# Transformations of functions

> AP Precalculus · Unit 1: Polynomial and Rational Functions
> Source: https://www.owlsprep.com/study/ap-precalculus-u1-transformations-of-functions/

This guide covers rigid (translations, reflections) and non-rigid (stretches, compressions) transformations, correct order for combined transformations, and applications to polynomial and rational functions for AP Precalculus Unit 1 exam prep.

**Prerequisites:** Function notation and evaluating functions for given inputs; Graphing basic polynomial and rational functions

## Learning objectives

- Identify rigid and non-rigid transformations of parent functions
- Apply transformation rules to write equations of transformed functions
- Determine the correct order of operations for combined transformations
- Apply transformations to polynomial, rational, and contextual functions

## What are Function Transformations?

Function transformations are a set of rules that modify the graph or equation of a known "parent function" $f(x)$ to produce a new related function. This skill lets you analyze complex functions by connecting them to simpler, familiar parents, rather than building graphs from scratch.

Transformations are a core assessed skill in AP Precalculus Unit 1, which makes up 27–30% of your total exam score, tested in both multiple-choice and free-response sections. Common exam questions ask you to match transformed equations to graphs, identify transformations from a given equation, or apply transformations to contextual functions.

> **info**
>
> This is one of the most frequently tested foundational skills in Unit 1, so mastering the sign and order rules will save you critical points on exam day.

## Rigid Transformations: Translations and Reflections

Rigid transformations only change the position of the parent function's graph, not its shape or size. The general form for a basic rigid transformation is:

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

**Rigid Transformation** — A transformation that preserves the shape and size of the parent function's graph, only changing its position on the coordinate plane. Includes translations and reflections.

The parameter $k$ controls vertical translation: adding $k$ to the output of $f(x)$ shifts every point $(x, y)$ on the original graph to $(x, y + k)$. If $k>0$, the graph shifts up $k$ units, and if $k<0$, it shifts down $|k|$ units. The parameter $h$ controls horizontal translation: replacing the input $x$ with $x-h$ shifts every point $(x, y)$ to $(x + h, y)$. A common point of confusion is the sign: if $h>0$, the graph shifts right $h$ units.

Rigid transformations also include reflections: $g(x) = -f(x)$ reflects the graph over the $x$-axis (flips all $y$-values), while $g(x) = f(-x)$ reflects over the $y$-axis (flips all $x$-values).

**Worked example:** The parent function $f(x) = x^2$ has a vertex at $(0,0)$. Give the equation of the function after a translation 4 units left, 1 unit up, then a reflection over the $x$-axis. What are the coordinates of the new vertex?

1. First, apply the translation: 4 units left means $h = -4$, so $x-h = x - (-4) = x+4$. 1 unit up means $k=1$, so after translation we get:
2. $$f(x+4) + 1 = (x+4)^2 + 1$$
3. Apply reflection over the $x$-axis: multiply the entire function by $-1$ to flip the sign of all outputs, giving:
4. $$g(x) = -\left((x+4)^2 + 1\right) = -(x+4)^2 - 1$$
5. The original vertex at $(0,0)$ is shifted 4 left and 1 up to $(-4, 1)$.
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.

## Non-Rigid Transformations: Stretches and Compressions

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:

$$g(x) = a f(bx)$$

**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 $bx$ 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.

**Worked example:** 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.

## Combined Transformations and Order of Operations

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.

**Worked example:** Rewrite the rational function $g(x) = \frac{2x + 1}{x + 3}$ 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: $2x + 1 = 2(x + 3) - 5$.
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.

**Check your understanding**

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

1. The graph of $f(x) = x^4 - 2x^2$ 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) $g(x) = (-x +1)^4 - 2(-x +1)^2 + 3$

   *Why:* 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.

**Worked example:** A local bakery models its monthly profit $P(t)$, in thousands of dollars, $t$ months after January 2023, as $P(t) = -0.4t(t - 10)$, 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, $f(3x + 9) = f(3(x + 3))$, which is a 3-unit left shift, not a 9-unit left shift.

## Common pitfalls

- **Wrong:** Interpreting $f(x + 4)$ as a 4-unit shift right instead of 4 units left
  - Why it fails: 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 $h$.
  - Correct: 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.
- **Wrong:** Interpreting $f(4x)$ as a horizontal stretch by a factor of 4 instead of a horizontal compression by a factor of $\frac{1}{4}$
  - Why it fails: Students directly match the coefficient $b$ to the scale factor, instead of using the reciprocal rule for horizontal transformations.
  - Correct: 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.
- **Wrong:** Calculating a 6-unit left shift for $f(2x + 6)$, instead of a 3-unit left shift
  - Why it fails: Students do not factor out the coefficient of $x$ before reading the shift value $h$.
  - Correct: Always factor the coefficient of $x$ inside the function: $f(2x + 6) = f(2(x + 3)) = f(2(x - (-3)))$, so $h=-3$, which is a 3-unit left shift.
- **Wrong:** Applying vertical shift before vertical stretch for $2f(x) + 3$, leading to incorrect output values
  - Why it fails: Students forget order of operations, and do addition before multiplication.
  - Correct: Always apply stretches/compressions/reflections (multiplication steps) before vertical shifts (addition steps), following PEMDAS order.
- **Wrong:** Confusing $-f(x)$ as a reflection over the $y$-axis instead of the $x$-axis
  - Why it fails: Students mix up whether the negative sign applies to the input (inside $f$) or output (outside $f$).
  - Correct: Negative outside $f$ flips $y$-values → reflection over $x$-axis; negative inside $f$ flips $x$-values → reflection over $y$-axis.

## Cheatsheet

| Category | Formula / Rule | Notes |
| --- | --- | --- |
| General Transformed Function | $g(x) = a f\left(b(x - h)\right) + k$ | $a$: 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 | $f(bx)$ | $\|b\|>1$: compression by $1/\|b\|$; $0<\|b\|<1$: stretch by $1/\|b\|$; negative $b$ adds $y$-axis reflection |
| Order of Transformations | 1. 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 |

## What's next

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.

- [Function model selection and assumption articulation](https://www.owlsprep.com/study/ap-precalculus-u1-function-model-selection-and-assumption/)
- [Function model construction and application](https://www.owlsprep.com/study/ap-precalculus-u1-function-model-construction-and-application/)
- [Exponential and Logarithmic Functions Overview](https://www.owlsprep.com/study/ap-precalculus-u2-overview/)

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