Stretching Graphs | 函数图像的拉伸

📚 Stretching Graphs | 函数图像的拉伸

Stretching a graph is one of the fundamental transformations in A-Level Mathematics, alongside translations and reflections. It involves scaling the shape of a function in the vertical or horizontal direction. In Edexcel’s specification, you are expected to understand how to apply and interpret stretches of the forms y = a f(x) and y = f(ax), including the effect of the scale factor a and how to combine these with other transformations. This article provides a comprehensive guide to stretching graphs, with clear explanations, worked examples, and common pitfalls to avoid.

图像拉伸是 A-Level 数学中的基本变换之一，与平移和反射并列。它涉及在垂直或水平方向上缩放函数的形状。在 Edexcel 的考试大纲中，你需要理解如何应用和解释形如 y = a f(x) 和 y = f(ax) 的拉伸，包括缩放因子 a 的影响，以及如何将这些变换与其他变换组合使用。本文全面介绍图像拉伸，提供清晰的解释、实例和常见错误提醒。

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1. Introduction to Stretches | 拉伸简介

A stretch transformation multiplies all x-coordinates or all y-coordinates of a graph by a constant factor, altering its shape but preserving its key features like intercepts in the direction perpendicular to the stretch. Unlike translations, stretches change distances from the axes. When the factor is greater than 1, the graph is ‘stretched’ away from the axis; when the factor is between 0 and 1, it is ‘compressed’ towards the axis. In Edexcel, we always describe these using the term stretch with the appropriate scale factor.

拉伸变换是将图形上所有 x 坐标或所有 y 坐标乘以一个常数因子，从而改变其形状，但保持与拉伸方向垂直的截距等关键特征不变。与平移不同，拉伸会改变点与轴的距离。当因子大于 1 时，图形远离坐标轴“拉伸”；当因子介于 0 和 1 之间时，图形向坐标轴“压缩”。在 Edexcel 中，我们总是用适当的缩放因子来描述这些拉伸。

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

A vertical stretch multiplies all the y-coordinates of the original function f(x) by a factor a. The transformation can be written as:

垂直拉伸将原函数 f(x) 的所有 y 坐标乘以因子 a。该变换可以写作：

y = a f(x)

If a > 1, the graph becomes steeper, with points moving further away from the x-axis. If 0 < a < 1, the graph becomes shallower, with points moving closer to the x-axis. If a is negative, there is also a reflection in the x-axis; however, the primary focus here is on positive a. For example, starting with f(x) = x², the graph of y = 2x² is a vertical stretch with scale factor 2, making the parabola narrower. The x-coordinates remain unchanged.

如果 a > 1，图形变陡，各点远离 x 轴。如果 0 < a < 1，图形变平缓，各点靠近 x 轴。如果 a 为负，还会同时发生关于 x 轴的反射；但这里主要关注正 a 的情况。例如，从 f(x) = x² 出发，y = 2x² 的图像是缩放因子为 2 的垂直拉伸，使抛物线变窄。x 坐标保持不变。

To identify the stretch on a specific point, (x, y) maps to (x, a y). The overall effect is that the graph is scaled parallel to the y-axis. Intercepts on the x-axis stay fixed because there y = 0 and a × 0 = 0.

对于特定点，(x, y) 映射为 (x, a y)。整体效果是图形沿 y 轴方向缩放。x 轴上的截距保持不变，因为此时 y = 0，a × 0 = 0。

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

A horizontal stretch modifies the x-coordinates. The transformation is given by:

水平拉伸改变 x 坐标。该变换由以下形式给出：

y = f(ax)

Here the stretch is parallel to the x-axis, and the scale factor is 1/a. If a > 1, the graph is compressed horizontally towards the y-axis (a stretch with factor 1/a < 1). If 0 < a < 1, the graph is stretched horizontally away from the y-axis. For instance, y = f(2x) means every x-coordinate is halved, so the graph squashes inward. Starting with y = sin x, the graph of y = sin(2x) completes one full cycle in 180° instead of 360°, demonstrating a horizontal compression with scale factor 1/2.

这里的拉伸平行于 x 轴，缩放因子是 1/a。如果 a > 1，图形水平地向 y 轴压缩（缩放因子 1/a < 1 的拉伸）。如果 0 < a < 1，图形水平地远离 y 轴拉伸。例如，y = f(2x) 意味着每个 x 坐标被减半，因此图形向中心压缩。从 y = sin x 开始，y = sin(2x) 的图像在 180° 内完成一个完整周期，而不是 360°，这展示了缩放因子为 1/2 的水平压缩。

The mapping of a point is (x, y) → (x/a, y). Intercepts on the y-axis are unaffected because x = 0 leads to f(0) remaining unchanged. Understanding that the inside multiplier a works inversely is a key skill.

点的映射为 (x, y) → (x/a, y)。y 轴上的截距不受影响，因为 x = 0 时 f(0) 保持不变。理解内部乘数 a 的反向作用是一项关键技能。

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4. Scale Factors and their Effects | 缩放因子及其影响

It is helpful to summarise the impact of the scale factor in a table. Let k be the multiplier that directly multiplies the coordinate being stretched.

将缩放因子的影响汇总成表会很有帮助。设 k 为直接乘以被拉伸坐标的乘数。

Type of Stretch	Form	Scale Factor for Coordinate	Effect on Graph
Vertical	y = a f(x)	a	Points move away from x-axis if a > 1; towards if 0 < a < 1
Horizontal	y = f(ax)	1/a	Points move towards y-axis if a > 1; away if 0 < a < 1

Remember that a horizontal stretch of factor 1/a is often more easily interpreted by reading the value of 1/a directly. For example, y = f(x/2) means a horizontal stretch of factor 2 (since a = 1/2, 1/a = 2).

请记住，因子为 1/a 的水平拉伸通常通过直接读取 1/a 的值更容易理解。例如，y = f(x/2) 表示因子为 2 的水平拉伸（因为 a = 1/2，1/a = 2）。

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5. Combining Stretches | 复合拉伸

A function can undergo both vertical and horizontal stretches simultaneously, for example:

一个函数可以同时经历垂直和水平拉伸，例如：

y = a f(bx)

Here, a represents the vertical scale factor and b the horizontal multiplier. The combined transformation maps (x, y) to (x/b, a y). The order of applying these stretches does not matter because they affect different coordinates independently. For instance, if f(x) = √x, then y = 3√(2x) represents a vertical stretch of factor 3 and a horizontal compression of factor 1/2. Point (4, 2) on √x would become (2, 6) on the new graph.

这里，a 代表垂直缩放因子，b 代表水平乘数。复合变换将 (x, y) 映射为 (x/b, a y)。应用这些拉伸的顺序无关紧要，因为它们独立地影响不同的坐标。例如，如果 f(x) = √x，那么 y = 3√(2x) 表示因子为 3 的垂直拉伸和因子为 1/2 的水平压缩。√x 上的点 (4, 2) 在新图上将变为 (2, 6)。

When sketching combined stretches, it is advisable to apply the horizontal transformation first to get the new x-coordinates, then apply the vertical transformation. This reduces errors in freehand drawing.

在草图绘制复合拉伸时，建议先应用水平变换得到新的 x 坐标，再应用垂直变换。这样可以减少徒手绘图中的错误。

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6. Stretches and Symmetry | 拉伸与对称性

Stretches can alter the symmetry of a graph. A function that is even, satisfying f(x) = f(-x), remains even after a vertical stretch y = a f(x) or a horizontal stretch y = f(ax) provided a is such that the domain symmetry is preserved. For example, y = 2 cos x is still even. However, a horizontal stretch can change the period of trigonometric functions, which affects periodic symmetry.

拉伸可能会改变图形的对称性。满足 f(x) = f(-x) 的偶函数，在垂直拉伸 y = a f(x) 或水平拉伸 y = f(ax) 后仍为偶函数，前提是 a 使得定义域的对称性得以保持。例如，y = 2 cos x 仍然是偶函数。然而，水平拉伸会改变三角函数的周期，从而影响周期对称性。

An odd function, where f(-x) = -f(x), remains odd under vertical stretches with positive a. Horizontal stretches also preserve oddness because the sign of x is maintained. For instance, y = 2 sin(3x) is odd. Understanding these properties can help verify the correctness of a transformed graph.

满足 f(-x) = -f(x) 的奇函数，在正 a 的垂直拉伸下仍保持奇函数性质。水平拉伸也保持奇偶性，因为 x 的符号得以保留。例如，y = 2 sin(3x) 是奇函数。理解这些性质有助于验证变换后图形的正确性。

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7. Identifying Stretches from Equations | 从方程识别拉伸

Given a transformed equation, you can identify the stretches by comparing it to the basic function. Look for multipliers outside the function for vertical stretches, and multipliers affecting x directly inside the function argument for horizontal stretches. For example, y = 4/x² can be seen as y = 4 f(x) with f(x) = 1/x², a vertical stretch factor 4. Alternatively, y = 1/(2x)² = 1/(4x²) shows a horizontal compression factor 1/2. Break down the equation carefully to avoid misinterpreting the stretch direction.

给定一个变换后的方程，你可以通过将其与基本函数进行比较来识别拉伸。寻找函数外部的乘数以获得垂直拉伸，以及在函数参数内部直接影响 x 的乘数以获得水平拉伸。例如，y = 4/x² 可以视为 y = 4 f(x)，其中 f(x) = 1/x²，即垂直拉伸因子为 4。另一种方式，y = 1/(2x)² = 1/(4x²) 表明水平压缩因子为 1/2。仔细分解方程，避免误解拉伸方向。

When the equation contains both additions and multiplications, remember the order of operations. For instance, y = 2 f(x + 3) indicates a horizontal translation left by 3, then a vertical stretch of factor 2. The stretch does not affect the shift inside the function. Always identify inner transformations first when analysing.

当方程同时包含加法和乘法时，记住运算顺序。例如，y = 2 f(x + 3) 表示先向左平移 3 个单位，然后进行因子为 2 的垂直拉伸。拉伸不会影响函数内部的平移量。分析时始终先识别内部的变换。

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8. Sketching Stretched Graphs | 绘制拉伸图像

To sketch a stretched graph accurately, start with the key points of the original function. Apply the horizontal stretch by changing each x-coordinate to x/a. Then apply the vertical stretch by multiplying y-coordinates by a. Plot the new points and draw a smooth curve through them. For example, sketch y = 0.5 f(3x) given f(x) = x³ – x. First, list points of f(x) like (-1,0), (0,0), (1,0), (-0.5, 0.375), etc. For horizontal stretch with a=3, scale x by 1/3: points become (-1/3,0), (0,0), (1/3,0). Then vertical stretch 0.5 multiplies y by 0.5. The overall shape is a compressed and flattened cubic.

要准确地绘制拉伸后的图形，先从原函数的关键点开始。通过将每个 x 坐标变为 x/a 来应用水平拉伸。然后通过将 y 坐标乘以 a 来应用垂直拉伸。标出新点，并通过它们画出平滑曲线。例如，给定 f(x) = x³ – x，绘制 y = 0.5 f(3x)。首先列出 f(x) 的点，如 (-1,0), (0,0), (1,0), (-0.5, 0.375) 等。对于 a=3 的水平拉伸，x 缩放 1/3：点变为 (-1/3,0), (0,0), (1/3,0)。然后垂直拉伸 0.5 将 y 乘以 0.5。整体形状是一个被压缩和压平的立方曲线。

Label axes clearly and indicate the important coordinates. Stretches can change the steepness significantly, so pay attention to the gradient at intersections. The shape remains recognisable but altered in proportion.

清晰地标记坐标轴，并标出重要的坐标。拉伸会显著改变陡峭程度，因此注意交点处的斜率变化。形状仍可辨认，但比例发生了改变。

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9. Stretches with Translations | 拉伸与平移

When combining stretches with translations, the order of operations becomes crucial. For example, consider the transformation that takes f(x) to y = 2 f(x + 1). If you apply the translation first (x + 1 shifts left by 1) and then the vertical stretch, you get the correct graph. If you apply the stretch first, you would incorrectly stretch the shift as well. The general rule is to follow the order of operations: horizontal shifts inside the function are applied before the multiplication of the function.

当拉伸与平移组合时，运算顺序变得至关重要。例如，考虑将 f(x) 变为 y = 2 f(x + 1) 的变换。如果先应用平移（x + 1 向左平移 1），然后应用垂直拉伸，你将得到正确的图形。如果先进行拉伸，你就会错误地将平移量也拉伸了。一般规则是遵循运算顺序：函数内部的水平平移应在函数乘法之前应用。

A tricky case is y = f(ax + b). Factor out a to get y = f(a(x + b/a)). This reveals a horizontal translation of -b/a followed by a horizontal stretch of factor 1/a. Many students mistakenly treat the translation as -b. Always factor inside the brackets to separate the transformations correctly.

一个棘手的情况是 y = f(ax + b)。提取因子 a 得到 y = f(a(x + b/a))。这揭示了水平平移 -b/a 以及随后因子为 1/a 的水平拉伸。许多学生错误地将平移视为 -b。务必在括号内提取因子，以正确分离变换。

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10. Common Mistakes | 常见错误

One frequent error is confusing the direction of a stretch. For y = f(2x), the graph is compressed horizontally, not stretched, because the x-values are halved. Remember the horizontal scale factor is 1/a.

一个常见错误是混淆拉伸的方向。对于 y = f(2x)，图形是水平压缩的，而不是拉伸的，因为 x 值被减半。记住水平缩放因子是 1/a。

Another mistake is applying the vertical stretch to x-coordinates or mixing up the mappings. Always apply vertical changes to y only, and horizontal changes to x only. A table mapping key points can prevent confusion.

另一个错误是将垂直拉伸应用于 x 坐标，或者混淆映射关系。始终只将垂直变化应用于 y，水平变化应用于 x。制表映射关键点可以避免混淆。

Students also forget to factor out coefficients when both a stretch and a translation are inside the function, leading to an incorrect translation vector. Practise writing y = f(ax + b) as y = f(a(x + b/a)) every time.

学生们还经常忘记在函数内部同时存在拉伸和平移时提取系数，从而导致错误的平移向量。每次都要练习将 y = f(ax + b) 写成 y = f(a(x + b/a))。

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11. Practice Questions | 练习题

1. The graph of y = f(x) is stretched vertically by a factor of 3. Write the equation of the new graph.

1. 将 y = f(x) 的图像垂直拉伸，因子为 3。写出新图像的方程。

2. Describe the transformation that maps y = cos x to y = cos(0.5x).

2. 描述将 y = cos x 映射到 y = cos(0.5x) 的变换。

3. The function f(x) = √x is transformed into y = 2√(4x). State the vertical and horizontal scale factors.

3. 函数 f(x) = √x 被变换为 y = 2√(4x)。说明垂直和水平缩放因子。

4. Find the coordinates of the point (2, 8) on y = x³ after the transformation y = 1/4 (2x)³.

4. 求点 (2, 8) 在 y = x³ 经过变换 y = 1/4 (2x)³ 后的坐标。

5. The graph of y = f(x) is first stretched horizontally by factor 2, then stretched vertically by factor 0.5. Write the resulting equation in terms of f.

5. 将 y = f(x) 的图像先水平拉伸因子 2，再垂直拉伸因子 0.5。用 f 写出结果方程。

Try solving these to test your understanding. Check that you can sketch each transformed graph by hand.

尝试解答这些问题以检验你的理解。确保你能手绘出每个变换后的图形。

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12. Summary | 总结

Stretching graphs is a powerful tool for modelling and understanding how functions behave under scaling. The key forms are y = a f(x) for a vertical stretch of scale factor a, and y = f(ax) for a horizontal stretch of scale factor 1/a. Combining stretches is straightforward as they work on different axes. When translations are involved, handle them in the correct order and always factor inside arguments. With consistent practice, you will be able to quickly recognise and sketch stretched graphs, a vital skill for the Edexcel A-Level Mathematics exam.

图像拉伸是一个强大的工具，用于建模和理解函数在缩放下的行为。关键形式是 y = a f(x)（缩放因子为 a 的垂直拉伸）和 y = f(ax)（缩放因子为 1/a 的水平拉伸）。由于拉伸作用于不同坐标轴，组合拉伸十分简单。当涉及平移时，务必按正确顺序处理，并在参数内部提取因子。通过持续练习，你将能够快速识别并绘制拉伸后的图形，这是 Edexcel A-Level 数学考试的一项重要技能。

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