Link to my project: Ferb

This is a picture of the artwork that I created using functions and relations on Desmos

This is a picture of the artwork that I created using functions and relations on Desmos.

This is the picture that I used for reference to create my project.

This is the picture that I used for reference to create my project.











For my Desmos graphing project, I chose to recreate a picture of Ferb, from Phineas and Ferb. I chose to recreate Ferb for a few reasons. First, I loved watching Phineas and Ferb as a kid, and recreating him using functions and relations was quite a nostalgic experience. That being said, I also chose to create Ferb because he is made up of many different types of lines and shapes that I thought would be challenging and enjoyable to try and replicate on Desmos. I feel as though my choice of drawing to recreate was appropriate for the time given and my skill level.

In order to keep my work organized I separated the different parts of my artwork into 5 categories: eyes, ear, head, hair, and clothes. I made the eyes solely using the (x-p)^2+(y-q)^2=r^2 relation . To replicate the ovular shape of the eyes, I multiplied the x value outside of the bracket to make the circles longer vertically.

To make the ear, I used the same relation, but this time I multiplied the y value to make one of the circles wider horizontally. Furthermore, I set the domain and range to cut the ellipses in half. I also used the function y=mx+b to create diagonal lines.

In order to make the shape of the head I used a variety of functions and relations. I used y=mx+b to create the diagonal lines on the back of the head and neck above the ear. I used x=y to the front of the neck and a small portion of the nose, and to create horizontal straight lines for the bottom of the lip as well as part of the nose I used y=x. I used the function y=a(x-p)^3 to create the bottom curve of the nose and the slightly slanted line between the lip and the bottom of the nose. By increasing or decreasing the value of , I could make each parabola wider or narrower. By increasing or decreasing the value of , I could move the parabola horizontally, and changing the value of  would move the parabola vertically. Furthermore, I used y=a(x-p)^2+q and y=-a(x-p)^2+q to create slightly diagonal lines to create the forehead, the top of the nose, and the bottom of the neck. I used the circle relation again to create the top curve of the nose as well as the curve of the lip. I used y=a√(x-p) +q to create the middle of the nose because of the slight curve. Similarly, multiplying x manipulates the width of the parabola, and changing the value of p and q will move the parabola vertically and horizontally

I initially struggled to decide what functions to use to create the hair. I used circle relations to create most of the strands of hair. I also used y=-a(x-p)^2+q to make some of the less curved lines. In some circumstances, two of the curved lines that I used didn’t meet, so I used y=mx+b to create diagonal lines that could connect the two curves. I used y=a√(x-p) +q on one of the strands of hair that didn’t curl under itself. I also used y=sinx to create part of one of the strands of hair. In order to do this, I divided sin by 16 to make the curve longer horizontally.

In order to create the collar of the shirt, I made straight lines by using the function y=mx+b and y=x. I used a circle relation to create the button of the shirt. Lastly, I used y=a(x-p)^3+q and y=a√(x-p) to create the curve of the shoulders.

I also spent some time working with inequalities, but I didn’t include them in my final project because I couldn’t figure out how to create an inequality with an empty circle inside of it (to make the eyes black without filling in the smaller white circle). I also struggled to learn how to make an inequality to go in between two functions without leaving any empty space or going out of the lines. I will continue to research and practice using inequalities in my spare time because I thoroughly enjoyed trying to learn how to use them.

Functions and Relations Used:

  1. x=y and y=x
  2. y=mx+b
  3. y=a(x-p)^2+q
  4. y=a(x-p)^3+q
  5. y=a√(x-p)
  6. y=sinx
  7. (x-p)^2+(y-q)^2=r^2


Before completing this project, I had very little experience with using functions and relations, especially on Desmos. As a result, I faced many challenges as I tried to complete this project. When I began working on my project, I didn’t know how to use sliders effectively. As a result, I wasted a lot of time trying to create accurate lines while manually adjusting them. With the help of my peers and Mr. Salisbury, I learned how to use sliders to increase my efficiency. This was definitely an ‘aha’ moment, because I was growing increasingly frustrated trying to plot lines without an accurate ‘starting point’ to work off of. Furthermore, at the beginning of this project, I struggled to remember the impact of each variable on the position and shape of my line. This was quite frustrating as I would accidentally move a line horizontally rather than vertically, ultimately furthering me from the place that I wanted it to be. That being said, through lots of repetition, moving my lines became muscle memory, which increased my efficiency and accuracy.

I feel as though my general ability to estimate the slopes and coordinates that I needed to use improved throughout this project. Initially, it took me a while to orient myself within my picture to choose where to put my lines. Through increased experience and practice, I improved in my ability to accurately guess the general area that I should place my line. Additionally, this project has allowed me to begin easily recognizing the formulas of basic functions and relations. at the beginning of this project I struggled to remember what each function looked like. I had to input each of them into Desmos in order to determine whether or not I should use them. Throughout this project, I gradually began to remember what each function looked like, which allowed me to become more efficient when creating my project. I believe this skill will also help me in future math classes.