A-Level Maths: Stretching and Compressing Graphs of Functions | A-Level 数学：函数图像的伸缩变换

📚 A-Level Maths: Stretching and Compressing Graphs of Functions | A-Level 数学：函数图像的伸缩变换

Graph transformations are a fundamental part of A-Level Mathematics. Among them, stretches and compressions allow us to scale a curve vertically or horizontally without changing its essential shape. This article explains the rules for stretch and compression transformations, with examples and exam tips to help you master the topic.

函数图像变换是 A-Level 数学的基础内容之一。其中，伸缩变换允许我们在不改变曲线基本形状的前提下，将图像沿纵向或横向缩放。本文讲解伸缩变换的规则，并通过例题和备考建议帮助你掌握这一考点。

1. Introduction to Transformations | 变换概述

Before focusing on stretches, recall that transforming a graph means applying a rule to the function so that every point on the original curve moves to a new position. The main types of transformations are translations (shifts), reflections, and stretches/compressions.

在聚焦伸缩变换之前，我们先回顾一下：对图像进行变换，就是将某种规则作用于函数，使原曲线上的每个点移动到新的位置。主要的变换类型包括平移、反射和伸缩。

For a function y = f(x), the stretch transformations are written as y = a f(x) for a vertical stretch and y = f(bx) for a horizontal stretch. The constants a and b control the scale factor.

对于函数 y = f(x)，伸缩变换可写为 y = a f(x) 表示纵向伸缩，y = f(bx) 表示横向伸缩。常数 a 和 b 控制缩放比例。

2. Vertical Stretch: y = af(x) | 纵向伸缩：y = af(x)

When we multiply the whole function by a constant a, each y-coordinate on the graph is multiplied by a. If a > 1, the graph stretches vertically away from the x-axis. If 0 < a < 1, the graph compresses vertically towards the x-axis.

当我们将整个函数乘以常数 a 时，图像上每个点的 y 坐标都会乘以 a。若 a > 1，图像沿纵向远离 x 轴伸展；若 0 < a < 1，图像沿纵向向 x 轴压缩。

Vertical stretch: y = a f(x), scale factor = a (relative to x-axis)

纵向伸缩：y = a f(x)，缩放因子为 a（相对于 x 轴）

The x-coordinates of all points remain unchanged.
Every y-coordinate is multiplied by a.
Points on the x-axis (where y = 0) stay fixed.

所有点的 x 坐标保持不变。
每个 y 坐标都乘以 a。
x 轴上的点（y = 0）保持不动。

3. Horizontal Stretch: y = f(bx) | 横向伸缩：y = f(bx)

For a horizontal stretch, we replace x by bx inside the function. The graph is stretched or compressed parallel to the x-axis. Crucially, the scale factor is 1/b, not b.

对于横向伸缩，我们将函数中的 x 替换为 bx。图像沿平行于 x 轴的方向被拉伸或压缩。关键点是，缩放因子是 1/b，而不是 b。

Horizontal stretch: y = f(bx), scale factor = 1/b (relative to y-axis)

横向伸缩：y = f(bx)，缩放因子为 1/b（相对于 y 轴）

If b > 1, then 1/b < 1, so the graph compresses horizontally.
If 0 < b < 1, then 1/b > 1, so the graph stretches horizontally.
All y-coordinates remain unchanged; the x-coordinates are divided by b.

若 b > 1，则 1/b < 1，图像在水平方向压缩。
若 0 < b < 1，则 1/b > 1，图像在水平方向拉伸。
所有 y 坐标不变；x 坐标除以 b。

4. Comparing Vertical and Horizontal Effects | 纵向与横向效果的对比

Students often confuse the two types of stretches. The table below summarises how the value of the multiplier changes a given point (x, y) on the original graph.

学生经常混淆这两种伸缩。下表总结了乘数取值如何改变原图像上给定点 (x, y) 的位置。

Transformation	Rule	Point mapping
Vertical stretch (a > 1)	y = a f(x)	(x, y) → (x, ay)
Vertical compression (0 < a < 1)	y = a f(x)	(x, y) → (x, ay)
Horizontal compression (b > 1)	y = f(bx)	(x, y) → (x/b, y)
Horizontal stretch (0 < b < 1)	y = f(bx)	(x, y) → (x/b, y)

Notice that the vertical stretch multiplies y while the horizontal stretch divides x by b. This inverse relationship for the horizontal case is a common source of mistakes.

注意，纵向伸缩是乘以 y，而横向伸缩是让 x 除以 b。横向情形这种“倒数”关系是常见的错误来源。

5. Mapping Points on a Graph | 图上点的映射

To apply a stretch, it is often helpful to track specific points. For example, take a point (p, q) on the graph of y = f(x).

进行伸缩变换时，跟踪具体点通常很有帮助。例如，取 y = f(x) 图像上的一点 (p, q)。

After a vertical stretch y = a f(x), the point becomes (p, aq). After a horizontal stretch y = f(bx), the point becomes (p/b, q). These rules apply to every point on the graph, including intercepts and turning points.

经过纵向伸缩 y = a f(x) 后，该点变为 (p, aq)。经过横向伸缩 y = f(bx) 后，该点变为 (p/b, q)。这些规则适用于图像上的每一个点，包括截距和顶点。

Vertical: (p, q) → (p, aq) Horizontal: (p, q) → (p/b, q)

纵向：(p, q) → (p, aq) 横向：(p, q) → (p/b, q)

This point-mapping view helps when sketching graphs without using a calculator: transform key points, then join them with a smooth curve.

这种点映射视角有助于在不用计算器的情况下画图：先变换关键点，再用平滑曲线连接它们。

6. Standard Graphs and Their Stretches | 常见函数图像的伸缩

Let us examine how stretches affect common functions you will meet in A-Level exams.

我们来看看伸缩如何影响 A-Level 考试中常见的函数图像。

Example 1: Quadratic f(x) = x². The graph of y = 2x² is a vertical stretch of y = x² by factor 2. Its vertex (0,0) stays fixed, but a point like (1,1) moves to (1,2). The graph becomes narrower in appearance, though the parabola opens upwards.

例 1：二次函数 f(x) = x²。 y = 2x² 是 y = x² 纵向拉伸 2 倍的图像。其顶点 (0,0) 保持不动，但像 (1,1) 这样的点移动到 (1,2)。图像看起来更窄，但抛物线仍开口向上。

Example 2: Sine f(x) = sin x. For y = sin(2x), the period becomes 2π/2 = π, so the graph compresses horizontally. The amplitude remains 1. For y = 3 sin x, the amplitude becomes 3, a vertical stretch.

例 2：正弦函数 f(x) = sin x。 对 y = sin(2x)，周期变为 2π/2 = π，因此图像横向压缩，振幅仍为 1。对 y = 3 sin x，振幅变为 3，属于纵向拉伸。

Example 3: Reciprocal f(x) = 1/x. The graph of y = 2/(x) is a vertical stretch of the hyperbola; the asymptotes at x = 0 and y = 0 do not move. For y = 1/(2x), which equals f(2x), the curve compresses horizontally, but the asymptotes stay the same.

例 3：反比例函数 f(x) = 1/x。 y = 2/x 是双曲线沿纵向的拉伸；渐近线 x = 0 和 y = 0 不动。对 y = 1/(2x)，它等于 f(2x)，曲线横向压缩，但渐近线仍保持不变。

7. Worked Example: Sketching Transformed Graphs | 例题：画变换后的图像

Suppose the graph of y = f(x) contains the points A(-2, 3), B(0, -1), and C(4, 5). Sketch the graph of (a) y = 2f(x) and (b) y = f(0.5x).

假设 y = f(x) 的图像经过点 A(-2, 3)、B(0, -1) 和 C(4, 5)。画出 (a) y = 2f(x) 和 (b) y = f(0.5x) 的图像。

Solution (a): Vertical stretch by factor 2. Multiply each y-coordinate by 2:

解 (a)：纵向拉伸 2 倍。 每个 y 坐标乘以 2：

A(-2, 3) → A′(-2, 6)
B(0, -1) → B′(0, -2)
C(4, 5) → C′(4, 10)

A(-2, 3) → A′(-2, 6)
B(0, -1) → B′(0, -2)
C(4, 5) → C′(4, 10)

Plot these new points and connect them with the same shape as the original curve.

标出这些新点，并用与原曲线相同的形状连接起来。

Solution (b): Horizontal stretch by factor 1/0.5 = 2. Since 0.5 < 1, the graph stretches away from the y-axis. Divide each x-coordinate by 0.5 (equivalently multiply by 2):

解 (b)：横向拉伸，缩放因子为 1/0.5 = 2。 由于 0.5 < 1，图像沿远离 y 轴的方向拉伸。每个 x 坐标除以 0.5（等价于乘以 2）：

A(-2, 3) → A′(-4, 3)
B(0, -1) → B′(0, -1)
C(4, 5) → C′(8, 5)

A(-2, 3) → A′(-4, 3)
B(0, -1) → B′(0, -1)
C(4, 5) → C′(8, 5)

Notice that B lies on the y-axis, so it does not move in a horizontal stretch. This is a useful check.

注意点 B 位于 y 轴上，因此横向伸缩时它不动。这是一个有用的检查方法。

8. Combining Stretches with Translations | 伸缩与平移的组合

In exams, a graph may be transformed by both stretches and translations. The general form is y = a f(b(x + c)) + d, where a is the vertical stretch factor, 1/b is the horizontal stretch factor, c is the horizontal shift, and d is the vertical shift.

在考试中，图像可能同时经历伸缩和平移。一般形式为 y = a f(b(x + c)) + d，其中 a 是纵向伸缩因子，1/b 是横向伸缩因子，c 是水平位移，d 是垂直位移。

The order of operations matters. For example, y = f(2x + 4) is not the same as y = f(2(x + 2)) in terms of intermediate steps, although they represent the same final graph. Always rewrite the argument of the function as b(x + c) to identify the horizontal shift correctly.

运算顺序很重要。例如，y = f(2x + 4) 与 y = f(2(x + 2)) 的中间步骤不同，尽管它们代表相同的最终图像。始终将函数参数改写为 b(x + c) 的形式，才能正确确定水平位移。

y = f(2x + 4) = f(2(x + 2)) → horizontal stretch by 1/2, then shift left by 2

y = f(2x + 4) = f(2(x + 2)) → 先横向压缩 1/2，再向左平移 2

For a vertical combination like y = 2f(x) + 3, multiply the y-values by 2 first, then add 3. This corresponds to a vertical stretch followed by a translation upwards.

对于纵向组合，如 y = 2f(x) + 3，先让 y 值乘以 2，再加上 3。这对应先纵向拉伸，再向上平移。

9. Common Pitfalls and Exam Tips | 常见错误与备考建议

Here are some mistakes that frequently appear in examiners’ reports:

以下是考官报告中经常出现的错误：

Mixing up the scale factor for horizontal stretches: remember y = f(bx) has scale factor 1/b, not b.
Forgetting that points on the axis of the stretch are fixed. For vertical stretches, points with y = 0 stay put; for horizontal stretches, points with x = 0 stay put.
Applying translations before stretches when the function is of the form y = f(bx + c). Always factor out b first.

混淆横向伸缩的缩放因子：记住 y = f(bx) 的缩放因子是 1/b，而不是 b。
忘记伸缩轴上的点是不动的：纵向伸缩时 y = 0 的点不动；横向伸缩时 x = 0 的点不动。
当函数形式为 y = f(bx + c) 时，先平移后拉伸。一定要先提出因子 b。

To avoid these errors, always write the transformation in the standard form and track at least two or three key points.

为了避免这些错误，请务必将变换写成标准形式，并跟踪两到三个关键点。

Exam tip: When asked to describe a transformation from a graph or equation, be precise. Say “vertical stretch with scale factor 3” or “horizontal compression with scale factor 1/2”, and always state the direction relative to an axis.

备考建议： 当被要求描述图像或方程所对应的变换时，要表述准确。例如说“纵向拉伸，缩放因子为 3”或“横向压缩，缩放因子为 1/2”，并始终说明相对于哪条轴的方向。

10. Conclusion | 结语

Stretches and compressions are straightforward once you remember two core rules: y = a f(x) multiplies y by a, and y = f(bx) divides x by b. Always interpret horizontal changes carefully because of the reciprocal scale factor.

只要记住两条核心规则，伸缩变换就变得简单：y = a f(x) 将 y 乘以 a，而 y = f(bx) 将 x 除以 b。由于倒数缩放因子的存在，解释横向变化时务必小心。

Practice by sketching transformed versions of familiar graphs, and check your work using key points. Combining stretches with translations is a high-weight topic in A-Level exams, so take time to master the order of operations.

通过画出熟悉函数图像的伸缩版本进行练习，并用关键点检查你的结果。将伸缩与平移组合是 A-Level 考试中的重点，请花时间掌握运算顺序。

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A-Level Maths: Stretching and Compressing Graphs of Functions | A-Level 数学：函数图像的伸缩变换