Link: https://www.desmos.com/calculator/bg4llgqali

Write Up:

This project has undoubtedly helped me gain an increased visual awareness, understanding, familiarity with all of the different functions I have learned since September; throughout the entire process, I physically had the chance and freedom to explore and manipulate the various functions in a creative manner. At the same time, I also became increasingly capable and proficient at creating specific domain and range restrictions for the various functions in a sophisticated fashion so different graphs would not overlap each other. When I began the project, I was rather unfamiliar with graphing on Desmos, yet I familiarized myself with the website by drawing a few practice graphs; one graph for each of the six functions. I also practiced performing transformations and reflections on these graphs on a separate Desmos graph.

Designing the self-portrait was a fairly difficult task for me in the beginning and it was a long and tedious process throughout. For inspiration, I looked at my annual school photos in picture frames throughout my house as well as several cartoon figures on Google Images. I finally made a rough sketch of what I wanted myself to look like on a piece of blank paper using pencil after being inspired by the images I had seen.

I began to graph myself on Desmos starting with the hair and face; these were the most difficult parts of the project. I had trouble coming up with equation for certain sections of my hairstyle; yet, after reviewing the graphs in the math workbook and experimenting with different graphs on Desmos that I believed were to be of relevance, I finally completed the hair in a style that closely resembled the design I had pre-drawn on paper. Throughout this process, the functions I had to review included: logarithmic, radical, and exponential functions. These were also some functions that I had less experience with. Nonetheless, graphing the hair had definitely warmed me up to graph the rest of the body! Coming up with different domain restrictions for the functions were also fairly confusing at first, though they became more obvious as I had more practice. This was especially important when I was graphing the hair as there could not be any overlapping lines. I separated each of the pre-drawn sections into separate functions with specific domains and ranges. Each separate function began where the previous function terminated. Using this method, I could accurately write out the specific domain restrictions for each part. This method was used when I graphed the hair as well as the other body parts and decorations.

The ears were also fairly challenging to graph as its shape was fairly irregular and it also needed to be in line with the sideburns I had already graphed as part of the hair. Yet, through experimenting with somequadratic, exponential, and radical functions, I finally graphed them in a shape that closely resembled an ear.

Yet, I found I could not use too many transformations or reflections with the hair and head portions of my graph due to its unique overall shape, so I decided to use these techniques commonly on the other parts of my portrait such as the X-mas lights, eyes, body, and neck.

Then, I continued to draw the neck and body. These parts were fairly easy to draw as by this point, I began to get really familiar with Desmos itself as well as the variety of functions, especially logarithmic, rational and exponential functions.  Later, I used red and green coloured trigonometric functions to embellish the X-mas sweater in which I decided would have a high collar.

I then went on to graph the eyes; the process of thinking what I wanted the eyes to look like was rather difficult, yet the graphing of them was much simpler. For the first time in my project, I used the semi-circle function to make the irises and shading to colour them. An aha moment occurred to me at this moment when I realized that using semi-circles functions were more useful in contrast to ellipse functions which I originally thought about using that were both more difficult to shade as well as manage.

Shading was another aspect on Desmos that I was not completely sure of how to do, at first. Accordingly, I watched a short Youtube tutorial to help me out. Though it seemed complicated at first, after a few attempts, I finally understood the steps in shading on Desmos: they were the results of creating inequalities using different functions. Nevertheless, I had some difficulties shading the hair. Due to its unique style I had to separate each of the functions that outlined the hair into even smaller sections. Without it, it would’ve been impossible for me to shade the hair perfectly, especially in the spikey sections. This process took some time as I needed to create new domain restrictions for each of the even smaller sections and create many new shading restrictions.

Once I had a clear idea of what the nose and lips were going to look like on paper, they were also fairly easy to graph. Again, I drew what I imagined the body part would look like on paper before blindly trying to graph it on Desmos; this technique really helped me out throughout the project as it always helped me recognize the appropriate function to use while graphing. I would always highlight different sections on my notebook sketch with different colours splitting up the entire shape into smaller, separate components which were then translated into each unique function. This way, I was more organized and efficient, the functions I were to use were easier to identify, and the domain restrictions were more obvious.

Graphing the Christmas lights to embellish my holiday card was also a tedious task at first, yet I quickly found an efficient way to graph them. The lights which consisted of semi-circle functions were attached directly on the minimum values of a repeating cosine graph. At first, I inserted each semicircle function separately, but then I realized that I could use also use transformations to move only one of the functions I had initially used to graph the lights and translate it both left or right according to the period of transformed cosine graph. This made the work much more efficient. I also experimented with slider button on Desmos to make the shaded lights blink; this was very interesting to play with and it made my card more active and high-spirited.

Overall, this project has deepened my understanding of the shapes of different functions as well as the predominant concept of transforming functions by giving me the opportunity to experiment with the unique properties of each of the six functions on such a simple, clear and intuitive graphing software. The project was also of great help by providing me with a chance to review all of the functions I had learned throughout the Precalculus 12 in such a memorable and creative approach