📚 Exercise 3G.1: Transformations of Functions | 练习3G.1：函数变换
In IB Mathematics, understanding how to shift, reflect and stretch graphs is essential for analyzing functions and solving equations. Exercise 3G.1 focuses on applying a sequence of transformations to basic functions such as polynomials, exponentials and trigonometric curves. This article revisits the key rules, explores common combinations, and works through IB-style examples to help you build confidence in graph transformations.
在 IB 数学中，理解如何平移、反射和伸缩函数图像是分析函数和求解方程的基础。练习 3G.1 重点关注如何将一系列变换应用于多项式、指数和三角函数等基本函数。本文回顾核心规则，探讨常见组合，并通过 IB 风格的例题帮助你建立对图像变换的信心。
1. Understanding Function Transformations | 理解函数变换
r controls a specific effect, and learning to interpret them is a key skill in IB Analysis and Approaches as well as Applications and Interpretation.
变换会改变图像的位置、大小或方向，但不会改变其基本形状。我们将这些变换作用于基本函数 y = f(x)，生成形如 y = a f(b(x - h)) + k 的新函数。每个参数控制一种特定的效果，学会解读它们是 IB 分析与方法和应用与解释科目中的关键技能。
2. Vertical Translations | 垂直平移
lly. The transformation y = f(x) + k moves every point up by k units if k > 0, and down by |k| fect the domain or the x-coordinates of any features like asymptotes or stationary points.
给 f(x) 加上常数 k 会将图像垂直平移。变换 y = f(x) + k 当 k > 0 时使每个点向上移动 k 个单位；当 k < 0 时向下移动 |k| 个单位。这不会影响定义域，也不会改变渐近线或驻点的 x 坐标。
Example: y = x² + 3 shifts the basic parabola up by 3 units.
3. Horizontal Translations | 水平平移
he graph horizontally. If h > 0, the graph translates to the right by h units; if h < 0, it translates to the left by |h| et: (x - h) produces a shift in the positive x-direction when h is positive. This is a frequent source of sign errors, so always double-check the direction.
变换 y = f(x - h) 将图像水平移动。若 h > 0，图像向右平移 h 个单位；若 h < 0，则向左平移 |h| 个单位。注意括号内的形式：(x - h) 在 h 为正时产生正 x 方向的平移。这是常见的符号错误来源，因此务必反复确认方向。
If f(x) = √x, then y = √x - 2 shifts the graph 2 units to the right.
4. Reflections Across the Axes | 关于坐标轴的反射
, reflects the graph in the y-axis. This swaps the left and right sides while keeping y-values unchanged.
将 f(x) 替换为 -f(x) 会将图像关于 x 轴反射。每个 y 坐标变号，点 (x, y) 变为 (x, -y)。将函数内部的自变量 x 替换为 -x，即 y = f(-x)，会将图像关于 y 轴反射。这会交换左右两侧，而 y 值保持不变。
| Transformation | Equation | Effect |
|---|---|---|
| Reflection in x-axis | y = -f(x) | (x, y) → (x, -y) |
| Reflection in y-axis | y = f(-x) | (x, y) → (-x, y) |
5. Vertical Stretches and Compressions | 垂直伸缩
Multiplying the whole function by a constant a gives y = a f(x). When |a| > 1, the graph stretches vertically away from the x-axis; when 0 < |a| xis. Negative values of a also combine a vertical stretch with a reflection in the x-axis.
将整个函数乘以常数 a 得到 y = a f(x)。当 |a| > 1 时，图像沿垂直方向远离 x 轴拉伸；当 0 < |a| < 1 时，则向 x 轴压缩。a 取负值还会同时包含垂直拉伸和关于 x 轴的反射。
For f(x) = sin x, y = 2 sin x doubles the amplitude, and y = ½ sin x halves it.
6. Horizontal Stretches and Compressions | 水平伸缩
The transformation y = f(bx) operates horizontally. If |b| , the graph compresses horizontally toward the y-axis (points get closer together). If 0 < |b| dinates are divided by b, so a factor of b = 2 means the period of a trig function halves.
变换 y = f(bx) 作用于水平方向。若 |b| > 1，图像水平向 y 轴压缩（点靠得更近）。若 0 < |b| < 1，则图像水平远离 y 轴拉伸。请记住，x 坐标除以 b，因此当 b = 2 时，三角函数的周期减半。
7. Combining Transformations (Order Matters) | 组合变换（顺序至关重要）
x - h)) + k, the transformations must be applied in a specific order to avoid mistakes. The recommended sequence is: first scale horizontally (b), then translate horizontally (h), then scale vertically (a), and finally translate vertically (k). Reflections can be treated as scaling with a negative factor and performed at the appropriate step.
当函数写成 y = a f(b(x - h)) + k 的形式时，必须按特定顺序应用变换才能避免错误。推荐的顺序是：先水平缩放 (b)，再水平平移 (h)，然后垂直缩放 (a)，最后垂直平移 (k)。反射可视为带负因子的缩放，并在相应步骤执行。
- Step 1: Horizontal stretch/compression by 1/|b| (include reflection in y-axis if b < 0)
- Step 2: Horizontal shift by h (right if h > 0)
- Step 3: Vertical stretch/compression by |a| (include reflection in x-axis if a < 0)
- Step 4: Vertical shift by k
- 步骤 1：水平伸缩因子 1/|b|（若 b < 0 则含 y 轴反射）
- 步骤 2：水平平移 h（若 h > 0 则向右）
- 步骤 3：垂直伸缩因子 |a|（若 a < 0 则含 x 轴反射）
- 步骤 4：垂直平移 k
8. Examples from IB-Style Questions | IB 风格例题示范
ift by 4, vertical stretch by factor 2 with reflection in x-axis, and finally down by 1. The point (0,0) on f maps to (-4, -1), and the overall shape is a vertically stretched, inverted cubic.
设 f(x) = x³。描述并绘制 y = -2 f(½(x + 4)) - 1 的图像。按顺序处理：水平拉伸因子 2（因 b = ½），然后左移 4，垂直拉伸因子 2 并关于 x 轴反射，最后下移 1。f 上的点 (0,0) 映射到 (-4, -1)，整体形状为垂直拉伸且反转的三次函数。
g able to track asymptotes through transformations is often tested in Paper 1 and Paper 2.
另一道典型的 IB 题目要求给出变换后渐近线的新方程。若 g(x) = 1/x 有垂直渐近线 x = 0，则应用 y = 3 + 1/(x - 5) 会将垂直渐近线移至 x = 5，水平渐近线移至 y = 3。能够通过变换追踪渐近线是试卷一和试卷二中常考的内容。
9. Common Pitfalls and How to Avoid Them | 常见错误及避免方法
the rightward shift. Another error is applying transformations in the wrong order, especially when dealing with both horizontal stretches and translations. Always respect the order: scale first, then shift.
一个经典错误是混淆水平平移的方向。学生经常把 y = f(x - 3) 向左移，而非向右。通过测试已知点来提醒自己：如果 f(0) 是特征点，令 x - 3 = 0 得 x = 3，即可确认右移。另一个错误是以错误的顺序应用变换，特别是在同时处理水平拉伸和平移时。务必遵守顺序：先缩放，后平移。
st be rewritten as f(2(x + 3)) to see a horizontal compression by factor ½ followed by a left shift of 3. Missing this step often leads to incorrect exam answers.
当函数同时包含水平缩放和平移时，平移量并不总是简单的 h；必须先提取公因子 b。例如，y = f(2x + 6) 必须重写为 f(2(x + 3))，才能看出水平压缩因子 ½，然后左移 3。忽略这一步往往导致考试答案错误。
10. Summary and Key Takeaways | 总结与关键要点
applying them in the right sequence. Do regular practice with parent functions such as x², √x, 1/x, sin x, and eˣ to build intuition. Use point-by-point mapping to verify your results, and always factor horizontally before translating. With careful attention, Exercise 3G.1 will strengthen your graphing skills for the entire IB course.
掌握变换意味着能识别 y = a f(b(x - h)) + k 中每个参数的作用，并按正确顺序应用它们。定期用 x²、√x、1/x、sin x 和 eˣ 等基本函数进行练习，以培养直觉。使用逐点映射来验证结果，并且始终在平移之前进行水平因式分解。只要细心，练习 3G.1 将巩固你在整个 IB 课程中的作图能力。
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