Transformations of Graphs | 函数图像变换

📚 Transformations of Graphs | 函数图像变换

Graph transformations are one of the most frequently tested topics in the Edexcel IGCSE Mathematics syllabus. Given a function y = f(x), you must be able to sketch and describe the effect of four basic transformations: vertical translation y = f(x) + a, horizontal translation y = f(x − a), reflection in the x-axis y = −f(x), and reflection in the y-axis y = f(−x). Mastering these rules allows you to plot new graphs quickly without calculating every coordinate, and it builds a foundation for solving equations graphically and for later advanced study.

函数图像变换是 Edexcel IGCSE 数学考试中最常考的知识点之一。给定一个函数 y = f(x)，你需要能够绘制并描述四种基本变换的效果：垂直平移 y = f(x) + a、水平平移 y = f(x − a)、关于 x 轴的对称变换 y = −f(x) 和关于 y 轴的对称变换 y = f(−x)。掌握这些规则可以帮助你无需逐点计算即可快速绘制新函数图像，同时为图解方程和后续进阶学习奠定坚实基础。

1. Vertical Translation: y = f(x) + a | 垂直平移：y = f(x) + a

The first transformation to understand is the vertical translation. For any function y = f(x), the transformed expression y = f(x) + a produces a graph shifted strictly vertically. If a > 0, every point on the original graph moves upward by a units. If a < 0, every point moves downward. Crucially, the shape of the graph never changes, only its vertical position. The x-coordinates of all key points remain identical; only the y-coordinates are adjusted by adding a.

首先需要理解的是垂直平移。对于任意函数 y = f(x)，变换后的式子 y = f(x) + a 使图像沿垂直方向整体平移。如果 a > 0，原图像上的每个点都向上移动 a 个单位；如果 a < 0，所有点都向下移动。关键在于，图像的形状完全不变，只有垂直位置改变。所有关键点的横坐标保持不变，纵坐标统一加上 a。

For example, take the simple quadratic f(x) = x². The transformed function f(x) + 2 = x² + 2 shifts the parabola upward by 2 units. Its vertex moves from (0, 0) to (0, 2), while the axis of symmetry remains x = 0. Likewise, f(x) − 3 = x² − 3 moves the parabola downward, placing the vertex at (0, −3). In vector notation, this transformation is written as a translation by the column vector (0, a).

例如，以简单二次函数 f(x) = x² 为例。变换后的函数 f(x) + 2 = x² + 2 将抛物线整体向上移动 2 个单位，顶点从 (0, 0) 移到 (0, 2)，对称轴仍然是 x = 0。类似地，f(x) − 3 = x² − 3 将抛物线向下移动，顶点位于 (0, −3)。用向量记号表示，这个变换对应平移向量 (0, a)。

y = f(x) + a: translate up by a units if a > 0; translate down by |a| units 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

2. Horizontal Translation: y = f(x − a) | 水平平移：y = f(x − a)

The horizontal translation is the most counterintuitive rule in this topic. For a function y = f(x), the expression y = f(x − a) shifts the graph to the right when a is positive, and to the left when a is negative. In other words, f(x − 2) moves the graph right by 2 units, while f(x + 2) moves it left by 2 units. This appears backwards to many students because the minus sign inside the brackets suggests a leftward movement, but the opposite is true.

水平平移是本主题中最容易让人困惑的规则。对于函数 y = f(x)，表达式 y = f(x − a) 在 a 为正数时使图像向右平移，在 a 为负数时使图像向左平移。换句话说，f(x − 2) 将图像向右移动 2 个单位，而 f(x + 2) 将图像向左移动 2 个单位。许多同学会感到别扭，因为括号内的减号似乎暗示向左移动，但事实恰恰相反。

Consider f(x) = x² again. The function f(x − 3) = (x − 3)² is a parabola with its vertex at (3, 0), whereas the original vertex was at (0, 0). Thus the entire graph slides right by 3 units. The rule works because the graph must attain the same y-value at x = 3 as the original graph attains at x = 0, which effectively moves the graph forward along the x-axis.

再次以 f(x) = x² 为例。函数 f(x − 3) = (x − 3)² 的抛物线顶点在 (3, 0)，而原抛物线的顶点在 (0, 0)。因此整个图像向右滑动 3 个单位。这个规则之所以成立，是因为新图像在 x = 3 处取得的 y 值与原图像在 x = 0 处取得的 y 值相同，相当于将图像沿 x 轴正向推移。

y = f(x − a): translate right by a units if a > 0; translate left by |a| units if a < 0

y = f(x − a)：若 a > 0 向右平移 a 个单位；若 a < 0 向左平移 |a| 个单位

3. Reflection in the x-axis: y = −f(x) | 关于 x 轴的对称变换：y = −f(x)

The next transformation is a reflection in the x-axis. When you replace f(x) by −f(x), every point (x, y) on the original graph maps to the point (x, −y). Geometrically, this flips the graph upside down over the horizontal axis, like the image in a mirror placed along the x-axis. All x-intercepts remain fixed because y = 0 maps to itself, but all other points swap their positive and negative y-values.

下一个变换是关于 x 轴的对称变换。当你把 f(x) 替换为 −f(x) 时，原图像上的每个点 (x, y) 都映射为点 (x, −y)。从几何上看，这相当于将图像沿水平轴上下翻转，就像一面放在 x 轴上的镜子所成的像。所有与 x 轴的交点都保持不变，因为 y = 0 映射到自身，但其他所有点的 y 值正负互换。

Take the parabola y = x² as an example. Its reflection in the x-axis is

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