📚 Exercise 4C.1: Transformations of Functions | 函数变换练习4C.1
In IB Mathematics, Exercise 4C.1 focuses on the fundamental transformations of function graphs – translations, reflections, stretches, and compressions. Mastering these allows you to quickly sketch and analyse functions, a skill that underpins much of the analysis and approaches syllabus. This article provides a clear, section-by-section breakdown of the key concepts, common pitfalls, and step-by-step examples that will help you work through Exercise 4C.1 with confidence.
在 IB 数学中，练习 4C.1 聚焦于函数图像的基本变换——平移、反射、拉伸和压缩。掌握这些变换能让你快速绘制和分析函数图像，这也是分析与方法和应用与解释课程的重要基础。本文将逐节梳理关键概念、常见易错点，并配有分步示例，帮助你自信地完成练习 4C.1。
1. Understanding the Basic Function | 认识基本函数
bout the parent function y = f(x). Common parent functions in IB include f(x) = x², f(x) = √x, f(x) = 1/x, f(x) = |x| ons like sin x, cos x. Recognising their basic shape, domain, and key points is essential.
在应用任何变换之前，你必须清楚地知道父函数 y = f(x) 。IB 中常见的父函数包括 f(x) = x² 、 f(x) = √x 、 f(x) = 1/x 、 f(x) = |x| 、 f(x) = x³ 以及 sin x 、 cos x 等三角函数。认识它们的基本形状、定义域和关键点是至关重要的。
f(x) = x² is a parabola with vertex (0,0) and axis of symmetry x = 0. The graph of f(x) = √x h (0,0) and (1,1). When a question asks to transform such a graph, you adjust these reference points.
例如， f(x) = x² 的图像是一条以 (0,0) 为顶点、对称轴为 x = 0 的抛物线。 f(x) = √x 的图像仅在 x ≥ 0 时有定义，且经过 (0,0) 和 (1,1) 。当题目要求变换此类图像时，你需要调整这些参考点。
2. Vertical Translations | 垂直平移
f(x) + d moves the graph d units vertically: if d > 0, the graph shifts upwards; if d < 0, it shifts downwards. Every y-coordinate is increased by d.
垂直平移使图像向上或向下移动。变换 y = f(x) + d 将图像垂直移动 d 个单位：若 d > 0 ，图像上移；若 d < 0 ，图像下移。每个点的 y 坐标都增加 d 。
Then y = x² + 3 shifts the parabola up by 3 units, making the vertex (0,3). Similarly, y = √x − 2 shifts the square root graph down by 2 units, so the point (0,0) becomes (0,−2). This does not affect the shape or the x-coordinates.
以 f(x) = x² 为例， y = x² + 3 将抛物线上移 3 个单位，顶点变为 (0,3) 。类似地， y = √x − 2 将平方根图像下移 2 个单位，点 (0,0) 变为 (0,−2) 。这不会影响图像的形状和点的 x 坐标。
3. Horizontal Translations | 水平平移
f(x − c) shifts the graph c units to the right if c > 0, and c units to the left if c < 0.
水平平移稍显棘手，因为图像向左或向右移动的方向与函数内部符号相反。变换 y = f(x − c) 在 c > 0 时将图像向右平移 c 个单位，若 c < 0 则向左平移 c 个单位。
For instance, y = |x − 4| the graph shifts 2 units to the left. Always rewrite the bracket to see the shift clearly.
例如， y = |x − 4| 将 V 形的绝对值图像向右平移 4 个单位，顶点落在 (4,0) 。对于 y = (x + 2)³ ，由于 x + 2 可写作 x − (−2) ，图像向左平移 2 个单位。始终将括号改写为 x − c 的形式便能看清平移方向。
4. Reflections in the Axes | 关于坐标轴的反射
n flips the graph across an axis. The transformation y = −f(x) reflects the graph in the x‑ the axis go below and vice versa. The transformation y = f(−x) reflects the graph in the y‑axis, swapping left and right.
反射将图像以坐标轴为镜面翻转。变换 y = −f(x) 使图像关于 x 轴对称：轴上方的点翻到下方，反之亦然。变换 y = f(−x) 使图像关于 y 轴对称，左右互换。
For example, if f(x) = √x, then y = −√x reflects the square root curve across the x‑ ) becomes (1,−1). For f(x) = x³, the graph y = (−x)³ reflects the cubic curve across the y‑axis, although odd functions like x³ already show symmetry.
例如，若 f(x) = √x ，则 y = −√x 将平方根曲线关于 x 轴对称翻转，点 (1,1) 变为 (1,−1) 。对于 f(x) = x³ ， y = (−x)³ 将三次曲线关于 y 轴对称，但由于 x³ 本身是奇函数，图像变化不大。
5. Vertical Stretches and Compressions | 垂直拉伸与压缩
Vertical scaling changes the y‑values by a factor a. The transformation y = a f(x) stretches the graph vertically if |a| > 1, and compresses it vertically if 0 < |a| < 1. When a is negative, a reflection in the x‑axis is combined with the scaling.
垂直伸缩按因子 a 改变 y 值。变换 y = a f(x) 在 |a| > 1 时将图像垂直拉伸，在 0 < |a| < 1 时垂直压缩。当 a 为负时，垂直缩放的同时还进行关于 x 轴的反射。
ng it narrower, with the point (1,1) moving to (1,3). The vertex stays at (0,0). For y = ½√x, the square root graph is compressed vertically, so (4,2) becomes (4,1). This affects only y‑coordinates.
对于 f(x) = x² ， y = 3x² 将抛物线拉伸，使其更陡，点 (1,1) 移动到 (1,3) ，顶点仍在 (0,0) 。对于 y = ½√x ，平方根图像被垂直压缩，点 (4,2) 变为 (4,1) 。这仅改变 y 坐标。
6. Horizontal Stretches and Compressions | 水平拉伸与压缩
the function: y = f(bx). Unlike vertical scaling, the effect on the graph is reversed: if |b| > 1, the graph is compressed horizontally by a factor of 1/|b|; if 0 < |b| < 1, the graph is stretched horizontally by 1/|b|.
水平伸缩涉及函数内部的因子 b ： y = f(bx) 。与垂直伸缩不同，对图像的作用是反过来的：若 |b| > 1 ，图像在水平方向以 1/|b| 的倍数压缩；若 0 < |b| < 1 ，则以 1/|b| 的倍数水平拉伸。
Take f(x) = √x. The graph of y = √2x tched horizontally by a factor of 2, doubling its period. Always isolate b inside the function.
以 f(x) = √x 为例， y = √2x 水平压缩为原来的 ½ ，点 (4,2) 移到 (2,2) 。对于 y = sin(½x) ，正弦波水平拉伸 2 倍，周期翻倍。务必先将 b 单独提取在函数内部。
7. Combined Transformations | 组合变换
rm is y = a f(b(x − c)) + d. This includes horizontal shift by c, horizontal scaling by 1/|b|, vertical scaling by |a|, possible reflections, and vertical shift by d.
练习 4C.1 中的多数 IB 题目要求对图像应用多种变换。常见形式为 y = a f(b(x − c)) + d ，其中包含水平平移 c 、水平伸缩 1/|b| 、垂直伸缩 |a| 、可能的反射以及垂直平移 d 。
ce, y = −2(x + 3)² + 4 can be seen as f(x) = x² transformed by: shift left 3, reflect in x‑ tor 2, then shift up 4. Breaking transformations down step by step is the key to accuracy.
例如， y = −2(x + 3)² + 4 可看作 f(x) = x² 依次经过：左移 3、关于 x 轴反射并垂直拉伸 2 倍，最后上移 4。将变换逐步分解是确保准确的关键。
8. Order of Transformations | 变换的顺序
When multiple transformations are present, the order matters for horizontal changes. A safe sequence is: (1) horizontal scaling (b factor) first, (2) then horizontal translation (c), (3) then vertical scaling and reflection (a), and (4) finally vertical translation (d).
当存在多种变换时，水平变换的顺序至关重要。一个稳妥的步骤是：(1) 先进行水平伸缩（因子 b ），(2) 再进行水平平移（ c ），(3) 然后垂直伸缩与反射（ a ），(4) 最后进行垂直平移（ d ）。
) + 1, the correct order is: compress horizontally by factor ½, shift right by π/4, stretch vertically by 3, and finally shift up by 1. Reversing horizontal steps often leads to a wrong graph.
对于函数 y = 3 sin(2(x − π/4)) + 1 ，正确顺序为：水平压缩 ½ 倍，右移 π/4 ，垂直拉伸 3 倍，最后上移 1 。若颠倒水平步骤，很容易画错图像。
9. Typical Exam-Style Questions | 典型考题举例
Exam questions often ask: "Describe the sequence of transformations that maps the graph of y = f(x) onto y = g(x)." asked to sketch the graph, or to find the image of a specific point after transformations.
考试常问：“描述将 y = f(x) 的图像映射到 y = g(x) 的变换序列。” 或者，给出变换后的方程要求绘制草图，或求出某个特定点在变换后的像。
For example, "The graph of y = 1/x is translated 2 units right and 3 units down, then reflected in the x‑axis. Write the resulting equation." e answer is y = −1/(x − 2) − 3. Practice rewriting the transformations in the given order.
例如，“将 y = 1/x 的图像右移 2 个单位、下移 3 个单位，然后关于 x 轴反射，写出所得方程。” 答案是 y = −1/(x − 2) − 3 。要多练习按给定顺序写出变换。
10. Common Pitfalls and Tips | 常见错误与提示
ph to the right, not left. Always think in terms of what value of x makes the inside zero.
错误一：混淆水平平移方向。牢记 y = f(x − 3) 将图像向右移，而非向左。始终思考 x 取何值能使括号内为零。
f(2x − 6) as y = f(2(x − 3)) to see that compression happens first, then shift right by 3.
错误二：组合水平伸缩与平移时遗漏括号。将 y = f(2x − 6) 改写为 y = f(2(x − 3)) ，便可看出先压缩，再右移 3。
Pitfall 3: Applying vertical transformations before their turn. The coefficient a only stretches y‑values; the constant d is added at the very end. Sketch intermediate graphs step by step to avoid mistakes.
错误三：在不当的步骤提前进行垂直变换。系数 a 只对 y 值进行缩放；常数 d 要到最后才加。分步绘制中间图像可避免错误。
11. Step-by-Step Worked Example | 分步详解例题
Let's work through a complete example: Starting from f(x) = √x, apply these transformations in order: reflection in the y‑axis, horizontal compression by factor ½, shift up by 4, vertical compression by factor ⅓. Write the final equation and sketch key points.
我们来完整演练一个例题：从 f(x) = √x 开始，依次进行：关于 y 轴的反射、水平压缩 ½ 倍、上移 4、垂直压缩 ⅓ 倍。写出最终方程并标出关键点。
Step 1: Reflect in y‑axis → y = √−x y 2x inside the square root, but careful with the negative: y = √(−(2x)) simplifies to y = √−2x. Step 3: Shift up 4 units → y = √−2x 4. Step 4: Vertical compression by ⅓ multiplies the whole function output by ⅓, so y = ⅓[√−2x + 4] or written as y = (1/3)√−2x + 4/3. Note the constant is also affected.
第 1 步：关于 y 轴反射 → y = √−x 。第 2 步：水平压缩 ½ 倍，需将 x 替换为 2x ，但注意负号： y = √(−(2x)) 化简为 y = √−2x 。第 3 步：上移 4 个单位 → y = √−2x + 4 。第 4 步：垂直压缩 ⅓ 倍，即整个函数输出乘以 ⅓ ，得 y = ⅓[√−2x + 4] 也即 y = (1/3)√−2x + 4/3 。注意常数项也受影响。
Original point (4,2) under these transformations: (−4,2) after y‑ after vertical compression, because 6 × ⅓ = 2. Plotting such points helps check your work.
原有点 (4,2) 经这些变换后： y 轴反射后为 (−4,2) ，水平压缩后为 (−2,2) ，上移后为 (−2,6) ，垂直压缩后为 (−2,2) ，因为 6 × ⅓ = 2 。绘制这些点有助于检验结果。
12. Summary and Practice Advice | 小结与练习建议
To master Exercise 4C.1, always identify the parent function, separate the transformations inside and outside the brackets, and respect the order: horizontal scaling, horizontal shift, vertical scaling/reflection, vertical shift. Sketching each intermediate stage on the same axes builds intuition.
要掌握练习 4C.1 ，务必先识别父函数，区分括号内外的变换，并遵循顺序：水平伸缩、水平平移、垂直伸缩/反射、垂直平移。在同一坐标系中绘制每个中间阶段有助于培养直觉。
Re‑attempt the exercise after reading this guide, and check each answer by testing a few key points. If a point does not map correctly, backtrack and verify the order. Consistent practice with functions like quadratics, radicals, reciprocals, and trigonometric graphs will make transformations second nature.
阅读本文后重新尝试练习，并用几个关键点检验每个答案。如果某点映射不正确，反向检查顺序。持续练习二次函数、根式函数、倒数函数和三角函数的图像变换，你就能很快掌握变换技巧。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图