📚 Transformations of Function Graphs: Translations & Stretches | IB数学：函数图像的平移与伸缩变换
Understanding how function graphs transform under translations and stretches is one of the most essential skills in IB Mathematics. Whether you are studying Analysis & Approaches (AA) or Applications & Interpretation (AI), transformations appear in almost every topic—from quadratic functions to trigonometric graphs, from exponential models to calculus problems involving curve sketching.
理解函数图像在平移和伸缩变换下的变化，是IB数学中最核心的技能之一。无论你学习的是分析与方法（AA）还是应用与解释（AI），变换几乎出现在所有专题中——从二次函数到三角函数图像，从指数模型到涉及曲线绘制的微积分问题。
1. Why Transformations Matter in IB | 为什么变换在IB中如此重要
In the IB syllabus, transformations of graphs are explicitly listed under the Functions topic (SL & HL). You are expected to recognise the effect of parameters on the graph of a function, write down the equation of a transformed graph, and apply transformations to sketch curves quickly and accurately.
在IB教学大纲中，函数图像的变换被明确列在函数专题（标准级别和高级级别）之下。你需要识别参数对函数图像的影响，写出变换后图像对应的方程，并运用变换快速、准确地绘制曲线。
Transformations also appear in exam questions involving composite functions, inverse functions, and modelling. A solid grasp of this topic will save you significant time in Paper 1 (no calculator) and reduce errors in Paper 2.
变换还会出现在涉及复合函数、反函数和数学建模的考试题目中。扎实掌握这一专题，将帮助你在Paper 1（不允许使用计算器）中节省大量时间，并在Paper 2中减少错误。
2. The Four Basic Transformations | 四种基本变换
gories: translations (shifts) and stretches (dilations). Reflections can be viewed as special cases of stretches with a scale factor of −1.
有四种基本变换可以应用于任何函数 y = f(x)。它们分为两类：平移（位移）和伸缩（缩放）。反射可以视为比例因子为 −1 的特殊伸缩情况。
- Vertical translation: y = f(x) + k — shifts the graph up or down.
- Horizontal translation: y = f(x − h) — shifts the graph left or right.
- Vertical stretch: y = a·f(x) — stretches or compresses the graph vertically.
- Horizontal stretch: y = f(b·x) — stretches or compresses the graph horizontally.
关键规则：垂直变换作用于函数值（在括号外），水平变换作用于自变量（在括号内）。
3. Vertical Translation: y = f(x) + k | 垂直平移：y = f(x) + k
When we add a constant k to a function, every point on the graph moves vertically by k units. If k > 0, the graph shifts upward; if k < 0, the graph shifts downward. The shape of the graph is completely unchanged—only its position along the y-axis changes.
当我们给函数加上一个常数 k 时，图像上的每一个点都会垂直移动 k 个单位。如果 k > 0，图像向上平移；如果 k < 0，图像向下平移。图像的形状完全不变——只有它在y轴上的位置发生变化。
rtex moves from (0, 0) to (0, 3). Similarly, y = x² − 2 shifts the vertex down to (0, −2).
例如，考虑 f(x) = x²。y = x² + 3 的图像是同样的抛物线，但向上平移了3个单位。其顶点从 (0, 0) 移动到 (0, 3)。类似地，y = x² − 2 将顶点向下移动到 (0, −2)。
| Original function | Transformed function | Effect |
| f(x) = sin x | y = sin x + 1 | Shift up by 1 |
| f(x) = 2ˣ | y = 2ˣ − 4 | Shift down by 4 |
4. Horizontal Translation: y = f(x − h) | 水平平移：y = f(x − h)
the graph of y = f(x) horizontally by h units. Crucially: if h > 0, the graph moves to the right; if h < 0, the graph moves to the left. This is the opposite of what many students initially expect.
水平平移常常是困惑的根源。y = f(x − h) 的图像将 y = f(x) 的图像水平移动 h 个单位。关键点：如果 h > 0，图像向右移动；如果 h < 0，图像向左移动。这与许多学生最初的直觉相反。
Why does this happen? For the graph to "look the same" maximum when x − h = 0, i.e. when x = h. Thus the graph appears shifted to the right by h.
为什么会这样？为了使图像在新x值处"看起来相同"，输入 x − h 必须产生与之前 x 相同的输出。因此，如果原始图像在 x = 0 处达到最大值，平移后的图像将在 x − h = 0 时达到最大值，即 x = h 处。因此图像看起来向右平移了 h 个单位。
y = f(x − h): shift right by h (if h > 0) | y = f(x − h)：当 h > 0 时向右平移 h 个单位
r hand, y = (x + 3)² = (x − (−3))² is shifted 3 units to the left, with vertex at (−3, 0).
示例：y = (x − 2)² 是将 y = x² 向右平移2个单位的抛物线。其顶点在 (2, 0)。另一方面，y = (x + 3)² = (x − (−3))² 是向左平移3个单位，顶点在 (−3, 0)。
5. Vertical Stretch: y = a·f(x) | 垂直伸缩：y = a·f(x)
Multiplying the entire function by a constant a produces a vertical stretch or compression. If |a| > 1, the graph stretches vertically—points move away from the x-axis. If 0 < |a| < 1, the graph compresses vertically—points move toward the x-axis. If a is negative, the graph also reflects across the x-axis.
将整个函数乘以常数 a 会产生垂直伸缩或压缩。如果 |a| > 1，图像垂直拉伸——点远离x轴。如果 0 < |a| < 1，图像垂直压缩——点靠近x轴。如果 a 为负，图像还会关于x轴反射。
Important: points on the x-axis (where f(x) = 0) do not move. These are called invariant points. All other points have their y-coordinates multiplied by a.
重要：x轴上的点（即 f(x) = 0 处）不移动。这些点称为不变点。所有其他点的y坐标都乘以 a。
h y = 3, and the troughs descend to y = −3. The x-intercepts remain at the same positions.
例如，y = 3 sin x 的振幅从1变为3。原本在 y = 1 处的峰值现在达到 y = 3，波谷下降到 y = −3。x轴截距保持在相同位置。
| Value of a | Effect on y = f(x) |
| a > 1 | Vertical stretch (points move away from x-axis) |
| 0 < a < 1 | Vertical compression (points move toward x-axis) |
| a < 0 | Vertical stretch/compression + reflection across x-axis |
6. Horizontal Stretch: y = f(b·x) | 水平伸缩：y = f(b·x)
involves multiplying the input by b. This produces a horizontal stretch or compression. If |b| > 1, the graph compresses horizontally (toward the y-axis). If 0 < |b| ph stretches horizontally (away from the y-axis). The scale factor is actually 1/b, not b.
变换 y = f(b·x) 涉及将自变量乘以 b。这会产生水平伸缩或压缩。如果 |b| > 1，图像水平压缩（向y轴靠近）。如果 0 < |b| < 1，图像水平拉伸（远离y轴）。实际比例因子是 1/b，而不是 b。
very point on the original graph moves to half its original x-coordinate, because the input is being "sped up" by a factor of 2. The graph appears squashed toward the y-axis.
这是最违反直觉的变换。考虑 y = f(2x)：原始图像上的每个点都移动到原来x坐标的一半，因为输入被"加速"了2倍。图像看起来被压缩向y轴。
y = f(b·x): horizontal scale factor = 1/b | y = f(b·x)：水平比例因子 = 1/b
of 2π. The period has been halved. Similarly, y = sin(x/2) = sin(0.5x) has a period of 4π—the graph stretches horizontally.
示例：y = sin(2x) 在一个长度为 π 的区间内完成一个完整周期。周期被减半。类似地，y = sin(x/2) = sin(0.5x) 的周期为 4π——图像水平拉伸。
7. Combined Transformations: Order Matters | 组合变换：顺序至关重要
When multiple transformations are applied simultaneously, the order of operations must follow the same logic as the order of operations in arithmetic. The general form is:
当同时应用多个变换时，运算顺序必须遵循算术中运算顺序的逻辑。一般形式为：
y = a·f(b(x − h)) + k
This expression combines four transformations in a specific order:
该表达式以特定顺序组合了四种变换：
- Step 1: Horizontal translation — x is replaced by (x − h).
- Step 2: Horizontal stretch — x is multiplied by b.
- Step 3: Vertical stretch — f(x) is multiplied by a.
- Step 4: Vertical translation — k is added.
步骤1：水平平移——x 被替换为 (x − h)。步骤2：水平伸缩——x 乘以 b。步骤3：垂直伸缩——f(x) 乘以 a。步骤4：垂直平移——加上 k。
ions in the wrong order. The key principle is that for horizontal transformations, we work "inside out" 2)). This reveals a horizontal stretch by factor 1/2 followed by a shift right by 2 units.
一个常见错误是错误地应用水平变换的顺序。关键原则是，对于水平变换，我们"由内向外"处理：先处理 (x − h)，再乘以 b。换句话说，先提取x的系数。例如，给定 y = f(2x − 4)，将其重写为 y = f(2(x − 2))。这揭示了先进行1/2比例因子的水平压缩，然后向右平移2个单位。
8. Reflections as Special Cases | 反射作为特殊情形
Reflections are simply stretches with a scale factor of −1. They can be classified into two types:
反射只是比例因子为 −1 的伸缩。它们可以分为两类：
- Reflection across the x-axis: ordinate changes sign. The graph is flipped vertically. The x-intercepts remain invariant.
- Reflection across the y-axis: dinate changes sign. The graph is flipped horizontally. The y-intercept remains invariant.
关于x轴的反射：y = −f(x)。每个y坐标变号。图像垂直翻转。x轴截距保持不变。关于y轴的反射：y = f(−x)。每个x坐标变号。图像水平翻转。y轴截距保持不变。
oss the x-axis (they produce the same result because odd functions satisfy f(−x) = −f(x)).
对于偶函数，如 f(x) = x² 或 f(x) = cos x，关于y轴的反射使图像保持不变。对于奇函数，如 f(x) = x³ 或 f(x) = sin x，关于y轴的反射等同于关于x轴的反射（它们产生相同的结果，因为奇函数满足 f(−x) = −f(x)）。
9. Working from a Graph to Find the Equation | 从图像反求方程
IB exam questions often give you a graph and ask you to find the equation. The systematic approach is to identify the parent function first, then compare key features of the given graph with the parent graph to determine the transformations.
IB考试题目通常给出图像并要求你求出方程。系统的方法是先识别母函数，然后比较给定图像与母函数图像的关键特征，以确定变换。
he narrower shape suggests a vertical stretch. So the equation might be y = 2(x − 2)² − 3.
例如，假设你看到一个顶点在 (2, −3)、开口向上、且比 y = x² 更窄的抛物线。顶点告诉你存在向右平移2个单位和向下平移3个单位的变换。更窄的形状表明存在垂直拉伸。因此方程可能是 y = 2(x − 2)² − 3。
Key features to check when identifying transformations from a graph:
从图像识别变换时需要检查的关键特征：
- Vertex / turning point — gives both horizontal and vertical translations.
- Asymptotes — give vertical and horizontal translations for rational and exponential functions.
- Amplitude and period — give vertical and horizontal stretches for trigonometric functions.
- Intercepts — help verify the parameters after a tentative equation is written.
顶点/转向点——给出水平和平移变换。渐近线——为有理函数和指数函数给出平移变换。振幅和周期——为三角函数给出伸缩变换。截距——在写出初步方程后帮助验证参数。
10. Invariant Points & Key Coordinates | 不变点与关键坐标
Tracking specific points through transformations is a reliable strategy for sketching transformed graphs. The five key points of a sine graph (maximum, minimum, and three intercepts) or the vertex and intercepts of a parabola can be transformed individually and then connected smoothly.
通过变换追踪特定点是绘制变换后图像的可靠策略。正弦图像的五个关键点（最大值、最小值和三个截距）或抛物线的顶点和截距可以逐个变换，然后平滑连接。
g a point (x, y) on y = f(x) to a point on y = a·f(b(x − h)) + k, the new coordinates are:
当将 y = f(x) 上的点 (x, y) 变换为 y = a·f(b(x − h)) + k 上的点时，新坐标为：
(x, y) → (x/b + h, a·y + k)
(1, 2) on f(x) transformed by y = 3f(2(x − 1)) + 4 becomes (1/2 + 1, 3×2 + 4) = (1.5, 10).
这个公式在考试中非常有用。例如，f(x) 上的点 (1, 2) 经过 y = 3f(2(x − 1)) + 4 变换后变为 (1/2 + 1, 3×2 + 4) = (1.5, 10)。
invariant. Under y = f(b·x), the points where x = 0 (i.e., the y-intercept) are invariant.
不变点是那些在变换下不移动的点。在 y = f(x) + k 下，除非 k = 0，否则没有不变点。在 y = a·f(x) 下，f(x) = 0 的点是不变点。在 y = f(b·x) 下，x = 0 的点（即y轴截距）是不变点。
11. Common IB Exam Mistakes & How to Avoid Them | IB常见考试错误及如何避免
Many students lose marks on transformation questions for preventable reasons. Here are the most common pitfalls:
许多学生因可避免的原因在变换题目上失分。以下是最常见的陷阱：
- Confusing horizontal shift direction: member y = f(x − 2) shifts right, not left. Replace x with (x − h) and the shift is always "h units in the positive direction of the axis."
- Forgetting to factor b first: y = f(2x − 6) is NOT a shift of 6 units right. Factor to get f(2(x − 3)) — a compression by 1/2 followed by a shift right of 3 units.
- Mixing up the order of transformations: Apply horizontal transformations before vertical ones; within horizontal, do translation after stretching; within vertical, do stretching before translation.
- Sign errors in the point transformation formula: Double-check whether you are adding h or subtracting h, and whether the vertical stretch factor multiplies before the translation is added.
混淆水平移动方向：记住 y = f(x − 2) 是向右移动，不是向左。将 x 替换为 (x − h)，移动方向总是"沿轴的正方向 h 个单位"。忘记先提取 b：y = f(2x − 6) 不是向右平移6个单位。提取因子得到 f(2(x − 3))——先压缩1/2，再向右平移3个单位。混淆变换顺序：先应用水平变换再应用垂直变换；在水平变换内部，先伸缩后平移；在垂直变换内部，先伸缩后平移。点变换公式中的符号错误：仔细检查是在加 h 还是减 h，以及垂直伸缩因子是否在平移之前相乘。
+ k before attempting any sketch or calculation. This single habit prevents most mistakes.
为了避免这些错误，在尝试任何绘图或计算之前，始终将变换写成标准形式 y = a·f(b(x − h)) + k。这个单一习惯可以防止大多数错误。
12. Practice Questions with Solutions | 练习题与解答
Let us consolidate your understanding with two typical IB-style questions.
让我们通过两道典型的IB风格题目来巩固你的理解。
Practice Question 1: = f(x). Find the coordinates of the corresponding point on the graph of y = 2f(x − 4) + 5.
练习1：点 (3, −2) 在 y = f(x) 的图像上。求该点在 y = 2f(x − 4) + 5 图像上对应点的坐标。
Solution: Using the point transformation rule (x, y) → (x + h, a·y + k), with h = 4, a = 2, k = 5:
解答：使用点变换规则 (x, y) → (x + h, a·y + k)，其中 h = 4，a = 2，k = 5：
(3, −2) → (3 + 4, 2×(−2) + 5) = (7, 1)
The corresponding point is (7, 1).
对应点为 (7, 1)。
Practice Question 2: y = f(x) has a vertical asymptote at x = 1. Find the vertical asymptote of y = f(2x − 6).
练习2：y = f(x) 的图像在 x = 1 处有一条垂直渐近线。求 y = f(2x − 6) 的垂直渐近线。
Solution: equals the original asymptote value. We need 2(x − 3) = 1, so x − 3 = 0.5, giving x = 3.5.
解答：首先将 f(2x − 6) 重写为 f(2(x − 3))。当自变量等于原始渐近线值时，渐近线出现。我们需要 2(x − 3) = 1，因此 x − 3 = 0.5，得到 x = 3.5。
2(x − 3) = 1 → x = 3.5
The vertical asymptote of the transformed graph is x = 3.5.
变换后图像的垂直渐近线为 x = 3.5。
Mastering transformations is not just about memorising rules—it is about understanding the relationship between algebraic expressions and geometric behaviour. Once you internalise the logic of "inside the function affects x, outside affects y," and "opposite direction for horizontal, same direction for vertical," you will find transformation questions straightforward and even enjoyable. Practise by sketching the same graph under multiple transformations, and always verify your results by testing one or two key points.
掌握变换不仅仅是记忆规则——而是理解代数表达式与几何行为之间的关系。一旦你内化了"函数内部影响x，函数外部影响y"以及"水平方向反方向，垂直方向同方向"的逻辑，你会发现变换题目变得直接甚至有趣。通过在同一图像上应用多种变换来练习绘图，并始终通过测试一两个关键点来验证你的结果。
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