📚 B – Translations | 函数平移变换
In IB Mathematics, the study of function transformations begins with translations, often labelled as Topic B. Translations shift a graph horizontally, vertically, or both, while preserving its size and shape. Mastering translations is essential for understanding more complex transformations like stretches and reflections.
在 IB 数学中，函数变换的学习始于平移，常标记为主题 B。平移将图像水平、垂直或同时移动，同时保持其大小和形状不变。掌握平移对于理解更复杂的伸缩和反射变换至关重要。
1. What Is a Translation? | 什么是平移？
A translation is a rigid transformation that moves every point of a function's graph by the same distance in a given direction. If we start with a parent function f(x), a translated function can be written as g(x) = f(x − h) + k. The parameters h and k control the horizontal and vertical shift respectively.
平移是一种刚性变换，将函数图像上的每一个点按给定方向移动相同的距离。如果从母函数 f(x) 开始，平移后的函数可写作 g(x) = f(x − h) + k。参数 h 和 k 分别控制水平与垂直方向的移动。
The shape, orientation, and steepness of the graph do not change; only its position in the coordinate plane is altered. This makes translations one of the easiest transformations to recognise and apply.
图像的形状、方向和陡峭程度都不变；唯一改变的是其在坐标平面中的位置。这使得平移成为最容易识别和应用的变换之一。
In function notation, the order of operations is important: first apply the horizontal shift inside the function's argument, then add the vertical shift to the output. However, because addition is commutative, the overall effect is simply shifting the graph to a new origin.
在函数记法中，运算顺序很重要：先在函数的输入内部应用水平平移，再对输出加上垂直平移。然而，因为加法满足交换律，总体效果只是将图像平移到新的原点。
2. Vertical Translations (Shifts Up and Down) | 垂直平移（上下平移）
Vertical translations shift the graph up or down. The transformation is described by g(x) = f(x) + k. When k > 0, the graph moves upward by k units; when k < 0, it moves downward by |k| units.
垂直平移将图像上下移动。变换描述为 g(x) = f(x) + k。当 k > 0 时，图像向上移动 k 个单位；当 k < 0 时，图像向下移动 |k| 个单位。
For example, consider the parent function f(x) = x². Adding 3 gives g(x) = x² + 3. This shifts the parabola up by 3 units, moving the vertex from (0,0) to (0,3). The axis of symmetry remains x = 0, and the shape is unchanged.
例如，考虑母函数 f(x) = x²。加上 3 得到 g(x) = x² + 3。这将抛物线向上平移 3 个单位，顶点从 (0,0) 移至 (0,3)。对称轴仍为 x = 0，形状保持不变。
Similarly, h(x) = |x| − 2 shifts the absolute value graph down by 2 units. The vertex goes from (0,0) to (0,−2). In general, adding a constant to the output of a function moves the graph vertically without affecting its x-intercepts, although the y-intercept changes by k.
类似地，h(x) = |x| − 2 将绝对值图像向下平移 2 个单位。顶点从 (0,0) 移至 (0,−2)。通常，在函数输出上加一个常数会垂直移动图像，不影响其 x 截距，但 y 截距会改变 k。
3. Horizontal Translations (Shifts Left and Right) | 水平平移（左右平移）
Horizontal translations move the graph left or right and are given by g(x) = f(x − h). A positive h shifts the graph to the right by h units, while a negative h shifts it to the left by |h| units. This direction often surprises students because it appears opposite to the sign inside the brackets.
水平平移将图像左右移动，其形式为 g(x) = f(x − h)。正数 h 将图像向右平移 h 单位，负数 h 向左平移 |h| 单位。这个方向经常令学生感到惊讶，因为它与括号内符号相反。
To understand why, think about what value of x produces the same output as the original function at some reference point. For f(x) = x², the function f(x − 2) = (x − 2)² requires x = 2 to give the same output as f(0). Hence the vertex shifts to (2,0) — a shift to the right.
要理解其原因，可以思考 x 取何值时能产生与原函数在某个参考点相同的输出。对于 f(x) = x²，函数 f(x − 2) = (x − 2)² 需要 x = 2 才能给出与 f(0) 相同的输出。因此顶点移到了 (2,0) —— 向右平移。
Common examples: if f(x) = √x, then f(x + 5) = √x + 5 shifts the graph 5 units to the left. The domain changes from x ≥ 0 to x ≥ −5. Remember: x − h with h positive shifts right; x + h shifts left.
常见例子：若 f(x) = √x，则 f(x + 5) = √x + 5 将图像向左平移 5 个单位。定义域从 x ≥ 0 变为 x ≥ −5。记住：x − h（h 正）向右移；x + h 向左移。
4. Combined Translations: g(x) = f(x − h) + k | 组合平移
When both horizontal and vertical shifts are present, we combine them in the general form g(x) = f(x − h) + k. The graph of f is shifted horizontally by h units and vertically by k units. The order in which you perform the shifts does not matter — the result is the same.
当水平与垂直平移同时存在时，我们将其组合为一般形式 g(x) = f(x − h) + k。f 的图像先水平移动 h 单位，再垂直移动 k 单位。执行的顺序并不重要——结果相同。
For example, starting from f(x) = x³, the function g(x) = (x + 3)³ − 4 shifts the cubic graph 3 units to the left and 4 units down. The inflection point moves from (0,0) to (−3,−4). The overall behavior remains unchanged.
例如，从 f(x) = x³ 开始，函数 g(x) = (x + 3)³ − 4 将三次函数图像向左平移 3 单位、向下平移 4 单位。拐点从 (0,0) 移到 (−3,−4)。整体性态保持不变。
When describing the translation, you can say "the graph of f(x) is translated by the vector (h, k)". This vector notation is particularly useful in coordinate geometry and IB exam questions that ask you to identify the transformation from a given equation.
描述平移时，你可以说“f(x) 的图像按向量 (h, k) 平移”。这种向量记法在坐标几何和 IB 考题中特别有用，常要求根据给定方程识别变换。
It is helpful to note that the horizontal shift is always applied to the x-values before any other transformation inside the argument, while the vertical shift is the final addition outside the function. This understanding prevents mistakes when dealing with stretches later on.
值得留意的是，水平平移总是先于输入内部的其他变换作用于 x 值，而垂直平移则是函数外部的最后加法。这一理解有助于避免日后处理伸缩变换时出现错误。
5. Translating Quadratic Functions | 二次函数的平移
Quadratic functions written in vertex form f(x) = a(x − h)² + k explicitly show the translations from the basic parabola y = x². The vertex is at (h, k), and the entire graph is a translation of y = x² by h units right and k units up (adjusting for sign).
写成顶点式 f(x) = a(x − h)² + k 的二次函数明确显示了从基本抛物线 y = x² 的平移。顶点为 (h, k)，整个图像是 y = x² 向右平移 h 单位、向上平移 k 单位（注意符号调整）。
If a = 1, the graph has the same width as y = x²; if a is not 1, the graph also has a vertical stretch or compression, but that is not a translation. Translating a quadratic simply relocates the vertex and the axis of symmetry.
若 a = 1，图像与 y = x² 宽度相同；若 a ≠ 1，图像还带有垂直伸缩或压缩，但这不属于平移。二次函数的平移仅改变顶点和对称轴的位置。
Example: f(x) = 2(x + 1)² − 5. This is derived from y = x² by stretching vertically by factor 2, then translating 1 unit left and 5 units down. The vertex is (−1, −5). The translation alone is vector (−1, −5).
例子：f(x) = 2(x + 1)² − 5。这由 y = x² 先垂直拉伸 2 倍，再向左平移 1 单位、向下平移 5 单位得到。顶点为 (−1, −5)。单纯就平移而言，向量为 (−1, −5)。
6. Translating Trigonometric Functions | 三角函数的平移
Translations are frequently applied to sine and cosine graphs. The general form y = a sin(b(x − h)) + k includes a horizontal shift h (phase shift) and a vertical shift k. The amplitude and period are not affected by pure translations.
平移常用于正弦和余弦图像。一般形式 y = a sin(b(x − h)) + k 包含水平平移 h（相位平移）和垂直平移 k。单纯的平移不影响振幅和周期。
For instance, y = sin(x − π/2) + 1 shifts the basic sine wave right by π/2 and up by 1. The midline moves from y = 0 to y = 1, and the maximum point originally at (π/2, 1) moves to (π, 2).
例如，y = sin(x − π/2) + 1 将基本正弦波向右平移 π/2、向上平移 1。中线从 y = 0 移到 y = 1，原本在 (π/2, 1) 的最大值点移至 (π, 2)。
When the argument contains a coefficient b, be careful: the horizontal shift is given by h in (x − h), not bh. Factor the inside if necessary. For y = cos(2x + 4), rewrite as cos(2(x + 2)) so the phase shift is −2 (2 units left).
当输入中含有系数 b 时要小心：水平平移由 (x − h) 中的 h 决定，而非 bh。必要时提取公因子。对 y = cos(2x + 4) 改写为 cos(2(x + 2))，可知相位平移为 −2（左移 2 单位）。
7. Translating Exponential and Logarithmic Functions | 指数与对数函数的平移
Exponential functions of the form f(x) = bˣ can be translated into g(x) = b⁽ˣ⁻ʰ⁾ + k. The horizontal asymptote y = 0 of the parent function shifts to y = k. The horizontal translation moves the graph without affecting the asymptote vertically.
形如 f(x) = bˣ 的指数函数可平移到 g(x) = b⁽ˣ⁻ʰ⁾ + k。母函数的水平渐近线 y = 0 移至 y = k。水平平移在移动图像时不影响渐近线的垂直位置。
For example, y = 2⁽ˣ⁻³⁾ − 1 has graph that is the result of translating y = 2ˣ three units to the right and one unit down. Its horizontal asymptote becomes y = −1, and the y-intercept changes from (0,1) to (0, 2⁻³ − 1) ≈ (0, −0.875).
例如，y = 2⁽ˣ⁻³⁾ − 1 的图像是由 y = 2ˣ 向右平移 3 单位、向下平移 1 单位得到的。其水平渐近线变为 y = −1，y 截距从 (0,1) 变为 (0, 2⁻³ − 1) ≈ (0, −0.875)。
Logarithmic functions behave similarly. The parent f(x) = log2(x) has a vertical asymptote at x = 0. After translation, g(x) = log2(x + 5) − 2 shifts the graph left 5 units and down 2 units, moving the vertical asymptote to x = −5.
对数函数类似。母函数 f(x) = log2(x) 有垂直渐近线 x = 0。平移后，g(x) = log2(x + 5) − 2 将图像向左平移 5 单位、向下平移 2 单位，垂直渐近线移至 x = −5。
8. Effects on Asymptotes and Key Features | 对渐近线和关键特征的影响
Translations directly affect key features of a function's graph. The following table summarises the shifts for a generic point (x0, y0) and for asymptotes:
平移直接影响函数图像的关键特征。下表总结了任意点 (x0, y0) 和渐近线的变化：
| Feature / 特征 | Under g(x) = f(x − h) + k |
|---|---|
| Point (x0, y0) | Moves to (x0 + h, y0 + k) |
| Vertical asymptote x = a | Becomes x = a + h |
| Horizontal asymptote y = b | Becomes y = b + k |
| Intercepts / 截距 | Shift horizontally and vertically according to h and k |
This means that the entire set of points moves in a unified way. For rational functions, vertical asymptotes move with horizontal translations, and horizontal asymptotes move with vertical translations. The domain and range shift accordingly.
这意味着整个点集以统一的方式移动。对于有理函数，垂直渐近线随水平平移移动，水平渐近线随垂直平移移动。定义域和值域也随之改变。 Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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