📚 Miscellaneous Transformations of Functions | 函数综合变换
In IB Mathematics, understanding how a function's graph changes under different algebraic operations is essential. These transformations—translation, stretching, reflection, and more—allow you to predict and sketch new graphs from a parent function without plotting every point. This article covers the full range of common and miscellaneous transformations that appear across the syllabus.
在 IB 数学中，理解函数图像在不同代数运算下的变化方式至关重要。这些变换——平移、伸缩、反射等等——能让你根据母函数预测和绘制新图像，而无需逐点描图。本文涵盖了课程中出现的所有常见及综合变换。
1. What Are Function Transformations? | 什么是函数变换？
A transformation modifies a function' applying operations to its equation. The usual parent functions include f(x) = x², f(x) = √x, f(x) = |x| e and size, like translation and reflection) or non-rigid (changing size, like stretches).
变换通过对函数方程施加运算来修改其图像。常见的母函数包括 f(x) = x²、f(x) = √x、f(x) = |x|、f(x) = 1/x 以及三角函数。变换可分为刚性的（保持形状和大小，如平移和反射）和非刚性的（改变大小，如伸缩）。
2. Vertical Translation | 垂直平移
Adding a constant k to a function shifts its graph vertically. The equation is
y = f(x) + k
parabola up by 3 units. The shape remains unchanged, but each output value increases by 3.
给函数加上常数 k 会使图像垂直移动。若 k 为正，图像向上移；若为负，则向下移。例如，y = x² + 3 将抛物线向上平移 3 个单位。形状不变，但每个输出值增加了 3。
3. Horizontal Translation | 水平平移
Replacing x by (x − h) shifts the graph horizontally. The form is
y = f(x − h)
es to the right; if h is negative, it moves left. Beware of the counterintuitive sign: y = √x − 2 shifts the square root graph 2 units right, not left. Think of finding the 'new zero' to confirm direction.
将 x 替换为 (x − h) 会使图像水平移动。如果 h 为正，图像向右平移；若 h 为负，则向左。注意符号可能反直觉：y = √x − 2 将平方根图像向右移动 2 个单位，而非向左。可通过寻找“新零点”来确认方向。
4. Vertical Stretch and Compression | 垂直伸缩
Multiplying the function by a constant a gives vertical stretching or compression.
y = a · f(x)
If |a| > 1, the graph is stretched vertically (narrower); if 0 < |a| s the amplitude compared to sin(x). If a is negative, it also reflects the graph across the x-axis.
将函数乘以常数 a 会产生垂直拉伸或压缩。若 |a| > 1，图像垂直拉伸（更窄）；若 0 < |a| < 1，则垂直压缩（更宽）。例如，y = 2 sin(x) 的振幅是 sin(x) 的两倍。若 a 为负，图像还会关于 x 轴反射。
5. Horizontal Stretch and Compression | 水平伸缩
Replacing x by bx causes a horizontal stretch or compression.
y = f(bx)
If 0 < |b| < 1, the graph stretches horizontally away from the y-axis; if |b| values are scaled by 1/b. For example, y = sin(2x) completes one cycle in π instead of 2π.
将 x 替换为 bx 会导致水平拉伸或压缩。若 0 < |b| < 1，图像水平拉伸远离 y 轴；若 |b| > 1，则向 y 轴压缩。这可能看起来相反，因为 x 值实际上按 1/b 缩放。例如，y = sin(2x) 在 π 内完成一个周期，而非 2π。
6. Reflections Across Axes | 关于坐标轴的反射
A reflection flips the graph over a line. The two primary reflections are over the x-axis and y-axis.
y = −f(x) reflects across the x-axis.
y = f(−x) reflects across the y-axis.
the graph unchanged. For odd functions, f(−x) = −f(x), implying symmetry about the origin.
反射使图像关于某条直线翻转。两个主要反射是关于 x 轴和关于 y 轴。y = −f(x) 关于 x 轴反射；y = f(−x) 关于 y 轴反射。对于偶函数，f(x) = f(−x)，因此关于 y 轴的反射保持图像不变。对于奇函数，f(−x) = −f(x)，意味着关于原点对称。
7. Combining Transformations: Order Matters | 组合变换：顺序很重要
When multiple transformations are applied, the sequence is crucial. A general combined form is
y = a · f(b(x − h)) + k
left 3, then square, then multiply by −2 (vertical stretch and reflect), then add 4 (shift up).
当施加多种变换时，顺序至关重要。一般组合形式为 y = a · f(b(x − h)) + k。推荐的操作顺序：先进行水平平移 h，再进行水平伸缩 b，然后是反射和垂直伸缩 a，最后是垂直平移 k。例如，将 y = x² 变换为 y = −2(x + 3)² + 4，先左移 3 个单位，平方后乘以 −2（垂直拉伸并反射），最后加 4（上移）。
8. Absolute Value Transformations | 绝对值变换
Two specific transformations involve the absolute value: y = |f(x)| and y = f(|x|).
y = |f(x)|
e graph below the x-axis and reflects it upward, making all y-values non-negative. For y = |x − 2|, the V-shape keeps its vertex at (2,0).
y = f(|x|)
ss the y-axis. This creates symmetry about the y-axis, regardless of the original function's parity. For example, y = (|x|)² yields a parabola always symmetric about the y-axis.
两种特定的变换涉及绝对值：y = |f(x)| 和 y = f(|x|)。y = |f(x)| 将 x 轴下方的任何部分反射到上方，使得所有 y 值非负。对于 y = |x − 2|，V 形顶点保持在 (2,0)。y = f(|x|) 使得 x ≥ 0 部分的图像保持不变，而左侧 (x < 0) 由右侧部分关于 y 轴的镜像替代。这将产生关于 y 轴的对称性，无论原函数是奇是偶。例如，y = (|x|)² 产生一个始终关于 y 轴对称的抛物线。
9. Other Miscellaneous Transformations (Square Roots and Reciprocals) | 其他综合变换（平方根与倒数）
Two less common but important transformations involve the square root and reciprocal of a function.
y = √(f(x))
alleys, because the square root reduces larger values more significantly. For example, y = √x² − 1 exists for |x| ≥ 1 and has a different curve than the original quadratic.
y = 1 / f(x)
r instance, y = 1 / sin(x) yields the cosecant function with asymptotes at multiples of π.
两种不太常见但重要的变换涉及平方根和倒数。y = √(f(x)) 仅定义在 f(x) ≥ 0 的区域。所得图像通常会使波峰变平、波谷变尖，因为平方根对较大值的缩减更显著。例如，y = √x² − 1 在 |x| ≥ 1 时存在，曲线与原二次函数不同。y = 1 / f(x) 的倒数变换将较大的 f(x) 值映射为小正数，小正数映射为大正数。垂直渐近线出现在 f(x) = 0 处。例如，y = 1 / sin(x) 产生余割函数，在 π 的倍数处有渐近线。
10. Describing Transformations from Given Graphs | 根据图像描述变换
IB exam questions often ask you to describe the sequence of transformations that map one graph to another. Start by identifying the parent function. Then compare key points: the vertex, intercepts, or asymptotes. Write transformations in a logical order—horizontal shifts first, then scaling, then reflections, then vertical shifts. Use clear language such as "translation 3 units left," "vertical stretch by factor 2," or "reflection in the x-axis."
IB 考题常要求描述将一个图像映射到另一个的一系列变换。首先识别母函数。然后比较关键点：顶点、截距或渐近线。按逻辑顺序写出变换——先水平移动，再伸缩，然后反射，最后垂直移动。使用清晰的语言，如“向左平移 3 个单位”、“垂直拉伸 2 倍”或“关于 x 轴反射”。
11. Transformations Applied to Specific Functions | 对特定函数应用变换
, amplitude, and phase shift: y = A sin(B(x − C)) + D. In exponential functions, y = a · bx − h y = k. Logarithmic functions y = a log_b(x − h) + k shift the vertical asymptote to x = h.
某些函数有其典型的变换任务。对于二次函数，配方后可得顶点式 a(x − h)² + k。对于三角函数，变换影响周期、振幅和相移：y = A sin(B(x − C)) + D。在指数函数中，y = a · bx − h + k 将水平渐近线平移到 y = k。对数函数 y = a log_b(x − h) + k 将垂直渐近线移到 x = h。
12. Summary and Key Points | 总结与要点
Mastering function transformations means recognizing that a single algebraic change produces a predictable geometric effect. Remember: f(x) + k translates vertically, f(x − h) translates horizontally, a · f(x) stretches vertically, f(bx) stretches horizontally, negative signs reflect, and absolute values fold the graph. Always consider domain restrictions when applying square roots or reciprocals. When sequences are given, perform operations on x before applying the main function, then apply vertical scaling and shifting. Practicing with a variety of functions—polynomials, exponentials, trig—will build the intuition needed to tackle any miscellaneous transformation question on the IB exam.
掌握函数变换意味着认识到单一的代数变化会产生可预测的几何效果。记住：f(x) + k 垂直平移，f(x − h) 水平平移，a · f(x) 垂直伸缩，f(bx) 水平伸缩，负号反射，绝对值使图像翻折。在应用平方根或倒数时，始终要考虑定义域限制。当变换序列给出时，先在 x 上进行操作再应用主函数，然后进行垂直缩放和平移。练习各种函数——多项式、指数、三角——将培养你在 IB 考试中应对任何综合变换问题所需的直觉。
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