📚 Graphs 6: Transformations of Graphs | 图像变换 6
Graph transformations are a fundamental topic in IGCSE Mathematics. When a function f(x) is modified algebraically, its graph shifts, flips, or stretches in predictable ways. Understanding these rules allows you to sketch complex graphs quickly and solve problems involving transformed functions with confidence.
图像变换是 IGCSE 数学中的一个基础专题。当函数 f(x) 经过代数变形后，其图像会以可预测的方式平移、翻转或伸缩。理解这些规则，你就能快速绘制复杂函数的草图，并自信地解决与变换函数相关的问题。
1. Vertical Translation: y = f(x) + a | 垂直平移：y = f(x) + a
Adding a constant a outside the function shifts the graph vertically. If a > 0, the graph moves upward by a units; if a < 0, it moves downward by |a| units. The shape of the graph remains identical, and only the y-coordinates of every point change by adding a.
在函数外部加上常数 a 会使图像垂直平移。若 a > 0，图像向上移动 a 个单位；若 a < 0，图像向下移动 |a| 个单位。图像的形状保持不变，每个点的 y 坐标都加上 a。
For example, starting from f(x) = x², the graph of y = x² + 3 is the parabola shifted up by 3 units. Its vertex moves from (0, 0) to (0, 3).
例如，从 f(x) = x² 出发，y = x² + 3 的图像是将抛物线向上平移 3 个单位，顶点从 (0, 0) 移动到 (0, 3)。
y = f(x) + a → vertical translation by vector (0, a)
2. Horizontal Translation: y = f(x – a) | 水平平移：y = f(x – a)
Replacing x with (x – a) inside the function shifts the graph horizontally. The key rule is counter-intuitive: y = f(x – a) moves the graph to the right by a units if a > 0, and y = f(x + a) moves it to the left by a units. This is because the input x must now take a larger value to produce the same output.
在函数内部用 (x – a) 替换 x 会使图像水平平移。这里的关键规则有些反直觉：当 a > 0 时，y = f(x – a) 将图像向右移动 a 个单位；而 y = f(x + a) 将图像向左移动 a 个单位。这是因为输入 x 需要取更大的值才能产生相同的输出。
For a concrete example, consider f(x) = √x. The graph of y = √(x – 2) is the same curve shifted 2 units to the right, with domain x ≥ 2. Similarly, y = √(x + 1) is shifted 1 unit to the left, with domain x ≥ -1.
举一个具体例子，考虑 f(x) = √x。y = √(x – 2) 的图像是该曲线向右平移 2 个单位，定义域为 x ≥ 2。类似地，y = √(x + 1) 向左平移 1 个单位，定义域为 x ≥ -1。
y = f(x – a) → horizontal translation by vector (a, 0)
3. Reflection in the x-axis: y = -f(x) | 关于 x 轴的反射：y = -f(x)
Multiplying the entire function by -1 reflects the graph in the x-axis. Every point (x, y) on the original graph maps to (x, -y). Points that lie on the x-axis remain fixed, while all other points are mirrored vertically.
将整个函数乘以 -1 会使图像关于 x 轴反射。原图像上的每个点 (x, y) 都会映射到 (x, -y)。位于 x 轴上的点保持不动，而所有其他点都被垂直镜像。
For example, the graph of y = -x² is an upside-down parabola compared to y = x². Its vertex stays at the origin, but the curve now opens downward instead of upward.
例如，与 y = x² 相比，y = -x² 的图像是一只倒置的抛物线。它的顶点仍在原点，但曲线现在向下开口而不是向上开口。
y = -f(x) → reflection in the x-axis
4. 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 reflects the graph in the y-axis. Every point (x, y) maps to (-x, y). This transformation preserves the y-coordinates and mirrors the graph horizontally.
在函数内部用 -x 替换 x 会使图像关于 y 轴反射。每个点 (x, y) 映射到 (-x, y)。此变换保持 y 坐标不变，将图像水平镜像。
Notice that not all functions change under this reflection. If f(-x) = f(x) for all x, the function is even (symmetric about the y-axis), such as f(x) = x² or f(x) = cos x. If f(-x) = -f(x), the function is odd (symmetric about the origin), such as f(x) = x³ or f(x) = sin x.
注意，并非所有函数在此反射下都会改变。如果对所有 x 都有 f(-x) = f(x)，则该函数是偶函数（关于 y 轴对称），例如 f(x) = x² 或 f(x) = cos x。如果 f(-x) = -f(x)，则该函数是奇函数（关于原点对称），例如 f(x) = x³ 或 f(x) = sin x。
y = f(-x) → reflection in the y-axis
5. Vertical Stretch: y = a·f(x) | 垂直伸缩：y = a·f(x)
Multiplying the function by a constant a > 0 stretches or compresses the graph vertically. If a > 1, the graph is stretched vertically by a factor of a; if 0 < a < 1, the graph is compressed vertically by a factor of a. Every y-coordinate is multiplied by a, while x-coordinates stay unchanged.
将函数乘以正常数 a 会使图像垂直拉伸或压缩。若 a > 1，图像在垂直方向拉伸为原来的 a 倍；若 0 < a < 1，图像在垂直方向压缩为原来的 a 倍。每个 y 坐标都乘以 a，x 坐标保持不变。
Taking f(x) = sin x as an example, the graph of y = 2 sin x has an amplitude of 2, so it oscillates between -2 and 2. The graph of y = ½ sin x has an amplitude of ½, oscillating between -½ and ½. In both cases, the period remains 360°.
以 f(x) = sin x 为例，y = 2 sin x 的图像振幅为 2，在 -2 和 2 之间振荡。y = ½ sin x 的图像振幅为 ½，在 -½ 和 ½ 之间振荡。在两种情况下，周期都保持为 360°。
y = a·f(x) → vertical stretch, factor a (a > 0)
6. Horizontal Stretch: y = f(kx) | 水平伸缩：y = f(kx)
Replacing x with kx inside the function, where k > 0, stretches or compresses the graph horizontally by a factor of 1/k. If k > 1, the graph is compressed horizontally (squashed towards the y-axis); if 0 < k < 1, the graph is stretched horizontally (pulled away from the y-axis).
在函数内部用 kx 替换 x（其中 k > 0）会使图像在水平方向拉伸或压缩，伸缩因子为 1/k。若 k > 1，图像在水平方向被压缩（向 y 轴挤压）；若 0 < k < 1，图像在水平方向被拉伸（远离 y 轴）。
For example, y = sin(2x) completes a full cycle in 180° instead of 360°, so its period is halved. Meanwhile, y = sin(½x) completes a full cycle in 720°, so its period is doubled.
例如，y = sin(2x) 在 180° 内完成一个完整周期，因此其周期减半。而 y = sin(½x) 在 720° 内完成一个完整周期，因此其周期加倍。
y = f(kx) → horizontal stretch, factor 1/k (k > 0)
7. Combining Transformations | 组合变换
Exam questions often require applying multiple transformations in sequence. When combining transformations, the order matters. A common format is y = a·f(b(x – c)) + d, where d gives the vertical shift, a gives the vertical stretch, c gives the horizontal shift, and b gives the horizontal stretch.
考试题目常常要求依次应用多个变换。组合变换时，顺序至关重要。一种常见的形式是 y = a·f(b(x – c)) + d，其中 d 表示垂直平移，a 表示垂直伸缩，c 表示水平平移，b 表示水平伸缩。
A recommended strategy is to apply horizontal transformations first, then vertical transformations. More precisely: first apply the horizontal stretch/compression (b), then the horizontal translation (c), then the vertical stretch (a), and finally the vertical translation (d).
推荐的策略是先进行水平变换，再进行垂直变换。更精确地说：先进行水平伸缩 (b)，再进行水平平移 (c)，然后进行垂直伸缩 (a)，最后进行垂直平移 (d)。
Consider the transformation from f(x) to y = 2f(x – 1) + 3. Starting from f(x), shift the graph 1 unit to the right to get f(x – 1); then stretch vertically by a factor of 2 to get 2f(x – 1); then shift up by 3 units to obtain the final graph.
考虑从 f(x) 到 y = 2f(x – 1) + 3 的变换。从 f(x) 开始，先将图像向右平移 1 个单位得到 f(x – 1)；然后垂直拉伸为原来的 2 倍得到 2f(x – 1)；最后向上平移 3 个单位得到最终图像。
8. Identifying Transformations from Equations | 从方程识别变换
To determine what transformations are applied, rewrite the transformed function in the standard order: y = a·f(b(x – c)) + d. Each parameter then tells you exactly one transformation.
要确定应用了什么变换，请将变换后的函数改写为标准顺序：y = a·f(b(x – c)) + d。每个参数都精确地告诉你一种变换。
| Parameter 参数 | Effect 效果 |
|---|---|
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