📚 IB Math AA Exercise 4H: Functions and Transformations - Essential Skills | IB 数学 AA 练习 4H：函数与变换必备技能
Exercise 4H in the IB Mathematics: Analysis and Approaches (AA) course is a pivotal collection of problems designed to deepen your understanding of functions and their transformations. It covers translations, stretches, reflections, inverse functions, and composite functions, requiring both algebraic precision and graphical insight. Mastering this exercise builds the foundation for higher-level calculus and problem-solving skills.
IB 数学分析与方法 (AA) 课程中的练习 4H 是一组关键问题，旨在加深你对函数及其变换的理解。它涵盖了平移、伸缩、反射、反函数以及复合函数，要求你既要掌握代数精度，又要具备图像洞察力。精通此练习将为高等微积分和解题能力奠定基础。
1. Overview of Exercise 4H | 练习 4H 概述
Exercise 4H brings together various function concepts that students have encountered throughout the chapter. You will be asked to sketch graphs, find algebraic expressions, determine domains and ranges, and interpret transformations. The problems often combine two or more skills, making them an excellent benchmark of your comprehension.
练习 4H 将整个章节中遇到的各种函数概念整合在一起。题目要求你绘制图像、求代数表达式、确定定义域和值域，并解释变换。这些问题通常结合了两到三种技能，是检验你理解程度的绝佳标尺。
2. Key Concepts Review | 关键概念回顾
Before tackling the exercise, review the basic function toolkit: linear, quadratic, reciprocal, exponential, logarithmic, and trigonometric functions. Remember that a function must map each input to exactly one output. The vertical line test can quickly tell you whether a graph represents a function.
在动笔之前，请回顾基本函数工具箱：一次函数、二次函数、倒数函数、指数函数、对数函数和三角函数。请记住，函数必须将每个输入对应到唯一输出。竖线检测可以快速判断一个图像是否表示函数。
- Domain: All allowable x-values (inputs).
- Range: All resulting y-values (outputs).
- One-to-one function: Each y-value comes from exactly one x-value, essential for having an inverse.
- 定义域：所有允许的 x 值（输入）。
- 值域：所有得到的 y 值（输出）。
- 一一对应函数：每个 y 值仅由一个 x 值产生，这是存在反函数的必要条件。
3. Vertical and Horizontal Translations | 垂直与水平平移
ally: to the right if h > 0, to the left if h < 0. Combining these gives y = f(x - h) + k.
平移在不改变形状的情况下移动图像。在函数外部加上常数，即 y = f(x) + k，将使图像垂直移动：若 k > 0 则向上，若 k < 0 则向下。在函数内部，y = f(x - h) 将使图像水平移动：若 h > 0 则向右，若 h < 0 则向左。两者结合即为 y = f(x - h) + k。
Vertical translation: y = f(x) + 3 | Horizontal translation: y = f(x - 2)
4. Stretches and Compressions | 伸缩与压缩
compressions) are caused by multiplying the whole function by a constant a: y = a·f(x). If |a| > 1, the graph is stretched vertically away from the x-axis; if 0 < |a| t is compressed. Horizontal stretches involve scaling inside the argument: y = f(bx). When |b| > 1, the graph compresses horizontally; when 0 < |b| < 1, it stretches horizontally.
垂直伸缩（或压缩）是由常数 a 乘以整个函数引起的：y = a·f(x)。若 |a| > 1，图像将沿竖直方向远离 x 轴拉伸；若 0 < |a| < 1，则压缩。水平伸缩涉及对自变量进行缩放：y = f(bx)。当 |b| > 1 时，图像水平压缩；当 0 < |b| < 1 时，图像水平拉伸。
Vertical stretch by factor 2: y = 2f(x) | Horizontal compression by factor ½: y = f(2x)
5. Reflections Across Axes | 关于坐标轴的反射
y = f(-x). These transformations preserve the size but alter the orientation of the graph.
将图像关于 x 轴对称反射可通过取函数的负数来实现：y = -f(x)，这将上下翻转图像。关于 y 轴的反射则是用 -x 替换函数内部的 x：y = f(-x)。这些变换保持图像大小不变，但改变了其方向。
- y = -√x reflects the square root graph across the x-axis.
- y = √-x reflects it across the y-axis (for x ≤ 0).
- y = -√x 将平方根图像关于 x 轴反射。
- y = √-x 将平方根图像关于 y 轴反射（在 x ≤ 0 时）。
6. Finding Inverse Functions | 求反函数
e range of f⁻¹, and vice versa. Graphically, f and f⁻¹ are symmetric about the line y = x.
反函数反转了原函数的映射。要求 f⁻¹(x)，可在方程 y = f(x) 中交换 x 和 y 并解出 y。请记住，只有在所选定义域内 f 是一一对应的，反函数才存在。f 的定义域变为 f⁻¹ 的值域，反之亦然。从图像上看，f 与 f⁻¹ 关于直线 y = x 对称。
Example: f(x) = 2x + 3 → f⁻¹(x) = (x - 3)/2
7. Composite Functions | 复合函数
The composite function (f ∘ f. Always consider the domain restrictions: the input x must be in the domain of g, and g(x) must be in the domain of f. When evaluating, work from the inside out. Composite functions are not generally commutative: f(g(x)) is usually different from g(f(x)).
复合函数 (f ∘ g)(x) = f(g(x)) 先应用 g，再应用 f。务必考虑定义域的限制：输入 x 必须在 g 的定义域内，且 g(x) 必须在 f 的定义域内。求值时要从内向外进行。复合函数通常不满足交换律：f(g(x)) 一般与 g(f(x)) 不同。
- If f(x) = √x and g(x) = 2x - 1, then (f ∘ g)(x) = √2x - 1, domain x ≥ 0.5.
- 若 f(x) = √x，g(x) = 2x - 1，则 (f ∘ g)(x) = √2x - 1，定义域为 x ≥ 0.5。
8. Solving Equations Graphically | 利用图像解方程
finding intersection points. This approach links algebraic solutions to visual understanding. Be precise with key features: intercepts, asymptotes, and turning points. Technology may be used, but you must still interpret the results correctly.
练习 4H 常要求你通过绘制两个图像并寻找交点来解方程 f(x) = g(x)。这种方法将代数解与直观理解联系起来。要精确标示关键特征：截距、渐近线和转折点。你可能会使用技术工具，但仍需正确解读结果。
| Equation | Graphical Approach |
| x² = 2x + 3 | Sketch y = x² and y = 2x + 3; find intersections. |
| sin x = ln x | Plot both curves and approximate solutions. |
9. Importance of Domain and Range | 定义域与值域的重要性
Many marks in Exercise 4H are lost by ignoring domain and range. Always state them explicitly for transformed, inverse, and composite functions. For even-root functions, the radicand must be non-negative; for logarithmic functions, the argument must be positive. These restrictions cascade through transformations, so re-evaluate after each step.
练习 4H 中许多失分源于忽略了定义域和值域。对于变换后的函数、反函数以及复合函数，务必明确写出其定义域和值域。对于偶次根式函数，被开方数必须非负；对于对数函数，真数必须为正。这些限制会在变换中层层传递，因此每一步后都要重新评估。
For y = √x - 4, domain is x ≥ 4, range is y ≥ 0.
10. Typical Mistakes and How to Avoid Them | 典型错误与规避方法
x + 3) moves the graph left, not right. Another is neglecting the order of transformations – es when rewriting. Also, check whether a function is truly one-to-one before finding its inverse; restrict the domain if necessary.
一个常见错误是混淆水平平移：y = f(x + 3) 将图像向左移动，而非向右。另一个错误是忽略变换顺序——重写表达式时应先进行水平平移、伸缩和反射，再进行垂直变换。此外，求反函数前要检查函数是否确实是一一对应的，必要时需限制定义域。
- Mistake: Thinking f(bx) with b=2 stretches the graph horizontally.
- Correction: It compresses horizontally by a factor of 1/2.
- 错误：认为 b = 2 时的 f(bx) 会拉伸图像。
- 纠正：它会使图像水平压缩为原来的 1/2。
11. Worked Example: Combining Transformations | 例题：组合变换
The sequence is: horizontal compression by factor 1/3, shift left by 2 units, vertical stretch by factor 2, reflection in the x-axis, and shift up by 1 unit. Always apply horizontal adjustments before vertical ones.
问题：已知 y = f(x) 的图像。绘制 y = -2f(3x + 6) + 1，并依次描述变换。首先将自变量因式分解：3x + 6 = 3(x + 2)。变换顺序为：水平压缩为原图的 1/3、左移 2 个单位、垂直拉伸为 2 倍、关于 x 轴反射、上移 1 个单位。始终先进行水平调整，再作垂直调整。
y = -2f(3(x + 2)) + 1
12. Summary and Final Tips | 总结与备考建议
Exercise 4H is a synthesis task: success comes from clear systematic thinking, thorough notation, and constant awareness of domain and range. Practice identifying transformations in both function notation and graphical sketches. Use checkpoints: after each transformation step, verify a key point’s new coordinates to catch errors early.
练习 4H 是一项综合任务：成功来自于清晰有条理的思维、完整的符号书写，以及对定义域和值域的持续关注。练习从函数表达式和图像草图中识别变换。设置检查点：每次变换步骤之后，验证一个关键点的新坐标，以及早发现错误。
- Revise basic parent functions and their shapes.
- Always write the final function and state its domain and range.
- Double-check the order of transformations with a test point.
- 复习基本母函数及其形状。
- 务必写出最终函数并标明其定义域和值域。
- 用检测点再次核对变换顺序。
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