Stretching Graphs | 图像的拉伸变换

📚 Stretching Graphs | 图像的拉伸变换

In A-Level mathematics, understanding how functions transform is essential for sketching curves and solving problems. This article focuses on one specific family of transformations: stretching graphs, both vertically and horizontally, as required by the Edexcel Pure Mathematics specification.

在 A-Level 数学中，理解函数的变换对于绘制曲线和解题至关重要。本文将聚焦于一类特定的变换：图像的拉伸，包括垂直拉伸与水平拉伸，这也是 Edexcel 纯数学考纲的明确要求。

1. What Does Stretching a Graph Mean? | 拉伸图像的含义

A stretch is a transformation that enlarges or shrinks a graph in one direction only, either parallel to the y-axis (vertical stretch) or parallel to the x-axis (horizontal stretch). Unlike a translation, which slides a graph without changing its shape, a stretch changes the proportions of the curve while preserving its fundamental character.

拉伸是一种仅沿一个方向放大或缩小图像的变换，要么沿 y 轴方向（垂直拉伸），要么沿 x 轴方向（水平拉伸）。与平移不同，平移只滑动图像而不改变其形状；拉伸则改变曲线的比例，但保留其基本特征。

There are two standard forms you must know for Edexcel Pure Mathematics:

你必须在 Edexcel 纯数学中掌握两种标准形式：

The vertical stretch: y = a·f(x), where a is the scale factor.

垂直拉伸：y = a·f(x)，其中 a 为比例系数。
The horizontal stretch: y = f(ax), where the scale factor is 1/a.

水平拉伸：y = f(ax)，其中比例系数为 1/a。

Notice immediately the asymmetry: a vertical stretch multiplies y-values by a, while a horizontal stretch divides x-values by a. Getting this the right way round is the single most important skill in this topic.

请立即注意这种不对称性：垂直拉伸将 y 值乘以 a，而水平拉伸将 x 值除以 a。把这个方向弄正确是本主题中最重要的技能。

2. Vertical Stretch: y = a·f(x) | 垂直拉伸：y = a·f(x)

For y = a·f(x), every y-coordinate of the original graph y = f(x) is multiplied by a. The x-coordinates remain completely unchanged. If a > 1, the graph is stretched away from the x-axis; if 0 < a < 1, the graph is compressed towards the x-axis.

对于 y = a·f(x)，原图 y = f(x) 上每个点的纵坐标都乘以 a，横坐标保持不变。若 a > 1，图像沿远离 x 轴的方向拉伸；若 0 < a < 1，图像则向 x 轴压缩。

Example: Consider f(x) = x². Under the transformation y = 3f(x) = 3x², the point (2, 4) on y = x² moves to (2, 12). Every point’s y-value is tripled, so the parabola appears narrower and steeper.

示例：考虑 f(x) = x²。在变换 y = 3f(x) = 3x² 下，y = x² 上的点 (2, 4) 移动到 (2, 12)。每一点的纵坐标都变为原来的三倍，因此抛物线看起来更窄、更陡。

If a is negative, for instance y = −2f(x), the graph is first stretched by scale factor 2 and then reflected in the x-axis. This is a combined transformation; the stretch component follows exactly the same rule as before.

若 a 为负数，例如 y = −2f(x)，图像先按比例系数 2 拉伸，再关于 x 轴反射。这是一种复合变换，其中拉伸部分遵循与之前完全相同的规则。

3. Horizontal Stretch: y = f(ax) | 水平拉伸：y = f(ax)

For y = f(ax), every x-coordinate is divided by a. The y-coordinates remain unchanged. The scale factor is 1/a. This result is counter-intuitive: if a > 1, the graph is compressed horizontally, not stretched.

对于 y = f(ax)，每个点的横坐标都除以 a，纵坐标保持不变。比例系数为 1/a。这个结果略带反直觉：若 a > 1，图像是在水平方向被压缩，而非拉伸。

Example: Take f(x) = x². The transformation y = f(2x) = (2x)² = 4x². The point (2, 4) on y = x² moves to (1, 4). The x-coordinate is halved, so the horizontal scale factor is 1/2.

示例：取 f(x) = x²。变换 y = f(2x) = (2x)² = 4x²。y = x² 上的点 (2, 4) 移动到 (1, 4)。横坐标减半，因此水平比例系数为 1/2。

It is vital to distinguish y = f(2x) from y = 2f(x). The first compresses the graph horizontally; the second stretches it vertically. For the special function f(x) = x², the two results coincide algebraically, but for most functions — such as f(x) = x³ or f(x) = sin x — they are completely different.

务必区分 y = f(2x) 与 y = 2f(x)。前者水平压缩图像，后者垂直拉伸图像。对于特殊函数 f(x) = x²，两者代数结果一致，但对于大多数函数——如 f(x) = x³ 或 f(x) = sin x——它们完全不同。

4. Invariant Points | 不动点

An invariant point is a point on the graph that does not move under a given transformation. Identifying invariant points is a powerful strategy for sketching stretched graphs accurately.

不动点是指在给定变换下位置保持不变的图像上的点。识别不动点是准确绘制拉伸图像的有力策略。

For a vertical stretch y = a·f(x), any point with y = 0 is invariant, because multiplying 0 by a still gives 0. These points are precisely the roots of the function f(x) = 0.

对于垂直拉伸 y = a·f(x)，任何纵坐标为 0 的点都是不动的，因为 0 乘以 a 仍为 0。这些点正是函数 f(x) = 0 的零点。

For a horizontal stretch y = f(ax), any point with x = 0 is invariant, because f(a × 0) = f(0). This includes the y-intercept, provided it exists.

对于水平拉伸 y = f(ax)，任何横坐标为 0 的点都是不动的，因为 f(a × 0) = f(0)。这包括 y 截距（若存在）。

Example: The graph of y = (x − 1)(x − 3) has roots at x = 1 and x = 3. Under y = 4f(x), these roots remain at x = 1 and x = 3. Under y = f(2x) = (2x − 1)(2x − 3), the roots become x = 1/2 and x = 3/2.

示例：y = (x − 1)(x − 3) 的图像在 x = 1 和 x = 3 处有零点。在 y = 4f(x) 下，零点仍位于 x = 1 和 x = 3。在 y = f(2x) = (2x − 1)(2x − 3) 下，零点变为 x = 1/2 和 x = 3/2。

5. Effects on Key Coordinates | 拉伸对关键坐标的影响

When a graph is vertically stretched by factor a, the turning points (maxima and minima) have their y-coordinates multiplied by a, but their x-coordinates are unchanged. When the graph is horizontally stretched by scale factor 1/a, the x-coordinates of turning points are multiplied by 1/a, while the y-coordinates stay the same.

当图像垂直拉伸 a 倍时，极值点（最大值与最小值）的纵坐标乘以 a，但横坐标不变。当图像水平拉伸 1/a 倍时，极值点的横坐标乘以 1/a，纵坐标保持不变。

Consider the cubic function f(x) = x³ + 2. Its point of inflection is at (0, 2). Under a vertical stretch y = 5f(x) = 5x³ + 10, the point becomes (0, 10). Under a horizontal stretch y = f( (1/2) x ) = (x/2)³ + 2 = x³/8 + 2, the point remains at (0, 2) because it lies on the y-axis.

考虑三次函数 f(x) = x³ + 2。其拐点位于 (0, 2)。在垂直拉伸 y = 5f(x) = 5x³ + 10 下，拐点变为 (0, 10)。在水平拉伸 y = f( (1/2) x ) = (x/2)³ + 2 = x³/8 + 2 下，拐点仍位于 (0, 2)，因为它落在 y 轴上。

When analysing a transformed equation, always track three sets of points: roots, y-intercepts, and turning points. Together, these give you enough information to sketch the new graph with confidence.

在分析变换后的方程时，始终跟踪三类点：零点、y 截距和极值点。这三类点合在一起，足以让你自信地绘制出新图像。

6. Worked Example: Stretching y = x³ − x | 例题一：拉伸 y = x³ − x

The cubic function f(x

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Stretching Graphs | 图像的拉伸变换