Understanding Function Graph Translations | 理解函数图像平移法则

📚 Understanding Function Graph Translations | 理解函数图像平移法则

Function graph translation is one of the most intuitive yet frequently misunderstood topics in IB Mathematics. Whether you are taking Analysis and Approaches (AA) or Applications and Interpretation (AI), understanding how horizontal and vertical shifts transform a function’s graph is essential for solving problems efficiently and accurately.

函数图像的平移是 IB 数学中既直观又常被误解的主题之一。无论你学习的是分析与方法（AA）还是应用与解释（AI），理解水平和垂直平移如何改变函数图像，对于高效准确地解题都至关重要。

1. The Basic Translation Rules | 基本平移法则

Before we dive into examples, let us establish the foundational rules. For a given function \( y = f(x) \), the following transformations apply:

在深入例子之前，我们先建立基础法则。对于给定函数 \( y = f(x) \)，以下变换成立：

Vertical translation upward by \( k \) units: \( y = f(x) + k \) — the entire graph shifts up.

垂直向上平移 \( k \) 个单位： \( y = f(x) + k \) — 整个图像向上移动。
Vertical translation downward by \( k \) units: \( y = f(x) – k \) — the entire graph shifts down.

垂直向下平移 \( k \) 个单位： \( y = f(x) – k \) — 整个图像向下移动。
Horizontal translation right by \( a \) units: \( y = f(x – a) \) — the graph moves to the right.

水平向右平移 \( a \) 个单位： \( y = f(x – a) \) — 图像向右移动。
Horizontal translation left by \( a \) units: \( y = f(x + a) \) — the graph moves to the left.

水平向左平移 \( a \) 个单位： \( y = f(x + a) \) — 图像向左移动。

Notice the key pattern: for horizontal shifts, the sign inside the function is opposite to the direction of movement. This counterintuitive rule is where many students lose marks.

注意关键规律：对于水平平移，函数内部的符号与移动方向相反。这个反直觉的法则正是许多学生丢分的地方。

2. Vertical Translations — The Intuitive Start | 垂直平移 — 直观的起点

Vertical translations are straightforward because they directly add to or subtract from the output value of the function. If we take the graph of \( y = x^2 \) and form \( y = x^2 + 3 \), every point on the original parabola moves up by 3 units.

垂直平移非常直接，因为它直接对函数的输出值进行加减。如果我们取 \( y = x^2 \) 的图像并构成 \( y = x^2 + 3 \)，原抛物线上的每一个点都向上移动 3 个单位。

In general, adding a constant \( k \) to \( f(x) \) produces a vertical shift of that magnitude. The domain of the function remains unchanged, but the range is shifted by \( k \).

一般来说，在 \( f(x) \) 上加上常数 \( k \) 会产生相应大小的垂直平移。函数的定义域保持不变，但值域会上下移动 \( k \) 个单位。

For example, the range of \( y = x^2 \) is \( y \ge 0 \), while the range of \( y = x^2 – 4 \) becomes \( y \ge -4 \). This direct correspondence makes vertical shifts easy to verify and apply.

例如，\( y = x^2 \) 的值域是 \( y \ge 0 \)，而 \( y = x^2 – 4 \) 的值域变为 \( y \ge -4 \)。这种直接的对应关系使得垂直平移易于验证和应用。

3. Horizontal Translations — The Counterintuitive Rule | 水平平移 — 反直觉法则

Horizontal translations are where the famous “opposite sign” rule comes into play. The transformation \( y = f(x – a) \) shifts the graph to the right by \( a \) units, while \( y = f(x + a) \) shifts it to the left.

水平平移正是著名的“符号相反”法则发挥作用的地方。变换 \( y = f(x – a) \) 将图像向右平移 \( a \) 个单位，而 \( y = f(x + a) \) 则将其向左平移。

Why is this the case? Consider a specific point. If the original function has a point at \( x = c \), then in \( y = f(x – a) \), this same output value now occurs at \( x = c + a \). In other words, to produce the same \( y \)-value, the input must be larger by \( a \), meaning the graph has moved to the right.

为什么会这样？考虑一个具体的点。如果原函数在 \( x = c \) 处有一个点，那么在 \( y = f(x – a) \) 中，这个相同的输出值现在发生在 \( x = c + a \) 处。换句话说，要产生相同的 \( y \) 值，输入必须增大 \( a \)，这意味着图像向右移动了。

Let us verify with a simple example. Compare \( y = x^2 \) and \( y = (x – 2)^2 \). The vertex of \( y = x^2 \) is at \( (0, 0) \). For \( y = (x – 2)^2 \), setting \( x – 2 = 0 \) gives \( x = 2 \), so the vertex is at \( (2, 0) \) — indeed shifted 2 units right.

让我们用一个简单例子验证。比较 \( y = x^2 \) 和 \( y = (x – 2)^2 \)。\( y = x^2 \) 的顶点在 \( (0, 0) \)。对于 \( y = (x – 2)^2 \)，令 \( x – 2 = 0 \) 可得 \( x = 2 \)，所以顶点在 \( (2, 0) \) — 确实是向右平移了 2 个单位。

4. The Algebraic Derivation | 代数推导

To fully internalise the horizontal shift rule, let us derive it algebraically. Suppose we want to shift the graph of \( y = f(x) \) to the right by \( a \) units. Every point \( (x, y) \) on the original graph becomes \( (x + a, y) \) on the new graph.

为了充分理解水平平移法则，让我们从代数角度进行推导。假设我们想将 \( y = f(x) \) 的图像向右平移 \( a \) 个单位。原图上的每个点 \( (x, y) \) 在新图上变为 \( (x + a, y) \)。

If we call the new horizontal coordinate \( X = x + a \), then \( x = X – a \). Since the original point satisfies \( y = f(x) \), the new point satisfies:

如果我们称新的横坐标为 \( X = x + a \)，那么 \( x = X – a \)。由于原满足 \( y = f(x) \)，新点满足：

y = f(X − a)

Thus, the new function is \( y = f(x – a) \). This derivation confirms why we replace \( x \) with \( x – a \) to move right: the subtraction inside the function compensates for the increased input.

因此，新函数是 \( y = f(x – a) \)。这个推导证实了为什么我们向右移动时要将 \( x \) 替换为 \( x – a \)：函数内部的减法是为了补偿增大的输入。

The same logic applies for a left shift: \( y = f(x + a) \). The pattern is consistent — to move the graph in the positive \( x \)-direction, subtract inside; to move it in the negative direction, add inside.

同样的逻辑适用于向左平移：\( y = f(x + a) \)。这一规律是一致的 — 要沿 \( x \) 轴正方向移动图像，就在函数内部做减法；要向负方向移动，就做加法。

5. Combining Translations | 组合平移

In IB examinations, you will frequently encounter functions that require both horizontal and vertical translations simultaneously. The general form is:

在 IB 考试中，你经常会遇到同时需要水平和垂直平移的函数。一般形式为：

y = f(x − a) + k

This represents a shift of \( a \) units horizontally (right if \( a > 0 \)) and \( k \) units vertically (up if \( k > 0 \)). The order in which you apply these shifts does not matter — the final graph is the same.

这表示水平平移 \( a \) 个单位（若 \( a > 0 \) 则向右）和垂直平移 \( k \) 个单位（若 \( k > 0 \) 则向上）。应用这些平移的先后顺序无关紧要 — 最终图像是相同的。

Consider the quadratic function \( y = (x – 3)^2 + 5 \). Starting from \( y = x^2 \), we shift right by 3 units and up by 5 units. The vertex moves from \( (0, 0) \) to \( (3, 5) \).

考虑二次函数 \( y = (x – 3)^2 + 5 \)。从 \( y = x^2 \) 出发，我们向右平移 3 个单位并向上平移 5 个单位。顶点从 \( (0, 0) \) 移动到 \( (3, 5) \)。

For IB exam preparation, practice identifying the translation from vertex form \( y = a(x – h)^2 + k \), where the vertex is at \( (h, k) \). This directly applies the translation rules.

在 IB 备考中，练习从顶点式 \( y = a(x – h)^2 + k \) 识别平移，其中顶点在 \( (h, k) \)。这直接运用了平移法则。

6. The Notation Trap — \( y = f(x + a) \) vs \( y = f(x) + a \) | 记法陷阱 — \( y = f(x + a) \) 与 \( y = f(x) + a \) 的区别

One of the most common mistakes in IB exams is confusing \( f(x + a) \) with \( f(x) + a \). Although they look similar, their effects are completely different:

IB 考试中最常见的错误之一是将 \( f(x + a) \) 与 \( f(x) + a \) 混淆。虽然它们看起来相似，但效果完全不同：

Expression 表达式	Type 类型	Effect 效果
\( y = f(x + a) \)	Horizontal 水平	Shifts left by \( a \) 向左平移 \( a \)
\( y = f(x) + a \)	Vertical 垂直	Shifts up by \( a \) 向上平移 \( a \)

The key is to observe exactly where the constant appears. If the constant is added to \( x \) inside the function, it is horizontal; if it is added to the entire function outside, it is vertical.

关键在于观察常数出现的位置。如果常数加在函数内部的 \( x \) 上，则是水平平移；如果加在函数外部的整体上，则是垂直平移。

When you encounter a transformed function in an exam, first identify whether the transformation is inside or outside the function notation. This simple step prevents the majority of graph transformation errors.

当你在考试中遇到变形的函数时，首先判断变换是在函数记号内部还是外部。这个简单的步骤可以避免绝大部分图像变换错误。

7. The Complete Picture — y = f(x − a) + k | 完整图像 — y = f(x − a) + k

Let us apply the combined rule to a concrete IB-style problem. Given the graph of \( y = \sqrt{x} \), sketch the graph of \( y = \sqrt{x + 2} – 3 \).

让我们将组合法则应用于一个具体的 IB 风格问题。给定 \( y = \sqrt{x} \) 的图像，画出 \( y = \sqrt{x + 2} – 3 \) 的图像。

Step 1: Identify the horizontal shift. Since \( x + 2 \) = \( x − (−2) \), we shift the graph left by 2 units. Step 2: Identify the vertical shift. The −3 outside shifts the graph down by 3 units.

步骤 1：识别水平平移。由于 \( x + 2 \) = \( x − (−2) \)，我们将图像向左平移 2 个单位。步骤 2：识别垂直平移。外部的 −3 将图像向下平移 3 个单位。

The original point \( (0, 0) \) on \( y = \sqrt{x} \) moves to \( (−2, −3) \). The domain shifts from \( x \ge 0 \) to \( x \ge −2 \), since we must have \( x + 2 \ge 0 \).

\( y = \sqrt{x} \) 上的原点 \( (0, 0) \) 移动到 \( (−2, −3) \)。定义域从 \( x \ge 0 \) 变为 \( x \ge −2 \)，因为我们必须有 \( x + 2 \ge 0 \)。

This systematic approach — separating horizontal and vertical components — works for any function type: polynomials, trigonometric functions, exponentials, and logarithms.

这种系统的方法 — 将水平和垂直分量分开 — 适用于任何函数类型：多项式、三角函数、指数函数和对数函数。

8. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧

Based on years of IB exam marking, the following pitfalls account for the majority of lost marks in graph transformation questions:

根据多年 IB 阅卷经验，以下陷阱是图像变换问题中丢分的主要原因：

Applying the wrong direction for horizontal shifts: Remember that \( x – a \) moves right, not left. Write the transformation as \( x − a \) to avoid sign confusion.

水平平移方向搞反： 记住 \( x − a \) 是向右移动，不是向左。将变换写成 \( x − a \) 的形式以避免符号混淆。
Forgetting to adjust the domain: When a graph shifts horizontally, the domain changes accordingly. Always recalculate the new domain rather than carrying over the original one.

忘记调整定义域： 当图像水平平移时，定义域相应改变。务必重新计算新定义域，而不是沿用原定义域。
Confusing the vertex or key point: When translating, every key point (vertex, intercept, asymptote) moves by the same vector. Apply the translation to all key points for an accurate sketch.

混淆顶点或关键点： 平移时，每个关键点（顶点、截距、渐近线）都沿相同向量移动。对图像上的所有关键点应用平移，才能画出准确的草图。

A reliable exam strategy is to track a few characteristic points of the original graph and verify their new positions. If these points are correctly placed, you can confidently sketch the entire transformed graph.

可靠的考试策略是追踪原图的几个特征点并验证它们的新位置。如果这些点的位置正确，你就能自信地画出整个变换后的图像。

9. Worked Example — IB Exam Style | 典型例题 — IB 考试风格

Let us work through a complete IB-style question. The function \( f(x) = x^2 – 4x + 5 \) is translated to form \( g(x) = x^2 – 10x + 26 \). Describe the transformation.

让我们完成一道完整的 IB 风格题目。函数 \( f(x) = x^2 – 4x + 5 \) 经过平移得到 \( g(x) = x^2 – 10x + 26 \)。描述这个变换。

First, rewrite \( f(x) \) in vertex form by completing the square:

首先，通过配方法将 \( f(x) \) 改写为顶点式：

f(x) = (x − 2)² + 1

Now rewrite \( g(x) \) in the same form:

现在将 \( g(x) \) 改写为相同形式：

g(x) = (x − 5)² + 1

Comparing the two forms, the vertex has moved from \( (2, 1) \) to \( (5, 1) \). This is a horizontal translation of 3 units to the right.

比较这两种形式，顶点从 \( (2, 1) \) 移动到 \( (5, 1) \)。这是向右水平平移 3 个单位。

Note that we did not start with a simple \( y = x^2 \) graph here — the same translation rules apply regardless of the original position of the graph.

注意这里我们并不是从简单的 \( y = x^2 \) 图像开始的 — 无论图像的原始位置如何，相同的平移法则都适用。

In the general case, translating the graph of \( y = f(x) \) by \( a \) units right and \( b \) units up yields:

在一般情况下，将 \( y = f(x) \) 的图像向右平移 \( a \) 个单位并向上平移 \( b \) 个单位，得到：

y − b = f(x − a)，即 y = f(x − a) + b

This is the complete translation formula that governs all function graph shifts in the IB syllabus.

这就是 IB 课程大纲中支配所有函数图像平移的完整公式。

10. Translation vs Reflection — A Critical Distinction | 平移与反射 — 关键区分

Finally, it is essential to distinguish translations from reflections, as IB questions often combine them. A translation slides the graph without changing its shape or orientation. A reflection, on the other hand, flips the graph across an axis.

最后，区分平移与反射至关重要，因为 IB 题目经常将两者结合。平移是滑动图像而不改变其形状或方向。而反射则是将图像翻转越过某个轴。

\( y = −f(x) \) is a reflection across the \( x \)-axis, not a translation.

\( y = −f(x) \) 是关于 \( x \) 轴的反射，不是平移。
\( y = f(−x) \) is a reflection across the \( y \)-axis, not a translation.

\( y = f(−x) \) 是关于 \( y \) 轴的反射，不是平移。
Mixing these up with \( y = f(x) + k \) or \( y = f(x + a) \) will lead to incorrect sketches and lost marks.

将它们与 \( y = f(x) + k \) 或 \( y = f(x + a) \) 混淆会导致图像画错并丢失分数。

A good test: after identifying a transformation, check whether the shape of the graph has been flipped or merely shifted. This will confirm your choice of transformation rule.

一个有效的检验方法：识别变换后，检查图像形状是被翻转还是仅仅被平移。这将确认你选择的变换法则是否正确。

11. Preparing for Your IB Exam — Final Strategies | 备战 IB 考试 — 最终策略

To master function graph translations for your IB examination, follow this systematic revision plan:

为了在 IB 考试中掌握函数图像平移，请遵循以下系统复习计划：

Memorise the four basic rules with a clear understanding of why the signs behave as they do.

牢记四个基本法则，并清楚理解符号变化的原因。
Practise rewriting functions into forms that reveal the translation, especially completing the square for quadratics and rewriting trigonometric functions in phase-shift form.

练习将函数改写为能揭示平移的形式，尤其是二次函数的配方和将三角函数改写为相移形式。
Sketch graphs by hand — start from the base function and apply translations one step at a time.

手动画图 — 从基础函数出发，一步一步地应用平移。

Function graph translation is not just a topic to memorise — it is a visual, intuitive concept. Once you can picture the graph moving on a coordinate plane, the algebra becomes secondary and your exam performance will improve significantly.

函数图像平移不仅仅是一个需要记忆的课题 — 它是一个可视化、直观的概念。一旦你能在脑海中看到图像在坐标平面上移动，代数就变得次要了，你的考试成绩将显著提升。

For further practice, explore past IB paper questions involving graph transformations. You will notice a recurring pattern: exams reward students who can connect the algebraic transformation to its geometric interpretation.

如需进一步练习，请探索涉及图像变换的 IB 历年真题。你会发现一个反复出现的规律：考试奖励那些能够将代数变换与其几何解释联系起来的学生。

To sum up, mastering graph translations involves knowing the four core rules, recognising the opposite sign law for horizontal shifts, and practising with IB-style problems until the process becomes automatic. With consistent practice, you will tackle any translation problem with confidence and precision.

总而言之，掌握图像平移需要了解四个核心法则，识别水平平移的符号相反定律，并通过 IB 风格题目练习直至过程自动化。通过持续练习，你将以自信和精准应对任何平移问题。

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Understanding Function Graph Translations | 理解函数图像平移法则