IB Math: Function Transformations Exam Guide | IB数学：函数变换考点梳理

📚 IB Math: Function Transformations Exam Guide | IB数学：函数变换考点梳理

Function transformations are one of the most frequently tested topics in IB Mathematics, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI), at both Standard Level (SL) and Higher Level (HL). Mastering this topic requires understanding how changes to the algebraic form of a function affect its graph.

函数变换是IB数学中最高频的考点之一，在分析与方法（AA）和应用与解释（AI）两个方向的Standard Level（SL）与Higher Level（HL）中都会出现。掌握这一主题的关键，是理解函数代数形式的变化如何影响其图像。

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1. The Four Core Transformations | 四种核心变换

Every transformation of a function graph can be classified into four types: translation (vertical or horizontal shift), reflection (across the x-axis or y-axis), vertical stretch/compression, and horizontal stretch/compression. IB questions often combine two or more of these in a single function.

函数图像的每一种变换都可以归为四类：平移（垂直或水平移动）、反射（关于x轴或y轴）、垂直拉伸/压缩、水平拉伸/压缩。IB考题经常将两种或多种变换结合在同一个函数中。

• Vertical translation: y = f(x) + k, shift up if k > 0, down if k < 0.
• Horizontal translation: y = f(x – h), shift right if h > 0, left if h < 0.
• Reflection in x-axis: y = -f(x).
• Reflection in y-axis: y = f(-x).

• 垂直平移：y = f(x) + k，k > 0时向上移，k < 0时向下移。
• 水平平移：y = f(x – h)，h > 0时向右移，h < 0时向左移。
• 关于x轴的反射：y = -f(x)。
• 关于y轴的反射：y = f(-x)。

y = a·f(b(x – h)) + k

This is the general form for a transformed function. The parameter a controls vertical stretch/compression and possibly reflection, b controls horizontal stretch/compression, h controls horizontal shift, and k controls vertical shift.

这是变换后函数的一般形式。参数a控制垂直拉伸/压缩并可能产生反射，b控制水平拉伸/压缩，h控制水平平移，k控制垂直平移。

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2. Vertical Transformations: Stretch and Reflection | 垂直变换：拉伸与反射

For y = a·f(x), the graph is stretched vertically by a factor of |a|. If a is negative, the graph is also reflected across the x-axis. Every y-coordinate of the original function is multiplied by a, while x-coordinates remain unchanged.

对于y = a·f(x)，图像在垂直方向拉伸|a|倍。如果a为负数，图像还会关于x轴反射。原函数每个点的纵坐标都乘以a，横坐标保持不变。

Key point: (x, y) → (x, ay)

关键点：(x, y) → (x, ay)

Example: If f(x) = x², then y = 2f(x) = 2x² is twice as tall as y = x² at every point. The vertex (0,0) remains fixed, but the point (1,1) moves to (1,2).

例如：若f(x) = x²，则y = 2f(x) = 2x²在每个点的高度都是y = x²的两倍。顶点(0,0)保持不变，但点(1,1)移动到(1,2)。

• |a| > 1: vertical stretch (graph becomes taller)
• 0 < |a| < 1: vertical compression (graph becomes shorter)
• a < 0: reflection across x-axis plus stretch/compression

• |a| > 1：垂直拉伸（图像变高）
• 0 < |a| < 1：垂直压缩（图像变矮）
• a < 0：关于x轴反射并伴随拉伸/压缩

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3. Horizontal Transformations: The Reversal Rule | 水平变换：反向法则

For y = f(bx), the graph is stretched horizontally by a factor of 1/|b|. Notice the reciprocal relationship: if b = 2, the graph is compressed by a factor of 1/2, not stretched. If b is negative, the graph is reflected across the y-axis.

对于y = f(bx)，图像在水平方向拉伸1/|b|倍。注意倒数关系：若b = 2，图像压缩为原来的1/2，而不是拉伸。若b为负数，图像关于y轴反射。

Key point: (x, y) → (x/b, y)

关键点：(x, y) → (x/b, y)

Example: If f(x) = sin(x), then y = f(2x) = sin(2x) has period π instead of 2π. The horizontal compression factor is 1/2, meaning the graph completes one full cycle in half the distance.

例如：若f(x) = sin(x)，则y = f(2x) = sin(2x)的周期为π而不是2π。水平压缩因子为1/2，意味着图像在原来一半的距离内完成一个完整周期。

Students often confuse horizontal and vertical transformations because horizontal changes feel “reversed”: to shift right, you subtract inside the function; to stretch horizontally by a factor of 2, you multiply x by 1/2 inside the function.

学生经常混淆水平和垂直变换，因为水平变化感觉上是”反向”的：要向右平移，在函数内部做减法；要将图像水平拉伸2倍，在函数内部将x乘以1/2。

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4. Order of Transformations | 变换的顺序

When multiple transformations are applied, the order matters, especially for horizontal transformations. The correct sequence is: first apply horizontal stretch/compression and reflection (the b parameter), then horizontal shift (the h parameter), then vertical stretch/compression and reflection (the a parameter), and finally vertical shift (the k parameter).

当多种变换同时应用时，顺序至关重要，尤其是水平变换。正确顺序是：先进行水平拉伸/压缩和反射（参数b），再进行水平平移（参数h），然后进行垂直拉伸/压缩和反射（参数a），最后进行垂直平移（参数k）。

General form: y = a·f(b(x – h)) + k

一般形式：y = a·f(b(x – h)) + k

Notice that the horizontal shift is written as x – h inside the function, after the b has been factored out. If you see y = f(2x – 4), you must first rewrite it as y = f(2(x – 2)) before identifying the horizontal shift. The shift is 2 units right, not 4 units right.

注意水平平移写在函数内部，写作x – h的形式，并且b已经从括号中提出。如果你看到y = f(2x – 4)，必须先将其改写为y = f(2(x – 2))才能确定水平平移量。平移是向右2个单位，而不是向右4个单位。

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5. Finding the Image of a Point | 求点的像

A common IB exam question gives you a specific point on the original function and asks for its image after a transformation. The method is straightforward: substitute the original coordinates into the transformation rules.

IB考试中一个常见题型是：给出原函数上的一个特定点，要求求其在某种变换后的对应点。方法非常直接：将原坐标代入变换规则中。

Example: The point (2, 3) lies on y = f(x). Find the image of this point on y = 2f(x – 1) + 5.

例题：点(2, 3)在y = f(x)上。求该点在y = 2f(x – 1) + 5上的对应点。

Step 1: The inside transformation is (x – 1), so the x-coordinate satisfies x_original = x_new – 1. Since x_original = 2, we have 2 = x_new – 1, so x_new = 3.

步骤1：内部变换为(x – 1)，因此原横坐标满足x_original = x_new – 1。由于x_original = 2，我们有2 = x_new – 1，所以x_new = 3。

Step 2: The outside transformation is 2f(…) + 5, so y_new = 2(3) + 5 = 11. The image is (3, 11).

步骤2：外部变换为2f(…) + 5，所以y_new = 2(3) + 5 = 11。对应点为(3, 11)。

General rule: (x, y) → (x + h, ay + k) for y = a·f(x – h) + k

一般规则：对于y = a·f(x – h) + k，(x, y) → (x + h, ay + k)

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6. Transformations of Asymptotes | 渐近线的变换

Asymptotes are critical in IB questions involving rational functions and exponential or logarithmic functions. Vertical asymptotes follow horizontal transformations; horizontal asymptotes follow vertical transformations.

渐近线在IB考试中对于有理函数、指数函数和对数函数的问题至关重要。垂直渐近线跟随水平变换；水平渐近线跟随垂直变换。

Example: The function f(x) = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. Under the transformation y = 2/(x – 3) + 1 = 2f(x – 3) + 1, the vertical asymptote moves to x = 3, and the horizontal asymptote moves to y = 1.

例题：函数f(x) = 1/x有一条垂直渐近线x = 0和一条水平渐近线y = 0。在变换y = 2/(x – 3) + 1 = 2f(x – 3) + 1下，垂直渐近线移动到x = 3，水平渐近线移动到y = 1。

For exponential functions of the form y = a·bˣ + k, the horizontal asymptote is y = k. For logarithmic functions y = log_b(x – h) + k, the vertical asymptote is x = h.

对于形如y = a·bˣ + k的指数函数，水平渐近线为y = k。对于形如y = log_b(x – h) + k的对数函数，垂直渐近线为x = h。

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7. Finding the Equation from a Graph | 根据图像求函数方程

IB questions sometimes present a transformed graph and ask you to determine its equation. The strategy is to compare the given graph with a known parent function and identify the transformations step by step.

IB考试有时会给出一个变换后的图像，要求你确定其方程。策略是将给定图像与已知的母函数进行比较，逐步识别出各种变换。

Step 1: Identify the parent function (linear, quadratic, exponential, sinusoidal, etc.).

步骤1：识别母函数类型（线性、二次、指数、正弦等）。

Step 2: Compare a key point (such as the vertex of a parabola, the asymptote of an exponential, or the amplitude/period of a sine curve) between the parent and the transformed graph.

步骤2：比较母函数与变换后图像之间的关键点（如抛物线的顶点、指数函数的渐近线或正弦曲线的振幅/周期）。

Step 3: Use the general form y = a·f(b(x – h)) + k and solve for the parameters.

步骤3：使用一般形式y = a·f(b(x – h)) + k，解出各个参数。

Example: The point (3, -1) is the vertex of a parabola with parent function f(x) = x². Since the parabola opens downward, a < 0. If the graph passes through (4, -3), then -3 = a(4 - 3)² - 1, so -3 = a - 1, giving a = -2. Thus the equation is y = -2(x - 3)² - 1.

例题：某抛物线以(3, -1)为顶点，母函数为f(x) = x²。由于抛物线开口向下，a < 0。若图像经过(4, -3)，则-3 = a(4 - 3)² - 1，所以-3 = a - 1，得a = -2。因此方程为y = -2(x - 3)² - 1。

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8. Composite Transformations in Exam Questions | 考试中的复合变换题

IB exams frequently combine transformations with other topics, such as finding the inverse function, solving equations graphically, or analysing trigonometric functions. A deep understanding of transformations allows you to sketch complicated graphs quickly and accurately.

IB考试经常将变换与其他主题结合，例如求反函数、用图像解方程或分析三角函数。深入理解变换可以帮助你快速且准确地画出复杂函数的草图。

Common composite patterns include:

常见的复合模式包括：

• y = f(x) reflected and shifted: y = -f(x – 2) + 3
• y = f(x) stretched in both directions: y = 2f(x/3)
• y = f(|x|): the graph to the right of the y-axis is reflected to the left
• y = |f(x)|: the portion below the x-axis is reflected upward

• y = f(x)反射并平移：y = -f(x – 2) + 3
• y = f(x)在水平和垂直方向都拉伸：y = 2f(x/3)
• y = f(|x|)：y轴右侧的图像被反射到左侧
• y = |f(x)|：x轴下方的部分被向上反射

For y = f(|x|), the original graph for x ≥ 0 is kept, and its mirror image is drawn for x < 0. For y = |f(x)|, the parts of the graph below the x-axis (where f(x) < 0) are reflected above the x-axis, leaving the rest unchanged.

对于y = f(|x|)，保留x ≥ 0的原始图像，并将它在x < 0的区域画出镜像。对于y = |f(x)|，将x轴下方（f(x) < 0）的部分关于x轴向上反射，其余部分保持不变。

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9. Table of Transformation Rules | 变换规则速查表

The following table summarises all the essential transformation rules. Memorise these patterns — they are the foundation for solving transformation problems efficiently in the IB exam.

下表总结了所有关键变换规则。请牢记这些模式——它们是在IB考试中高效解决变换问题的基础。

Transformation | 变换	Equation | 方程	Effect on Graph | 对图像的影响
Vertical shift | 垂直平移	y = f(x) + k	Move up (k>0) or down (k<0) | 向上（k>0）或向下（k<0）移动
Horizontal shift | 水平平移	y = f(x – h)	Move right (h>0) or left (h<0) | 向右（h>0）或向左（h<0）移动
Reflection in x-axis | 关于x轴反射	y = -f(x)	Flip vertically | 垂直翻转
Reflection in y-axis | 关于y轴反射	y = f(-x)	Flip horizontally | 水平翻转
Vertical stretch/compression | 垂直拉伸/压缩	y = a·f(x)	Stretch by |a| if |a|>1, compress if 01时拉伸，0
Horizontal stretch/compression | 水平拉伸/压缩	y = f(bx)	Compress by |b| if |b|>1, stretch if 01时压缩，0

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10. Common Pitfalls and Exam Tips | 常见误区与考试技巧

Many students lose marks on transformation questions due to predictable errors. Being aware of these pitfalls will help you avoid them under exam pressure.

许多学生因可预测的错误而在变换题中失分。了解这些陷阱能帮助你在考试压力下避免它们。

• Pitfall 1: Using the original point instead of the image when working backwards. Always check whether the problem gives you a point on f(x) or a point on the transformed graph.
• Pitfall 2: Forgetting to factor out the coefficient before finding the horizontal shift. Write y = f(2x – 4) as y = f(2(x – 2)) first.
• Pitfall 3: Confusing horizontal stretch with horizontal compression. Remember: y = f(2x) compresses the graph, it does not stretch it.
• Pitfall 4: Ignoring reflections when a or b is negative. Always check the sign.

• 误区1：倒推时误用原始点而非对应点。务必检查题目给的是f(x)上的点还是变换后图像上的点。
• 误区2：求水平平移量之前忘记提取公因子。先将y = f(2x – 4)写成y = f(2(x – 2))。
• 误区3：混淆水平拉伸与水平压缩。记住：y = f(2x)是压缩图像，不是拉伸。
• 误区4：忽略a或b为负时的反射。务必检查正负号。

Exam tip: Always sketch a quick graph — even a rough one — to verify your algebraic answer. This is especially helpful for questions involving both transformations and domain/range analysis.

考试技巧：始终快速画一个草图——哪怕是粗略的——来验证你的代数答案。这对于同时涉及变换与定义域/值域分析的题目尤为重要。

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11. Worked Example | 完整例题

Let us work through a full IB-style question step by step.

让我们逐步完成一道完整的IB风格例题。

Question | 题目: The function f(x) = x² is transformed to g(x) = -2(x + 1)² + 4. Describe the transformations applied to f to obtain g, and find the image of the vertex.

问题：函数f(x) = x²经过变换得到g(x) = -2(x + 1)² + 4。描述从f得到g所应用的变换，并求出顶点的对应点。

Solution | 解答:

Rewrite g(x) = -2(x + 1)² + 4 in the general form y = a·f(b(x – h)) + k. Here a = -2, b = 1, h = -1, k = 4.

将g(x) = -2(x + 1)² + 4改写成一般形式y = a·f(b(x – h)) + k。这里a = -2，b = 1，h = -1，k = 4。

Step 1: b = 1 means no horizontal stretch or compression. The term (x + 1) = (x – (-1)) means a horizontal shift 1 unit to the left.

步骤1：b = 1意味着没有水平拉伸或压缩。(x + 1) = (x – (-1))表示水平向左平移1个单位。

Step 2: a = -2 means the graph is vertically stretched by a factor of 2 and reflected across the x-axis.

步骤2：a = -2表示图像在垂直方向拉伸2倍并关于x轴反射。

Step 3: k = 4 means a vertical shift 4 units upward.

步骤3：k = 4表示垂直向上平移4个单位。

The vertex of f(x) = x² is at (0, 0). Applying the transformations: first shift left 1 unit → (-1, 0); then stretch by factor 2 and reflect → (-1, 0); finally shift up 4 units → (-1, 4). Thus the image of the vertex is (-1, 4).

f(x) = x²的顶点在(0, 0)。依次应用变换：先向左平移1个单位 → (-1, 0)；然后拉伸2倍并反射 → (-1, 0)；最后向上平移4个单位 → (-1, 4)。因此顶点的对应点为(-1, 4)。

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12. Practice Questions | 练习题

Test your understanding with these exam-style questions. Work through them before checking the answers.

用以下考试风格题目测试你的理解。先尝试作答，再核对答案。

• Question 1: The point (3, -5) lies on y = f(x). Find its image on y = -f(x – 2) + 1.
• Question 2: The graph of y = f(x) is stretched horizontally by a factor of 3 and translated 2 units up. Write the equation of the transformed graph.
• Question 3: For the function f(x) = eˣ, write the equation of the graph after reflection in the y-axis, followed by a vertical shift down by 2.

• 练习1：点(3, -5)在y = f(x)上。求其在y = -f(x – 2) + 1上的对应点。
• 练习2：函数y = f(x)的图像水平拉伸3倍并向上平移2个单位。写出变换后图像的方程。
• 练习3：函数f(x) = eˣ的图像先关于y轴反射，再垂直向下平移2个单位。写出变换后图像的方程。

Answers: 1. (5, 6). 2. y = f(x/3) + 2. 3. y = e⁻ˣ – 2.

答案：1. (5, 6)。2. y = f(x/3) + 2。3. y = e⁻ˣ – 2。

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