A-Level Maths: Transformations of Graphs – Stretches | A-Level数学：函数图像伸缩变换

📚 A-Level Maths: Transformations of Graphs – Stretches | A-Level数学：函数图像伸缩变换

Stretching a graph is one of the key transformations you need to master for Edexcel A-Level Mathematics. Together with translations and reflections, stretches allow you to sketch complicated functions from a small set of basic graphs. You will also need to reverse the process: given a transformed graph, identify the stretch factor and write its equation.

伸缩变换是 Edexcel A-Level 数学必须掌握的关键图像变换之一。与平移和反射一起，伸缩变换可以让你从少数基本图像出发，画出复杂函数的图像。你还需要会逆向操作：根据变换后的图像，确定伸缩因子并写出其方程。

1. The Two Stretch Types | 两种伸缩类型

There are two independent ways to stretch a graph: vertically and horizontally. A vertical stretch multiplies every y-coordinate by a constant factor, keeping the x-coordinate unchanged. A horizontal stretch multiplies every x-coordinate by a factor, keeping the y-coordinate unchanged.

图像有两种互不相关的伸缩方式：垂直伸缩和水平伸缩。垂直伸缩将每个点的纵坐标 y 乘以一个常数倍，横坐标 x 保持不变；水平伸缩则将每个点的横坐标 x 乘以一个因子，纵坐标 y 保持不变。

The two definitions are written as:

y = a f(x) → vertical stretch, factor a, in the y-direction

y = f(ax) → horizontal stretch, factor 1/a, in the x-direction

Notice the asymmetry: when the multiplier appears outside the function, it affects the output (y). When it appears inside the function, it affects the input (x), and the image is stretched by the reciprocal factor.

请注意这种不对称性：倍数在函数外面时影响输出（y）；倍数在函数里面时影响输入（x），并且图像按倒数倍伸缩。

2. Vertical Stretch: y = a f(x) | 垂直伸缩：y = a f(x)

For a > 1, the graph of y = a f(x) is the graph of y = f(x) stretched vertically by factor a about the x-axis. Every point (x, y) moves to (x, a y).

当 a > 1 时，y = a f(x) 的图像是把 y = f(x) 的图像沿 y 轴方向按因子 a 伸长（以 x 轴为基准）。每个点 (x, y) 变为 (x, a y)。

Example: f(x) = x² and a = 2 gives y = 2x². The turning point (0, 0) stays fixed, while the point (1, 1) moves to (1, 2) and (2, 4) moves to (2, 8). The graph becomes narrower in appearance but its x-axis intercepts do not change.

例如：f(x) = x²，取 a = 2，得 y = 2x²。顶点 (0, 0) 不动，点 (1, 1) 移到 (1, 2)，点 (2, 4) 移到 (2, 8)。图像虽然看上去更“瘦”，但 x 轴交点不变。

Key points: x-axis intercepts are invariant under a vertical stretch. If a = 1, no change occurs. If 0 < a < 1, the graph is compressed vertically.

要点：x 轴交点在垂直伸缩下保持不变。若 a = 1，图像不变；若 0 < a < 1，图像沿垂直方向被压缩。

3. Horizontal Stretch: y = f(ax) | 水平伸缩：y = f(ax)

For a > 1, the graph of y = f(ax) is the graph of y = f(x) stretched horizontally by factor 1/a about the y-axis. Every point (x, y) moves to (x/a, y).

当 a > 1 时，y = f(ax) 的图像是把 y = f(x) 的图像沿 x 轴方向按因子 1/a 压缩（以 y 轴为基准）。每个点 (x, y) 变为 (x/a, y)。

Example: f(x) = sin x and a = 2 gives y = sin 2x. The period changes from 360° to 180° (or from 2π to π), so the graph completes two full cycles in the original interval. The y-axis intercept remains the same.

例如：f(x) = sin x，a = 2，得 y = sin 2x。周期从 360° 变为 180°（即从 2π 变为 π），因此在原来的区间内会出现两个完整周期。y 轴截距不变。

Important: y = f(2x) does not stretch the graph by a factor of 2 horizontally; it compresses it by a factor of 1/2. To stretch the graph horizontally by a factor of 3, you need a = 1/3, giving y = f(x/3).

注意：y = f(2x) 并不是把图像水平放大 2 倍，而是压缩为原来的 1/2。若要将图像水平拉长 3 倍，需要令 a = 1/3，即 y = f(x/3)。

4. Negative Scale Factors and Reflections | 负伸缩因子与反射

If the multiplying constant is negative, the same stretch factor is applied to the absolute value, and the graph is then reflected in the corresponding axis.

如果伸缩常数为负，则先按绝对值进行伸缩，再关于相应坐标轴进行反射。

For y = a f(x) with a < 0, the graph is first reflected in the x-axis and then stretched vertically by |a|. For example, y = -3 f(x) means a horizontal-axis reflection combined with a vertical stretch factor 3.

对于 a < 0 的 y = a f(x)，先关于 x 轴反射，再按 |a| 进行垂直伸缩。例如 y = -3 f(x) 表示关于 x 轴反射，同时垂直拉伸 3 倍。

For y = f(ax) with a < 0, the graph is reflected in the y-axis and then stretched horizontally by 1/|a|. For example, y = f(-2x) is a y-axis reflection together with a horizontal compression by factor 1/2.

对于 a < 0 的 y = f(ax)，先关于 y 轴反射，再进行水平伸缩，水平伸缩因子为 1/|a|。例如 y = f(-2x) 是 y 轴反射加上水平压缩 1/2。

A pure reflection is just a stretch with factor −1: y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis.

纯反射就是拉伸因子为 −1 的伸缩：y = −f(x) 关于 x 轴反射；y = f(−x) 关于 y 轴反射。

5. Stretches and Translations Combined | 伸缩与平移的组合

When translations and stretches appear together, the order matters. For expressions of the form y = a f(bx + c) + d, you must identify the correct sequence.

当伸缩和平移同时出现时，变换顺序至关重要。对于形如 y = a f(bx + c) + d 的表达式，必须确定正确的变换次序。

For the horizontal part, rewrite the argument as b(x + c/b). This shows that the graph is first stretched horizontally by factor 1/b (with reflection if b < 0), and then translated to the left by c/b (or to the right if c/b is negative).

对于水平部分，先把括号内改写为 b(x + c/b)。这说明图像先按水平因子 1/b 伸缩（若 b < 0 则还要反射），然后向左平移 c/b（若 c/b 为负则向右平移）。

Example: y = f(2x − 4) can be written as y = f(2(x − 2)). Start with y = f(x), compress horizontally by factor 1/2, then translate right by 2. If you translate first, you would get the wrong graph.

例如：y = f(2x − 4) 可以写成 y = f(2(x − 2))。先由 y = f(x) 水平压缩 1/2，再向右平移 2。如果先平移后压缩，就会得到错误的图像。

For the vertical part, y = a f(x) + d is interpreted as: stretch vertically by factor a, then translate up by d. Again, the order is fixed by the algebra.

对于垂直部分，y = a f(x) + d 的解读是：先垂直拉伸 a 倍，再向上平移 d。同样，顺序由代数结构决定。

6. How to Sketch a Stretched Graph | 如何绘制伸缩后的图像

To sketch a graph under a stretch, follow these steps:

Identify the basic graph y = f(x) and mark its key features: intercepts, turning points, asymptotes.

先确定基本图像 y = f(x)，标出关键特征：截距、顶点、渐近线。
Decide whether the constant multiplies f(x) or x. If it multiplies f(x), apply a vertical stretch; if it multiplies x, apply a horizontal stretch.

判断常数是乘以 f(x) 还是 x。如果乘以 f(x)，作垂直伸缩；如果乘以 x，作水平伸缩。
Choose a few reference points on the original graph, apply (x, y) → (x, ay) for a vertical stretch, or (x, y) → (x/a, y) for a horizontal stretch.

在原图上选择若干参考点，垂直伸缩时应用 (x, y) → (x, ay)，水平伸缩时应用 (x, y) → (x/a, y)。
Draw the new curve smoothly through the transformed points, preserving the overall shape.

将变换后的点用平滑曲线连接，保持图像的整体形状。
Label the coordinates of any intercepts and turning points that have moved.

标出移动后截距和顶点的坐标。

Always check whether the graph is stretched or compressed: for y = a f(x), a > 1 is an outward stretch; for y = f(ax), a > 1 is an inward compression.

始终检查图像是被拉伸还是被压缩：y = a f(x) 中 a > 1 是向外拉伸；y = f(ax) 中 a > 1 是向内压缩。

7. Finding the Equation from a Given Graph | 根据图像求方程

In exam questions you may be given the image of a graph after a stretch and asked to find its equation. You need to compare the coordinates of corresponding points.

考试中可能会给出一个经过伸缩变换的图像，要求你写出其方程。你需要比较对应点的坐标。

If a point (x₁, y₁) on the original graph maps to (x₂, y₂) on the transformed graph, then:

Vertical stretch: x₂ = x₁ and y₂ = a y₁, so a = y₂ / y₁.

垂直伸缩：x₂ = x₁，y₂ = a y₁，所以 a = y₂ / y₁。
Horizontal stretch: y₂ = y₁ and x₂ = x₁ / a, so a = x₁ / x₂.

水平伸缩：y₂ = y₁，x₂ = x₁ / a，所以 a = x₁ / x₂。

Example: If y = √x is transformed to y = √(kx) and the point (4, 2) on the original graph moves to (1, 2), then k = 4/1 = 4. The equation is y = √(4x).

例：若 y = √x 被变换为 y = √(kx)，原图上点 (4, 2) 移到 (1, 2)，则 k = 4/1 = 4，方程为 y = √(4x)。

For combined transformations, reverse the order using the algebraic structure. Write the new equation in the form y = a f(bx + c) + d and solve for a, b, c, d.

对于组合变换，根据代数结构反向拆分。将方程写成 y = a f(bx + c) + d 的形式，从而确定 a、b、c、d 的值。

8. Effect on Key Features | 对关键特征的影响

Different features of a graph respond differently to stretches:

Feature	Vertical stretch y = a f(x)	Horizontal stretch y = f(ax)
x-intercepts	unchanged	multiplied by 1/a
y-intercept	multiplied by a	unchanged (unless x=0 is outside domain)
Turning point (p, q)	(p, a q)	(p/a, q)
Vertical asymptote x = p	unchanged	becomes x = p/a
Horizontal asymptote y = q	becomes y = a q	unchanged

Notice that a vertical stretch changes all y-coordinates, so it affects horizontal asymptotes but not vertical ones. A horizontal stretch changes all x-coordinates, so it affects vertical asymptotes but not horizontal ones.

注意：垂直伸缩改变所有 y 坐标，因此影响水平渐近线而不影响垂直渐近线；水平伸缩改变所有 x 坐标，因此影响垂直渐近线而不影响水平渐近线。

9. Stretches of Trigonometric Graphs | 三角函数图像的伸缩

Trigonometric graphs are common in stretch questions. For y = A sin(Bx) + D, the vertical stretch factor A changes the amplitude, while the horizontal stretch changes the period.

三角函数图像是伸缩问题的常客。对于 y = A sin(Bx) + D，垂直伸缩因子 A 改变振幅，水平伸缩则改变周期。

The amplitude of y = A sin x is |A|. The period of y = sin(Bx) is 2π / |B|. If B > 1, the graph is compressed horizontally; if 0 < B < 1, it is stretched horizontally.

y = A sin x 的振幅为 |A|。y = sin(Bx) 的周期为 2π / |B|。若 B > 1，图像水平压缩；若 0 < B < 1，图像水平拉伸。

Example: y = 3 cos(2x) has amplitude 3 and period π. Compared to y = cos x, it is stretched vertically by 3 and compressed horizontally by a factor 1/2.

例：y = 3 cos(2x) 的振幅为 3，周期为 π。与 y = cos x 相比，它在垂直方向拉伸 3 倍，水平方向压缩 1/2。

The same rules apply to tan x, but vertical asymptotes move with horizontal stretches; for example, y = tan(2x) has asymptotes at x = π/4 + kπ/2 instead of x = π/2 + kπ.

同样的规律也适用于 tan x，但垂直渐近线会随水平伸缩移动。例如 y = tan(2x) 的渐近线为 x = π/4 + kπ/2，而不是 x = π/2 + kπ。

10. Common Mistakes and Exam Tips | 常见错误与应试提示

Students often confuse the direction of a horizontal stretch. Remember: y = f(2x) is narrower, not wider. A quick way to check is to use the point (1, f(1)): in y = f(2x), that point becomes (1/2, f(1)).

学生经常搞混水平伸缩的方向。记住：y = f(2x) 是变窄，而不是变宽。快速检验方法：看点 (1, f(1))，在 y = f(2x) 中该点变为 (1/2, f(1))。

Another common mistake is applying transformations in the wrong order. Always rewrite the function in the form a f(b(x − h)) + k before deciding the sequence.

另一个常见错误是变换顺序使用不当。务必先把函数改写成 a f(b(x − h)) + k 的形式，再决定变换顺序。

Do not assume that a horizontal stretch about the y-axis leaves turning points with respect to y unchanged; the x-coordinate will change unless the turning point lies on the y-axis.

不要以为水平伸缩以 y 轴为基准，顶点的 y 坐标就不变；除非顶点在 y 轴上，否则其 x 坐标一定会改变。

Finally, when a question says “stretch factor 2 parallel to the y-axis”, write y = 2 f(x). When it says “stretch factor 2 parallel to the x-axis”, write y = f(x/2).

最后，若题目说“沿 y 轴方向伸缩因子为 2”，应写 y = 2 f(x)；若说“沿 x 轴方向伸缩因子为 2”，应写 y = f(x/2)。

11. Worked Example | 典型例题

Given the graph of y = x² − 4x + 3, find the equation of the graph obtained by stretching it vertically with factor 2 about the x-axis, and then translating it up by 1 unit. Then describe the horizontal stretch that would make the resulting graph pass through (1, 0).

已知 y = x² − 4x + 3 的图像，先以 x 轴为基准垂直拉伸 2 倍，再向上平移 1 个单位，求所得图像的方程。然后再解释要使最终图像经过点 (1, 0)，应进行怎样的水平伸缩。

Vertical stretch: y = 2(x² − 4x + 3) = 2x² − 8x + 6. Then translate up by 1: y = 2x² − 8x + 7.

垂直拉伸后：y = 2(x² − 4x + 3) = 2x² − 8x + 6。再向上平移 1 单位：y = 2x² − 8x + 7。

Let a horizontal stretch factor k be applied: replace x by x/k, giving y = 2(x/k)² − 8(x/k) + 7. To pass through (1, 0), set x = 1, y = 0:

设水平伸缩因子为 k，即将 x 换为 x/k，得 y = 2(x/k)² − 8(x/k) + 7。要使图像过 (1, 0)，令 x = 1，y = 0：

0 = 2/k² − 8/k + 7

Multiplying by k² gives 7k² − 8k + 2 = 0. Using the quadratic formula, k = (8 ± √(64 − 56)) / 14 = (8 ± 2√2) / 14 = (4 ± √2) / 7. Both values are valid, so there are two possible horizontal stretch factors.

两边乘以 k² 得 7k² − 8k + 2 = 0。用求根公式，k = (8 ± √(64 − 56)) / 14 = (8 ± 2√2) / 14 = (4 ± √2) / 7。两个值都有效，因此存在两个可能的水平伸缩因子。

12. Practice Questions | 练习

Try these questions yourself:

试做以下题目：

Sketch y = 3√x and y = √(x/2) on separate axes, starting from y = √x.

从 y = √x 出发，分别画出 y = 3√x 和 y = √(x/2) 的图像。
The graph of y = 1/x is stretched horizontally by factor 3 about the y-axis. Write the new equation and find the new vertical asymptote.

将 y = 1/x 的图像以 y 轴为基准水平拉伸 3 倍。写出新方程，并求出新的垂直渐近线。
Describe the sequence of transformations to obtain y = 2 f(3x − 6) + 1 from y = f(x).

描述从 y = f(x) 得到 y = 2 f(3x − 6) + 1 的变换序列。
If a point (4, 9) on y = g(x) maps to (2, 18) on y = h(x), and h(x) can be obtained by one vertical and one horizontal stretch, find the stretch factors.

若 y = g(x) 上一点 (4, 9) 在 y = h(x) 上对应点为 (2, 18)，且 h(x) 由一次垂直伸缩和一次水平伸缩得到，求两个伸缩因子。

Answers: (1) vertical factor 3; horizontal factor 2 because y = √(x/2) = √(x × 1/2) uses a = 1/2, so stretch factor is 2. (2) y = 3/x; vertical asymptote x = 0 unchanged because horizontal stretch about the y-axis keeps x = 0 fixed. (3) Rewrite as y = 2 f(3(x − 2)) + 1: horizontal compression 1/3, translate right 2; vertical stretch 2, translate up 1. (4) Vertical stretch factor 2, horizontal stretch factor 2.

答案：（1）垂直因子 3；水平因子 2，因为 y = √(x/2) = √(x × 1/2) 中 a = 1/2，所以伸缩因子为 2。（2）y = 3/x；垂直渐近线 x = 0 不变，因为以 y 轴为基准的水平伸缩保持 x = 0 不动。（3）改写为 y = 2 f(3(x − 2)) + 1：水平压缩 1/3，右移 2；垂直拉伸 2，上移 1。（4）垂直伸缩因子为 2，水平伸缩因子为 2。

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A-Level Maths: Transformations of Graphs – Stretches | A-Level数学：函数图像伸缩变换