Combined Transformations of Function Graphs | 函数图像的综合变换

📚 Combined Transformations of Function Graphs | 函数图像的综合变换

Function graphs are one of the most visual topics in IB Mathematics. A solid grasp of how translations, reflections and stretches combine allows you to sketch complicated functions quickly and solve problems that would otherwise require lengthy algebra. In this revision article, we will break down the rules, reveal the common traps, and practise a systematic method that works for polynomials, exponentials, trigonometric functions and beyond.

函数图像是IB数学中最具视觉性的话题之一。透彻理解平移、反射与伸缩如何相互叠加，可以帮助你快速画出复杂函数的草图，并解决那些原本需要大量代数运算的问题。在这篇复习文章中，我们将拆解规则、揭示常见陷阱，并练习一套适用于多项式、指数函数、三角函数等各类函数的系统方法。

1. The Basic Transformations | 基础变换回顾

Before combining transformations, you must be fluent with the three basic types. A vertical translation moves the graph up or down; a horizontal translation moves it left or right; a stretch changes its shape in one direction; and a reflection flips it across an axis.

在组合变换之前，你首先必须熟练三种基本类型。垂直平移将图像上下移动；水平平移将图像左右移动；伸缩改变图像在某一方向上的形状；反射则将图像沿某条坐标轴翻转。

Transformation	Rule	Effect
Vertical shift	y = f(x) + k	Move up by k (k > 0) \| 向上平移 k
Horizontal shift	y = f(x – h)	Move right by h (h > 0) \| 向右平移 h
Vertical stretch	y = a·f(x)	Stretch vertically by factor a \| 纵向拉伸 a 倍
Horizontal stretch	y = f(bx)	Compress horizontally if b > 1 \| b > 1 时横向压缩
Reflection in x-axis	y = -f(x)	Flip over the x-axis \| 关于 x 轴翻转
Reflection in y-axis	y = f(-x)	Flip over the y-axis \| 关于 y 轴翻转

Notice that a negative value of a or b automatically includes a reflection. For example, y = -2·f(x) first stretches the graph vertically by factor 2 and then reflects it across the x-axis.

请注意，a 或 b 为负值时自动包含反射。例如，y = -2·f(x) 先将图像纵向拉伸2倍，再关于 x 轴反射。

2. The General Transformation Equation | 一般变换公式

Any combination of these transformations can be written in the standard form

任意组合变换都可以写成如下标准形式：

y = a·f(b(x – h)) + k

Here, a controls the vertical stretch and the possible x-axis reflection; k controls the vertical translation; b controls the horizontal stretch and the possible y-axis reflection; and h controls the horizontal translation. Once you identify a, b, h and k, you can describe every transformation systematically.

其中 a 控制纵向伸缩及可能的 x 轴反射；k 控制纵向平移；b 控制横向伸缩及可能的 y 轴反射；h 控制横向平移。一旦你确定了 a、b、h、k 的值，就能系统化地描述每一个变换。

For example, the expression y = 3·f(2(x – 4)) + 5 tells us that the graph is vertically stretched by 3, shifted up by 5, horizontally compressed by 2, and shifted right by 4.

例如，表达式 y = 3·f(2(x – 4)) + 5 说明图像被纵向拉伸3倍、向上平移5个单位、横向压缩为原来的二分之一、并向右平移4个单位。

3. The Correct Order of Transformations | 正确的变换顺序

Order matters when transformations share the same direction. For the y-direction, the sequence is always: stretch first, then translate. For the x-direction, when the expression is written as f(b(x – h)), the sequence is: translate first, then stretch.

当变换位于同一方向上时，顺序非常重要。对于 y 方向，顺序永远是：先伸缩，后平移。对于 x 方向，当表达式写成 f(b(x – h)) 的形式时，顺序是：先平移，后伸缩。

To see why, take y = -f(x) + 1. If you first reflect and then translate up by 1, you get y = -f(x) + 1, which is correct. If you translate first and then reflect, you get y = -(f(x) + 1) = -f(x) – 1, a completely different graph.

以 y = -f(x) + 1 为例。如果你先反射再向上平移1个单位，得到 y = -f(x) + 1，这是正确的。如果你先平移再反射，会得到 y = -(f(x) + 1) = -f(x) – 1，完全是另一条图像。

For the horizontal direction, always rewrite the expression in the form f(b(x – h)) before reading the translation. The expression f(2x – 6) should become f(2(x – 3)), which means shift right by 3, then compress horizontally by factor 2.

在水平方向上，务必先将表达式改写为 f(b(x – h)) 的形式，再读取平移量。表达式 f(2x – 6) 应该改写为 f(2(x – 3))，这表示先向右平移3个单位，再横向压缩为原来的二分之一。

4. From Transformations to Equations | 由变换描述求函数表达式

In exams, you may be given a description such as “the graph of y = f(x) is stretched vertically by factor 3, reflected in the x-axis, then translated left by 2 units” and asked to write the new equation.

在考试中，你可能会遇到这样的描述：“将 y = f(x) 的图像纵向拉伸3倍，关于 x 轴反射，再向左平移2个单位”，并请写出新的方程。

Apply the y-direction rules first. A vertical stretch by factor 3 gives y = 3·f(x). A reflection in the x-axis gives y = -3·f(x). A translation left by 2 units gives y = -3·f(x + 2).

先应用 y 方向的规则。纵向拉伸3倍得到 y = 3·f(x)。关于 x 轴反射得到 y = -3·f(x)。向左平移2个单位得到 y = -3·f(x + 2)。

If the description mixes directions, treat the x-direction independently. For example, “shift right by 4 and stretch horizontally by factor 2” becomes y = f(2(x – 4)).

如果描述混合了不同方向，请独立处理 x 方向。例如，“向右平移4个单位并横向拉伸2倍”应写成 y = f(2(x – 4))。

5. Identifying Transformations from a Graph | 从图像识别变换

Given the graph of y = f(x) and its transformed version, you should be able to read off the parameters. Start with a known point on the original graph and trace where it lands on the transformed graph.

已知 y = f(x) 的图像及其变换后的图像，你应该能够读出参数。先取原图像上的一个已知点，追踪它落在变换后图像上的位置。

Suppose f(x) has a key point at (0, 0) and its image appears at (5, 2). This suggests a horizontal shift right by 5 and a vertical shift up by 2: k = 2, h = 5. If another original point, say (2, 4), lands at (6, 6), the change in x relative to the first point is unchanged, so no horizontal stretch is present; the change in y is also consistent with k = 2 only.

假设 f(x) 有一个关键点 (0, 0)，它的像出现在 (5, 2)。这表明水平向右平移5个单位、垂直向上平移2个单位：k = 2, h = 5。如果另一个原始点 (2, 4) 落在 (6, 6)，相对于第一个点的 x 变化保持不变，说明没有横向伸缩；y 的变化也与 k = 2 一致。

To detect stretches, compare the distance between two transformed points with the distance between the corresponding original points. If the vertical distance doubles, then a = 2; if the horizontal distance halves, then b = 2.

为了检测伸缩，请比较两个变换后点之间的距离与两个原始点之间的距离。如果垂直距离变为原来的两倍，则 a = 2；如果水平距离变为原来的二分之一，则 b = 2。

6. Transformations and the Point Formula | 变换与点坐标公式

A powerful short-cut for solving transformed-point problems is the point formula. If the original graph y = f(x) contains the point (x₀, y₀), then the graph of y = a·f(b(x – h)) + k contains the corresponding point

解决变换点问题的一个强大捷径是点坐标公式。如果原图像 y = f(x) 经过点 (x₀, y₀)，那么图像 y = a·f(b(x – h)) + k 经过对应的点：

(h + x₀/b, a·y₀ + k)

Let us verify this with an example. If f contains (1, 2), then for y = 3·f(2(x – 5)) + 4 the new point is x = 5 + 1/2 = 5.5 and y = 3 × 2 + 4 = 10. Check the algebra: 2(x – 5) = 2(5.5 – 5) = 1, so f(1) = 2; then y = 3 × 2 + 4 = 10. The formula works.

让我们通过一个例子验证。如果 f 经过点 (1, 2)，那么对于 y = 3·f(2(x – 5)) + 4，新点的坐标为 x = 5 + 1/2 = 5.5，y = 3 × 2 + 4 = 10。验证代数：2(x – 5) = 2(5.5 – 5) = 1，所以 f(1) = 2；则 y = 3 × 2 + 4 = 10。公式成立。

7. Using Transformations to Solve Equations | 用变换思想解方程

Transformations can convert a difficult equation into a simpler one. If you need to solve f(2x – 4) = 0 and you already know the roots of f(x) = 0, you can first rewrite f(2x – 4) as f(2(x – 2)). If f has roots at x = -1 and x = 3, then each root x₀ maps to the new x-coordinate h + x₀/b = 2 + x₀/2.

变换可以将困难的方程转化为更简单的方程。如果需要解 f(2x – 4) = 0，且你已经知道 f(x) = 0 的根，可以先将 f(2x – 4) 改写为 f(2(x – 2))。如果 f 的根为 x = -1 和 x = 3，那么每个根 x₀ 映射到新的 x 坐标为 h + x₀/b = 2 + x₀/2。

Therefore the new roots are 2 + (-1)/2 = 1.5 and 2 + 3/2 = 3.5. In general, the roots of f(b(x – h)) = 0 are found from x = h + x₀/b for each original root x₀.

因此新根为 2 + (-1)/2 = 1.5 和 2 + 3/2 = 3.5。一般地，f(b(x – h)) = 0 的根可以通过公式 x = h + x₀/b 由每个原始根 x₀ 求得。

This approach also works for inequalities. The solution set of f(x) > 0 is a collection of open intervals; after transformation, every interval must be mapped by x ↦ h + x/b, and the inequality symbol remains unchanged as long as b > 0. If b < 0, the orientation of the intervals reverses.

这种方法同样适用于不等式。f(x) > 0 的解集是一些开区间；变换之后，每个区间都必须通过映射 x ↦ h + x/b 进行转换，只要 b > 0，不等号方向不变。如果 b < 0，区间的方向会反转。

8. Symmetry and Odd/Even Functions | 对称性与奇偶函数

Transformations interact with symmetry in predictable ways. An even function satisfies f(-x) = f(x) and is symmetric about the y-axis. An odd function satisfies f(-x) = -f(x) and has 180° rotational symmetry about the origin.

变换与对称性之间存在可预测的相互作用。偶函数满足 f(-x) = f(x)，关于 y 轴对称。奇函数满足 f(-x) = -f(x)，关于原点具有180°旋转对称性。

If an even function is translated horizontally by h, the symmetry axis moves from x = 0 to x = h. For example, y = (x – 3)² has its axis of symmetry at x = 3. Similarly, if an odd function is translated by (h, k), the point of rotational symmetry moves to (h, k).

如果偶函数经过水平平移 h，对称轴会从 x = 0 移动到 x = h。例如，y = (x – 3)² 的对称轴位于 x = 3。类似地，如果奇函数平移了 (h, k)，旋转对称中心会移动到 (h, k)。

A quick test for the axis of symmetry of a transformed quadratic or trigonometric graph: the axis is always x = h in the standard form. Never use the vertex alone; check that f(h + t) = f(h – t) for the transformed function.

快速判断变换后二次函数或三角函数图像的对称轴：标准形式下对称轴总是 x = h。不要只依赖顶点；请验证变换后的函数满足 f(h + t) = f(h – t)。

9. Trigonometric Transformations in Practice | 三角函数的综合变换

Trigonometric graphs are the most common place where all four transformations appear together. Consider

三角函数图像是四个变换同时出现的最常见场合。考虑：

y = 2·sin(3(x – π/4)) + 1

Starting from y = sin x, apply the x-transformations first: shift right by π/4, then compress horizontally by factor 3. The period becomes 2π/3. Then apply the y-transformations: stretch vertically by factor 2, then shift up by 1. The new amplitude is 2, the new midline is y = 1, and the new phase shift is π/4 to the right.Published by TutorHao | IB Mathematics Revision Series | aleveler.com

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Combined Transformations of Function Graphs | 函数图像的综合变换