📚 Stretching Graphs | 函数图像的伸缩变换
A stretch is a transformation that changes the scale of a graph in one direction only. In the Edexcel A-Level Pure Mathematics specification, stretches are described using the graphs of y = a f(x) and y = f(a x), where a is a positive constant.
伸缩变换是只在一个方向上改变图形比例的变换。在 Edexcel A-Level 纯数学考试中，伸缩变换用 y = a f(x) 和 y = f(a x) 来描述，其中 a 是正的常数。
1. What is a Stretch? | 什么是伸缩变换
A stretch moves every point on a graph closer to or further away from a fixed straight line, while keeping points on that line unchanged. The fixed line is called the invariant line of the stretch. Unlike a translation, a stretch changes distances in one direction only.
伸缩变换将图像上的每一个点拉近或拉远一条固定直线，同时保持该直线上的点不变。这条固定直线称为伸缩的不变线。与平移不同，伸缩只在一个方向上改变距离。
Vertical stretch: y = a f(x)
水平伸缩：y = a f(x)（垂直方向）
Horizontal stretch: y = f(a x)
垂直伸缩：y = f(a x)（水平方向）
Throughout this article we use y = f(x) as the original graph and assume a is a positive real constant, except where stated otherwise.
在本文中，我们用 y = f(x) 表示原图像，并假设 a 为正实数常数，除非另有说明。
2. Vertical Stretch: y = a f(x) | 垂直伸缩：y = a f(x)
For the transformation y = a f(x), the output value of f(x) is multiplied by a. This stretches the graph parallel to the y-axis by scale factor a. Every point (x, y) is mapped to (x, a y).
对于变换 y = a f(x)，函数值 f(x) 要乘以 a。这使图像沿 y 轴方向以比例因子 a 进行伸缩。每个点 (x, y) 都映射为 (x, a y)。
If a > 1, the graph is stretched away from the x-axis, so it becomes taller. If 0 < a < 1, the graph is compressed towards the x-axis, but Edexcel still describes this as a stretch with scale factor a.
如果 a > 1，图像会远离 x 轴拉伸，因此变得更高。如果 0 < a < 1，图像会向 x 轴压缩，但 Edexcel 仍将其描述为比例因子为 a 的伸缩。
Example: If f(x) = x², then y = 3 f(x) = 3x². The point (1, 1) on y = x² becomes (1, 3), and the point (2, 4) becomes (2, 12).
示例：如果 f(x) = x²，则 y = 3 f(x) = 3x²。原图像 y = x² 上的点 (1, 1) 变为 (1, 3)，点 (2, 4) 变为 (2, 12)。
3. Horizontal Stretch: y = f(a x) | 水平伸缩：y = f(a x)
For y = f(a x), the input x is multiplied by a before applying the function. This stretches the graph parallel to the x-axis by scale factor 1/a. Each point (x, y) is mapped to (x/a, y).
对于 y = f(a x)，输入 x 在代入函数之前先乘以 a。这使图像沿 x 轴方向以比例因子 1/a 进行伸缩。每个点 (x, y) 都映射为 (x/a, y)。
If a > 1, the x-coordinates are divided by a, so the graph is compressed horizontally towards the y-axis. We still call this a stretch with scale factor 1/a in the x-direction.
如果 a > 1，x 坐标要除以 a，因此图像向 y 轴方向水平压缩。我们仍称其为 x 方向比例因子为 1/a 的伸缩。
If 0 < a < 1, the graph is stretched away from the y-axis. For example, y = f(½ x) maps (x, y) to (2x, y), moving points further from the y-axis.
如果 0 < a < 1，图像会远离 y 轴拉伸。例如，y = f(½ x) 将 (x, y) 映射为 (2x, y)，使点远离 y 轴。
Example: If f(x) = x², then y = f(2x) = (2x)² = 4x². The point (1, 1) on y = x² becomes (½, 1), and (2, 4) becomes (1, 4).
示例：如果 f(x) = x²，则 y = f(2x) = (2x)² = 4x²。原图像 y = x² 上的点 (1, 1) 变为 (½, 1)，点 (2, 4) 变为 (1, 4)。
4. Stretch Factors and the Role of a | 伸缩因子与参数 a 的作用
The value of a determines both the direction and the size of the stretch. The table below summarises the two standard forms.
参数 a 的值决定了伸缩的方向和大小。下表总结了两种标准形式。
| Transformation | Type | Scale factor | Mapping of (x, y) |
|---|---|---|---|
| y = a f(x) | Vertical stretch | a | (x, y) → (x, a y) |
| y = f(a x) | Horizontal stretch | 1/a | (x, y) → (x/a, y) |
In Edexcel exam answers, you should state the direction, the scale factor, and whether the stretch is parallel to the x-axis or y-axis. For y = f(3x), the correct description is: ‘stretch by scale factor 1/3 in the x-direction’.
在 Edexcel 考试答案中，你应该说明方向、比例因子，以及伸缩是平行于 x 轴还是 y 轴。对于 y = f(3x)，正确的描述是：“在 x 方向以比例因子 1/3 进行伸缩”。
5. Invariant Lines and Invariant Points | 不变线与不变点
An invariant point is a point that stays exactly where it is after a transformation. For a vertical stretch y = a f(x), any point on the x-axis has y = 0, so its new coordinate is (x, a × 0) = (x, 0). Therefore the x-axis is invariant.
不变点是指经过变换后保持原位置不变的点。对于垂直伸缩 y = a f(x)，x 轴上的任意点都有 y = 0，因此其新坐标为 (x, a × 0) = (x, 0)。所以 x 轴是不变的。
For a horizontal stretch y = f(a x), any point on the y-axis has x = 0, so its new coordinate is (0/a, y) = (0, y). Therefore the y-axis is invariant.
对于水平伸缩 y = f(a x)，y 轴上的任意点都有 x = 0，因此其新坐标为 (0/a, y) = (0, y)。所以 y 轴是不变的。
If a = 1, the transformation is the identity transformation, and every point on the graph is invariant.
如果 a = 1，该变换就是恒等变换，图像上的每个点都是不变的。
6. Effect on Key Features | 对关键特征的影响
When a graph is stretched, its key features change in a predictable way. For y = a f(x), the y-coordinate of every point is multiplied by a, so the y-intercept is multiplied by a, but the x-intercepts do not move.
当图像被伸缩时，其关键特征会以可预测的方式变化。对于 y = a f(x)，每个点的 y 坐标都要乘以 a，因此 y 截距要乘以 a，但 x 截距保持不变。
For y = f(a x), the x-coordinate of every point is divided by a, so the x-intercepts are divided by a, but the y-intercept does not move.
对于 y = f(a x)，每个点的 x 坐标要除以 a，因此 x 截距要除以 a，但 y 截距保持不变。
Asymptotes: A vertical stretch y = a f(x) multiplies any horizontal asymptote y = k by a, giving y = a k. A horizontal stretch y = f(a x) divides any vertical asymptote x = c by a, giving x = c/a.
渐近线：垂直伸缩 y = a f(x) 会将水平渐近线 y = k 乘以 a，变为 y = a k。水平伸缩 y = f(a x) 会将垂直渐近线 x = c 除以 a，变为 x = c/a。
Turning points: A turning point at (p, q) becomes (p, a q) under a vertical stretch, and (p/a, q) under a horizontal stretch.
驻点：驻点 (p, q) 在垂直伸缩下变为 (p, a q)，在水平伸缩下变为 (p/a, q)。
7. Sketching Stretched Graphs | 绘制伸缩后的图像
To sketch a stretched graph, start by identifying the parent function and the type of stretch. Then calculate the new positions of key points such as intercepts, turning points, and asymptotes.
要绘制伸缩后的图像，首先要确定母函数和伸缩类型。然后计算截距、驻点和渐近线等关键点的新位置。
- Identify whether the transformation is y = a f(x) or y = f(a x).
- 识别变换是 y = a f(x) 还是 y = f(a x)。
- Write down the mapping of coordinates: (x, y) becomes (x, a y) or (x/a, y).
- 写下坐标映射：(x, y) 变为 (x, a y) 或 (x/a, y)。
- Plot the new key points and draw a smooth curve with the same general shape.
- 标出新关键点，并画出具有相同大致形状的光滑曲线。
- Label the invariant axis and any new intercepts or asymptotes.
- 标出不变轴以及任何新的截距或渐近线。
A stretched graph should look like the original graph pulled or squeezed, not shifted sideways or vertically unless a translation is also applied.
伸缩后的图像应该看起来像是原图像被拉伸或压缩，而不是向旁边或上下移动，除非同时施加了平移。
8. Recognising Stretches from Equations | 从方程识别伸缩
In an exam, you may be given an equation and asked to describe the transformation fully. Compare the given equation with the original y = f(x).
在考试中，你可能会看到给定的方程，并被要求完整描述变换。将给定方程与原方程 y = f(x) 进行比较。
If the entire function is multiplied by a constant, such as y = 3sin x, it is a vertical stretch by scale factor 3. If only the input x is multiplied, such as y = cos(2x), it is a horizontal stretch by scale factor ½.
如果整个函数乘以一个常数，例如 y = 3sin x，这就是在垂直方向以比例因子 3 伸缩。如果只有输入 x 乘以常数，例如 y = cos(2x)，这就是在水平方向以比例因子 ½ 伸缩。
Be careful with expressions like y = (2x)². This can be viewed as y = f(2x) where f(x) = x², so the graph is a horizontal stretch by scale factor ½, not a vertical stretch by 4, even though the expanded equation is y = 4x².
对于 y = (2x)² 这样的表达式要小心。它可以看作 y = f(2x)，其中 f(x) = x²，因此图像是水平方向比例因子为 ½ 的伸缩，而不是比例因子为 4 的垂直伸缩，尽管展开后方程是 y = 4x²。
9. Combining Stretches with Translations | 伸缩与平移的组合
When a function contains both a stretch and a translation, the order of transformations can affect the result, especially when both act on the x-variable. Always read the transformation sequence carefully.
当函数同时包含伸缩和平移时，变换的顺序可能会影响结果，尤其是当两者都作用于 x 变量时。要始终仔细读取变换顺序。
For example, to transform f(x) into f(2x + 6), rewrite 2x + 6 as 2(x + 3). One valid sequence is:
例如，要将 f(x) 变换为 f(2x + 6)，可将 2x + 6 改写为 2(x + 3)。一种有效的顺序是：
- First stretch horizontally by scale factor ½, giving y = f(2x).
- 先以比例因子 ½ 进行水平伸缩，得到 y = f(2x)。
- Then translate left by 3 units, giving y = f(2(x + 3)) = f(2x + 6).
- 然后向左平移 3
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