📚 Combined Transformations of Functions | 函数图像的综合变换
rms by applying shifts, stretches, and reflections. In Edexcel A-Level Pure Mathematics, you are required not only to apply single transformations but also to combine several transformations into a single mapping, and to describe the resulting graph in terms of the original function. Understanding the order of operations and the effect of each transformation is essential for success in exams.
图像变换使我们能够通过平移、伸缩和反射等操作，将 f(x)=x² 或 f(x)=sinx 等基本函数修改成更复杂的形式。在 Edexcel A-Level 纯数学中，你不仅需要运用单一变换，还要将多个变换组合成一个整体映射，并能基于原函数描述最终图像。理解变换的顺序以及每种变换的效果是考试取得成功的关键。
1. Introduction to Graph Transformations | 图像变换简介
ction. The basic graph y=f(x) can be altered by applying a series of rules to either the x‑coordinate, the y‑coordinate, or both. These rules become especially powerful when used in combination.
变换会改变函数图像的位置、大小或方向。基本图像 y=f(x) 可以通过对 x 坐标、y 坐标或两者同时应用一系列规则而改变。当这些规则组合使用时，它们的功能会变得尤为强大。
nt a, b, c, d corresponds to a specific transformation: vertical stretch/reflection, horizontal stretch/reflection, horizontal translation, and vertical translation, respectively.
当我们组合变换时，通常会将新函数写成 y = a f(b(x + c)) + d 的形式。常数 a、b、c、d 分别对应特定的变换：垂直拉伸/反射、水平拉伸/反射、水平平移以及垂直平移。
It is crucial to remember that transformations inside the function argument (affecting x) work in the opposite direction to what you might intuitively expect.
必须牢记，函数自变量内部的变换（影响 x）的作用方向与你直觉预期的相反。
2. Vertical Translations | 垂直平移
cally. The transformation y = f(x) + d moves the graph up by d units if d > 0, and down by |d| units if d < 0.
给函数加上常数 d 会使图像垂直平移。变换 y = f(x) + d 当 d > 0 时图像向上移动 d 个单位，当 d < 0 时则向下移动 |d| 个单位。
shifts the whole curve 4 units upwards. This does not change the shape of the graph; it simply relocates every point.
例如，若 f(x) = x³，则 y = x³ + 4 将整条曲线向上平移 4 个单位。这不会改变图像形状，仅仅重新定位每一个点。
Vertical translations are applied after the function has been evaluated, so they affect the y‑coordinates directly.
垂直平移是在函数求值之后施加的，因此它会直接影响 y 坐标。
3. Horizontal Translations | 水平平移
ft. The graph of y = f(x + c) moves to the left by c units when c > 0, and to the right by |c| units when c < 0. This 'opposite' behavior often confuses students.
将函数中的 x 替换为 (x + c) 会产生水平平移。y = f(x + c) 的图像当 c > 0 时向左移动 c 个单位，当 c < 0 时则向右移动 |c| 个单位。这种“反向”行为常常让学生感到困惑。
Consider f(x) = √x. Then y = √x + 2 is the square root graph shifted 2 units to the left. The movement is opposite to the sign of c.
考虑 f(x) = √x。则 y = √x + 2 是平方根图像向左平移 2 个单位。移动方向与 c 的符号相反。
Because this transformation alters the input before the function acts on it, we say it takes place 'inside the bracket'.
由于此变换在函数作用之前改变了输入，我们常称之为发生在“括号内部”。
4. Vertical Stretches | 垂直拉伸
Multiplying the whole function by a constant a gives y = a f(x). If |a| > 1, the graph is stretched vertically away from the x‑axis. If 0 < |a| < 1, it is compressed towards the x‑axis. A negative a reflects the graph in the x‑axis as well as stretching or compressing it.
将整个函数乘以常数 a 得到 y = a f(x)。若 |a| > 1，图像会沿垂直方向远离 x 轴拉伸。若 0 < |a| < 1，图像会向 x 轴压缩。负的 a 在拉伸或压缩的同时还会使图像关于 x 轴反射。
the sine wave so that its maximum and minimum values become 3 and −3, instead of 1 and −1.
例如，y = 3 sinx 将正弦波拉伸，使其最大值和最小值变为 3 和 −3，而不是 1 和 −1。
Vertical stretches multiply the y‑coordinates, keeping x‑coordinates unchanged.
垂直拉伸会乘以 y 坐标，而 x 坐标保持不变。
5. Horizontal Stretches | 水平拉伸
The transformation y = f(bx) stretches the graph horizontally. If |b| > 1, the graph is compressed towards the y‑axis (points get closer together). If 0 < |b| < 1, the graph is stretched away from the y‑axis. Again, the effect is the reverse of the multiplier's magnitude.
变换 y = f(bx) 会水平拉伸图像。若 |b| > 1，图像向 y 轴压缩（点变得更紧密）。若 0 < |b| < 1，图像远离 y 轴拉伸。同样，效果与乘数的量级是相反的。
ull cycle in π radians rather than 2π, because the graph is compressed horizontally by a factor of 1/2.
例如，y = sin(2x) 在 π 弧度内完成一个完整周期，而非 2π，因为图像在水平方向被压缩了 1/2 倍。
Because the constant b is inside the bracket with x, the transformation is often described as 'stretch by factor 1/b in the x‑direction'.
由于常数 b 与 x 同处括号内部，这种变换通常被描述为“在 x 方向上拉伸 1/b 倍”。
6. Reflections | 反射变换
cases of stretches with a factor of −1. The mapping y = −f(x) reflects the graph in the x‑axis, turning points (x, y) into (x, −y).
反射是拉伸因子为 −1 的特殊情况。映射 y = −f(x) 将图像关于 x 轴反射，把点 (x, y) 变为 (x, −y)。
Similarly, y = f(−x) reflects the graph in the y‑ tal reflection and occurs because the sign of x is changed before the function is applied.
同样，y = f(−x) 将图像关于 y 轴反射，将 (x, y) 变为 (−x, y)。这是一种水平反射，因为 x 的符号在函数作用之前发生了变化。
−x) represents a rotation by 180° about the origin, effectively applying both reflections.
像 y = −f(−x) 这样的组合相当于绕原点旋转 180°，实质上同时应用了两种反射。
7. Combining Transformations: The General Form | 组合变换：一般形式
ression encodes up to four transformations: a vertical stretch/reflection (a), a horizontal stretch/reflection (b), a horizontal translation (−c), and a vertical translation (d).
我们通常将变换后的函数写成 y = a f(b(x + c)) + d。这一表达式编码了多达四种变换：垂直拉伸/反射 (a)、水平拉伸/反射 (b)、水平平移 (−c) 和垂直平移 (d)。
The form is carefully structured: the horizontal translation is written as (x + c) so that the shift is in the correct direction after the horizontal stretch. This arrangement is the key to understanding the correct order of operations.
该形式的精心构造体现在：水平平移写为 (x + c)，以便在水平拉伸之后沿正确方向移动。这种排列正是理解正确操作顺序的关键。
Every combination of transformations can be rewritten in this canonical form to avoid mistakes.
任何变换组合都可以改写为此规范形式以避免错误。
8. Order of Transformations | 变换顺序
When applying multiple transformations to a point or graph, the order matters. The standard procedure is to work from the inside out: first apply horizontal transformations (inside the bracket with x), then vertical transformations (outside the function).
当对一个点或图像施加多个变换时，顺序至关重要。标准流程是从内向外操作：首先应用水平变换（与 x 同在括号内），然后是垂直变换（在函数外部）。
ontally by 1/b first, then translate by −c. Outside, translation (d) occurs after stretch/reflection a.
在括号内部，拉伸/反射先于平移。对于 y = f(b(x + c))，你会先进行水平方向 1/b 的拉伸，然后再平移 −c。在外部，平移 (d) 发生在拉伸/反射 a 之后。
Reversing the order can lead to a completely different graph, especially when translations and stretches are both present.
颠倒顺序会导致完全不同的图像，特别是在平移和拉伸同时存在时。
9. Describing Combined Transformations | 描述组合变换
identify each transformation fully, giving its direction, magnitude, and whether it is a stretch or a translation.
考试题目常要求你描述将 y = f(x) 映射到 y = g(x) 的变换序列。你必须完整识别每个变换，说明其方向、大小以及是拉伸还是平移。
For instance, to map f(x) to 3 f(2x − 4), you could say: first, a horizontal translation by +4, then a horizontal stretch by factor 1/2, and finally a vertical stretch by factor 3. But the order can be expressed differently if you rewrite the bracket: 3 f(2(x − 2)). Then the horizontal translation becomes −2 after the stretch, which is often the preferred method.
例如，要描述从 f(x) 到 3 f(2x − 4) 的映射，你可以说：先水平平移 +4，再水平拉伸 1/2 倍，最后垂直拉伸 3 倍。但如果将括号改写为 3 f(2(x − 2))，则水平平移在拉伸后变为 −2，这往往是更推荐的方法。
Always check whether the function inside has been factorised first, as this clarifies the actual translation needed.
务必先检查函数内部是否已完成因式分解，因为这将明确实际所需的平移量。
10. Worked Examples | 例题解析
Example 1: transformed into y = 2 f(x + 3) − 1. State the three transformations in the correct order.
例题 1： y = f(x) 的图像被变换为 y = 2 f(x + 3) − 1。请按正确顺序说明这三个变换。
Solution: Reading from inside out: first, horizontal translation by −3 (left 3); then vertical stretch by factor 2; finally, vertical translation by −1 (down 1).
解答：从内向外解读：首先，水平平移 −3（向左 3）；然后，垂直拉伸 2 倍；最后，垂直平移 −1（向下 1）。
Example 2: Describe the transformation that maps y = sinx to y = 4 sin(0.5x + π) − 2.
例题 2： 描述将 y = sinx 映射到 y = 4 sin(0.5x + π) − 2 的变换。
.5(x + 2π). Now the sequence is: horizontal translation by −2π, horizontal stretch by factor 2, vertical stretch by factor 4, vertical translation by −2. Note that the horizontal stretch happens after the translation inside the bracket as per factorised form.
首先改写括号：0.5x + π = 0.5(x + 2π)。现在序列为：水平平移 −2π，水平拉伸 2 倍，垂直拉伸 4 倍，垂直平移 −2。注意，根据因式分解形式，水平拉伸发生于括号内的平移之后。
These examples highlight the importance of factorising the coefficient of x before describing the translation.
这些例子凸显了在描述平移前将 x 的系数因式分解的重要性。
11. Common Mistakes | 常见错误
Many students forget to factorise the x‑ ot −6; after rewriting as f(2(x + 3)), it is clear the translation is −3, but the stretch factor remains 1/2.
许多学生忘记将括号内的 x 项因式分解，从而导致水平平移错误。对于 y = f(2x + 6)，平移量并不是 −6；改写为 f(2(x + 3)) 后，平移量显然是 −3，而拉伸因子依然为 1/2。
slation is outside the stretch. The order outside is not commutative: y = a f(x) + d is different from y = a (f(x) + d).
另一个错误是按错误顺序进行变换，比如当平移在拉伸外部时，先做垂直拉伸再做垂直平移。外部的顺序不可交换：y = a f(x) + d 与 y = a (f(x) + d) 是不同的。
Students also often confuse horizontal and vertical transformations, particularly reflections. Remember that changes inside f affect x, outside affect y.
学生还经常混淆水平与垂直变换，尤其是反射。记住，f 内部的改变影响 x，外部的改变影响 y。
12. Exam Tips | 考试技巧
) + d, even if the question gives it in a different layout. This canonical form prevents sign errors and order mistakes.
务必先将变换后的函数写成 y = a f(b(x + c)) + d 的形式，即使题目给出的形式不同。这一规范形式可防止符号错误和顺序错误。
When sketching, plot key points by applying transformations step by step to a few original coordinates. For example, take (0, 0), (1, 1), (−1, 1) for a quadratic and track them through the sequence.
作图时，通过对少数原始坐标逐步施加变换来绘制关键点。例如，对二次函数取 (0, 0)、(1, 1)、(−1, 1)，并跟踪它们在变换序列中的位置。
Use clear language in descriptions: "translation by vector (a, b)", "stretch parallel to the x‑axis with scale factor k", "reflection in the y‑axis". Avoid vague terms.
描述时使用清晰的语言：“平移向量 (a, b)”、“沿 x 轴方向伸缩，比例因子为 k”、“关于 y 轴反射”。避免模糊表述。
Finally, remember that combined transformations are a favourite topic for higher‑mark questions on papers, so mastering this concept thoroughly will repay you in the exam.
最后，请记住组合变换是试卷中高分题目的热门话题，因此彻底掌握这一概念将在考试中为你带来回报。
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