# Exploring Transformations of Function Families

- Author:
- Diana Bowen

## INTRODUCTION

The first part of this activity allows students to make sense of vertical transformations, horizontal transformations, dilations, and reflections across the function families by utilizing multiple representations and the parent function for each family. The second part of this activity allows students to make sense of even and odd functions and their connections across multiple representations.
This addresses the following Common Core State Standard in Mathematics:
CCSS.MATH.CONTENT.HSF.BF.B.3
Identify the effect on the graph of replacing

*f*(*x*) by*f*(*x*) +*k*,*k**f*(*x*),*f*(*kx*), and*f*(*x*+*k*) for specific values of*k*(both positive and negative); find the value of*k*given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.## LAUNCH: Defining and Identifying Parent Functions

A **parent function** is the simplest function in a family of functions. Each of the functions below can be classified as either a **parent function** or a **transformed function**. Mark all of the responses that represent a **parent function **for a function family.

How did you decide if a function was a parent function? What did you notice about the parent functions? What did you notice about the functions that are transformations of the parent functions? What would you say if someone asked you to explain what is meant by the word "simplest" in the definition above?

What function families are represented in the choices above?

What is the parent function for the cubic function family?

## ACTIVITY: Using Multiple Representations to Identify Transformations of Parent Functions

For this next section, you will be asked to predict and identify the effect on the graph of a function given changes in its equation.

## Exploring Shifts

For this activity, you will be examining how shifts such as f(x+k) and f(x)+k lead to changes in the graph.