📚 Transformations of Functions | 函数变换
In IGCSE Mathematics, understanding transformations of functions is essential for interpreting and sketching graphs. A transformation changes the position or shape of the graph of a function without changing its fundamental nature. In this revision guide, we will explore translations, reflections, and stretches, and learn how to apply them systematically.
在 IGCSE 数学中，理解函数变换对于解读和绘制图像至关重要。变换会改变函数图像的位置或形状，但不会改变其基本性质。在本复习指南中，我们将探讨平移、反射和伸缩，并学习如何系统地应用它们。
1. Vertical Translation: y = f(x) + a | 垂直平移：y = f(x) + a
The graph of y = f(x) + a is obtained by shifting the graph of y = f(x) vertically. If a is positive, the graph moves upward by a units. If a is negative, the graph moves downward by |a| units.
函数 y = f(x) + a 的图像是通过将 y = f(x) 的图像垂直移动得到的。若 a 为正，图像向上移动 a 个单位；若 a 为负，图像向下移动 |a| 个单位。
For example, if f(x) = x², then y = x² + 2 is the same parabola shifted up by 2 units. The vertex moves from (0, 0) to (0, 2).
例如，若 f(x) = x²，则 y = x² + 2 是将同一抛物线向上移动 2 个单位的结果。顶点从 (0, 0) 移动到 (0, 2)。
Key point: The y-intercept changes by a, but the x-intercepts may change. Vertical translations affect the output values directly.
要点：y 截距会改变 a，而 x 截距可能改变。垂直平移直接影响输出值。
2. Horizontal Translation: 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. If a is positive, the graph moves left by a units. If a is negative, the graph moves right by |a| units. This is the opposite of what many students expect.
函数 y = f(x + a) 的图像是通过将 y = f(x) 的图像水平移动得到的。若 a 为正，图像向左移动 a 个单位；若 a 为负，图像向右移动 |a| 个单位。这与许多学生的直觉相反。
For example, if f(x) = x², then y = (x – 3)² is the parabola shifted right by 3 units. Notice the form y = f(x – 3), which corresponds to a = -3.
例如，若 f(x) = x²，则 y = (x – 3)² 是将抛物线向右移动 3 个单位的结果。注意形式 y = f(x – 3)，对应 a = -3。
Think of it this way: to make the graph move right, we replace x with x – a. To make it move left, we replace x with x + a.
可以这样理解：要使图像向右移动，我们用 x – a 替换 x；要使图像向左移动，我们用 x + a 替换 x。
3. Reflection in the x-axis: y = -f(x) | 关于 x 轴的反射：y = -f(x)
The graph of y = -f(x) is obtained by reflecting the graph of y = f(x) in the x-axis. Every point (x, y) becomes (x, -y).
函数 y = -f(x) 的图像是通过将 y = f(x) 的图像关于 x 轴反射得到的。每个点 (x, y) 变为 (x, -y)。
For example, if f(x) = sin x, then y = -sin x is the sine wave flipped upside down. The maximum and minimum values are interchanged.
例如，若 f(x) = sin x，则 y = -sin x 是上下翻转的正弦波。最大值和最小值相互交换。
Key point: The x-intercepts remain unchanged because y = 0 stays y = 0 under reflection.
要点：x 截距保持不变，因为在反射下 y = 0 仍然为 0。
4. Reflection in the y-axis: y = f(-x) | 关于 y 轴的反射：y = f(-x)
The graph of y = f(-x) is obtained by reflecting the graph of y = f(x) in the y-axis. Every point (x, y) becomes (-x, y).
函数 y = f(-x) 的图像是通过将 y = f(x) 的图像关于 y 轴反射得到的。每个点 (x, y) 变为 (-x, y)。
For example, if f(x) = 2x + 1, then y = f(-x) = -2x + 1. The line has the same y-intercept but the slope changes sign.
例如，若 f(x) = 2x + 1，则 y = f(-x) = -2x + 1。该直线有相同的 y 截距，但斜率变号。
Key point: If a function is even, that is f(-x) = f(x), then its graph is symmetric about the y-axis, and the reflection leaves it unchanged.
要点：如果函数是偶函数，即 f(-x) = f(x)，则其图像关于 y 轴对称，反射后保持不变。
5. Vertical Stretch: y = af(x) | 垂直伸缩：y = af(x)
The graph of y = af(x) is a vertical stretch of y = f(x) by a factor a. If a > 1, the graph is stretched away from the x-axis. If 0 < a < 1, the graph is compressed towards the x-axis. Every point (x, y) becomes (x, ay).
函数 y = af(x) 的图像是 y = f(x) 的垂直伸缩，伸缩因子为 a。若 a > 1，图像远离 x 轴拉伸；若 0 < a < 1，图像向 x 轴压缩。每个点 (x, y) 变为 (x, ay)。
For example, if f(x) = sin x, then y = 2 sin x has amplitude 2, while y = ½ sin x has amplitude ½.
例如，若 f(x) = sin x，则 y = 2 sin x 的振幅为 2，而 y = ½ sin x 的振幅为 ½。
Key point: If a is negative, combine the stretch by |a| with a reflection in the x-axis.
要点：若 a 为负，则先将图像按 |a| 伸缩，再关于 x 轴反射。
6. Horizontal Stretch: y = f(ax) | 水平伸缩：y = f(ax)
The graph of y = f(ax) is a horizontal stretch of y = f(x) by a factor 1/a. If a > 1, the graph is compressed towards the y-axis. If 0 < a < 1, the graph is stretched away from the y-axis. Every point (x, y) becomes (x/a, y).
函数 y = f(ax) 的图像是 y = f(x) 的水平伸缩，伸缩因子为 1/a。若 a > 1，图像向 y 轴压缩；若 0 < a < 1，图像远离 y 轴拉伸。每个点 (x, y) 变为 (x/a, y)。
For example, if f(x) = sin x, then y = sin(2x) has period π instead of 2π. The wave completes two full cycles in the same space.
例如，若 f(x) = sin x，则 y = sin(2x) 的周期为 π 而不是 2π。该波在同一空间内完成两个完整周期。
Key point: Horizontal stretches are ‘opposite’ in effect: a larger a means a smaller graph width.
要点：水平伸缩的效果是“相反”的：a 越大，图像的宽度越小。
7. Combined Transformations | 组合变换
When multiple transformations are applied, the order of operations matters. The general strategy is to follow the order inside the function first, then outside.
当应用多个变换时，运算顺序很重要。一般策略是：先处理函数内部的变换，再处理函数外部的变换。
For example, to sketch y = -2f(x) + 3 from y = f(x): first stretch by factor 2 vertically, then reflect in the x-axis, then translate up by 3 units.
例如，要从 y = f(x) 绘制 y = -2f(x) + 3：先垂直拉伸 2 倍，然后关于 x 轴反射，最后向上平移 3 个单位。
More generally, y = af(x + b) + c: shift left by b, stretch vertically by a, then shift up by c. But note that vertical stretch and translation do not commute: y = af(x) + c is not the same as y = a(f(x) + c).
更一般地，y = af(x + b) + c：先向左平移 b，再垂直拉伸 a，最后向上平移 c。但注意垂直拉伸和平移不可交换：y = af(x) + c 与 y = a(f(x) + c) 不同。
For horizontal changes, y = f(ax + b) can be rewritten as y = f(a(x + b/a)), so the shift is b/a units left, then a horizontal compression by 1/a.
对于水平变化，y = f(ax + b) 可以改写为 y = f(a(x + b/a))，因此先向左平移 b/a 个单位，然后按 1/a 进行水平压缩。
8. Identifying Transformations from Graphs | 从图像识别变换
In exam questions, you may be given two graphs and asked to describe
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