📚 Exercise 3E: Transformations of Functions | 练习3E：函数的变换
In IB Mathematics, Exercise 3E typically opens the door to one of the most visually powerful topics: transformations of functions. Whether you are studying Analysis and Approaches (AA) or Applications and Interpretation (AI), mastering shifts, stretches and reflections allows you to decode complex graphs and build sophisticated models from simple parent functions.
在IB数学中，练习3E通常开启了最具视觉力量的主题之一：函数的变换。无论你学习的是分析与方法（AA）还是应用与解释（AI），掌握平移、伸缩和反射能让你从简单的母函数出发，解读复杂图形并建立精密的模型。
1. The Parent Function and Notation | 母函数与记法
Every transformation begins with a parent function, such as f(x) = x², f(x) = √x nput (inside the bracket) and to the output (outside) to obtain a new function y = A f(B(x – C)) + D. The notation is the key to unlocking the geometry.
每一种变换都从一个母函数开始，例如 f(x) = x²、f(x) = √x、f(x) = sin x 或 f(x) = 1/x。我们对输入（括号内）和输出（外部）施加一系列操作，得到一个新函数 y = A f(B(x – C)) + D。这个记法是开启几何意义的钥匙。
Always identify the type of the parent graph before describing transformations. The parameters A, B, C and D control vertical stretch, horizontal stretch, horizontal shift and vertical shift respectively, though the exact order of operations matters.
在描述变换之前，务必先识别母图的类型。参数 A、B、C 和 D 分别控制垂直伸缩、水平伸缩、水平平移和垂直平移，但操作的精确顺序非常重要。
2. Vertical Shifts: y = f(x) + k | 垂直平移：y = f(x) + k
Adding a constant k to the entire function moves the graph vertically. If k > 0, the graph slides up; if k < 0, it moves down. The sign is intuitive and does not invert.
给整个函数加上常数 k 会垂直移动图形。若 k > 0，图形向上平移；若 k < 0，则向下移动。符号是直观的，并不反转。
up, so the vertex shifts from (0, 0) to (0, 3). In function notation, every y-coordinate is increased by k.
例如，f(x) = x² 变换为 g(x) = x² + 3 将抛物线向上提升3个单位，因此顶点从 (0, 0) 移至 (0, 3)。在函数记法中，每个 y 坐标都增加了 k。
Vertical shift: y = f(x) + k
3. Horizontal Shifts: y = f(x – h) | 水平平移：y = f(x – h)
Horizontal shifts are counter‑intuitive. The transformation y = f(x – e left when h < 0. Think of it as the graph chasing the value that makes the bracket zero.
水平平移与直觉相反。变换 y = f(x – h) 当 h > 0 时将图形向右移动 h 个单位，当 h < 0 时向左移动。可以将其理解为图形在追逐使括号为零的那个值。
If f(x) = √x, then g(x) = √x – 2 shifts the root graph 2 units right, starting at (2, 0). The point (0, 0) maps to (2, 0), which is a rightward shift.
若 f(x) = √x，则 g(x) = √x – 2 将平方根图形向右移动2个单位，起点在 (2, 0)。点 (0, 0) 映射到 (2, 0)，这是一次向右平移。
Horizontal shift: y = f(x – h)
4. Reflections: y = –f(x) and y = f(–x) | 反射：y = –f(x) 与 y = f(–x)
A reflection across the x‑axis is produced by multiplying the output by –1: y = –f(x). Every point (x, y) turns into (x, –y). For y = f(–x), the input sign changes, reflecting the graph in the y‑axis.
关于 x 轴的反射是通过将输出乘以 –1 产生的：y = –f(x)。每个点 (x, y) 变为 (x, –y)。而 y = f(–x) 改变了输入的符号，将图形反射于 y 轴。
Consider f(x) = eˣ. Then y = –eˣ pulls the exponential graph below the x‑axis, while y = e⁻ˣ turning growth into decay. Note that both reflections preserve shape but flip orientation.
考虑 f(x) = eˣ。那么 y = –eˣ 将指数图形拉到 x 轴下方，而 y = e⁻ˣ 将其水平反射，使增长变为衰减。注意两种反射都保持形状但翻转了朝向。
5. Vertical Stretches: y = a f(x) | 垂直伸缩：y = a f(x)
Multiplying the whole function by a factor a (where a ≠ 1) vertically stretches or compresses the graph. If |a| > 1, the graph stretches away from the x‑axis; if 0 < |a| < 1, it compresses towards the x‑axis. A negative a also reflects in the x‑axis.
将整个函数乘以因子 a（a ≠ 1）会垂直拉伸或压缩图形。若 |a| > 1，图形离 x 轴越远；若 0 < |a| < 1，则向 x 轴压缩。负的 a 还会在 x 轴上产生反射。
d. The transformation y = 0.5 sin x halves the amplitude, so the wave sits closer to the x‑axis.
对于 f(x) = sin x，y = 2 sin x 使每个正弦值翻倍，振幅变为 2。周期保持不变。变换 y = 0.5 sin x 将振幅减半，波形更贴近 x 轴。
Vertical stretch: y = a f(x), scale factor |a| from the x‑axis
6. Horizontal Stretches: y = f(bx) | 水平伸缩：y = f(bx)
Changing the input by a factor b stretches the graph horizontally by a factor of 1/|b|. When |b| > 1, the graph shrinks towards the y‑axis; when 0 < |b| < 1, it expands away from the y‑axis. This is the inverse of what you might expect.
以因子 b 改变输入会以 1/|b| 的比例水平拉伸图形。当 |b| > 1 时，图形向 y 轴收缩；当 0 < |b| < 1 时，则远离 y 轴扩展。这与你可能预料的正好相反。
Take f(x) = x². The graph of y = f(2x) is a narrower parabola— , 4). If b = 1/3, the graph stretches horizontally by a factor of 3, spreading out widely.
以 f(x) = x² 为例。y = f(2x) 的图形是一个更窄的抛物线——水平压缩了 ½ 倍。点 (2, 4) 移到 (1, 4)。若 b = 1/3，图形水平拉伸3倍，向外扩散。
Horizontal stretch: y = f(bx), scale factor 1/|b| from the y‑axis
7. Combining Transformations – The Correct Order | 组合变换 – 正确的顺序
When a function is written as y = a f(b(x – h)) + k, the order of operations follows 'inside first, then outside' , but with careful attention to horizontal scaling and shifting. The recommended sequence is: 1) Horizontal stretch by 1/|b|, 2) Horizontal shift by h, 3) Vertical stretch by |a| (and reflection if a < 0), 4) Vertical shift by k.
当一个函数写成 y = a f(b(x – h)) + k 时，操作的顺序遵循“先内部后外部”，但需小心处理水平缩放和平移。推荐的顺序是：1) 水平伸缩 1/|b|，2) 水平平移 h，3) 垂直伸缩 |a|（若 a < 0 还要反射），4) 垂直平移 k。
For instance, transform f(x) = √x to g(x) = –2 √(0.5(x + 3)) + 4. First, horizontal stretch by factor 2 (since b=0.5), giving √0.5x. Then shift left 3, yielding √(0.5(x+3)). Next, vertical stretch by 2 and reflect in x‑axis: –2√.... Finally, shift up 4 units. Reversing the order distorts the graph.
例如，将 f(x) = √x 变换为 g(x) = –2 √(0.5(x + 3)) + 4。首先，水平拉伸2倍（因为 b=0.5），得到 √0.5x。然后左移3个单位，得到 √(0.5(x+3))。接着，垂直拉伸2倍并在 x 轴上反射：–2√...。最后，上移4个单位。顺序颠倒会导致图形失真。
8. Mapping Points Under Transformations | 变换下点的映射
nt (x, y) on the parent graph moves to a new location under the transformation y = a f(b(x – is a powerful tool for sketching graphs or checking key features like axes intersections.
母图上的任意点 (x, y) 在经过变换 y = a f(b(x – h)) + k 后移动到新位置。像点坐标为 (x/b + h，a y + k)。该公式是绘制草图或检查坐标轴交点等关键特征的强大工具。
Example: The vertex of y = x² is (0,0). After y = –2(x + 4)² + 3, which is of the form a f(b(x – h)) + k with a = –2, b = 1, h = –4, k = 3, the image is (0/1 + (–4), –2·0 + 3) = (–4, 3). The parabola opens downward, stretched, and has vertex at (–4, 3).
示例：y = x² 的顶点是 (0,0)。经过 y = –2(x + 4)² + 3（属于 a f(b(x – h)) + k 形式，其中 a = –2, b = 1, h = –4, k = 3），像点为 (0/1 + (–4), –2·0 + 3) = (–4, 3)。该抛物线开口向下、被拉伸且顶点位于 (–4, 3)。
Image point: (x', y') = (x/b + h, a y + k)
9. Finding the Original Function from a Transformed Graph | 从变换后的图形反求原函数
the parent function or the transformation parameters. Reverse the mapping formula: x = b(x' – h) and y = (y' – k)/a. Then check if the reversed points satisfy a known basic function.
有时候你会面对一条变换后的曲线，并被要求反推母函数或变换参数。只需逆转映射公式：x = b(x' – h) 且 y = (y' – k)/a。然后验证反推后的点是否满足某个已知的基本函数。
(5, 2) and appears to be a cosine curve with amplitude 3 and period π. Using y = A cos(B(x – C)) + D, you can solve for A=3, D = –1 ? he parent cosine max is at (0,1). Image (5,2) = (C, A+D) if period adapted. Carefully back‑substituting yields the equation.
假如一条变换后的图形在 (5, 2) 处有最大值，且看似振幅为3、周期为π的余弦曲线。利用 y = A cos(B(x – C)) + D，可求解 A=3，D 可能为 -1 ？实际上从映射角度思考：母余弦最大值在 (0,1)。像点 (5,2) = (C, A+D) 若周期相应调整。仔细回代就能得出方程。
This reverse process reinforces the mapping logic and is frequently tested in IB paper 2 with technology, or even paper 1 without GDC.
这一逆向过程强化了映射逻辑，在IB试卷2中常借助技术考查，甚至在试卷1中不依赖计算器也可能出现。
10. Applying Transformations to Trigonometric Functions | 应用于三角函数的变换
The general sine or cosine function y = A sin(B(x – C)) + D or y = A cos(B(x – C)) + D directly illustrates all four transformation types. Here, |A| is the amplitude, |B| determines the period as 2π/|B| (or 360°/|B| degrees), C is the horizontal (phase) shift, and D is the vertical shift (principal axis).
一般的正弦或余弦函数 y = A sin(B(x – C)) + D 或 y = A cos(B(x – C)) + D 直接展示了全部四种变换类型。其中 |A| 是振幅，|B| 决定周期 2π/|B|（或以度数为 360°/|B|），C 为水平（相位）平移，D 为垂直平移（主轴线）。
For y = 3 sin(2(x – is centred on y = 1. Always factorise the argument to reveal the correct horizontal shift.
对于 y = 3 sin(2(x – π/4)) + 1，振幅为 3，周期为 π，相位平移为向右 π/4，图形以 y = 1 为中心线。务必对自变量的表达式进行因式提取以揭示正确的水平平移量。
Trigonometric transformation: y = A sin(B(x – C)) + D
11. Graph Sketching – Step-by-Step Strategy | 图形草图 – 分步策略
Begin with the basic shape of the parent graph. Draw a light sketch or imagine key points. Apply the transformations in the correct order: horizontal stretch, horizontal shift, then vertical stretch/reflection, and finally vertical shift. Plot the images of critical points such as intercepts, stationary points and asymptotes.
从母图形的基本形状入手。画出一个轻浅的草图或在脑中构想关键点。按正确顺序应用变换：水平伸缩、水平平移，然后是垂直伸缩/反射，最后是垂直平移。标出截距、驻点和渐近线等关键点的像点。
For rational functions like f(x) = 1/x, the hyperbola's asymptotes translate predictably: under g(x) = a/(x – alter the curvature but not the asymptote positions unless combined with other operations.
对于像 f(x) = 1/x 这样的有理函数，双曲线的渐近线可预测地平移：在 g(x) = a/(x – h) + k 下，垂直渐近线变为 x = h，水平渐近线变为 y = k。伸缩改变曲率但不改变渐近线位置，除非与其他操作组合。
Practise sketching using the mapping method: select 3–5 anchor points, transform them numerically, and then join them preserving the parent shape. This reduces errors and helps when a GDC is not available.
练习使用映射方法绘制草图：选择3–5个锚点，对其进行数值变换，然后保留母图形形状将其连接起来。这能减少错误，在没有图形计算器时也很有帮助。
12. Common Pitfalls and Exam Tips | 常见错误与考试提示
Many students confuse horizontal shift direction, forgetting the 'opposite sign' rule. Remember: y = f(x – tal stretch factor is the reciprocal of b, so y = f(3x) compresses, rather than stretches.
许多学生混淆水平平移方向，忘记了“符号相反”规则。记住：y = f(x – 2) 向右移动而非向左。类似地，水平伸缩因子是 b 的倒数，所以 y = f(3x) 是压缩而非拉伸。
hen the function is not written with factored brackets. Always rewrite to the form a f(b(x – h)) + k by extracting b from the argument. For instance, y = sin(3x – π) must be written as sin(3(x – π/3)) to reveal the true phase shift.
另一个常见错误是先应用垂直变换再应用水平变换，特别是当函数未写成带有分解因式的括号形式时。务必通过从自变量中提取 b 重新写成 a f(b(x – h)) + k 的形式。例如，y = sin(3x – π) 必须写成 sin(3(x – π/3)) 才能显示真正的相位平移。
In IB exams, clearly label key points and asymptotes on your sketches. Show step‑by‑step reasoning if asked to 'Describe the sequence of transformations.' Use precise language: 'horizontal translation 2 units right' rather than 'move right by 2'.
在IB考试中，在草图上清晰标出关键点和渐近线。如果被要求“描述变换序列”，要展示逐步推理。使用精确语言：“水平平移2个单位向右”而不是“向右移2”。
Finally, check your final graph by substituting a simple point from the parent function into the transformed equation. If the image matches your sketch, your order is likely correct.
最后，将母函数中的一个简单点代入变换后的方程进行检验。如果像点与你的草图相符，那么你的操作顺序很可能就是正确的。
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