📚 Mastering Stretches and Compressions of Function Graphs | 函数图像的伸缩变换要点
Graph transformations are one of the most frequently tested topics in A-Level mathematics. Among them, stretches and compressions often confuse students because the rules are counter-intuitive, especially when dealing with horizontal transformations. This guide gives you a complete, exam-focused breakdown of how to master these transformations with confidence.
函数图像的变换是A-Level数学中最高频的考点之一。其中，伸缩变换尤其容易让学生困惑，因为它的规则有时与直觉相反，特别是水平方向的处理。本指南将为你提供一个完整、紧扣考点的梳理，帮助你自信掌握这一内容。
1. What Are Stretches and Compressions? | 什么是伸缩变换？
Stretches and compressions (also called dilations) are transformations that change the size of a graph in a given direction, without altering its basic shape. There are two types: vertical (along the y-axis) and horizontal (along the x-axis). These transformations are written directly inside or outside the function notation, and their rules are strictly defined.
伸缩变换（stretches and compressions，也称伸缩变形或伸缩）是一种沿特定方向改变图像尺寸、但保留其基本形状的变换。它分为两类：垂直方向（沿 y 轴）和水平方向（沿 x 轴）。它们的写法取决于变换是位于函数表达式内部还是外部，规则也非常明确。
The two basic forms you must remember are:
你必须牢记两种基本形式：
y = a f(x) (vertical stretch/compression) | y = f(kx) (horizontal stretch/compression)
Read on to understand exactly what each parameter does and how to apply these rules in exams.
接下来我们将逐一解释每个参数的作用，以及如何在考试中正确运用这些规则。
2. Vertical Stretch and Compression: y = a f(x) | 垂直伸缩变换：y = a f(x)
When you multiply the whole function by a constant a, every y-coordinate is multiplied by a. The x-coordinates remain unchanged, so each point (x, y) maps to (x, ay).
当你将整个函数乘以常数 a 时，所有 y 坐标都会被乘以 a；x 坐标保持不变，因此每个点 (x, y) 变为 (x, ay)。
If a > 1: om the graph is stretched vertically away from the x-axis by a factor of a. For example, y = 3f(x) makes the graph three times taller.
如果 a > 1：图像沿垂直方向远离 x 轴伸展，拉伸倍数为 a。例如，y = 3f(x) 使图像变为原来的三倍高。
If 0 < a < 1: s by a factor of a. For example, y = ½f(x) squeezes the graph to half its original height.
如果 0 < a < 1：图像沿垂直方向向 x 轴压缩，压缩倍数为 a。例如，y = ½f(x) 把图像高度压缩为原来的一半。
Key point: any point where f(x) = 0 (x-intercepts) remains fixed, because 0 × a = 0.
关键点：任何满足 f(x) = 0 的点（x 截距）都保持不变，因为 0 × a = 0。
ample: For f(x) = x², y = 2f(x) = 2x² has vertex (0,0) but the point (1,1) moves to (1,2).
In general, vertical stretches only affect the "height" of a graph; the "width" and horizontal position stay the same.
总之，垂直伸缩只影响图像的“高度”，宽度和水平位置保持不变。
3. Horizontal Stretch and Compression: y = f(kx) | 水平伸缩变换：y = f(kx)
tion y = f(kx) replaces every x with kx. As a result, each point (x, y) moves to (x/k, y).
水平变换与许多学生的直觉恰好相反。函数 y = f(kx) 相当于把每个 x 替换为 kx，因此每个点 (x, y) 都移动到 (x/k, y)。
If k > 1: owards the y-axis by a factor of 1/k. For example, y = f(2x) makes the graph half as wide.
如果 k > 1：图像沿水平方向向 y 轴压缩，压缩倍数为 1/k。例如，y = f(2x) 使图像宽度变为原来的一半。
If 0 < k < 1: from the y-axis by a factor of 1/k. For example, y = f(½x) makes the graph twice as wide.
如果 0 < k < 1：图像沿水平方向远离 y 轴伸展，伸展倍数为 1/k。例如，y = f(½x) 使图像宽度变为原来的两倍。
Critical rule: the horizontal scale factor is always the reciprocal of the multiplier inside the function.
关键规则：水平缩放倍数始终是函数内部乘数的倒数。
x, y = sin(2x) completes one full cycle in π radians instead of 2π. The period is halved.
the y-intercept (where x = 0) does not move, because replacing 0 with k × 0 still gives 0.
注意：y 截距（即 x = 0 处）不会移动，因为 0 替换为 k × 0 仍然是 0。
4. Vertical vs Horizontal: Key Differences | 垂直与水平伸缩的核心区别
Students often confuse the two directions. The table below summarises the essential differences:
学生经常混淆两个方向。下表总结了核心区别：
| Feature | y = a f(x) | y = f(kx) |
| Direction | Vertical (y-axis direction) | Horizontal (x-axis direction) |
| Coordinates affected | y-coordinates multiplied by a | x-coordinates divided by k |
| Scale factor | a (directly) | 1/k (reciprocal) |
| If the multiplier is 2 | Stretch taller by 2 | Compress narrower by ½ |
| Fixed points | x-intercepts stay fixed | y-intercept stays fixed |
A helpful memory: "outside the brackets changes y; inside the brackets changes x, and take the reciprocal for x."
一个有用的记忆技巧：“括号外面影响 y；括号里面影响 x，并且影响 x 时要用倒数。”
5. Identifying the Type from the Equation | 从方程识别伸缩类型
When given an equation, look at where the constant sits relative to the function symbol.
当你拿到一个方程时，先看常数相对于函数符号的位置。
If it multiplies the entire function (e.g. y = 4f(x)), it is a vertical stretch by factor 4. The graph becomes taller.
如果它乘在整个函数外面（例如 y = 4f(x)），这就是垂直伸缩，伸展倍数为 4，图像变得更高。
If it sits inside the argument (e.g. y = f(4x)), it is a horizontal compression by factor ¼. The graph becomes narrower.
如果它位于函数自变量内部（例如 y = f(4x)），这就是水平压缩，压缩倍数为 ¼，图像变得更窄。
ives a vertical stretch by 3; the 2 inside gives a horizontal compression by ½. The amplitude becomes 3 and the period becomes π.
以 y = 3 sin(2x) 为例。外面的 3 表示垂直伸展 3 倍；里面的 2 表示水平压缩 ½。因此振幅变为 3，周期变为 π。
Similarly, y = 5e3x is a vertical stretch by 5 combined with a horizontal compression by ⅓.
同理，y = 5e3x 是垂直伸展 5 倍与水平压缩 ⅓ 的组合。
with a reference point: plug in x = 0 or find an intercept to verify your identification.
始终用参考点验证：代入 x = 0 或找一个截距来检查你的判断。
6. Combining Stretches with Translations | 伸缩与平移的组合
In exams, stretches rarely appear alone. You must know the correct order of operations when combining with translations.
在考试中，伸缩很少单独出现。你必须知道伸缩与平移组合时的正确顺序。
graph of y = 2f(x) + 1 is obtained by first doubling all y-values, then shifting up by 1.
对于形如 y = af(x) + b 的垂直变换，应“先伸缩，后平移”。例如，从 f(x) = x² 出发，y = 2f(x) + 1 的方法是：先把所有 y 值加倍，然后上移 1 个单位。
et y = 2(f(x) + 1) = 2f(x) + 2, which is a different graph. Always double-check the order.
如果你先平移，就会得到 y = 2(f(x) + 1) = 2f(x) + 2，这是完全不同的图像。务必检查运算顺序。
e correct sequence is: first apply the horizontal compression by ½, then shift right by 2.
水平变换的顺序更复杂。考虑 y = f(2x − 4)。把它改写为 y = f(2(x − 2))。正确的顺序是：先作水平压缩 ½，再向右平移 2 个单位。
or y = f(kx + c), rewrite as f(k(x + c/k)); compress by 1/k first, then shift left by c/k.
The reason is that horizontal transformations act on x directly; extracting the coefficient first reveals the true shift.
原因是水平变换直接作用于 x；先把系数提取出来，才能看到真正的平移量。
7. Special Cases: Quadratic and Trigonometric Functions | 特殊情形：二次函数与三角函数
Certain functions behave in surprising ways under stretches, and exam questions often exploit this.
某些函数在伸缩变换下会呈现特殊行为，考试题目常常利用这一点。
Quadratic functions: tal compression by ½, but it looks identical to a vertical stretch by 4. This is because squaring amplifies the horizontal effect. You must decide based on the wording of the question.
二次函数：考虑 y = x²。变换 y = (2x)² = 4x² 从技术上说是水平压缩 ½，但它看起来与垂直伸展 4 完全相同。这是因为平方放大了水平效果。你必须根据题目的表述来判断。
Trigonometric functions: onstant A is the vertical stretch (amplitude) and B controls the horizontal stretch/compression through the period. The period is given by 2π/B (in radians) or 360°/B (in degrees).
三角函数：对于 y = A sin(Bx) 和 y = A cos(Bx)，常数 A 是垂直伸缩（振幅），B 通过周期控制水平伸缩。周期为 2π/B（弧度制）或 360°/B（角度制）。
Exponential functions: y = e2x can be seen as a horizontal compression of y = ex by ½, but also as a vertical stretch because e2x = (ex)². Questions may ask you to state "the transformation that maps y = ex to y = e2x" — the expected answer is usually a horizontal compression by ½.
指数函数：y = e2x 可以看作 y = ex 的水平压缩 ½，但同时因为 e2x = (ex)²，它也可看作垂直伸展。题目如果问“将 y = ex 变换为 y = e2x 的变换”，通常期望的答案是水平压缩 ½。
8. Invariant Points and Symmetry | 不变点与对称性
Invariant points are points that do not move under a transformation. They are extremely useful for sketching and checking answers.
不变点是指在变换下不发生移动的点。它们在绘图和验证答案时非常有用。
Under y = af(x): every point where y = 0 (i.e. all x-intercepts) remains fixed.
在 y = af(x) 下：所有 y = 0 的点（即所有 x 截距）保持不变。
Under y = f(kx): the point where x = 0 (the y-intercept) remains fixed.
在 y = f(kx) 下：x = 0 的点（即 y 截距）保持不变。
Additionally, vertical stretches preserve the x-coordinates of turning points, although their y-values change. Horizontal compressions preserve y-coordinates of all points, while x-coordinates scale by 1/k.
此外，垂直伸缩保持极值点的 x 坐标不变，但 y 值改变。水平压缩保持所有点的 y 坐标不变，而 x 坐标按 1/k 缩放。
Symmetry is also preserved: a vertical stretch preserves even symmetry (about the y-axis) and odd symmetry (about the origin). A horizontal compression also preserves both types of symmetry, provided the scale factor is applied symmetrically about the origin.
对称性同样得到保持：垂直伸缩保持偶对称（关于 y 轴对称）和奇对称（关于原点对称）。水平压缩也保持这两类对称性，前提是缩放关于原点对称进行。
9. Common Mistakes to Avoid | 常见错误与避坑指南
Many marks are lost each year due to a few recurring mistakes. Here are the most important ones to avoid.
每年都有大量分数因为几个反复出现的错误而丢失。下面是最需要避免的几点。
- Confusing stretch with compression: ntally; it does NOT stretch it. Always check whether the multiplier is greater than or less than 1.
- 混淆伸展与压缩：y = f(2x) 是水平压缩，不是水平伸展。始终检查乘数是大于 1 还是小于 1。
- Forgetting the reciprocal for horizontal: the scale factor for y = f(kx) is 1/k, not k.
- 忘记水平方向的倒数：y = f(kx) 的缩放倍数是 1/k，而不是 k。
- Applying horizontal translations incorrectly: hift is left, not right. For y = f(x − 3), the shift is right. Many students reverse this.
- 水平平移方向搞反：y = f(x + 3) 向左平移，y = f(x − 3) 向右平移。很多学生把方向弄反。
- Ignoring the order of combined transformations: h before translating when the transformation is written as y = af(x) + b or y = f(kx + c).
- 忽略组合变换的顺序：当表达式为 y = af(x) + b 或 y = f(kx + c) 时，务必先伸缩再平移。
- Forgetting to state the scale factor: exam answers must include "scale factor ¼ horizontally" or "stretch by 3 in the y-direction". Give the full description.
- 忘记写清缩放倍数：考试答案必须写明“水平缩放 ¼”或“沿 y 方向伸缩 3 倍”，要给出完整描述。
10. Exam Practice Tips and Summary | 备考建议与总结
To master stretches and compressions, build a quick checklist before answering any transformation question.
要掌握伸缩变换，请在回答任何变换题之前建立一个快速检查清单。
- Identify whether the constant is outside (vertical) or inside (horizontal) the function.
- 判断常数是在函数外面（垂直）还是里面（水平）。
- State the scale factor: a for vertical; 1/k for horizontal.
- 写明缩放倍数：垂直方向为 a；水平方向为 1/k。
- Pay attention to the direction: a > 1 means vertical stretch; k > 1 means horizontal compression.
- 注意方向：a > 1 表示垂直伸展；k > 1 表示水平压缩。
- When combining with translations, determine the order carefully and rewrite horizontal expressions like f(kx + c) as f(k(x + c/k)).
- 与平移组合时，仔细判断顺序，并把水平表达式 f(kx + c) 改写为 f(k(x + c/k))。
- Use invariant points as anchors when sketching the transformed graph.
- 画图时用不变点作为锚点。
Remember the golden rule: "outside changes y, inside changes x — and for x, use the reciprocal." With practice, stretches and compressions will become one of the easiest transformation questions on your paper.
请记住这条黄金法则：“外面影响 y，里面影响 x——而且处理 x 时要用倒数。”多加练习，伸缩变换将成为你卷面上最容易拿分的变换题之一。
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