Mastering Stretches and Compressions of Function Graphs | 函数图像的伸缩变换要点

📚 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: 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: the graph is compressed vertically towards the x-axis 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。

Example: 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)

Horizontal transformations are the opposite of what many students expect. The function 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: the graph is compressed horizontally towards 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: the graph is stretched horizontally away 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。

Example: For f(x) = sin x, y = sin(2x) completes one full cycle in π radians instead of 2π. The period is halved.

Note that 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。

Consider y = 3 sin(2x). The 3 outside gives 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 = 5e^(3x) is a vertical stretch by 5 combined with a horizontal compression by ⅓.

同理，y = 5e^(3x) 是垂直伸展 5 倍与水平压缩 ⅓ 的组合。

Always test 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。

For vertical transformations in the form y = af(x) + b, apply the stretch first, then the translation. For example, starting with f(x) = x², the 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。

If you translated first, you would get 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。

For horizontal transformations, the order is trickier. Consider y = f(2x − 4). Rewrite it as y = f(2(x − 2)). The 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。

General rule: for 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: Consider y = x². The transformation y = (2x)² = 4x² is technically a horizontal 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: For y = A sin(Bx) and y = A cos(Bx), the constant 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 = e^(2x) can be seen as a horizontal compression of y = e^x by ½, but also as a vertical stretch because e^(2x) = (e^x)². Questions may ask you to state “the transformation that maps y = e^x to y = e^(2x)” — the expected answer is usually a horizontal compression by ½.

指数函数：y = e^(2x) 可以看作 y = e^x 的水平压缩 ½，但同时因为 e^(2x) = (e^x)²，它也可看作垂直伸展。题目如果问“将 y = e^x 变换为 y = e^(2x) 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: y = f(2x) compresses the graph horizontally; 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: for y = f(x + 3), the shift 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: always compress/stretch 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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Mastering Stretches and Compressions of Function Graphs | 函数图像的伸缩变换要点