📚 Transformation of Graphs | 图形变换
Graph transformations allow us to take a basic function and shift, stretch, or reflect its graph to model more complex situations without recalculating every point. In IB Mathematics, understanding these transformations is essential for sketching unfamiliar functions and solving equations graphically. By mastering a small set of standard rules, you can predict precisely how changing the equation affects the image of the graph. This topic underpins everything from quadratic modelling to trigonometric analysis, making it a core skill across both Analysis & Approaches and Applications & Interpretation courses.
图形变换使我们能够对一个基本函数进行平移、伸缩或反射，从而在不重新计算每一个点的情况下构建更为复杂的模型。在IB数学中，理解这些变换对绘制陌生函数的草图和利用图像求解方程至关重要。通过掌握少量标准规则，你可以准确预测方程的变化会如何影响图形。这一主题是二次函数建模、三角函数分析等诸多内容的基础，因此也是分析与方法、应用与解释两门课程中的核心技能。
1. Understanding the Parent Function | 理解母函数
function, which is the simplest form of a function in a family, such as f(x) = x², f(x) = √x s the first step to sketching or analysing its graph efficiently. For instance, g(x) = 2(x – (x) = x², with vertical stretch, horizontal translation, and vertical translation applied.
每一种变换都从母函数开始，母函数是某个函数族中最简单的形式，例如 f(x) = x²、f(x) = √x 或 f(x) = sin x。当我们施加变换时，我们修改这个母函数来产生新图形，它们保留了基本形状，但出现在不同的位置或具有不同的比例。在复杂表达式中识别出母函数是有效绘制或分析其图像的第一步。例如，g(x) = 2(x – 3)² + 1 的母函数是 f(x) = x²，同时应用了垂直伸缩、水平平移和垂直平移。
2. Vertical Translation: y = f(x) + k | 垂直平移：y = f(x) + k
ph vertically. If k > 0, the graph moves upward by k units; if k < 0, it moves downward by |k| ing with f(x) = x³, the graph of g(x) = x³ + 4 is the same cubic curve shifted 4 units up.
将常数 k 加到函数输出上会使整个图形垂直移动。若 k > 0，图像向上移动 k 个单位；若 k < 0，图像向下移动 |k| 个单位。图形形状保持不变，原图上每一点 (x, y) 变为 (x, y + k)。这一变换极为常见，通常是绘制变换后函数的最后一步。例如，从 f(x) = x³ 出发，g(x) = x³ + 4 的图像就是同一条三次曲线向上平移了 4 个单位。
3. Horizontal Translation: y = f(x + h) | 水平平移：y = f(x + h)
lly. Here, the graph shifts to the left by h units when h is positive, and to the right by |h| units when h is negative – ccurs because the input value must compensate for the addition inside the argument. So f(x – f setting x + h = 0 to find the new position of key features like the vertex or intercept.
在函数内部用 (x + h) 替换 x 会使图形水平移动。当 h 为正时，图像向左移动 h 个单位；当 h 为负时，图像向右移动 |h| 个单位——这与许多同学的直觉恰好相反。原因是输入值必须抵消参数内的加法。因此 f(x – 2) 将 f(x) 的图像向右平移 2 个单位，而 f(x + 5) 将其向左平移 5 个单位。可以尝试令 x + h = 0 来找到顶点或截距等关键特征的新位置，这样更直观。
4. Vertical Stretch and Compression: y = a f(x) | 垂直伸缩：y = a f(x)
Multiplying the entire function by a factor a changes the vertical scale. When |a| > 1, the graph stretches away from the x‑axis; when 0 < |a| < 1, it compresses toward the x‑axis. Each y‑coordinate is multiplied by a, so points on the x‑axis stay fixed. If a is negative, the graph is also reflected in the x‑ attens the parabola. In IB problems, you will often need to identify the value of a from a given diagram.
将整个函数乘以因子 a 会改变垂直比例。当 |a| > 1 时，图形沿 y 轴方向拉伸远离 x 轴；当 0 < |a| < 1 时，图形向 x 轴方向压缩。每一个 y 坐标都乘以 a，因此 x 轴上的点保持不变。若 a 为负，图形还会关于 x 轴反射。因此，y = 3 sin x 使正弦波的振幅变为原来的三倍，而 y = ½ x² 则使抛物线变得扁平。在 IB 考试题中，你经常需要根据给定图像确定 a 的值。
5. Horizontal Stretch and Compression: y = f(bx) | 水平伸缩：y = f(bx)
Replacing x with bx scales the graph horizontally. For |b| > 1, the graph compresses toward the y‑axis by a factor of 1/|b|; for 0 < |b| < 1, it stretches away from the y‑ and mirrors the horizontal translation rule. If b is negative, the graph reflects in the y‑ = sin(2x) completes a full period in 180° instead of 360°, effectively squeezing the wave.
用 bx 替换 x 会在水平方向上缩放图形。当 |b| > 1 时，图形向 y 轴方向压缩，比例为 1/|b|；当 0 < |b| < 1 时，图形沿水平方向拉伸远离 y 轴。这与直观感觉相反，与水平平移规则相似。如果 b 为负，图形将关于 y 轴反射。一个有用的检验方法是，原图上点 (x, y) 在变换后变为 (x/b, y)。例如，y = sin(2x) 在 180° 内完成一个完整周期，而不是 360°，仿佛把波形挤紧了。
6. Reflection in the x‑axis: y = –f(x) | 关于 x 轴的反射：y = –f(x)
Negating the function output flips the graph over the x‑axis. All y‑coordinates change sign, so a maximum becomes a minimum and vice versa. The x‑ d. This transformation is often combined with vertical stretches. In quadratic graphs, y = –x² is the familiar upside‑ s the wave, which is equivalent to a phase shift in some cases but geometrically distinct.
将函数输出取负会使图形关于 x 轴翻转。所有 y 坐标改变符号，因此最大值变为最小值，反之亦然。x 轴截距保持不变，但图形的整体朝向反转。这一变换经常与垂直伸缩组合使用。在二次函数图像中，y = –x² 就是我们熟悉的倒置抛物线。应用于三角函数时，它会使波形颠倒，这在某些情况下与相位移动等效，但在几何上是不同的变换。
7. Reflection in the y‑axis: y = f(–x) | 关于 y 轴的反射：y = f(–x)
Negating the input variable x reflects the graph across the y‑axis. Points to the right of the y‑axis move to the left and vice versa. The y‑ particularly important for testing whether a function is even or odd. For example, f(x) = √x cannot be reflected in the y‑ e domain, but f(x) = x² remains unchanged after reflection because it is an even function.
将输入变量 x 取负会使图形关于 y 轴反射。y 轴右侧的点移到左侧，反之亦然。y 轴截距保持不变，图像形状与原图全等。这一变换对于检验函数的奇偶性尤为重要。例如，f(x) = √x 若不限制定义域就无法关于 y 轴反射，而 f(x) = x² 反射后与原图相同，因为它是一个偶函数。
8. Combining Transformations: Order Matters | 组合变换：顺序至关重要
ly transformations horizontally first (from inside to outside), then vertically. Horizontal transformations work on the x‑ first reflection/stretch (b), then translation (h). Vertical transformations work on the y‑ e sequence and end up with an incorrect graph, so careful attention to order is essential.
当同一个母函数上施加了多种变换时，执行顺序至关重要。IB 的标准方法是先将变换函数写成 y = a f(b(x + h)) + k 的形式，然后先水平后垂直，水平方向从内到外进行。水平变换作用于 x 坐标：先反射/伸缩 (b)，再平移 (h)。垂直变换作用于 y 坐标：先伸缩/反射 (a)，再平移 (k)。同学们经常混淆顺序，导致图像绘制错误，因此必须格外注意先后次序。
For example, consider the function y = 2 f(3(x – r 2, and finally shift up by 4. Reversing the translation and stretch steps would place the vertex in the wrong location.
例如，对于函数 y = 2 f(3(x – 1)) + 4，你需要从母函数 f(x) 出发，先将水平方向压缩至 1/3，再向右平移 1，然后将垂直方向拉伸为 2 倍，最后上移 4。如果颠倒平移和伸缩的步骤，顶点就会出现在错误的位置。
9. Absolute Value Transformations | 绝对值变换
volving the absolute value, producing two distinctive graph shapes. The transformation y = |f(x)| keeps the parts of the graph that lie above the x‑axis unchanged and reflects any portion below the x‑axis upward, making the entire graph non‑negative. The transformation y = f(|x|) discards the left‑hand part of the graph (for x < 0) and replaces it with a mirror image of the right‑ gularly and are easily sketched by considering how negative outputs or inputs are handled.
IB 数学还包含涉及绝对值的变换，会产生两种独特的图形形状。变换 y = |f(x)| 将位于 x 轴上方部分的图像保持不变，而将 x 轴下方的任何部分向上反射，使得整个图形非负。变换 y = f(|x|) 则舍弃左侧图像（x < 0），并用右侧图像的镜像替代，从而得到一个偶函数。这两种变换都是常见考点，通过思考如何处理负值输出或输入，就可以轻松绘制草图。
For instance, starting with the line f(x) = x – 2, the graph of |f(x)| becomes a V‑shape with the vertex at (2,0), while f(|x|) = |x| – 2 produces a W‑like shape symmetric about the y‑axis.
例如，从直线 f(x) = x – 2 出发，|f(x)| 的图像变成以 (2,0) 为顶点的 V 形，而 f(|x|) = |x| – 2 则产生一个关于 y 轴对称的 W 形。
10. Transformations of Trigonometric Graphs | 三角函数的图形变换
b(x + c)) + d or y = a cos(b(x + c)) + d encapsulates all the basic transformations. Here, |a| controls the amplitude, |b| determines the period (period = 2π/|b| ask you to find the equation from a given wave or sketch one when parameters are provided.
三角函数是练习图形变换的绝佳对象，因为其周期性质使得伸缩和平移的效果格外明显。标准形式 y = a sin(b(x + c)) + d 或 y = a cos(b(x + c)) + d 包含了所有基本变换。其中，|a| 控制振幅，|b| 决定周期（正弦和余弦的周期为 2π/|b|），c 给出相位移（水平平移），而 d 设定主轴线（垂直平移）。考试题常常要求根据给定的波形求方程，或者在给出参数的情况下绘制草图。
For example, y = 3 cos(2(x – π/4)) – 1 has amplitude 3, period π, phase shift π/4 to the right, and principal axis y = –1. Visualising these step by step reinforces understanding of combined transformations.
例如，y = 3 cos(2(x – π/4)) – 1 的振幅为 3，周期为 π，相位移为向右 π/4，主轴线为 y = –1。通过逐步想象每一重变换，可以加深对组合变换的理解。
11. Using Transformations to Sketch Graphs | 利用变换绘制图形草图
Once you have broken a complicated function into its parent and a series of transformations, you can sketch the graph quickly without a table of values. Start by drawing the parent shape lightly. Then apply horizontal stretch/compression and reflection (b), followed by horizontal translation (h). Next, apply vertical stretch/compression and reflection (a), and finally vertical translation (k). Mark the new position of key points such as intercepts, vertices, and asymptotes. This systematic method reduces errors and is highly valued in IB exam solutions where you must show your reasoning.
一旦将复杂函数分解为母函数和一系列变换，你无需借助数值表就能快速绘制草图。首先轻轻画出母函数形状。然后依次应用水平伸缩和反射（b），再进行水平平移（h）。接下来，应用垂直伸缩和反射（a），最后进行垂直平移（k）。标出截距、顶点和渐近线等关键点的新位置。这种系统化的方法能减少错误，在 IB 考试中备受推崇，因为你需要展示推理过程。
A typical exam question might present g(x) = 1 – 2 f(–x + 3). Rewrite it as g(x) = –2 f(–(x – 3)) + 1 to see clearly: reflect in y‑axis, shift right 3, reflect in x‑ ly by factor 2, then shift up 1. Working through the sequence avoids misplacing the graph.
一个典型的考题可能给出 g(x) = 1 – 2 f(–x + 3)。将其改写为 g(x) = –2 f(–(x – 3)) + 1 即可清晰地看出：先关于 y 轴反射，右移 3，再关于 x 轴反射并垂直拉伸 2 倍，最后上移 1。按顺序操作可以避免图形位置出错。
12. Summary and Key Points | 总结与关键点
ns. Remember that vertical changes (adding to the output, multiplying the output) affect y‑coordinates directly, while horizontal changes (modifying the input) affect x‑ ply transformations from left to right, inside to outside. Practice linking each parameter to a geometric change, and double‑ ng this topic will strengthen your function analysis skills across the entire IB syllabus.
图形变换为你提供了解释和构建函数的强大工具组合。要记住，垂直变化（输出上加常数、输出乘以常数）直接影响 y 坐标，而水平变化（修改输入）则以相反方向影响 x 坐标。务必把函数写成标准形式 y = a f(b(x + h)) + k，并从左至右、由内向外地施加变换。练习将每个参数与具体的几何变化联系起来，并在不确定时用检验点进行复核。掌握这一主题将全面提升你在整个 IB 课程中的函数分析能力。
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