函数变换

AP 微积分预科· AP 微积分预科课程与考试说明 — 多项式与有理函数· 14 分钟阅读

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

📘 定义

刚体变换

保留母函数图像形状和大小，仅改变其在坐标平面上位置的变换，包括平移和反射。

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）。

📐 例题

母函数 $f (x) = x^{2}$ 的顶点在 $(0, 0)$ 。写出该函数经过向左平移4个单位、向上平移1个单位，再关于 $x$ -轴反射后的方程。新顶点的坐标是什么？

1
首先应用平移：向左平移4个单位意味着 $h = - 4$ ，因此 $x - h = x - (- 4) = x + 4$ 。向上平移1个单位意味着 $k = 1$ ，平移后得到：
2
$f(x+4) + 1 = (x+4)^2 + 1$
3
应用关于 $x$ -轴的反射：将整个函数乘以 $- 1$ 翻转所有输出的符号，得到：
4
$g(x) = -\left((x+4)^2 + 1\right) = -(x+4)^2 - 1$
5
原顶点 $(0, 0)$ 左移4、上移1后得到 $(- 4, 1)$ 。
6
Reflection over the $x$ -axis flips the sign of the $y$ -coordinate, so the new vertex is $(- 4, - 1)$ .

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)

📘 定义

Non-Rigid Transformation

A transformation that changes the shape and/or size of the parent function's graph. Includes vertical and horizontal stretches and compressions.

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 $b x$ 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.

📐 例题

Given parent function $f (x) = x^{3}$ , write the equation of the transformed function after a horizontal compression by a factor of $\frac{1}{2}$ and a vertical stretch by a factor of 4. What is the value of the transformed function at $x = 3$ ?

1
A horizontal compression by factor $\frac{1}{2}$ means the scale factor $\frac{1}{∣ b ∣} = \frac{1}{2}$ , so solving for $b$ gives $∣ b ∣ = 2$ . There is no reflection, so $b = 2$ .
2
A vertical stretch by factor 4 means $a = 4$ , with no reflection, so the general form is:
3
$g(x) = 4f(2x)$
4
Substitute $f (z) = z^{3}$ to get:
5
$g(x) = 4(2x)^3 = 4(8x^3) = 32x^3$
6
Evaluate at $x = 3$ :
7
$g(3) = 32(3)^3 = 32(27) = 864$

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.

📐 例题

Rewrite the rational function $g (x) = \frac{2 x + 1}{x + 3}$ as a transformation of the parent function $f (x) = \frac{1}{x}$ , then list all transformations in order.

1
Rewrite the numerator to separate the constant term: $2 x + 1 = 2 (x + 3) - 5$ .
2
Simplify the function:
3
$g(x) = \frac{2(x + 3) - 5}{x + 3} = 2 - \frac{5}{x + 3} = -5 \cdot \frac{1}{x + 3} + 2$
4
Write in standard form:
5
$g(x) = -5 \cdot f\left(1(x - (-3))\right) + 2$
6
So $a = - 5$ , $b = 1$ , $h = - 3$ , $k = 2$ . List transformations in order: (1) Shift $f (x) = \frac{1}{x}$ 3 units left (no horizontal scaling/reflect), (2) Stretch vertically by a factor of 5, reflect over the $x$ -axis, then shift up 2 units. Verifying: the point $(1, 1)$ on $f (x)$ becomes $(- 2, - 3)$ after transformations, and $g (- 2) = \frac{2 ( - 2 ) + 1}{- 2 + 3} = - 3$ , which matches.

✓ 快速检测

Test your understanding of combined transformations with this AP-style multiple-choice question:

The graph of $f (x) = x^{4} - 2 x^{2}$ is shifted 1 unit right, reflected over the $y$ -axis, then shifted 3 units down. Which of the following is the equation of the transformed function?
- A) $g (x) = (- x - 1)^{4} - 2 (- x - 1)^{2} + 3$
- B) $g (x) = - (x - 1)^{4} + 2 (x - 1)^{2} - 3$
- C) $g (x) = (- (x - 1))^{4} - 2 (- (x - 1))^{2} - 3$
- D) $g (x) = (- x + 1)^{4} - 2 (- x + 1)^{2} + 3$
显示答案
C —
Correct! We apply transformations step-by-step: shift 1 right ( $x \to x - 1$ ), reflect over y-axis ( $x \to - x$ ), shift 3 down (subtract 3 from the output), giving $g (x) = f (- (x - 1)) - 3$ , which matches option C. If you got a different answer, check that you substituted reflection into the input correctly.

📐 例题

A local bakery models its monthly profit $P (t)$ , in thousands of dollars, $t$ months after January 2023, as $P (t) = - 0.4 t (t - 10)$ , a quadratic function peaking at 5 months. In 2024, the bakery shifts its peak profit to 2 months earlier than in 2023, and its maximum profit decreases by 2 thousand dollars compared to 2023. Write the function $Q (s)$ that models monthly profit $s$ months after January 2024, and find the maximum profit for 2024.

1
A peak 2 months earlier means a horizontal shift 2 units left, so $h = - 2$ , and we replace $t$ with $s + 2$ . A maximum profit decrease of 2 thousand dollars means a vertical shift down 2 units, so subtract 2 from the output.
2
The transformed function is:
3
$Q(s) = -0.4(s + 2)(s + 2 - 10) - 2 = -0.4(s + 2)(s - 8) - 2$
4
The original 2023 maximum profit is $P (5) = - 0.4 (5) (- 5) = 10$ thousand dollars. After a 2 thousand dollar decrease, the new maximum profit is $10 - 2 = 8$ thousand dollars. In context, the bakery's maximum monthly profit in 2024 is $8,000, occurring 3 months after January 2024.

Exam tip:

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

5. 常见陷阱

错误做法:

Interpreting $f (x + 4)$ as a 4-unit shift right instead of 4 units left

原因:

Students associate positive numbers with right movement, and forget the standard form uses $f (x - h)$ , so a positive shift inside the function is actually negative $h$ .

正确做法:

Always rewrite the input as $x - h$ to confirm: $f (x + 4) = f (x - (- 4))$ , so $h = - 4$ , which is a 4-unit left shift.

错误做法:

Interpreting $f (4 x)$ as a horizontal stretch by a factor of 4 instead of a horizontal compression by a factor of $\frac{1}{4}$

原因:

Students directly match the coefficient $b$ to the scale factor, instead of using the reciprocal rule for horizontal transformations.

正确做法:

For input coefficient $b$ , the horizontal scale factor is always $\frac{1}{∣ b ∣}$ , so $b = 4$ gives a scale factor of $\frac{1}{4}$ , which is a compression.

错误做法:

Calculating a 6-unit left shift for $f (2 x + 6)$ , instead of a 3-unit left shift

原因:

Students do not factor out the coefficient of $x$ before reading the shift value $h$ .

正确做法:

Always factor the coefficient of $x$ inside the function: $f (2 x + 6) = f (2 (x + 3)) = f (2 (x - (- 3)))$ , so $h = - 3$ , which is a 3-unit left shift.

错误做法:

Applying vertical shift before vertical stretch for $2 f (x) + 3$ , leading to incorrect output values

原因:

Students forget order of operations, and do addition before multiplication.

正确做法:

Always apply stretches/compressions/reflections (multiplication steps) before vertical shifts (addition steps), following PEMDAS order.

错误做法:

Confusing $- f (x)$ as a reflection over the $y$ -axis instead of the $x$ -axis

原因:

Students mix up whether the negative sign applies to the input (inside $f$ ) or output (outside $f$ ).

正确做法:

Negative outside $f$ flips $y$ -values → reflection over $x$ -axis; negative inside $f$ flips $x$ -values → reflection over $y$ -axis.

6. 速查表

Category	Formula / Rule	Notes
General Transformed Function	$g (x) = a f (b (x - h)) + k$	$a$ : vertical scale/reflection; $b$ : horizontal scale/reflection; $h$ : horizontal shift; $k$ : vertical shift
Vertical Translation	$f (x) + k$	$k > 0$ : shift up $∣ k ∣$ units; $k < 0$ : shift down $∣ k ∣$ units
Horizontal Translation	$f (x - h)$	$h > 0$ : shift right $∣ h ∣$ units; $h < 0$ : shift left $∣ h ∣$ units
Reflection over $x$ -axis	$- f (x)$	Negates all $y$ -values; flips over horizontal axis
Reflection over $y$ -axis	$f (- x)$	Negates all $x$ -values; flips over vertical axis
Vertical Stretch/Compression	$a f (x)$	$∣ a ∣ > 1$ : stretch by $∣ a ∣$ ; $0 < ∣ a ∣ < 1$ : compression by $∣ a ∣$ ; negative $a$ adds $x$ -axis reflection
Horizontal Stretch/Compression	$f (b x)$	$∣ b ∣ > 1$ : compression by $1/∣ b ∣$ ; $0 < ∣ b ∣ < 1$ : stretch by $1/∣ b ∣$ ; negative $b$ adds $y$ -axis reflection
Order of Transformations	Horizontal shift 2. Horizontal scale/reflect 3. Vertical scale/reflect 4. Vertical shift	Always factor $b$ out of the input term before reading $h$ for combined transformations

真题中的出现

AI 根据考纲规律估算的考点位置,请对照官方真题核实准确性。仅作复习重点参考。

2024 · AP Precalc MCQ
将变换后的方程与图像匹配
2023 · AP Precalc FRQ
将二次函数改写为母函数的变换形式

下一步

Immediately after mastering transformations of functions, you will apply this skill to graphing and analyzing quadratic, cubic, and general polynomial functions in the remainder of Unit 1. Transformations let you quickly sketch the graph of any polynomial written in vertex or factored form by relating it to a simple parent function, which saves critical time on both multiple-choice and free-response sections of the exam. This topic also feeds into later units: when you study rational functions, you will use transformations to shift and scale parent reciprocal functions to graph complex rational expressions. Transformations are also a core foundation for studying trigonometric functions, where period and phase shift are just horizontal stretch and translation, respectively. Without mastering the order and sign rules for transformations, you will struggle to correctly identify key features like vertices, asymptotes, and extrema of more complex functions.