‘outside’ (on the whole f(x)). The order should be: (1) horizontal transformations in the natural order of operations – factor out the x‑coefficient, then translation, then stretch/compression; (2) vertical transformations, usually working from the inside outward: stretch, then reflection, then translation.
在组合平移、伸缩和反射时，一条有用的指南是：先处理作用于函数“内部”（即对 x 作用）的变换，再处理作用于“外部”（对整个 f(x) 作用）的变换。操作顺序应为：(1) 水平变换按运算自然顺序——提取 x 的系数，然后平移，再伸缩/压缩；(2) 垂直变换通常由内向外进行：先伸缩，再反射，最后平移。
For example, to obtain y = –2 f( ½ , then stretch horizontally by factor 2. Vertically: stretch by factor 2, reflect in the x‑ tice how the reflection applies to the stretched function, not to the original f(x) alone.
例如，要从 y = f(x) 得到 y = –2 f( ½ (x + 3)) + 4，水平次序为：向左平移 3，然后水平拉伸至 2 倍。垂直次序为：先拉伸 2 倍，再关于 x 轴反射，最后向上平移 4。请注意，反射是作用于已拉伸的函数，而不仅作用于原始的 f(x)。
8. Describing Transformations from One Curve to Another | 用变换描述从曲线 A 到曲线 B
Exam questions often ask, ‘ g the algebraic changes: look for additions or subtractions outside the function (vertical translations), multipliers outside (vertical stretches/reflections), additions or subtractions inside (horizontal translations) and multipliers inside (horizontal stretches/reflections). When a combination is present, factorise the x‑term first.
考试题目常会问：“请完整描述将 y = f(x) 的图像映射到 y = g(x) 的图像的变换。”首先找出代数上的变化：观察函数外部的加减项（垂直平移）、外部乘数（垂直拉伸/反射）、内部的加减项（水平平移）以及内部的乘数（水平拉伸/反射）。当存在组合时，要先对 x 项进行因式分解。
onto y = sin(2x + 60°), write the target as y = sin[2(x + 30°)]. The description is then: ‘ izontal stretch of scale factor 1/2’. The order inside the bracket must be clearly stated.
例如，要将 y = sin x 映射为 y = sin(2x + 60°)，先把目标写成 y = sin[2(x + 30°)]。描述为：“先向左平移 30°，再进行水平拉伸，比例因子为 1/2（即水平压缩）”。括号内的顺序必须表述清晰。
9. Combined Transformations Involving Asymptotes | 涉及渐近线的组合变换
to the same transformation rules. If y = 1/x has asymptotes x = 0 and y = 0, then y = 3/(x – y = 1. Always track the asymptotes when sketching, because they define the boundaries of the curve.
对于倒数函数及对数函数，渐近线遵循相同的变换规则移动。若 y = 1/x 的渐近线为 x = 0 和 y = 0，则 y = 3/(x – 2) + 1 可通过以下变换得到：向右平移 2，垂直拉伸为 3 倍，再向上平移 1。新的渐近线变为 x = 2 和 y = 1。绘图时一定要追踪渐近线，因为它们界定了曲线的边界。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
ing the direction of horizontal translation: y = f(x + 5) shifts left, not right. Remember ‘inside opposite’.
混淆水平平移的方向：y = f(x + 5) 是向左移动，而非向右。记住“内部取反”。
should be written as f(2(x + 3)) before describing the stretch and translation. Otherwise the translation value is misread.
忘记因式分解：在描述拉伸与平移之前，应将 y = f(2x + 6) 写成 f(2(x + 3))，否则平移量会被误读。
+3 happens first, then the ×2 vertically. If you stretch first, you get a different graph.
在未检查影响的情况下先应用垂直变换：对于 y = 2 f(x + 3)，先进行 +3 的水平平移，再进行垂直的 ×2。如果先拉伸，就会得到不同的图像。
Ignoring the order of multiple stretches: y = f(–2x) involves a reflection in the y‑ are both horizontal and act independently, but you must still describe the correct scale.
忽视多次伸缩的顺序：y = f(–2x) 包含关于 y 轴的反射和水平压缩。这两者之间顺序无关紧要，因为它们都是水平变换且各自独立，但你仍需描述正确的比例因子。
11. Worked Examples in an Exam Style | 考试风格例题
Example 1: The graph of y = f(x) is transformed to give y = 3 f(½ x – 2) – 1. Describe the series of transformations step by step.
例题 1：将 y = f(x) 的图像经过变换得到 y = 3 f(½ x – 2) – 1。逐步描述变换系列。
Rewrite horizontal part: ½ x – 2 = ½ (x – 4). Therefore:
把水平部分重写：½ x – 2 = ½ (x – 4)。因此：
Step 1: Horizontal translation 4 units right. / 步骤1：向右平移 4 个单位。
tep 2: Horizontal stretch by factor 2 (since ½ x corresponds to stretch factor 1/½ = 2). / 步骤2：水平拉伸至 2 倍（因为 ½ x 对应的伸缩因子为 2）。
Step 3: Vertical stretch by factor 3. / 步骤3：垂直拉伸 3 倍。
Step 4: Vertical translation 1 unit down. / 步骤4：向下平移 1 个单位。
Example 2: The curve y = √x is first reflected in the y‑axis, then translated 3 units right and finally stretched vertically by a factor of 2. Find the equation of the resulting curve.
例题 2：曲线 y = √x 先关于 y 轴反射，再向右平移 3 个单位，最后垂直拉伸 2 倍。求所得曲线的方程。
Reflection in y‑axis: y = √–x / y 轴反射：y = √–x
Translation right 3: replace x by (x – 3) → y = √(–(x – 3)) = √–x + 3 / 向右平移 3：用 (x – 3) 替代 x → y = √(–(x – 3)) = √–x + 3
Vertical stretch ×2: y = 2√–x + 3. / 垂直拉伸 ×2：y = 2√–x + 3。
Domain check: original y = √x After reflection, domain x ≤ 0; after translation right 3, domain becomes x ≤ 3. This step‑by‑step approach avoids algebraic mistakes.
定义域检查：原 y = √x 要求 x ≥ 0。反射后定义域为 x ≤ 0；再向右平移 3 后，定义域变为 x ≤ 3。这种逐步处理能避免代数错误。
12. Summary and Quick Reference | 总结与快速参考
stematically. Remember the golden rule: for y = a f(b(x + c)) + d, the transformations are – horizontally: translate by – retch by factor 1/b (including reflection if b is negative); vertically: stretch by factor |a| trigonometric, exponential and reciprocal functions, will make these skills second nature.
只要你有条理地处理水平和垂直变化，图像的组合变换就是可预测的。请记住黄金法则：对于 y = a f(b(x + c)) + d，变换步骤是——水平方向：平移 –c，然后伸缩，比例因子为 1/b（若 b 为负则包含反射）；垂直方向：伸缩 |a| 倍，若 a 为负则反射，最后平移 d。用各种母函数（特别是三角函数、指数函数和倒数函数）多加练习，这些技能就会成为你的第二天性。
Quick reference:
快速参考：
y = f(x) + c: vertical translation / 垂直平移
y = f(x + c): horizontal translation opposite sign / 水平平移（符号相反）
y = a f(x): vertical stretch by a / 垂直拉伸 a 倍
y = f(ax): horizontal stretch by 1/a / 水平拉伸 1/a 倍
y = –f(x): reflect in x‑axis / 关于 x 轴反射
y = f(–x): reflect in y‑axis / 关于 y 轴反射
Combine inside‑out: horizontal, then vertical / 由内向外组合：先水平，再垂直
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