The link to my rendition of Mike Wazowski doing the sine function dance: https://www.desmos.com/calculator/cxicbkswjz
For this project, I decided to choose Mike Wazowski. I drew inspiration from childhood movie characters and thought Mike would be a fun challenge. I loved his monster appearance and also knew that creating him would require the use of several different functions. I chose not to use an image so I could fully recreate Mike with my own style, meaning I could modify certain aspects of his appearance to fit my liking and to employ the use of a variety of graphs.
Circle function of form x2 + y2 = r2
Rational function of form x^2 + 2 / x – a rational function uses polynomials in the numerator and denominator – represented as rational fraction
Square root function of form f(x)=√x
Quadratic function of form y = ax2 + bx + c,
Exponential function of form y = abx
Linear function of form y=mx+b
Sine function of form y=sinx
My strategy is as follows:
First, I wanted to create the main shapes and outlines of his body. I knew that creating the larger shapes ar first would allow me to set some boundaries for the smaller details within his face and arms. I began with the head, which was a simple circle function. I discovered that manipulating the radius would allow me to modify the size of the head. I positioned the head in the middle of the graph about the origin to make future reflections easier, adding values to the x and y values as needed. After creating the head, I moved on the outer outline of his eyeball. I used another circle function with a smaller radius. I then used an inequality to shade in his iris, then proceeded to manipulate the coefficients to create the pupil. I discovered that the coefficients modified the length and width of the circle, meaning that I could also make it an oval.
To make the horns, I used rational functions rather than using quadratic functions. I knew rational functions would give me a softer curve, and I wanted to experiment with using an unconventional function. I added a coefficient to the input within the function in order to perform a horizontal compression. To reflect the horn on the other side of the head, I used a negative reflection of the y-axis by adding a negative sign to the input. I then added restrictions
Now onto the arms (part 1). Though his arms were quite linear and straight in the movie, I wanted to make a realistic arm that was somewhat curved. To do this, I used square root functions to make the bottom half of his arms. I added a coefficient to the input to make his arm the steepness I wanted it to be, using a horizontal compression. I then subtracted a constant from the entirety of the function, vertically shifting the bottom line of his arm to the position I wanted it to be on his body. To create the second line above the bottom line, I manipulated the vertical shift so it would be higher than the previous lines. To make things simple, I reflected both these lines to the other side of his body and used the necessary restrictions.
The elbows were an interesting, albeit ridiculous addition. To make Mike even more lifelike, I wanted to give him monstrous, pointy elbows. I used quadratic equations, as they produce parabolas with vertexes/points. I added a coefficient of 13 to the input to horizontally compress the parabolas. I then reflected the elbow to the other side and used inequalities to shade them in.
The second part of the arms were simple linear equations. I modified the y-intercept to create parallel lines above the initial lines I created. To add the same arms to both sides of his body, I reflected across the y-axis. For his hands, I also used quadratic functions.
For Mike’s legs, I used exponential functions, knowing they have a long, sweeping curve. I didn’t want to repeat the arms process again for the legs, although they are straight in the movie. I wanted to experiment with exponential functions and their potential to create even creepier legs. The first line was relatively straight, but the lower part of the leg needed to be curvier, so I manipulated the number and power in front of the input (which in this case, lies in the exponent). I reflected the leg across the y-axis. I used inequalities to shade this area in, but it was difficult to shade the entirety of the leg in with inequalities without the inequality not covering a certain section or extending too far into his head. For his feet, I used a quadratic equation (not function) by switching the inputs and outputs in the quadratic equation to make it sideways. I reflected this across the y-axis to create both feet.
In order to make the smile, I knew I could restrict a sine function to use its curve/dips. I inputted a sine function and restricted the x value so I ended up with a portion of the function. For the upper line of the smile, I used a quadratic function with a small slope in order to make it flat. Creating the teeth was a painful process. I used a large number of linear functions, creating new lines by manipulating the slopes from positive to negative repeatedly to create triangles and restricting them as needed.
Just for fun, I added a blue hat (representation of his MonstersU hat) and rainbow eyelashes. I liked the creative freedom of not using a reference photo. For the eyelashes, I used linear equations. For the hat, I used a quadratic equation.
This project was a challenging but rewardable experience. The hardest aspect was making sure small details of the lines were neat and positioned at the right places. Shading the arm was also incredibly difficult, as I had to use individual inequalities for the elbows and different sections of the arm to shade in the arm as much as I could. I loved the experimentation process and discovering how manipulating values would change the shapes of graphs. These skills will undoubtedly help me in future math classes. I improved my ability to visualize mathematical functions and the relationship between an equation and its visual image.