Forging Connections
The Reflect Family

1 Introduction

  • Create an independent variable using the Point tool.
  • Create a function rule using the Mirror tool.
  • Create a dependent variable using the Reflect tool.

Hint videos are not yet available.

2 Reflect

Using this sketch, you’ll construct and explore reflect functions. You’ll manipulate them, observe them, and learn how this function family behaves.


Unable to display the worksheet

Hint videos: Step 1 Step 5 Step 11 Download the student worksheet.

3 Interlude

From I Love Lucy Season 4 Episode 27. Downloaded from

✏️ How does this video relate to the reflect functions you just constructed?

4 Reflection Designs

Each page of this sketch has a faint background image showing traces left by an independent variable x and its reflected image `r_j(x)`. But the mirror is hidden!

Construct your own reflect function and adjust it so you can drag x to make the same pattern as the one shown faintly in the background.

✏️ What did you learn as you solved these challenges? Describe your method so another student could use it.


5 Reflect Games

Each of these three games has a different purpose:

  1. Given independent variable `x` and mirror `j`, can you find `r_j(x)`?
  2. Given mirror `j` and `r_j(x)`, can you find independent variable `x`?
  3. Given independent variable `x` and `r_j(x)`, can you find mirror `j`?

For each game play a few times at easier levels, and work your way up to level 5. (Once you start a game at one level, you must press Reset to change to a different level.)

✏️ For each game, how many hits can you score in ten tries at level 5?

✏️ What methods did you invent to make it easier to get a high score?


6 Reflect Dance

In a group of four or five students, mark a mirror on the classroom floor with a string or rope and invent a reflect-family dance

One person should be the independent variable, another should be the dependent variable, a third should be the videographer, and the remaining group members are choreographers in charge of making sure the dancers are correctly positioned relative to the mirror.

Take turns so everyone in the group has a chance to be the independent or dependent dancer.

Then choreograph a reflect-family dance for your entire group, and ask someone from another group to video your group dance.

NY Times 25 October 2016 Suzanne Farrell Ballet at Kennedy Center

7︎ Objectives

This page supports the Reflect Family lesson from the Introducing Geometric Transformations as Functions unit. You will construct reflect functions, manipulate and observe them, and learn how this function family behaves.

By the end of this lesson, you’ll be able to perform these actions and answer these questions:

  • Construct independent variable x and reflect it across a mirror.
  • Label the dependent variable using meaningful function notation.
  • Drag independent variable x, trace both variables, and describe their relative motion.
  • Move the mirror to form and investigate a different member of the family.
  • Restrict the domain of x to a polygon and describe the resulting range.
  • Identify fixed points of the function and describe their relationship to the mirror.
  • Solve challenges that involve finding a hidden mirror.

The Fine Print


These activities require web access using a browser that supports HTML5 and JavaScript. (That means almost any current browser.) No purchase is required, and there’s no advertising anywhere.

Release Information

Update History:

Record every major revision, in reverse chronological order

04 Mar 2018: Converted to new lesson format.
06 Mar 2017: Fixed a bug on page 5 where the x tool didn’t work.
16 May 2016: Updated websketch page 8 to eliminate tracing, which was confusing.
31 Oct 2015: Updated this page to the website format.