For a long time, I had this idea that no “decent function”, apart from the absolute value function, could have multiple abrupt changes. By “decent,” I mean functions on my calculator that are not piecewise functions. Once I started messing around with the “math” tab on my calculator, I realized how wrong I was.

One of the functions I found is the “min” function. I put in two values, e.g. 2 and 5, and the function gives back the smaller value, 2. If I put in min (x,5), I will get a line until y becomes 5. This is essentially 2 functions, but I am not using piecewise notation.

This is good, but I want more bends in the function. Right now, this function is just a rotated absolute value graph. I could make things more interesting by having a polynomial or sine wave in the min function.

 

 

This made me wonder: could I make the Super-Man logo with a handful of functions? The “S” would take 3 functions, but if I made the pentagon correctly, I would only need 2 more. I could make the logo with 5 total functions.

The top portion of the pentagon is a function, but to express it in an equation, I have to use the min function to give an otherwise normal line two bends. I cannot accomplish this by inputting a line into the min function because that will only give me one bend. To create a function with two bends, I need to input a function that already has a bend, like the absolute value function.

Just putting in min(abs(x),5)) doesn’t work, however, as shown below.

If I use the negative absolute value function, however, and shift it up a few units, I get an entirely different function that looks exactly like the top of the pentagon.

Once I have the basic function, I can cut off the values for the domain that I do not need by tricking the calculator.

 

 

By putting that group of square roots at the end, I ensure that no x value less than -2 will be used. I can use this method to just make certain values go away, I can also shift and stretch the function until I get what I need.

This was the most complex part of the logo. Everything else just fell into place. I used square root graphs and a tangent graph for the “S,” and I used an absolute value graph for the bottom portion of the Pentagon.