Transformations of Graphs | 函数图像变换

📚 Transformations of Graphs | 函数图像变换

In this teacher-focused revision guide, we examine the core techniques for transforming the graphs of functions, a key topic in the IGCSE Mathematics syllabus. We will cover translations, reflections, and stretches, with clear notation, worked examples, and common pitfalls to highlight in class.

本篇教师用书聚焦 IGCSE 数学中的核心考点——函数图像变换。我们将系统讲解平移、反射与伸缩变换，给出清晰的符号记法、典型例题及课堂中需要特别强调的常见错误。

1. The Coordinate Plane and Function Notation | 坐标平面与函数记号

Before applying transformations, students must be fluent in reading points on the Cartesian plane. A point is written as (x, y), where x is the horizontal coordinate and y is the vertical coordinate. For a function y = f(x), each input x produces exactly one output y, and the graph of f is the set of all such points.

在讲解变换之前，学生必须熟练掌握笛卡尔平面上的点坐标。点的记法为 (x, y)，其中 x 为横坐标，y 为纵坐标。对于函数 y = f(x)，每一个输入 x 都对应唯一输出 y，函数 f 的图像就是所有这些点的集合。

Teachers should remind students that transforming a graph means applying the same algebraic operation to every point on the graph. This idea connects the symbolic form of a function to its geometric picture, and it is the foundation for every rule in this chapter.

教师应提醒学生：变换图像，就是对图像上的每一个点施加相同的代数操作。这一思想将函数的符号形式与其几何图像联系起来，是本章所有规则的基础。

2. Vertical Translations: y = f(x) + a | 垂直平移：y = f(x) + a

For a function y = f(x), the graph of y = f(x) + a is obtained by shifting every point on the original graph vertically by a units. If a is positive, the graph moves upward; if a is negative, it moves downward. The shape of the graph is unchanged.

对于函数 y = f(x)，函数 y = f(x) + a 的图像是将原图像上的每一个点垂直移动 a 个单位得到的。若 a 为正，则图像向上移动；若 a 为负，则图像向下移动。图像的形状保持不变。

Key teaching point: adding a constant outside the function only affects the y-coordinates. Students sometimes confuse this with horizontal shifts, so it is useful to test with a specific point, such as (0, f(0)).

教学要点：在函数外侧添加常数只会影响 y 坐标。学生常将它与水平平移混淆，因此用一个具体点如 (0, f(0)) 来验证会非常有帮助。

y = f(x) + a: shift up by a if a > 0, shift down by |a| if a < 0.

y = f(x) + a：若 a > 0 向上平移 a 个单位；若 a < 0 向下平移 |a| 个单位。

3. Horizontal Translations: y = f(x + a) | 水平平移：y = f(x + a)

The graph of y = f(x + a) is obtained by shifting the graph of y = f(x) horizontally. A critical point for students to remember is the sign direction: y = f(x + a) shifts the graph to the left by a units when a is positive, while y = f(x – a) shifts it to the right by a units when a is positive.

函数 y = f(x + a) 的图像是由 y = f(x) 的图像水平平移得到的。学生需要特别注意符号方向：当 a 为正时，y = f(x + a) 将图像向左移动 a 个单位；而 y = f(x − a) 将图像向右移动 a 个单位。

This is a common source of errors. A useful classroom analogy is to consider solving f(x + a) = 0: the x-value that satisfies it is shifted compared to solving f(x) = 0. For example, if f(2) = 0, then for y = f(x + 3), we need x + 3 = 2, so x = -1. This shows the graph moved left.

这是常见的错误来源。一个有效的课堂类比是考虑方程 f(x + a) = 0 的求解：满足该方程的 x 值与 f(x) = 0 相比发生了平移。例如，若 f(2) = 0，则对于 y = f(x + 3)，需要 x + 3 = 2，即 x = −1。这正说明图像向左移动了。

y = f(x + a): shift left by a (a > 0); y = f(x − a): shift right by a (a > 0).

y = f(x + a)：当 a > 0 时向左平移 a；y = f(x − a)：当 a > 0 时向右平移 a。

4. Vertical Stretch and Compression: y = k·f(x) | 垂直伸缩：y = k·f(x)

Multiplying the function by a constant k transforms the graph vertically. If k > 1, every y-coordinate is multiplied by k, producing a vertical stretch. If 0 < k < 1, the graph is compressed vertically toward the x-axis. If k is negative, a reflection across the x-axis is also involved, which we discuss separately in Section 6.

将函数乘以常数 k 会对图像进行垂直变换。若 k > 1，每个 y 坐标都乘以 k，产生垂直拉伸。若 0 < k < 1，则图像向 x 轴方向垂直压缩。若 k 为负数，还会同时涉及关于 x 轴的反射，我们将在第 6 节单独讨论。

Students should note that points on the x-axis, where y = 0, remain fixed under this transformation because multiplying zero by k still gives zero. This provides a quick way to check whether a transformation has been applied correctly.

学生应注意，x 轴上 y = 0 的点在该变换下保持不变，因为零乘以 k 仍为零。这为检验变换是否正确提供了快速方法。

y = k·f(x): vertical stretch by factor k when k > 1; vertical compression by factor k when 0 < k < 1.

y = k·f(x)：当 k > 1 时垂直拉伸 k 倍；当 0 < k < 1 时垂直压缩至 k 倍。

5. Horizontal Stretch and Compression: y = f(kx) | 水平伸缩：y = f(kx)

For y = f(kx), the graph is stretched or compressed horizontally. If k > 1, the graph is compressed toward the y-axis by a factor of 1/k. If 0 < k < 1, the graph is stretched away from the y-axis by a factor of 1/k. The multiplier acts inversely on the x-direction, which is a frequent source of confusion.

对于 y = f(kx)，图像进行水平方向的拉伸或压缩。若 k > 1，图像向 y 轴方向压缩至原来的 1/k；若 0 < k < 1，图像沿远离 y 轴的方向拉伸至 1/k 倍。这个倍数在 x 方向上是倒数关系，是学生最容易困惑的地方。

Teachers can clarify this by tracking a specific x-intercept. If the original graph crosses the x-axis at x = 2, then for y = f(3x), we need 3x = 2, so x = 2/3. The point has moved closer to the y-axis, confirming a horizontal compression.

教师可以通过跟踪一个具体的 x 截距来厘清这一点。若原图像在 x = 2 处穿过 x 轴，那么对于 y = f(3x)，需要 3x = 2，即 x = 2/3。该点移到了更靠近 y 轴的位置，说明发生了水平压缩。

y = f(kx): horizontal compression by factor 1/k when k > 1; horizontal stretch by factor 1/k when 0 < k < 1.

y = f(kx)：当 k > 1 时水平压缩至 1/k；当 0 < k < 1 时水平拉伸至 1/k 倍。

6. Reflections: y = −f(x) and y = f(−x) | 反射：y = −f(x) 与 y = f(−x)

The reflection of y = f(x) in the x-axis is given by y = −f(x). Every point (x, y) becomes (x, −y). This transformation flips the graph vertically. On the other hand, y = f(−x) reflects the graph in the y-axis, changing each point (x, y) to (−x, y).

y = f(x) 关于 x 轴的反射为 y = −f(x)。每个点 (x, y) 变为 (x, −y)，图像上下翻转。而 y = f(−x) 是 y = f(x) 关于 y 轴的反射，每个点 (x, y) 变为 (−x, y)。

Students must be able to distinguish these two cases quickly. A helpful classroom check is to use the point (1, f(1)): under y = −f(x) it moves to (1, −f(1)); under y = f(−x) it moves to (−1, f(1)). This single-point test usually resolves ambiguity.

学生必须快速区分这两种情况。一个有效的课堂检验是使用点 (1, f(1))：在 y = −f(x) 下它变为 (1, −f(1))；在 y = f(−x) 下它变为 (−1, f(1))。单点测试通常能消除疑惑。

7. Combined Transformations | 组合变换

When multiple transformations are applied, the order matters. For example, y = 2f(x) + 1 first stretches the graph vertically by factor 2, then translates it upward by 1 unit. In general, outside operations (those applied to the output y) are performed after inside operations (those applied to the input x).

当多个变换同时应用时，顺序至关重要。例如，y = 2f(x) + 1 先将图像垂直拉伸 2 倍，再向上平移 1 个单位。一般来说，外部运算（作用于输出 y）在内部运算（作用于输入 x）之后执行。

Teachers should explicitly state this order: for expressions like y = af(bx + c) + d, first handle the horizontal shift and stretch/compression involving bx + c, then apply the vertical stretch/compression and shift. Breaking the process into steps prevents sign errors and misapplication.

教师应明确说明顺序：对于形如 y = af(bx + c) + d 的表达式，先处理涉及 bx + c 的水平平移与伸缩，再应用垂直伸缩与平移。将过程分步进行，可以避免符号错误和误用。

Order of operations: horizontal transformations first, then vertical transformations.

操作顺序：先完成水平变换，再进行垂直变换。

8. Recognizing Transformations from a Graph | 从图像识别变换

A typical IGCSE question gives two graphs and asks students to identify the transformation. Students should compare corresponding points on the two graphs. If all y-values are multiplied by a constant, it is a vertical stretch; if all x-values are shifted by a constant, it is a horizontal translation.

IGCSE 的典型题目会给出两个图像，要求学生识别变换。学生应比较两个图像上的对应点。如果所有 y 值都乘以一个常数，则为垂直伸缩；如果所有 x 值都平移一个常数，则为水平平移。

It is useful to adopt a systematic method:

采用系统化的方法非常有用：

Select a few recognizable points on the original graph.
Find their images on the transformed graph.
Determine whether the change is in x, y, or both.
Write the transformation in the form y = af(bx + c) + d.

在原图像上选取几个特征点。
在变换后的图像上找到它们的对应点。
判断变化发生在 x、y 还是两者。
将变换写成 y = af(bx + c) + d 的形式。

9. Worked Example | 典型例题精讲

Let f(x) = x². Find the equation of the graph obtained by translating f(x) 3 units to the left and 2 units downward, then stretching it vertically by factor 4.

设 f(x) = x²。求将 f(x) 向左平移 3 个单位、向下平移 2 个单位，再纵向拉伸 4 倍后所得图像的方程。

Step 1: Translate 3 units left: replace x by x + 3, giving y = (x + 3)².

第 1 步：向左平移 3 个单位，将 x 替换为 x + 3，得到 y = (x + 3)²。

Step 2: Translate 2 units down: subtract 2 from the function, giving y = (x + 3)² − 2.

第 2 步：向下平移 2 个单位，函数整体减 2，得到 y = (x + 3)² − 2。

Step 3: Stretch vertically by factor 4: multiply the whole function by 4, giving y = 4[(x + 3)² − 2].

第 3 步：纵向拉伸 4 倍，整个函数乘以 4，得到 y = 4[(x + 3)² − 2]。

Final answer: y = 4(x + 3)² − 8

最终答案：y = 4(x + 3)² − 8

10. Common Mistakes and Teaching Tips | 常见错误与教学建议

The most frequent error students make is treating y = f(x + a) as a shift to the right. This misconception arises from an intuitive but incorrect reading of the plus sign. Teachers should repeatedly use the point-by-point method to correct this.

学生最常犯的错误是将 y = f(x + a) 认为是向右平移。这一误解源于对加号的直觉性但错误的理解。教师应反复使用逐点法来纠正。

Another common issue is confusing vertical and horizontal stretches. Remembering that y = k·f(x) affects y and y = f(kx) affects x can be reinforced by testing the point (1, f(1)). Additionally, when reflecting, students may forget to reverse the sign of every y-coordinate or every x-coordinate, so routine practice with simple functions such as f(x) = x² is recommended.

另一个常见问题是混淆垂直与水平伸缩。记住 y = k·f(x) 影响 y，y = f(kx) 影响 x，可以通过测试点 (1, f(1)) 来强化。此外，在做反射时，学生可能忘记反转每个 y 坐标或 x 坐标的符号，因此建议使用简单函数如 f(x) = x² 进行常规练习。

Teaching Tip: Use graphing software or dynamic geometry tools to show transformations live. Visual confirmation helps students internalize the algebraic rules more deeply than static diagrams alone.

教学建议：使用绘图软件或动态几何工具现场演示变换。视觉确认能帮助学生比静态图形更深刻地内化代数规则。

11. Practice Questions for Classroom Use | 课堂练习建议

The following exercises are suitable for a revision lesson and can be adapted to different ability levels.

以下练习适合复习课使用，可根据学生水平进行适当调整。

Question / 题目	Expected Answer / 参考答案
Given g(x) = √x, write the equation after shifting 2 units up.	y = √x + 2
Given h(x) = x³, find the equation after reflecting in the y-axis.	y = (−x)³ = −x³
Given f(x) = \|x\|, describe the transformation to y = \|x − 4\|.	Horizontal translation 4 units to the right.
Given p(x) = 2ˣ, write the equation after vertical stretch by factor 3.	y = 3 · 2ˣ

Teachers should ask students to explain their reasoning verbally as well as in writing. This encourages deeper understanding and reveals any remaining misconceptions.

教师应要求学生在书面作答之外，也用口头方式解释他们的推理过程。这有助于深化理解，并暴露仍然存在的误解。

12. Summary and Key Takeaways | 总结与核心要点

Transformations of graphs are a fundamental topic that connects algebra, geometry, and functions. The four families of transformations—vertical translation, horizontal translation, reflection, and stretch—can be understood through the effect they have on individual points.

函数图像变换是一个将代数、几何与函数联系起来的基础性主题。四种变换类型——垂直平移、水平平移、反射与伸缩——都可以通过对具体点的影响来理解。

The essential rules can be summarized as follows:

核心规则可总结如下：

y = f(x) + a moves the graph vertically by a units.
y = f(x + a) moves the graph horizontally by −a units.
y = −f(x) reflects in the x-axis.
y = f(−x) reflects in the y-axis.
y = k·f(x) stretches or compresses vertically.
y = f(kx) stretches or compresses horizontally by factor 1/k.

y = f(x) + a 将图像垂直平移 a 个单位。
y = f(x + a) 将图像水平平移 −a 个单位。
y = −f(x) 关于 x 轴反射。
y = f(−x) 关于 y 轴反射。
y = k·f(x) 垂直拉伸或压缩。
y = f(kx) 水平拉伸或压缩至 1/k 倍。

By mastering the point-by-point logic behind each transformation, students will be equipped to handle both sketch-based and equation-based IGCSE questions confidently.

掌握了每种变换背后的逐点逻辑，学生就能自信地应对 IGCSE 中基于草图或基于方程的两类问题。

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Transformations of Graphs | 函数图像变换