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  2. AP Pre Calculus
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Transformations of Functions

Tom Green

Tom Green

8 min read

Next Topic - Function Model Selection and Assumption Articulation

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Study Guide Overview

This study guide covers transformations of functions in AP Precalculus, including additive transformations (translations) and multiplicative transformations (dilations and reflections). It explains vertical and horizontal shifts, stretches, shrinks, and reflections, along with how to combine these transformations. The guide also discusses how transformations affect the domain and range of functions and provides practice questions and exam tips focusing on identifying transformations, applying them to functions, and understanding their impact on domain and range.

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

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)g(x) = f(x + h)g(x)=f(x+h) shifts the graph of f(x)f(x)f(x) horizontally by hhh units.
    • If h>0h > 0h>0, the graph moves to the left.
    • If h<0h < 0h<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.
Horizontal Translation Graph

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

Key Concept
  • Key takeaway: Additive transformations (translations) o...
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Previous Topic - Equivalent Representations of Polynomial and Rational ExpressionsNext Topic - Function Model Selection and Assumption Articulation

Question 1 of 13

If f(x)=x2f(x) = x^2f(x)=x2, what is the function g(x)g(x)g(x) if the graph of f(x)f(x)f(x) is shifted 3 units upwards? 🚀

g(x)=x2−3g(x) = x^2 - 3g(x)=x2−3

g(x)=(x−3)2g(x) = (x-3)^2g(x)=(x−3)2

g(x)=x2+3g(x) = x^2 + 3g(x)=x2+3

g(x)=(x+3)2g(x) = (x+3)^2g(x)=(x+3)2