📚 Exercise 2I: Transformations of Functions | 练习2I：函数变换
In IB Mathematics, Exercise 2I typically focuses on the transformations of functions — a foundational topic that enables students to manipulate graphs and understand the relationship between algebraic expressions and their geometric representations. This exercise covers translations, stretches, reflections, and combinations of these, providing essential skills for both Analysis & Approaches (AA) and Applications & Interpretation (AI) courses. Mastering function transformations is not only key for graph sketching but also for modelling real-world scenarios, solving equations graphically, and understanding inverse and composite functions later in the curriculum.
在IB数学中，练习2I通常围绕函数变换展开——这是一个基础主题，使学生能够操作图形并理解代数表达式与其几何表示之间的关系。该练习涵盖平移、伸缩、反射及其组合，为分析与方法(AA)和应用与解释(AI)课程提供必要技能。掌握函数变换不仅是图形绘制的关键，对于后期课程中的现实情境建模、方程图解求解以及理解反函数与复合函数也至关重要。
1. Understanding Function Transformations | 理解函数变换
A transformation of a function modifies its graph in the coordinate plane. Starting from a parent function f(x), we can apply a series of rules to obtain a new function g(x). The fundamental transformations are vertical shifts, horizontal shifts, vertical stretches/compressions, horizontal stretches/compressions, and reflections. Each transformation can be identified by the structural change in the function's equation.
函数的变换会修改其在坐标平面上的图形。从父函数 f(x) 出发，我们可以应用一系列规则得到新函数 g(x)。基本的变换包括垂直平移、水平平移、垂直伸缩、水平伸缩以及反射。每种变换都可以通过函数方程式的结构变化来识别。
nslation upward by 3 units. Recognizing these patterns allows us to sketch complex graphs quickly without plotting numerous points.
例如，如果 f(x) = x²，那么 g(x) = x² + 3 表示向上垂直平移 3 个单位。识别这些模式使我们能够快速绘制复杂图形，而无需描出大量点。
2. Vertical Translations | 垂直平移
(x) + k. If k > 0, the graph moves up; if k < 0, it moves down. This transformation affects the y-coordinates directly, adding k to each output value. The shape of the graph remains unchanged.
垂直平移将图形向上或向下移动。一般形式为 y = f(x) + k。若 k > 0，图形上移；若 k < 0，图形下移。该变换直接影响 y 坐标，给每个输出值加上 k。图形形状保持不变。
hifts the entire sine wave 2 units upward. All maximum and minimum points, intercepts, and the midline are raised accordingly. This is a common manipulation in trigonometric modelling.
考虑 f(x) = sin x。那么 y = sin x + 2 将整个正弦波向上移动 2 个单位。所有最大点、最小点、截距和中线都相应抬高。这是三角函数建模中的常见操作。
3. Horizontal Translations | 水平平移
- h). Note the sign reversal: if h > 0, the graph shifts to the right; if h < 0, it shifts to the left. This is because replacing x by x - h means that to achieve the same output as f at x, we need a larger x when h > 0, effectively moving the graph to the right.
水平平移将图形向左或向右移动，形式为 y = f(x - h)。注意符号反转：若 h > 0，图形向右移动；若 h < 0，向左移动。这是因为用 x - h 替换 x，意味着要达到与 f 在 x 处相同的输出，当 h > 0 时需要更大的 x，从而将图形向右移动。
For instance, y = |x - 3| ). Many students initially confuse the direction, so memorise: f(x - h) shifts right by h.
例如，y = |x - 3| 将绝对值图形向右平移 3 个单位，顶点从 (0,0) 移到 (3,0)。许多学生最初会混淆方向，所以记住：f(x - h) 向右移 h。
4. Vertical Stretches and Compressions | 垂直伸缩
Vertical scaling multiplies the function output: y = a·f(x). If |a| > 1, the graph stretches vertically, making it taller; if 0 < |a| laborate later. The x-intercepts remain fixed because f(x) = 0 multiplied by a is still 0.
垂直缩放将函数输出相乘：y = a·f(x)。若 |a| > 1，图形垂直拉伸，变得更高；若 0 < |a| < 1，则垂直压缩。当 a 为负值时，还会发生关于 x 轴的反射，我们稍后再详述。x 截距保持不变，因为 f(x) = 0 乘以 a 仍为 0。
ider, compressing points towards the x-axis. Understanding this helps in quadratic modelling and parabola analysis.
以 f(x) = x² 为例，y = 3x² 的图形更窄，各点远离 x 轴拉伸。相反，y = 0.5x² 更宽，各点向 x 轴压缩。理解这一点有助于二次函数建模和抛物线分析。
5. Horizontal Stretches and Compressions | 水平伸缩
Horizontal scaling is given by y = f(b·x). If |b| > 1, the graph compresses horizontally towards the y-axis; if 0 < |b| he function squashes the graph. The y-intercept remains unchanged since f(0) is invariant.
水平缩放的形式为 y = f(b·x)。若 |b| > 1，图形向 y 轴水平压缩；若 0 < |b| < 1，则远离 y 轴拉伸。这具有反直觉性：函数内部乘以一个大的因子会挤压图形。y 截距保持不变，因为 f(0) 不受影响。
Example: y = sin(2x) compresses the sine wave, doubling its frequency — cy, and the period becomes 4π. This is crucial for trigonometric function transformations.
示例：y = sin(2x) 压缩正弦波，频率加倍——周期变为 π 而非 2π。相反，y = sin(x/2) 拉伸波浪，频率减半，周期变为 4π。这对于三角函数变换至关重要。
6. Reflections in the Axes | 坐标轴反射
, flipping x-coordinates. These are special cases of vertical and horizontal scaling with factor -1.
反射将图形相对于坐标轴进行镜像。变换 y = -f(x) 将图形关于 x 轴反射，翻转所有 y 坐标。而 y = f(-x) 关于 y 轴反射，翻转 x 坐标。这些是因子为 -1 的垂直和水平缩放的特殊情况。
for odd functions, f(-x) = -f(x), so reflection in both axes is equivalent to a rotation.
当同时应用两种反射时，顺序无关紧要：y = -f(-x) 是关于原点的 180° 旋转，相当于在两个坐标轴上反射。对于偶函数，f(-x) = f(x)，因此 y 轴反射不会改变图形；对于奇函数，f(-x) = -f(x)，因此两个轴反射等效于旋转。
7. Combined Transformations: Order Matters | 混合变换：顺序很重要
Typically, when a function is written as y = a·f(b(x - h)) + k, the transformations should
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