For my Pre-Calculus 12 this semester, my class was tasked with recreating an image of our choosing using equations of functions and relations. I am proud to have brought to life one of the most iconic cartoon couples to date: Marge and Homer Simpson.

Did you have any challenges?
Unfortunately, applying function notation was the one thing I couldn’t get working. Following my teacher’s advice, I attempted to incorporate it into a few equations as opposed to all of them. However, upon any additions, the left pane would immediately display ALL colours randomized. Further integrating function notations to every equation caused the picture itself to warp colours as well.

Another area that proved particularly challenging was maneuvering around the finer details, such as facial features and Marge’s beehive hair. When one obstacle (e.g. a wrinkle above Homer’s right eye or a crease along Homer’s shirt collar) was finally completed, another problem would suddenly emerge, directly impacting the previous equation’s results. To overcome this, I manipulated different lengths and widths of ellipses, circles, and logarithms. Through much trial and error, I found that certain equations work better as restrictions while others are more effective as functions.

How you figured out what equations to use?
Through tedious experimentation, I discovered the best-suited equations to use for each section of the image. For example, parabolas and ellipses were useful for creating most of Marge and Homer. However, some detailed shapes called for other equations. More specifically, logs and exponentials were better suited to making the pants, shoulders, and Marge’s dress.

Did you get help?
Help in doing this project came in several forms. Aside from my teacher, I also found helpful information through YouTube videos, which provided good insight into working around certain parts of each body. For example, colour-shading on a large area appeared simple (Homer’s shirt, filling in Marge’s hair). However, the edges of the shapes must be carefully considered due to possible overlapping features or conflicting equations. Instead of tackling the area all at once, the Desmos YouTubers recommended dividing it into smaller sections to fully colour-shade everything. This divide-and-conquer approach worked well for my design.

In addition to researching online, I also engaged in dialogue with classmates to understand their thought processes in tackling various problems they faced in their projects. This was mutually beneficial as sharing our wins and struggles allowed us to build on our collective knowledge and experiences.

Did you use any strategies?
There was one strategy that I used quite consistently, namely pattern recognition. If one equation for a particular line or curve at a specific angle worked elsewhere, I would duplicate it, repeating this process as often as I saw fit. While the strategy was sound in principle, it quickly became a double-edged sword since prompt organization was neglected. As a result, I had to dedicate extra time to sift through all the equations once the picture was completed, re-sorting and re-organizing them into their respective folders.

Any aha moments?
I enjoyed how this project forced me to constantly shift perspectives, tackling issues from different angles when my usual methods weren’t working. A particularly memorable instance was when I was shading the top portion of Marge’s dress. I had originally hoped to colour a bright green horizontal stripe across the chest, but there were several curved regions where the green exceeded the outline. Finally, after racking my brain for a while, it clicked. Instead of thinking about it through linear equations, I set the curve equation as a starting point and adjusted the range and domain to shade Marge’s dress section by section, as I did with her hair.

How did this assignment help you understand more about transformations of functions and relations?
While our textbook provided me with the fundamentals surrounding transformations of functions and relations, this assignment allowed me to visualize the equations and truly understand how to manipulate them through hands-on experience. In addition, the sliders for the individual variables helped immensely in grasping what variable controlled what part of the graph and how the domain and range played a crucial role in the restrictions. In conclusion, all these features ultimately gave me control over what the graph would look like, whether to move, shorten, lengthen, expand or compress structures.