函数变换 — AP 微积分预科

1. 什么是函数变换？ ★★☆☆☆ ⏱ 3 min

函数变换是一组规则，用于修改已知"母函数" $f(x)$ 的图像或方程，得到一个新的相关函数。这项技能让你可以通过将复杂函数与更简单、熟悉的母函数关联起来分析，无需从头绘制图像。

函数变换是AP微积分预科第一单元的核心考核技能，该单元占考试总分的27–30%，在选择题和自由作答题部分都会考察。常见考题要求你将变换后的方程与图像匹配、从给定方程识别变换，或将变换应用到实际情境函数中。

2. 刚体变换：平移与反射 ★★☆☆☆ ⏱ 4 min

刚体变换仅改变母函数图像的位置，不改变其形状和大小。基础刚体变换的一般形式为：

g(x) = f(x - h) + k

参数 $k$ 控制竖直平移：在 $f(x)$ 的输出上加 $k$，会将原图像上的每个点 $(x, y)$ 移动到 $(x, y + k)$。若 $k>0$，图像向上平移 $k$ 个单位；若 $k<0$，图像向下平移 $|k|$ 个单位。参数 $h$ 控制水平平移：将输入 $x$ 替换为 $x-h$，会将每个点 $(x, y)$ 移动到 $(x + h, y)$。符号是常见的易错点：若 $h>0$，图像向右平移 $h$ 个单位。

刚体变换还包括反射：$g(x) = -f(x)$ 是图像关于 $x$-轴的反射（翻转所有 $y$-值），而 $g(x) = f(-x)$ 是关于 $y$-轴的反射（翻转所有 $x$-值）。

Exam tip: Always rewrite horizontal translations in the standard $f(x-h)$ form to confirm the shift direction. For example, rewrite $f(x+5)$ as $f(x - (-5))$ to avoid misreading it as a right shift.

3. Non-Rigid Transformations: Stretches and Compressions ★★★☆☆ ⏱ 5 min

Non-rigid transformations change the shape and size of the parent function's graph, rather than just its position. The general form for scaling transformations is:

g(x) = a f(bx)

For vertical scaling: multiplying the output of $f(x)$ by $a$ scales every $y$-value by $a$. If $|a|>1$, this is a vertical stretch by a factor of $|a|$ (the graph gets taller). If $0<|a|<1$, this is a vertical compression by a factor of $|a|$ (the graph gets shorter). If $a$ is negative, the scaling also includes a reflection over the $x$-axis.

For horizontal scaling: replacing the input $x$ with $bx$ scales every $x$-value by $\frac{1}{|b|}$. If $|b|>1$, this is a horizontal compression by a factor of $\frac{1}{|b|}$ (the graph gets narrower horizontally). If $0<|b|<1$, this is a horizontal stretch by a factor of $\frac{1}{|b|}$ (the graph gets wider horizontally). If $b$ is negative, the scaling also includes a reflection over the $y$-axis. The reciprocal rule for horizontal scaling is the most commonly tested rule for this subtopic.

Exam tip: Remember the reciprocal rule for horizontal scaling: the scale factor is always the reciprocal of the coefficient of $x$ inside the function. Never directly use the coefficient as the scale factor for horizontal transformations.

4. Combined Transformations and Order of Operations ★★★★☆ ⏱ 6 min

When multiple transformations are applied to a parent function, they must be applied in the correct order to get the right equation and graph. The standard form of any fully transformed function is:

g(x) = a \cdot f\left(b(x - h)\right) + k

Order of operations follows the same PEMDAS rules you use to evaluate $g(x)$ for a given input: first process operations on the input $x$ (inside the function), then process operations on the output of $f(x)$ (outside the function). The correct sequence is: 1. Horizontal transformations: shift by $h$, then scale/reflect by $b$ (because you subtract $h$ before multiplying by $b$ inside the parentheses). 2. Vertical transformations: scale/reflect by $a$, then shift by $k$ (because you multiply the output by $a$ before adding $k$). The most common mistake here is failing to factor out $b$ from the input term before identifying $h$, which leads to incorrect horizontal shift values.

Exam tip: Always factor the coefficient of $x$ out of the input term before identifying the horizontal shift. For example, $f(3x + 9) = f(3(x + 3))$, which is a 3-unit left shift, not a 9-unit left shift.