Translations of Graphs: y = f(x) + b and y = f(x - a) | 函数图像的平移变换：y = f(x) + b 与 y = f(x - a)

📚 Translations of Graphs: y = f(x) + b and y = f(x – a) | 函数图像的平移变换：y = f(x) + b 与 y = f(x – a)

Graph translations are one of the most fundamental transformations in IB Mathematics. The functions y = f(x) + b and y = f(x – a) shift the entire graph of f without altering its shape, size, or orientation. Understanding these translations is essential for sketching functions, solving equations, and analysing real-world models. In this article, we will explore vertical and horizontal translations, their combination, effect on domain and range, vector notation, and common pitfalls.

图像平移是 IB 数学中最基本的变换之一。函数 y = f(x) + b 与 y = f(x − a) 可以在不改变形状、大小和方向的情况下整体位移 f 的图像。理解这些平移对于绘制函数草图、解方程以及分析实际模型至关重要。本文将深入探讨垂直平移和水平平移、它们的组合、对定义域与值域的影响、向量表示法以及常见错误。

1. What is a Translation? | 什么是图像平移？

A translation is a rigid transformation that slides every point of a graph the same distance in the same direction. For a given function y = f(x), applying a translation produces a new function whose graph is congruent to the original. No stretching, compressing, or reflecting occurs; only the position changes. In IB Mathematics, translations are described either by modifying the function equation or by using a translation vector.

平移是一种刚体变换，它将图像上的每一点沿相同方向移动相同距离。对于给定的函数 y = f(x)，施加平移会生成一个新函数，其图像与原图像全等。平移过程中不发生拉伸、压缩或反射，仅位置改变。在 IB 数学中，平移既可通过修改函数解析式来描述，也可借助平移向量来表示。

2. Vertical Translation: y = f(x) + b | 垂直平移：y = f(x) + b

Adding a constant b to the output of a function shifts its graph vertically. The equation becomes y = f(x) + b. If b > 0, the graph moves upward by b units; if b < 0, it moves downward by |b| units. For example, transforming f(x) = x² to y = x² + 3 lifts the parabola 3 units up, while y = x² − 2 drops it 2 units down. The mapping of points is (x, y) → (x, y + b).

在函数输出上加一个常数 b 可使图像上下平移。方程变为 y = f(x) + b。若 b > 0，图像向上移动 b 个单位；若 b < 0，则向下移动 |b| 个单位。例如，将 f(x) = x² 变换为 y = x² + 3 会把抛物线向上抬升 3 个单位，而 y = x² − 2 使其向下平移 2 个单位。点的映射关系为 (x, y) → (x, y + b)。

Vertical translation: (x, y) → (x, y + b) | 垂直平移：(x, y) → (x, y + b)

3. Horizontal Translation: y = f(x – a) | 水平平移：y = f(x − a)

A horizontal translation is performed by replacing x with (x − a) inside the function argument. The new function is y = f(x − a). Despite what the minus sign might suggest at first glance, when a > 0 the graph shifts to the right by a units, and when a < 0 it shifts to the left by |a| units. This counter‑intuitive direction arises because the value f(0) – originally located at x = 0 – now occurs when x − a = 0, i.e. at x = a. Hence the whole graph moves right. The mapping of points is (x, y) → (x + a, y).

水平平移是通过将函数自变量 x 替换为 (x − a) 来实现的。新函数为 y = f(x − a)。尽管减号可能让人第一眼误解，但当 a > 0 时图像向右平移 a 个单位，当 a < 0 时图像向左平移 |a| 个单位。这种反直觉的方向源于：原来的 f(0) 值出现在 x = 0 处，现在则需要 x − a = 0，即 x = a 处才能取到，因此图像整体右移。点的映射关系为 (x, y) → (x + a, y)。

Horizontal translation: (x, y) → (x + a, y) | 水平平移：(x, y) → (x + a, y)

4. Sign Conventions and Direction | 符号惯例与移动方向

For newcomers, remembering the direction of shift can be challenging. The rule is:

Vertical: y = f(x) + b → shift up if b > 0, down if b < 0 (sign matches the direction).
Horizontal: y = f(x − a) → shift right if a > 0, left if a < 0 (sign opposite to the direction).

An easy mnemonic: “Inside the bracket, opposite to what you expect; outside the bracket, same as what you expect.” The table below summarises the four cases.

对于初学者，记住平移方向有时比较棘手。法则如下：

垂直平移：y = f(x) + b → b > 0 时上移，b < 0 时下移（符号与方向一致）。
水平平移：y = f(x − a) → a > 0 时右移，a < 0 时左移（符号与方向相反）。

一个简单的口诀：“括号里面，符号与直觉相反；括号外面，符号与直觉一致。”下表总结了四种情形。

Transformation	Equation	Direction
Up	y = f(x) + b, b > 0	↑ upwards
Down	y = f(x) − b, b > 0	↓ downwards
Right	y = f(x − a), a > 0	→ right
Left	y = f(x + a), a > 0	← left

5. Combining Vertical and Horizontal Translations | 组合垂直与水平平移

A function can undergo both vertical and horizontal translations simultaneously. The general form is y = f(x − a) + b. This represents a translation by the vector (a, b): the graph moves a units horizontally (right if a > 0) and b units vertically (up if b > 0). The order of operations does not matter – applying the horizontal shift then the vertical shift gives the same result as the reverse. For example, starting with f(x) = √x, the equation y = √(x − 4) + 3 translates the graph 4 units right and 3 units up, placing the starting point at (4, 3).

函数可同时进行垂直和水平平移。一般形式为 y = f(x − a) + b，它表示沿向量 (a, b) 的平移：图像水平移动 a 个单位（a > 0 右移）且垂直移动 b 个单位（b > 0 上移）。操作的先后顺序不影响结果——先水平后垂直与先垂直后水平所得图像相同。例如，从 f(x) = √x 出发，方程 y = √(x − 4) + 3 将图像右移 4 个单位、上移 3 个单位，起始点变为 (4, 3)。

Combined translation: y = f(x − a) + b | 组合平移：y = f(x − a) + b

6. Effect on Domain and Range | 对定义域和值域的影响

Translations systematically alter the domain and range of a function. A vertical translation y = f(x) + b leaves the domain unchanged but shifts the range: if the original range is R, the new range becomes {y + b | y ∈ R}. A horizontal translation y = f(x − a) leaves the range unchanged but shifts the domain: the new domain is {x + a | x ∈ D}, where D is the original domain. For instance, f(x) = ln x has domain x > 0 and range all real numbers. Under y = ln(x − 2) + 1, the domain becomes x > 2 and the range stays all reals, but every output is increased by 1. Paying attention to these shifts is particularly important when dealing with piecewise functions or inverse functions in IB exams.

平移会系统地改变函数的定义域与值域。垂直平移 y = f(x) + b 保持定义域不变，但将值域平移：若原始值域为 R，则新值域为 {y + b | y ∈ R}。水平平移 y = f(x − a) 保持值域不变，但将定义域平移：新定义域为 {x + a | x ∈ D}，其中 D 为原定义域。例如，f(x) = ln x 的定义域为 x > 0，值域为全体实数。在 y = ln(x − 2) + 1 变换下，定义域变为 x > 2，值域仍为全体实数，但每个输出值增加了 1。处理 IB 考试中的分段函数或反函数问题时，特别需要关注这些平移效应。

7. Translations of Specific Function Families | 特定函数族的平移

Applying translations to common parent functions yields predictable results:

Linear: f(x) = mx + c. Translating by (a, b) gives y = m(x − a) + c + b = mx + (c − ma + b), still a line with same slope.
Quadratic: f(x) = x². y = (x − a)² + b moves the vertex from (0,0) to (a, b).
Sine/Cosine: y = sin(x − a) + b shifts the wave horizontally and vertically; amplitude and period remain unchanged.
Exponential: y = 2ˣ → y = 2ˣ⁻ᵃ + b shifts the horizontal asymptote y = 0 to y = b and moves the curve right by a.
Rational: y = 1/x → y = 1/(x − a) + b translates both vertical and horizontal asymptotes.

对常见的基本函数施加平移会得到可预测的结果：

线性函数：f(x) = mx + c，经向量 (a, b) 平移后变为 y = m(x − a) + c + b = mx + (c − ma + b)，仍为斜率相同的直线。
二次函数：f(x) = x²，y = (x − a)² + b 将顶点从 (0,0) 移至 (a, b)。
正弦/余弦函数：y = sin(x − a) + b 将波形水平和垂直平移，振幅与周期保持不变。
指数函数：y = 2ˣ → y = 2ˣ⁻ᵃ + b 将水平渐近线 y = 0 平移至 y = b，曲线右移 a 个单位。
有理函数：y = 1/x → y = 1/(x − a) + b 同时平移垂直渐近线和水平渐近线。

8. Translations and Asymptotes | 平移与渐近线

When a function possesses asymptotes, these also undergo the same translation. For a vertical asymptote originally at x = c, the transformation x → x − a moves it to x = c + a. For a horizontal asymptote originally at y = d, adding b to the function shifts it to y = d + b. Consider the reciprocal function y = 1/x. Its asymptotes are x = 0 and y = 0. The translated function y = 1/(x − 3) + 2 has a vertical asymptote at x = 3 and a horizontal asymptote at y = 2. IB questions often require stating the equations of asymptotes after translations, especially for rational and logarithmic functions.

如果函数具有渐近线，这些渐近线也会经历相同的平移。原本在 x = c 处的垂直渐近线，经变换 x → x − a 后移至 x = c + a 处。原本在 y = d 处的水平渐近线，经函数加 b 后移至 y = d + b 处。以反比例函数 y = 1/x 为例，其渐近线为 x = 0 和 y = 0。平移后的函数 y = 1/(x − 3) + 2 的垂直渐近线为 x = 3，水平渐近线为 y = 2。IB 试题中，经常要求写出平移后渐近线的方程，尤其是针对有理函数和对数函数。

Asymptote translation: (x = c) → (x = c + a); (y = d) → (y = d + b)

9. Vector Notation for Translations | 平移的向量表示

In IB Mathematics, translations can be described compactly using column vectors. A translation vector (a, b)ᵀ or written as (a, b) moves every point (x, y) to (x + a, y + b). In function notation, this corresponds to replacing y by y − b and x by x − a, leading to y − b = f(x − a), i.e. y = f(x − a) + b. Thus, the translation vector (a, b) directly gives the parameters in the combined translation equation. This vector approach is particularly useful when dealing with composite transformations or when describing the image of a specific point. For example, the point (2, 5) on the graph of y = f(x) is mapped to (2 + a, 5 + b) on the translated graph.

在 IB 数学中，平移可用列向量简明表示。平移向量 (a, b)ᵀ 或记作 (a, b)，将每一点 (x, y) 映射到 (x + a, y + b)。用函数符号表示，相当于将 y 替换为 y − b、将 x 替换为 x − a，从而得到 y − b = f(x − a)，即 y = f(x − a) + b。因此，平移向量 (a, b) 直接给出了组合平移方程中的参数。这种向量方法在处理复合变换或描述具体点的像时特别有用。例如，y = f(x) 图像上的点 (2, 5) 在平移后图像上的像为 (2 + a, 5 + b)。

10. Distinguishing Translations from Stretches and Reflections | 区别平移与拉伸、反射

Students often confuse translations with other transformations because the algebraic forms can look similar. The key identifiers are:

y = f(x) + b → translation (vertical shift, b added outside).
y = k f(x) → vertical stretch (multiplication outside), not a translation.
y = f(x − a) → translation (horizontal shift, subtraction inside).
y = f(qx) → horizontal stretch (multiplication inside).
y = −f(x) → reflection in the x-axis.
y = f(−x) → reflection in the y-axis.

Remember: translations add or subtract a constant, whereas stretches multiply the variable or the whole function. In IB questions, you may be asked to describe fully a transformation that is given in the form y = 2f(x − 3) + 1; you must recognise the order: translate right by 3, then vertically stretch by factor 2, then translate up by 1. Horizontal shift is always applied before stretch/reflection when the transformation is inside the argument.

学生常将平移与其他变换混淆，因为它们的代数形式有时很相似。主要识别方法：

y = f(x) + b → 平移（外部加减，垂直移动）。
y = k f(x) → 垂直拉伸（外部乘法），非平移。
y = f(x − a) → 平移（内部加减，水平移动）。
y = f(qx) → 水平拉伸（内部乘法）。
y = −f(x) → 关于 x 轴的反射。
y = f(−x) → 关于 y 轴的反射。

记住：平移是加或减一个常数，而拉伸则是对变量或整个函数做乘法。在 IB 试题中，可能要求完整描述 y = 2f(x − 3) + 1 所表示的变换；你必须识别顺序：先向右平移3个单位，然后垂直拉伸至原来的2倍，再向上平移1个单位。当变换位于自变量内部时，水平平移总是在拉伸/反射之前应用。

11. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Even experienced students slip up on translations. Watch out for these pitfalls:

Horizontal direction error: Thinking y = f(x − 2) moves left instead of right. Remind yourself: substituting x = 2 gives f(0), which was originally at x = 0 – so the point has moved to the right.
Forgetting brackets: Writing f(x − a) + b as f(x − a + b) – the vertical shift must be added outside the function, not inside the argument.
Domain mistakes: When the parent function has a restricted domain (e.g. square root, logarithm), failing to adjust the domain after horizontal translation can lead to incorrect sketches or invalid inputs.
Vector sign confusion: A vector (−3, 2) means left 3 and up 2, which corresponds to y = f(x + 3) + 2. Some students mistakenly write x − 3 for a left shift.
Misapplying translation to asymptotes: Assuming only the visible curve moves but not the asymptotes – always translate asymptotes as if they were part of the graph.

Prevent these mistakes by writing the mapping clearly and checking with a test point.

即使经验丰富的学生也会在平移上出错。请防范以下常见陷阱：

水平方向错误：以为 y = f(x − 2) 是向左平移，而非向右。提醒自己：代入 x = 2 时得到 f(0)，这原本在 x = 0 处——因此该点右移了。
遗漏括号：将 y = f(x − a) + b 误写成 f(x − a + b)——垂直平移必须加在函数外部，而非自变量内部。
定义域错误：当母函数有受限定义域（如平方根、对数）时，水平平移后若未相应调整定义域，会导致绘图错误或输入值无效。
向量符号混淆：向量 (−3, 2) 表示左移 3、上移 2，对应的方程为 y = f(x + 3) + 2。有些学生会错误地写成 x − 3 表示左移。
对渐近线错误应用平移：以为只有可见曲线移动而渐近线不动——应始终将渐近线视为图像的一部分进行平移。

通过清晰写出映射关系并用测试点检验，可以避免这些错误。

12. Summary and IB Exam Tips | 总结与 IB 考试技巧

Translations y = f(x) + b and y = f(x − a) preserve the shape of a graph while repositioning it. The vertical translation is straightforward: add b to the y-coordinates. The horizontal translation requires careful attention to sign: f(x − a) shifts right when a > 0. A combined translation y = f(x − a) + b corresponds to vector (a, b). Always state the new domain, range, and asymptote equations where relevant. For exam success:

Identify the parent function first, then list the sequence of transformations.
When asked to sketch, plot key points after applying the translations.
Use the point‑mapping approach (x, y) → (x + a, y + b) as a foolproof check.
Watch out for transformed inverse functions, where domain/range swaps matter.

Mastering these translations will give you a solid foundation for more complex transformations involving stretches and reflections.

平移变换 y = f(x) + b 与 y = f(x − a) 在保留图像形状的同时改变其位置。垂直平移非常直观：直接对 y 坐标加 b。水平平移则需特别注意符号：当 a > 0 时 f(x − a) 向右平移。组合平移 y = f(x − a) + b 对应于向量 (a, b)。记住在相关情况下给出新的定义域、值域和渐近线方程。IB 考试技巧：

首先识别母函数，然后列出变换顺序。
要求作草图时，先对关键点施加平移再描点。
利用点映射法 (x, y) → (x + a, y + b) 作为万无一失的检验方式。
当涉及反函数变换时，注意定义域与值域的对调关系。

掌握这些平移变换将为你进一步学习包含拉伸和反射的更复杂变换打下坚实基础。

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Translations of Graphs: y = f(x) + b and y = f(x – a) | 函数图像的平移变换：y = f(x) + b 与 y = f(x – a)