#AP Pre-Calculus: Transformations of Functions - Your Ultimate Review 🚀

Hey there! Let's make sure you're totally prepped for transformations. This guide is designed to be your best friend tonight, making everything crystal clear and helping you feel confident for the exam tomorrow. Let’s dive in!

#1.12 Transformations of Functions

Transformations are all about how we can move, stretch, or flip functions. Think of it like playing with Play-Doh – you can shift it, squish it, or mirror it! We’ll cover additive (translations) and multiplicative (dilations and reflections) transformations. Let's get started!

# Additive Transformations (Translations) 🚶‍♀️🚶‍♂️

Additive transformations involve adding or subtracting constants, which causes the function to shift its position on the graph. These are also known as translations.

#1️⃣ Vertical Translations ⬆️⬇️

• Concept: $g(x) = f(x) + k$ shifts the graph of $f(x)$ vertically by $k$ units.
  • If $k > 0$, the graph moves up.
  • If $k < 0$, the graph moves down.
• Key Point: The shape of the graph stays the same; only its position changes.

Image Courtesy of Cuemath

Caption: A graph of the function f(x) and its new graph g(x) that’s shifted 3 units upward

#2️⃣ Horizontal Translations ↔️

• Concept: $g(x) = f(x + h)$ shifts the graph of $f(x)$ horizontally by $h$ units.
  • If $h > 0$, the graph moves to the left.
  • If $h < 0$, the graph moves to the right.
• Key Point: Remember, it's the opposite of what you might expect with the sign inside the function.

Image Courtesy of Quora

Caption: A graph of the function f(x) and its new graph g(x) that’s shifted six units to the left