Graph Transformations: Translations, Reflections & Stretches | 图像变换：平移、反射与伸缩

📚 Graph Transformations: Translations, Reflections & Stretches | 图像变换：平移、反射与伸缩

Graph transformations are a central topic in IGCSE Mathematics. Once you understand how a function’s equation is related to its graph, you can sketch new graphs quickly without plotting every point. In this article, we will explore three main families of transformations: translations, reflections, and stretches, together with their algebraic rules and common pitfalls.

图像变换是 IGCSE 数学的核心内容。一旦你理解了函数方程与其图像之间的对应关系，你就能无需逐点描图而快速画出新图像。在本文中，我们将探讨三大类变换：平移、反射和伸缩，以及它们所对应的代数规则和常见易错点。

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1. Why Study Transformations? | 为什么要学习变换？

Graph transformations allow us to take a known graph, such as y = x² or y = sin x, and move, flip, or stretch it to obtain a related graph. This skill is tested in both Paper 2 and Paper 4 of IGCSE Mathematics, and it appears in questions about coordinates, asymptotes, and turning points.

图像变换使我们能够以已知图像为基础，例如 y = x² 或 y = sin x，通过移动、翻折或拉伸来得出相关图像。这一技能在 IGCSE 数学的 Paper 2 和 Paper 4 中都会考到，并常与坐标、渐近线和极值点的问题一起出现。

We use function notation f(x) throughout. If a point (x, y) lies on y = f(x), then after a transformation the new coordinates can often be expressed as a simple change. The key is: always pay attention to what happens to x inside the function, and what happens to y outside the function.

我们全程使用函数记号 f(x)。如果点 (x, y) 在 y = f(x) 的图像上，那么经过一次变换后，新坐标通常可以表示为简单变化。关键在于：始终注意函数内部 x 发生了什么变化，以及函数外部 y 发生了什么变化。

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2. Vertical Translations: y = f(x) + a | 纵向平移：y = f(x) + a

Adding a constant a to the whole function moves the graph up if a is positive, and down if a is negative. Every point (x, y) on the original graph becomes (x, y + a).

在整个函数后加上常数 a，如果 a 为正，图像向上平移；如果 a 为负，图像向下平移。原图像上的每一个点 (x, y) 都会变为 (x, y + a)。

For example, y = x² has its vertex at (0, 0). The graph y = x² + 3 has its vertex at (0, 3). The shape is identical, only the vertical position changes.

例如，y = x² 的顶点在 (0, 0)。图像 y = x² + 3 的顶点在 (0, 3)。图像形状完全不变，只是纵向位置发生了改变。

y = f(x) + a → translate by a units in the positive y-direction

y = f(x) + a → 沿 y 轴正方向平移 a 个单位

Remember that vertical translations do not affect the x-coordinates of any points. The domain of the function stays the same, and only the range changes.

请记住，纵向平移不会影响任何点的 x 坐标。函数的定义域保持不变，只有值域发生改变。

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3. Horizontal Translations: y = f(x − a) | 横向平移：y = f(x − a)

This is the most counter-intuitive rule. The graph y = f(x − a) is the graph of y = f(x) shifted right by a units if a > 0. The minus sign inside the bracket means the graph moves to the right, not to the left.

这是最容易让人感到困惑的规则。当 a > 0 时，y = f(x − a) 的图像是 y = f(x) 的图像向右平移 a 个单位。括号内的减号意味着图像向右移动，而不是向左移动。

Why? To produce the same y-value as before, x must now be larger by a. For instance, the vertex of y = x² is at x = 0. The vertex of y = (x − 2)² satisfies x − 2 = 0, so x = 2. Hence the graph shifts right by 2.

为什么呢？为了得到与原来相同的 y 值，x 现在必须比原来大 a。例如，y = x² 的顶点在 x = 0。y = (x − 2)² 的顶点满足 x − 2 = 0，即 x = 2。因此图像向右平移了 2 个单位。

y = f(x − a) → translate by a units in the positive x-direction (right)

y = f(x − a) → 沿 x 轴正方向（向右）平移 a 个单位

Similarly, y = f(x + a) shifts the graph left by a units. A common exam trap is to write “left for minus, right for plus” — this is wrong. Always think: x − a means the original x must increase to compensate, so the whole graph moves right.

类似地，y = f(x + a) 将图像向左平移 a 个单位。考试中常见的陷阱是认为“减号向左、加号向右”——这是错误的。请这样理解：x − a 意味着原来的 x 需要增大来补偿，所以整个图像向右移动。

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4. Reflection in the x-axis: y = −f(x) | 关于 x 轴的反射：y = −f(x)

Multiplying the whole function by −1 flips the graph over the x-axis. Every point (x, y) becomes (x, −y). Points on the x-axis itself remain fixed.

将整个函数乘以 −1，会使图像关于 x 轴翻转。每一个点 (x, y) 都变为 (x, −y)。位于 x 轴上的点保持不变。

For example, y = sin x oscillates between −1 and 1. The graph y = −sin x is the mirror image: where the original reaches a peak, the new graph reaches a trough, and vice versa.

例如，y = sin x 在 −1 和 1 之间振荡。图像 y = −sin x 是它的镜像：原图像达到波峰处，新图像则为波谷，反之亦然。

y = −f(x) → reflection in the x-axis

y = −f(x) → 关于 x 轴反射

This transformation changes the sign of all y-coordinates. Roots of the function (where y = 0) do not move, because −0 = 0. The maxima and minima swap roles.

该变换改变所有 y 坐标的符号。函数的根（即 y = 0 处）不会移动，因为 −0 = 0。最大值与最小值互换角色。

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5. Reflection in the y-axis: y = f(−x) | 关于 y 轴的反射：y = f(−x)

Replacing x with −x inside the function flips the graph over the y-axis. Every point (x, y) becomes (−x, y). This time the y-coordinates are unchanged, but the x-coordinates change sign.

将函数内部的 x 替换为 −x，会使图像关于 y 轴翻转。每一个点 (x, y) 都变为 (−x, y)。这一次 y 坐标保持不变，而 x 坐标改变符号。

For instance, y = 2ˣ is an increasing exponential. Its reflection y = 2⁻ˣ is a decreasing exponential. The two graphs meet at the y-intercept (0, 1) because 2⁰ = 2⁻⁰ = 1.

例如，y = 2ˣ 是递增指数函数。它的反射 y = 2⁻ˣ 是递减指数函数。两条图像在 y 轴截距 (0, 1) 处相交，因为 2⁰ = 2⁻⁰ = 1。

y = f(−x) → reflection in the y-axis

y = f(−x) → 关于 y 轴反射

Notice how the rules mirror each other: outside the function affects y, inside the function affects x. Always keep this inside/outside distinction clear in your mind.

请注意两条规则的对应关系：函数外部影响 y，函数内部影响 x。务必在心中明确区分“内部”与“外部”。

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6. Vertical Stretch: y = a·f(x) | 纵向伸缩：y = a·f(x)

Multiplying the whole function by a positive constant a stretches or compresses the graph vertically. If a > 1, every y-coordinate is multiplied by a, so the graph becomes taller (vertical stretch). If 0 < a < 1, the graph becomes flatter (vertical compression).

将整个函数乘以正数 a，会沿纵向拉伸或压缩图像。如果 a > 1，每个 y 坐标都乘以 a，因此图像变得更高（纵向拉伸）。如果 0 < a < 1，图像变得更扁平（纵向压缩）。

Consider y = 2x² versus y = x². At x = 3, the first gives y = 18 while the second gives y = 9. Every point is twice as far from the x-axis. The vertex at (0, 0) remains fixed.

对比 y = 2x² 与 y = x²。在 x = 3 处，前者给出 y = 18，而后者给出 y = 9。每一个点到 x 轴的距离都是原来的两倍。顶点 (0, 0) 保持不动。

y = a·f(x), a > 1 → vertical stretch by scale factor a

y = a·f(x)，a > 1 → 纵向伸长为原来的 a 倍

If a is negative, the graph is also reflected in the x-axis. For example, y = −2f(x) first stretches the graph by factor 2, then reflects it across the x-axis. Transformations can be combined in this way.

如果 a 为负，图像还会关于 x 轴反射。例如，y = −2f(x) 先将图像伸长 2 倍，再关于 x 轴反射。变换可以这样组合。

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7. Horizontal Stretch: y = f(ax) | 横向伸缩：y = f(ax)

Replacing x with ax inside the function stretches or compresses the graph horizontally. The scale factor is the reciprocal: 1/a. If a > 1, the graph becomes narrower (horizontal compression by factor 1/a). If 0 < a < 1, the graph becomes wider (horizontal stretch).

将函数内部的 x 替换为 ax，会沿横向拉伸或压缩图像。伸缩倍数是 a 的倒数，即 1/a。如果 a > 1，图像变得更窄（横向压缩为原来的 1/a）。如果 0 < a < 1，图像变得更宽（横向拉伸）。

Let us test with y = sin x. The graph of y = sin(2x) completes a full cycle between x = 0 and x = π, instead of between 0 and 2π. So it oscillates twice as fast — the period is halved, from 2π to π.

我们用 y = sin x 来测试。y = sin(2x) 的图像在 x = 0 到 x = π 之间就完成一个完整周期，而不是 0 到 2π。因此它振荡频率是原来的两倍——周期从 2π 减半为 π。

y = f(ax), a > 1 → horizontal compression by scale factor 1/a

y = f(ax)，a > 1 → 横向压缩为原来的 1/a

This is the inverse of intuition: multiplying x by a larger number makes the graph narrower, not wider. Remember “inside is backwards”: inside the function, adding shifts left, and multiplying compresses.

这与直觉相反：x 乘以更大的数会使图像变窄，而不是变宽。请记住“函数内部一切反向”：函数内部做加法时图像左移，内部做乘法时图像压缩。

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8. Summary Table | 汇总表

The table below summarises all six basic transformations. Learn this table thoroughly — it is the quickest way to secure marks.

下表总结了六种基本变换。请彻底掌握这张表，这是稳妥得分的最快途径。

Transformation | 变换	New equation | 新方程	Effect on graph | 对图像的影响
Vertical translation | 纵向平移	y = f(x) + a	Move up by a (a > 0) | 向上平移 a（a > 0）
Horizontal translation | 横向平移	y = f(x − a)	Move right by a (a > 0) | 向右平移 a（a > 0）
Reflection in x-axis | x 轴反射	y = −f(x)	Flip over x-axis | 关于 x 轴翻折
Reflection in y-axis | y 轴反射	y = f(−x)	Flip over y-axis | 关于 y 轴翻折
Vertical stretch | 纵向伸缩	y = a·f(x)	Stretch factor a from x-axis | 距 x 轴距离放大 a 倍
Horizontal stretch | 横向伸缩	y = f(ax)	Compress factor 1/a toward y-axis | 向 y 轴压缩为 1/a

A useful summary rule: changes outside the brackets act on y and are “normal”; changes inside the brackets act on x and are “reversed”.

一个实用的总结规则：括号外部的变化作用于 y，方向“正常”；括号内部的变化作用于 x，方向“相反”。

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9. Combining Transformations | 组合变换

Exam questions often ask for a sequence of two or three transformations. For example, to sketch y = 2f(x) − 3, you first stretch vertically by factor 2, then translate down by 3 units. The order matters, especially when translations and stretches are mixed.

考试题经常要求连续进行两次或三次变换。例如，要画 y = 2f(x) − 3，你应先纵向伸长 2 倍，再向下平移 3 个单位。当平移与伸缩混合时，变换顺序非常重要。

Consider y = f(x + 1) + 2. Starting from y = f(x), shift left by 1 (because of +1 inside), then shift up by 2. The turning point of y = x², originally at (0, 0), moves to (−1, 2).

考虑 y = f(x + 1) + 2。从 y = f(x) 出发，先向左平移 1 个单位（因为内部是 +1），再向上平移 2 个单位。y = x² 的极值点原本在 (0, 0)，现在移动到 (−1, 2)。

For multiple transformations inside the brackets, factorise first. For example, y = f(2x + 4) should be rewritten as y = f(2(x + 2)). This is a horizontal compression by factor ½ followed by a shift left by 2.

当括号内有多重变换时，应先分解因式。例如，y = f(2x + 4) 应改写为 y = f(2(x + 2))。这意味着先进行 ½ 倍的横向压缩，再向左平移 2 个单位。

When reflections and stretches combine, apply stretches first and reflections afterwards if the equation is written as y = −a·f(x). However, for reflections across the x-axis and y-axis, order often does not affect the final shape because they commute with each other.

当反射与伸缩结合时，若方程写作 y = −a·f(x)，应先伸缩再反射。不过，关于 x 轴和 y 轴的反射之间通常顺序不影响最终图形，因为它们满足交换律。

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10. Asymptotes and Key Points | 渐近线与关键点

When a graph is transformed, its asymptotes move too. For a vertical asymptote x = c, a horizontal shift changes c. For a horizontal asymptote y = k, a vertical shift or vertical stretch changes k.

当图像发生变换时，其渐近线也会随之移动。对于竖直渐近线 x = c，水平平移会改变 c。对于水平渐近线 y = k，纵向平移或纵向伸缩会改变 k。

Example: y = 1/x has vertical asymptote x = 0 and horizontal asymptote y = 0. Under y = 1/(x − 2) + 3, the vertical asymptote becomes x = 2 and the horizontal asymptote becomes y = 3.

例如：y = 1/x 有竖直渐近线 x = 0 和水平渐近线 y = 0。在 y = 1/(x − 2) + 3 下，竖直渐近线变为 x = 2，水平渐近线变为 y = 3。

Always identify three types of key points before transforming: intercepts (where x = 0 or y = 0), turning points, and asymptotes. Track each one through the transformation to sketch accurately.

在进行变换之前，务必先找出三类关键点：截距（x = 0 或 y = 0 处）、极值点和渐近线。逐一追踪它们在变换中的位置变化，才能准确画出图像。

For y = af(x), the x-intercepts stay fixed because a × 0 = 0. For y = f(ax), the y-intercept stays fixed because f(a × 0) = f(0). These fixed points are useful anchors when sketching.

对于 y = af(x)，x 轴截距保持不变，因为 a × 0 = 0。对于 y = f(ax)，y 轴截距保持不变，因为 f(a × 0) = f(0)。这些不动点在画图时是很有用的锚点。

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11. Common Mistakes | 常见错误

Mistake 1: Confusing the direction of horizontal translations. y = f(x − 3) moves right, not left. A mnemonic: “minus goes right” in graph transformations, because the x-coordinate must increase to compensate.

错误 1：混淆横向平移的方向。y = f(x − 3) 是向右移动，而不是向左。记忆口诀：“减号向右走”，因为 x 坐标必须增大来补偿。

Mistake 2: Mixing up stretches. y = f(3x) is a horizontal compression, not stretch. Many students write “stretch by 3” — the scale factor is actually 1/3. Always divide by the coefficient of x.

错误 2：混淆伸缩类型。y = f(3x) 是横向压缩，而不是拉伸。许多学生写成“拉伸 3 倍”——实际倍数是 1/3。请始终用 x 的系数去除。

Mistake 3: Forgetting order of operations. For y = f(x) + 4, translate up by 4; but for y = f(x + 4), translate left by 4. The operations are not interchangeable.

错误 3：忘记运算顺序。对于 y = f(x) + 4，是向上平移 4；但对于 y = f(x + 4)，是向左平移 4。二者不能互换。

Mistake 4: When a question gives the image graph and asks for the original function, students try to reverse the transformation wrongly. If the image is y = g(x), and the rule was y = f(x − 2), then the original is f(x) = g(x + 2). Reverse the sign.

错误 4：当题目给出变换后的图像并求原函数时，学生常常反向变换错误。如果变换后的图像为 y = g(x)，且规则是 y = f(x − 2)，那么原函数为 f(x) = g(x + 2)。需要把符号反过来。

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12. Practice Questions | 练习题

Try these four questions yourself before checking the answers below. They cover the most important transformation skills for IGCSE.

请先自己尝试以下四道题，然后再对照答案。它们涵盖了 IGCSE 最重要的变换技能。

• Question 1: The graph y = x² is transformed to y = (x + 3)² − 5. State the translation in the form (a, b).

  第 1 题：图像 y = x² 被变换为 y = (x + 3)² − 5。请以 (a, b) 的形式说明平移量。

• Question 2: The graph y = 2ˣ is reflected in the y-axis. Write down the equation of the new graph.

  第 2 题：图像 y = 2ˣ 关于 y 轴反射。写出新图像的方程。

• Question 3: The point (4, −6) lies on y = f(x). Find the coordinates of the image point after the transformation y = −2f(x).

  第 3 题：点 (4, −6) 在 y = f(x) 上。求其在变换 y = −2f(x) 后的像点坐标。

• Question 4: The graph y = sin x is stretched vertically by factor 3, then translated up by 2. Write the final equation and state the new maximum value.

  第 4 题：图像 y = sin x 先纵向伸长 3 倍，再向上平移 2。写出最终方程并说明新的最大值。

Answer 1: (a, b) = (−3, −5). The graph moves left 3 and down 5.

答案 1：(a, b) = (−3, −5)。图像向左移动 3 并向下移动 5。

Answer 2: y = 2⁻ˣ. Replacing x by −x inside gives y = 2⁻ˣ.

答案 2：y = 2⁻ˣ。将内部 x 换成 −x，得到 y = 2⁻ˣ。

Answer 3: The y-coordinate becomes −2 × (−6) = 12. The x-coordinate stays at 4. So the image point is (4, 12).

答案 3：y 坐标变为 −2 × (−6) = 12。x 坐标保持为 4。因此像点为 (4, 12)。

Answer 4: y = 3 sin x + 2. The maximum value of sin x is 1, so the new maximum is 3 × 1 + 2 = 5.

答案 4：y = 3 sin x + 2。sin x 的最大值是 1，因此新最大值为 3 × 1 + 2 = 5。

By mastering these six rules and their combinations, you have covered all graph transformation content required for IGCSE Mathematics. Practice sketching every transformation on the same set of axes, and you will build strong visual intuition for the exam.

通过掌握这六条规则及其组合，你已经覆盖了 IGCSE 数学中所有图像变换的内容。请在同一个坐标系上练习画出每一种变换，你将逐步培养出在考试中非常重要的直观图形感。

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