Stretching Graphs | 图像拉伸变换

📚 Stretching Graphs | 图像拉伸变换

Graph transformations are a key topic in IGCSE Mathematics. Among them, stretching graphs allows us to change the shape of a function while preserving its essential structure. This guide explains vertical and horizontal stretches, their algebraic forms, and how to apply them confidently in exams.

图像变换是IGCSE数学的核心内容之一。其中，图像的拉伸变换让我们在保持函数基本结构不变的前提下改变其形状。本指南将详细讲解纵向拉伸与横向拉伸、它们的代数形式，以及如何在考试中自信地运用这些知识。

1. What is a Graph Transformation? | 什么是图像变换？

A graph transformation is an operation that changes the position, size, or orientation of a graph. Translations move the graph, reflections flip it, and stretches expand or compress it along the x-axis or y-axis.

图像变换是指改变图像位置、大小或方向的操作。平移改变图像的位置，翻转使图像镜像，而拉伸则沿x轴或y轴方向将图像扩张或压缩。

When we stretch a graph, every point on the original curve moves to a new location. The shape of the curve is distorted in a regular way: distances from a fixed line (the axis of stretch) are multiplied by a constant factor.

当我们拉伸一个图像时，原曲线上的每一个点都会移动到新的位置。曲线的形状以规律的方式发生变化：到固定直线（拉伸轴）的距离被乘以一个常数因子。

Vertical stretch: distances from the x-axis are multiplied by a factor.
Horizontal stretch: distances from the y-axis are multiplied by a factor.

纵向拉伸：到x轴的距离乘以一个因子。
横向拉伸：到y轴的距离乘以一个因子。

2. Vertical Stretch: y = k f(x) | 纵向拉伸：y = k f(x)

Consider a function y = f(x). If we multiply the whole function by a constant k (where k > 0), we obtain y = k f(x). This is a vertical stretch with scale factor k.

考虑函数 y = f(x)。如果我们把整个函数乘以一个常数 k（其中 k > 0），得到 y = k f(x)。这就是比例因子为 k 的纵向拉伸。

For every point (x, y) on the original graph, the new point is (x, ky). The x-coordinate stays the same; the y-coordinate is multiplied by k. If k > 1, the graph stretches upwards away from the x-axis. If 0 < k < 1, the graph compresses towards the x-axis.

原图像上的每一点 (x, y) 变为新点 (x, ky)。x坐标不变，y坐标乘以 k。若 k > 1，图像远离x轴向上拉伸；若 0 < k < 1，图像向x轴压缩。

y = k f(x) ⟹ vertical stretch, scale factor k

Note that if k is negative, there is also a reflection in the x-axis, but a pure stretch usually assumes k > 0.

注意：如果 k 为负数，还包含关于x轴的翻转，但纯粹的拉伸通常假设 k > 0。

3. Horizontal Stretch: y = f(kx) | 横向拉伸：y = f(kx)

If we replace x in f(x) with kx, we get y = f(kx). This is a horizontal stretch with scale factor 1/k.

如果我们将 f(x) 中的 x 替换为 kx，得到 y = f(kx)。这是比例因子为 1/k 的横向拉伸。

Why 1/k? Because to keep the same y-value, the original x must be divided by k. For example, if k = 2, the point (2, f(2)) moves to (1, f(2)). The graph is compressed horizontally by a factor of 2, which is the same as a stretch with scale factor 1/2.

为什么是 1/k？因为为了保持相同的y值，原来的x必须除以k。例如，若 k = 2，点 (2, f(2)) 移动到 (1, f(2))。图像在水平方向被压缩为原来的1/2，这等价于比例因子为1/2的拉伸。

y = f(kx) ⟹ horizontal stretch, scale factor 1/k

Thus y = f(2x) compresses the graph horizontally, while y = f(x/2) stretches it horizontally by a factor of 2.

因此，y = f(2x) 将图像水平压缩，而 y = f(x/2) 将图像水平拉伸2倍。

4. Comparing Vertical and Horizontal Stretches | 纵向与横向拉伸对比

Many students confuse the two. The key is to look at where the multiplier acts: outside the function or inside the function.

很多学生容易混淆这两种拉伸。关键在于观察乘数作用在函数外还是函数内。

Transformation	Algebraic form	Scale factor	Direction
Vertical stretch	y = k f(x)	k	Away from / towards x-axis
Horizontal stretch	y = f(kx)	1/k	Away from / towards y-axis

For a vertical stretch, the factor k is exactly the number multiplying f(x). For a horizontal stretch, the factor is the reciprocal of the number multiplying x.

对于纵向拉伸，因子 k 就是乘以 f(x) 的那个数；对于横向拉伸，因子是乘以 x 的那个数的倒数。

Think of it this way: if you want to double the width of a graph, you need to halve the frequency inside the function. So y = f(x/2) stretches horizontally by 2.

可以这样理解：如果你想让图像的宽度加倍，就需要将函数内部的频率减半。所以 y = f(x/2) 将图像水平拉伸2倍。

5. Stretch Factors and Scale Factors | 拉伸因子与比例因子

In exam questions, you may be asked to find the scale factor of a stretch, or to write down the equation of a stretched graph. The scale factor is the number by which distances from the axis are multiplied.

考试中可能会要求你求拉伸的比例因子，或写出拉伸后图像的方程。比例因子就是到轴的距离所乘的数。

Given the original function y = f(x), the transformed function is:

已知原函数 y = f(x)，变换后的函数为：

Vertical stretch scale factor a: y = a f(x)
Horizontal stretch scale factor a: y = f(x/a)

纵向拉伸比例因子 a：y = a f(x)
横向拉伸比例因子 a：y = f(x/a)

Notice the asymmetry: horizontal stretch uses division by a inside the function. This is a common source of error, so remember it well.

注意这种不对称性：横向拉伸在函数内部使用除以 a。这是常见错误来源，请务必牢记。

6. Stretching Quadratic Graphs | 二次函数图像的拉伸

Let’s apply these rules to the quadratic function y = x². Its graph is a parabola with vertex at the origin.

让我们将这些规则应用于二次函数 y = x²。它的图像是顶点在原点的抛物线。

Vertical stretch: y = 3x². Every y-coordinate is tripled. The parabola becomes narrower because for a given x, y is larger. But note: the horizontal scale is unchanged. The vertex remains at (0,0).

纵向拉伸：y = 3x²。每个y坐标变为原来的3倍。因为对于给定的x，y更大，抛物线看起来更“窄”。但注意：水平尺度不变，顶点仍在(0,0)。

Horizontal stretch: y = (x/2)² = x²/4. This stretches the parabola horizontally by a factor of 2. The graph becomes wider, and the vertex stays at (0,0).

横向拉伸：y = (x/2)² = x²/4。这将抛物线水平拉伸2倍。图像变得更宽，顶点仍在(0,0)。

Observe that a vertical stretch with factor 4 (y = 4x²) produces the same graph as a horizontal stretch with factor 1/2 (y = (2x)² = 4x²). These two transformations are equivalent for this particular function.

观察发现，纵向拉伸4倍（y = 4x²）与横向拉伸1/2（y = (2x)² = 4x²）产生相同的图像。对于这个特定函数，两种变换是等价的。

7. Stretching Trigonometric Graphs | 三角函数图像的拉伸

Trigonometric functions are ideal for understanding stretches because their graphs are periodic.

三角函数图像具有周期性，非常适合用来理解拉伸变换。

For y = sin(x), a vertical stretch y = 2 sin(x) doubles the amplitude. The graph oscillates between -2 and 2 instead of -1 and 1.

对于 y = sin(x)，纵向拉伸 y = 2 sin(x) 使振幅加倍。图像在-2和2之间振荡，而不是在-1和1之间。

A horizontal stretch y = sin(2x) changes the period. The original period is 360° (or 2π radians). After replacing x with 2x, the period becomes 180° (or π). The graph is compressed horizontally.

横向拉伸 y = sin(2x) 改变周期。原周期为360°（或2π弧度）。将x替换为2x后，周期变为180°（或π）。图像在水平方向被压缩。

In general, for y = a sin(bx), the amplitude is a and the period is 360°/b (in degrees). The parameter a is the vertical stretch factor, and 1/b is the horizontal stretch factor.

一般地，对于 y = a sin(bx)，振幅为 a，周期为 360°/b（以度为单位）。参数 a 是纵向拉伸因子，1/b 是横向拉伸因子。

Similarly, y = tan(x) has no amplitude, but horizontal stretches change its period. For y = tan(x/2), the period increases from 180° to 360°.

类似地，y = tan(x) 没有振幅，但横向拉伸会改变其周期。对于 y = tan(x/2)，周期从180°增加到360°。

8. Stretching Cubic and Other Graphs | 三次函数及其他图像的拉伸

Cubic functions like y = x³ can also be stretched. A vertical stretch y = 2x³ multiplies all y-values by 2. A horizontal stretch y = (x/2)³ = x³/8 stretches the graph in the x-direction.

三次函数如 y = x³ 也可以被拉伸。纵向拉伸 y = 2x³ 将所有y值乘以2。横向拉伸 y = (x/2)³ = x³/8 将图像沿x方向拉伸。

For the reciprocal function y = 1/x, a vertical stretch y = 3/x multiplies y-values by 3. A horizontal stretch y = 1/(x/2) = 2/x is equivalent to a vertical stretch by 2. This shows that some functions can be transformed in different ways to achieve the same result.

对于反比例函数 y = 1/x，纵向拉伸 y = 3/x 将y值乘以3。横向拉伸 y = 1/(x/2) = 2/x 等价于纵向拉伸2倍。这表明有些函数可以通过不同方式变换得到相同结果。

When dealing with absolute value graphs or exponential functions, the same stretch rules apply. Always identify whether the multiplier is outside (vertical) or inside (horizontal) the function.

对于绝对值图像或指数函数，同样的拉伸规则适用。始终判断乘数是在函数外（纵向）还是函数内（横向）。

9. Combined Transformations | 组合变换

In more complex problems, a stretch may be combined with translations or reflections. For example, y = 2f(x) + 3 means a vertical stretch by factor 2 followed by a vertical translation of +3.

在更复杂的问题中，拉伸可能与平移或翻转组合。例如，y = 2f(x) + 3 表示纵向拉伸2倍后再向上平移3个单位。

The order of transformations is important. If you translate first, then stretch, the translation is also stretched. If you stretch first, the translation is unaffected by the stretch.

变换的顺序很重要。如果先平移再拉伸，平移量也会被拉伸；如果先拉伸再平移，平移量不受拉伸影响。

For example, compare y = 2(f(x) + 1) and y = 2f(x) + 1. In the first, the +1 is inside the parentheses, so it is applied first and then stretched. In the second, the stretch is applied first, then +1 is added outside.

例如，比较 y = 2(f(x) + 1) 和 y = 2f(x) + 1。第一个中，+1在括号内，先执行再被拉伸；第二个中，先拉伸再加1。

For horizontal transformations, the order also matters. y = f(2x + 4) is not the same as y = f(2(x + 2)). The latter is a horizontal stretch by 1/2 followed by a translation left by 2. Always rewrite expressions inside f in the form f(k(x + a)) to identify transformations clearly.

对于横向变换，顺序同样重要。y = f(2x + 4) 与 y = f(2(x + 2)) 不同。后者是横向压缩1/2后再向左平移2。始终将 f 内的表达式写成 f(k(x + a)) 的形式，以便清晰识别变换。

10. Common Mistakes and Tips | 常见错误与技巧

Many students lose marks on stretching graphs due to avoidable errors. Here are the most common pitfalls and how to avoid them.

许多学生在图像拉伸问题上失分，原因在于可避免的错误。以下是最常见的陷阱及避免方法。

Mistake: Confusing the horizontal scale factor. Remember y = f(kx) is a compression by k, not a stretch by k.
错误：混淆横向比例因子。记住 y = f(kx) 是压缩 k 倍，而不是拉伸 k 倍。

Mistake: Applying a stretch to both x and y coordinates when only one is meant. A vertical stretch only changes y-coordinates.
错误：在只针对一个方向时同时改变x和y坐标。纵向拉伸只改变y坐标。

Mistake: Forgetting that points on the axis of stretch remain fixed. For a vertical stretch, the x-axis (y=0) is the axis; all points with y=0 stay in place.
错误：忘记拉伸轴上的点保持不变。对于纵向拉伸，x轴（y=0）是轴；所有y=0的点位置不变。

Tip: Use a known point, such as (1, f(1)), to check your transformed graph.
技巧：使用一个已知点（如 (1, f(1))）来检查变换后的图像。

Tip: In the equation y = a f(bx), the parameter a controls vertical stretch, and the parameter b controls horizontal compression (scale factor 1/b).
技巧：在方程 y = a f(bx) 中，参数 a 控制纵向拉伸，参数 b 控制横向压缩（比例因子 1/b）。

11. Exam-Style Questions | 考试型例题

Let’s work through typical questions you might encounter in the Edexcel IGCSE exam.

让我们解答一些在Edexcel IGCSE考试中可能遇到的典型问题。

Question: The graph of y = x² + 2x is stretched vertically by scale factor 3. Write down the equation of the transformed graph.

题目：y = x² + 2x 的图像被纵向拉伸，比例因子为3。写出变换后图像的方程。

Solution: A vertical stretch multiplies the entire function by 3. So the new equation is y = 3(x² + 2x) = 3x² + 6x.

解答：纵向拉伸将整个函数乘以3。因此新方程为 y = 3(x² + 2x) = 3x² + 6x。

Question: The graph of y = sin(x) is transformed to y = sin(3x). Describe the transformation.

题目：y = sin(x) 的图像变换为 y = sin(3x)。描述该变换。

Solution: Since 3 multiplies x inside the function, it is a horizontal stretch with scale factor 1/3. In other words, the graph is compressed horizontally by a factor of 3. The period changes from 360° to 120°.

解答：由于3在函数内乘以x，这是比例因子为1/3的横向拉伸。换句话说，图像被水平压缩3倍。周期从360°变为120°。

Question: The graph of y = f(x) is stretched horizontally by factor 4. What is the new equation?

题目：y = f(x) 的图像被横向拉伸4倍。新方程是什么？

Solution: Horizontal stretch by factor 4 means we replace x with x/4. So the new equation is y = f(x/4).

解答：横向拉伸4倍意味着将x替换为x/4。因此新方程为 y = f(x/4)。

12. Summary | 总结

Stretching graphs is a fundamental skill in coordinate geometry. The two rules are simple:

图像拉伸是坐标几何中的基本技能。两条规则很简单：

y = k f(x) → Vertical stretch, scale factor k

y = f(kx) → Horizontal stretch, scale factor 1/k

Always pay attention to whether the multiplier is outside or inside the function. Practise with different families of functions—linear, quadratic, trigonometric, and cubic—to build confidence.

始终注意乘数是作用在函数外还是函数内。通过不同类型的函数（线性、二次、三角函数和三次函数）进行练习，以增强信心。

In the exam, draw a rough sketch if possible. Mark fixed points, check the direction of the stretch, and verify your equation with a known point. With careful reasoning, stretching graphs becomes a reliable source of marks.

在考试中，如果可能，画一个大致草图。标记不动点，检查拉伸方向，并用一个已知点验证方程。只要仔细推理，图像拉伸就能成为稳定的得分点。

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Stretching Graphs | 图像拉伸变换