📚 Investigation 2 – Translations | 探究2 – 函数图像的平移
In algebra and graphing, a translation is a transformation that shifts a graph horizontally, vertically, or both, without altering its shape or orientation. This investigation explores the rules and effects of translations on various IB Mathematics functions, guiding you through systematic discovery of horizontal and vertical shifts. Understanding translations is foundational for analysing sinusoidal models, exponential growth, inverse functions, and solving equations graphically.
在代数与图像绘制中，平移是一种在不改变形状或方向的情况下，将图像水平、垂直或同时移动的变换。本探究将探索平移对多种IB数学函数的影响规律，引导你系统地发现水平和垂直移动的规则。理解平移是分析正弦模型、指数增长、反函数以及用图像法解方程的基础。
1. What Is a Translation? | 什么是平移？
by a vector (h, k), meaning h units horizontally and k units vertically. The transformed function takes the form y – k = f(x – h) or equivalently y = f(x – lations do not squash or flip the curve; they simply slide it across the coordinate plane.
平移将图像上的每一个点沿给定方向移动相同的距离。若原函数为 y = f(x)，平移可用向量 (h, k) 描述，即水平移动 h 个单位、垂直移动 k 个单位。变换后的函数可写为 y – k = f(x – h)，或等价地 y = f(x – h) + k。与反射或伸缩不同，平移不会压扁或翻转曲线，只是让图像在坐标平面上滑动。
The term 'translation' in geometry means 'sliding'. Every point (x, y) on the original graph moves to a new point (x + h, y + k). This is why the argument inside the function becomes (x – h) when we isolate y: to undo the horizontal shift on the original input.
“平移”在几何中意为“滑动”。原图像上的每个点 (x, y) 移动到新点 (x + h, y + k)。这就是为什么当我们将 y 单独表示时，函数内部的自变量变为 (x – h)：相当于抵消了对原始输入的水平移动。
2. Horizontal Translations in Detail | 详细解析水平平移
For a horizontal translation, the graph of y = f(x – h) shifts right by h units when h > 0, and left by |h| when h < 0. Notice the counter-intuitive sign rule: f(x – mon source of errors, so it is worth investigating with a simple quadratic like f(x) = x².
对于水平平移，图像 y = f(x – h) 在 h > 0 时向右移动 h 个单位，在 h < 0 时向左移动 |h| 个单位。注意违反直觉的符号规则：f(x – 2) 将图像向右移动 2 个单位，这是因为新函数中 x = 2 对应的输出与原函数中 x = 0 处的输出相同。这是常见的错误来源，因此值得用简单的二次函数如 f(x) = x² 进行探究。
Investigation Step: Plot f(x) = |x|. Then on the same axes, sketch g(x) = |x – 3|. Observe that the vertex moves from (0, 0) to (3, 0). Next try h(x) = |x + 4| and see the graph shift left to (– Summarise the finding: the x-coordinate of every point on f increases by h when using f(x – h) with h positive.
探究步骤：画出 f(x) = |x| 的图像。然后在同一坐标系中勾画 g(x) = |x – 3|。观察顶点从 (0, 0) 移到 (3, 0)。再尝试 h(x) = |x + 4|，看到图像向左移到 (–4, 0)。总结发现：使用 f(x – h) 且 h 为正时，f 上每个点的 x 坐标增加了 h。
3. Vertical Translations in Detail | 详细解析垂直平移
e more straightforward: y = f(x) + k moves the graph up by k units when k > 0, and down by |k| . For example, if f(x) = x², then y = x² + 5 lifts the parabola 5 units upward, and y = x² – 3 sinks it 3 units downward. The shape remains identical; only the y-coordinates change.
垂直平移更为直观：y = f(x) + k 在 k > 0 时将图像向上移动 k 个单位，在 k < 0 时向下移动 |k| 个单位。与水平移位不同，符号与方向一致。例如，若 f(x) = x²，则 y = x² + 5 将抛物线向上抬高 5 个单位，而 y = x² – 3 将其向下降低 3 个单位。形状保持不变，只有 y 坐标改变。
se become (0,2), (1,3), (2,6). The entire set of points is translated upward, preserving the pattern of a parabola.
为了验证，选取原曲线上的样本点，如 y = x² 的 (0,0)、(1,1)、(2,4)。应用 +2 后，这些点变为 (0,2)、(1,3)、(2,6)。整个点集向上平移，保持了抛物线的形状。
4. Combining Horizontal and Vertical Shifts | 组合水平和垂直平移
Most practical translations involve both h and k. The general form y = f(x – lying the vector (h, k) in one go. For instance, starting with y = x³, the function y = (x – 1)³ – 4 shifts the graph right by 1 and down by 4.
大多实际平移同时涉及 h 和 k。一般形式 y = f(x – h) + k 表示水平移动 h 个单位、垂直移动 k 个单位。如果你逐步构建平移，操作顺序会有讲究，但最终结果等同于一次性应用向量 (h, k)。例如，从 y = x³ 开始，函数 y = (x – 1)³ – 4 将图像向右移动 1、向下移动 4。
new origin for the translated graph. For a basic quadratic y = x², the translation y = (x – um point. This concept is extremely useful for graphing and solving equations graphically.
将点 (h, k) 想象为平移后图像的新原点会很有帮助。对于基本二次函数 y = x²，平移 y = (x – 2)² + 3 将顶点放在 (2, 3) 处，直接给出最小值点的位置。这一概念对图像绘制和用图像解方程极为有用。
5. Effects on Key Points of Graphs | 对图像关键点的影响
Identifying how translations affect key points—intercepts, vertices, asymptotes, and turning points— s. Similarly, a horizontal asymptote y = c becomes y = c + k after a vertical translation.
弄清平移如何影响关键点——截距、顶点、渐近线和驻点——对草图绘制至关重要。如果原图像在 (a, 0) 处有 x 轴截距，经过 (h, k) 的平移后，新的 x 截距将位于 (a + h, k)，除非垂直移动使其脱离坐标轴。类似地，水平渐近线 y = c 在垂直平移后变为 y = c + k。
Consider y = 1/x with asymptotes x = 0 and y = 0. The translation y = 1/(x – hifts the vertical asymptote to x = 2 and the horizontal asymptote to y = 1. The hyperbola' (0,0) to (2,1). This shows how simple shifts can completely reposition rational functions.
考虑渐近线为 x = 0 和 y = 0 的 y = 1/x。平移 y = 1/(x – 2) + 1 将垂直渐近线移到 x = 2，水平渐近线移到 y = 1。双曲线的中心从 (0,0) 移到 (2,1)。这表明简单的移动可以完全重定有理函数的位置。
6. Translations of Trigonometric Functions | 三角函数的平移
izontal translations are often referred to as phase shifts. The general form y = a sin(b(x – c)) + d or y = a cos(b(x – d. This directly gives the new midline y = d and moves key points like maxima and minima.
对于正弦和余弦图像，水平平移常被称为相移。一般形式 y = a sin(b(x – c)) + d 或 y = a cos(b(x – c)) + d 将 c 作为相移参数。这里水平向右移动 c 个单位，垂直移动 d 个单位。这直接给出了新的中线 y = d，并移动了最大值和最小值等关键点。
hat the period and amplitude are unchanged. For example, compare y = sin(x) with y = sin(x – , 0). The shape cycles identically. When simplifying b ≠ 1, factor out b first: y = sin(2x – π) = sin(2(x – π/2)), revealing a phase shift of π/2, not π.
探究可以表明周期和振幅保持不变。例如，比较 y = sin(x) 与 y = sin(x – π/2) + 1。后者向右移动 π/2、向上移动 1，使起点位于 (π/2, 1) 而非 (0, 0)。形状完全相同地循环。当 b ≠ 1 时，先提取因子 b：y = sin(2x – π) = sin(2(x – π/2))，揭示相移为 π/2，而非 π。
7. Translating Exponential and Logarithmic Functions | 指数与对数函数的平移
Exponential functions of the form y = a · bˣ + k experience a vertical shift k, while a horizontal shift appears as y = a · bx – h + k. The horizontal asymptote y = 0 naturally moves to y = k. For instance, y = 2ˣ shifted left by 3 and down by 1 gives y = 2x + 3 – 1, with asymptote y = –1. The growth rate remains the same; only the baseline changes.
形如 y = a · bˣ + k 的指数函数经历垂直移动 k，而水平移动则表现为 y = a · bx – h + k。水平渐近线 y = 0 自然移动到 y = k。例如，y = 2ˣ 向左移动 3、向下移动 1 得到 y = 2x + 3 – 1，渐近线为 y = –1。增长率保持不变，仅基线改变。
arent y = log_b(x) has vertical asymptote x = 0. Translating right by h yields y = log_b(x – h), moving the asymptote to x = h. Adding k vertically gives y = log_b(x – ordingly: originally x > 0, after translation x > h. The intercept also moves predictably.
对数函数遵循类似规则。母函数 y = log_b(x) 的垂直渐近线为 x = 0。向右平移 h 得到 y = log_b(x – h)，渐近线移至 x = h。垂直加上 k 给出 y = log_b(x – h) + k。定义域相应移动：原定义为 x > 0，平移后为 x > h。截距也按预期移动。
8. Domain and Range After Translations | 平移后的定义域和值域
undary shifts, not in the fundamental nature of the set. For example, the domain of f(x) = √x is x ≥ 0. After a horizontal translation right by 2, y = √x – 2 does not alter the domain. Similarly, the range of f(x) = x² is y ≥ 0; after y = (x + 1)² – 4, the range becomes y ≥ –4.
平移仅通过边界移动影响定义域和值域，而不改变集合的基本性质。例如，f(x) = √x 的定义域为 x ≥ 0。向右平移 2 个单位后得到 y = √x – 2，定义域变为 x ≥ 2。垂直平移不会改变定义域。同样，f(x) = x² 的值域为 y ≥ 0；在 y = (x + 1)² – 4 后，值域变为 y ≥ –4。
accordingly. For instance, if a function is defined as f(x) = x for x < 0, then after f(x – 3), the condition becomes x – 3 < 0, i.e., x < 3. Keep the algebraic manipulation precise.
对于分段函数，平移会将分界点移动 (h, k)。相应地重写分段定义域条件至关重要。例如，若函数定义为当 x < 0 时 f(x) = x，则在 f(x – 3) 后，条件变为 x – 3 < 0，即 x < 3。务必保持代数处理的精确性。
9. Translating Inverse Functions | 反函数的平移
the line y = x, and translations interact interestingly. If f(x) is translated to y = f(x – inverse by (h, k). Instead, solving for the new inverse shows that the inverse of y = f(x – h) + k corresponds to x = f⁻¹(y – k) + h or y = f⁻¹(x – k) + h after swapping variables. This reveals a swap of roles between h and k.
反函数关于直线 y = x 反射，平移与之的交互颇为有趣。若 f(x) 被平移为 y = f(x – h) + k，其反函数并非原反函数简单地经过 (h, k) 平移。相反，求解新的反函数表明，y = f(x – h) + k 的反函数对应于 x = f⁻¹(y – k) + h，交换变量后为 y = f⁻¹(x – k) + h。这揭示了 h 和 k 的角色互换。
An illustrative case: f(x) = eˣ has inverse f⁻¹(x) = ln x. Translating the exponential: y = ex – 2 + 1. To find its inverse, swap x and y: x = ey – 2 + 1 → x – 1 = ey – 2 → ln(x – 1) = y – 2 → y = ln(x – investigation breaks the assumption that inverse translations mirror the original vector.
一个说明性案例：f(x) = eˣ 的反函数为 f⁻¹(x) = ln x。将指数函数平移：y = ex – 2 + 1。求其反函数，交换 x 和 y：x = ey – 2 + 1 → x – 1 = ey – 2 → ln(x – 1) = y – 2 → y = ln(x – 1) + 2。该反函数是 ln x 向右 1、向上 2 的平移，而非 2 和 1。这一探究打破了反函数平移会镜像原向量的假设。
10. Graphical vs. Algebraic Approach | 图解法与代数法
When solving equations like √x – 4 ognising the translation can provide a graphical shortcut. The equation can be seen as y = √x -coordinate of the intersection, which is 13. This builds conceptual linking between function transformations and equations.
当求解类似 √x – 4 + 2 = 5 的方程时，你可以采用代数操作：隔离根式、平方并求解。然而，识别平移可以提供图像上的捷径。该方程可看作 y = √x 向右平移 4、向上平移 2 后与 y = 5 相交。通过图像求解得到交点的 x 坐标为 13。这建立了函数变换与方程之间的概念联系。
Similarly, inequalities such as |x – 3| < 2 can be interpreted as the distance from x to 3 being less than 2, which is a translation of the basic |x| < 2 graph. The solution interval (1, 5) emerges directly from adding the shift to the original interval (–2, 2). This approach reinforces the geometric meaning of translation as sliding the number line or plane.
类似地，不等式 |x – 3| < 2 可以解释为 x 到 3 的距离小于 2，这是基本图像 |x| < 2 的平移。解区间 (1, 5) 可直接通过将移动量加到原区间 (–2, 2) 上得到。这种方法强化了平移作为滑动数轴或平面的几何意义。
11. Investigation Activities and Common Pitfalls | 探究活动与常见错误
Activity 1 – Discover the Sign Rule: Use graphing technology to plot f(x) = x² and h(x) = (x – sign inside the bracket to direction. Test your rule with f(x) = 1/x and g(x) = 1/(x + 5).
活动一 – 发现符号规则：使用绘图工具绘制 f(x) = x² 和 h(x) = (x – 3)²。记录顶点坐标。然后绘制 f(x + 2) 并注意移动方向。总结出括号内符号与方向关系的规则。用 f(x) = 1/x 和 g(x) = 1/(x + 5) 检验你的规则。
Activity 2 – Combined Shifts: Start with f(x) = |x|. Predict the position of the vertex for y = |x + 4| – the new coordinates of the vertex and explain how each translation component contributes.
活动二 – 组合移动：从 f(x) = |x| 出发。预测 y = |x + 4| – 2 的顶点位置。先手绘草图，再用计算器核对。写出顶点的新坐标，并解释每个平移分量是如何起作用的。
Common pitfalls include: forgetting to factor out the coefficient of x before identifying horizontal shift in trigonometric functions; mixing up the signs for horizontal translations; and assuming the inverse of a translated function follows the same (h, k) pattern. IB examiners often test these subtle points, so careful investigation pays off.
常见错误包括：在识别三角函数的水平移动时忘记先提取 x 的系数；混淆水平平移的符号；以及假设平移后函数的反函数遵循相同的 (h, k) 模式。IB 考官经常考察这些细微之处，因此仔细探究会大有裨益。
rule by substituting a known point and confirming the output matches the graph. If y = f(x – moves to (a + h, f(a) + k). Use this mapping to double-check your equations and sketches.
始终通过代入已知点并确认输出与图像匹配来验证你的平移规则。若 y = f(x – h) + k，则 f 上的点 (a, f(a)) 移动到 (a + h, f(a) + k)。使用这一映射来反复核对你的方程和草图。
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