Desmos Relation Explorations

Exploring Families of Functions with Desmos

As promised, I am writing a few blog posts about using Desmos to explore certain parts of mathematics. One part of the motivation for this is to show that "higher mathematics" can be explored and modeled. I've noticed that most of the explorations that are pre-made are often for lower mathematics and that once you reach past Algebra or Geometry the resources (especially online explorations) tend to become much more traditional. From my brief searches, exploratory resources for the mathematics of calculus and beyond are few and far between. 

In this and future posts, I will attempt to show how these mathematics can be explored with online software. While I don't claim to be an expert on this, I do hope that this can show that with some extensive work a teacher can make at least part of their curriculum of higher mathematics exploratory. 

With that disclaimer, today I wish to explore a family of functions, those of the form xⁿ + yⁿ  = 1, with n an integer greater than 0. These are a very interesting family of functions, but in my experience we rarely look at these functions past n = 2. For n = 1, we have the line x + y = 1, and for n = 2 we have a circle with radius 1 centered at the origin. In this exploration, we will focus on n greater than that.

The first observation that I will make is that 1 = 1ⁿ for all n positive. Our general form then turns to xⁿ + yⁿ  = 1ⁿ, which is a special case of Fermat's Last Theorem. In 1994, Andrew Wiles proved that there are no integer solutions for neither x nor y equal to zero when n is greater than 2. While this is not directly related to our Desmos exploration, it is an interesting piece of number theory. I include this observation to show that exploring a simple formula can bring some very interesting results. In this case, Fermat's Last Theorem brought about 358 years of mathematical research and was the basis of the creation of the field of algebraic number theory. Students can also learn to explore and build from simple problems and develop complex theories and understandings of interesting mathematics. 

Anyway, back to Desmos. When graphing this function, a suggestion pops up to ask whether you want to add a slider for n. After adding that slider, you can slide your n to be any value you wish, without having to type in the individual equation. Here are a few screenshots of our function at specific values of n:

Looking at the graphs, we notice that two points are always part of the function: (0,1) and (1,0). This makes sense, as 1ⁿ = 1 and 0ⁿ = 0 for all positive integers n. Thus xⁿ + yⁿ  = 0 + 1 = 1 + 0 = 1 for any n. Thus these two points must always be part of our family of functions.

From these simple graphs, another, more complex pattern starts to appear. It seems as though when n is odd, we have a line that is interrupted near the middle by a strange bump. When n is even though, we have a square-ish circle. Hmmm.... That's interesting. Students can continue searching and see whether the next few (or few hundred) functions follow this same pattern, or we could help them be able to recognize and justify this pattern mathematically as well. 

For the sake of convincing without proof, observe the following two graphs with n large [the graphs don't look much different for n any larger]:

Those look like right angles now, don't they? Another pattern has emerged.

How can we justify these patterns without just reverting to "look and see"? Well, let's first put our observations in a little bit more formal mathematical language: 

For n odd:

As n grows large, on the intervals (-∞, -1) and (1, ∞) it appears that our function is linear and is approximated by the line x + y = 1. Inside the interval (-1, 1) our function appears to be constant at y = 1. 

For n even:

As n grows large, our function appears to be approximately equal to the unit square.

We will treat these as two separate conjectures and attempt to justify each separately. 

For n odd:
Observe that for n odd, xⁿ and yⁿ are always positive for positive x, y, and always negative for negative x, y. Thus we can make x, y as large as we like, as long as the other is similarly large.
Solving xⁿ + yⁿ = 1 for y yields y = ⁿ√(1-xⁿ) [the nth root of (1 -xⁿ)], which equals (1 - xⁿ)⁽¹/ ⁿ⁾.
Taking the derivative of y with respect to x yields
dy/dx = 1/n × n xⁿ⁻¹ (1 - xⁿ)⁽¹/ ⁿ⁺¹⁾ = xⁿ⁻¹ × (1 - xⁿ)⁽¹/ ⁿ⁺¹⁾.  For |x| > 1, and n large, (1 - xⁿ) ≈ -xⁿ.
Thus (1 - xⁿ)⁽¹/ ⁿ⁺¹⁾ ≈ -xⁿ⁽¹/ ⁿ⁺¹⁾ = -x⁽ⁿ/ⁿ⁺¹⁾  ≈ -x. Thus dy/dx ≈ -1. For x ∊ (-1, 1) xⁿ ￫ 0, thus dy/dx ≈ 0.
Since we must go through (1,0) and (0,1), we are constrained to our right angle around x = 1 and the rest of the graph is approximately equal to our line x + y = 1.

For n even:
Observe that for n even, xⁿ and yⁿ are always positive for any x, y in the real numbers. Thus we are constrained to always have x, y ≤ 1. Thus we are constrained to have x ∊ (-1, 1) xⁿ ￫ 0, and thus dy/dx ≈ 0 as above. We are thus constrained to the unit square.



So now what? Students aren't expected to go as in depth as in this blog post. In fact, if you google these types of functions you will find a few explorations from college students. However, most attempt to justify their conclusions with visuals alone. Here are a few examples: here, and here. While this is extremely useful for students to know how to do, it is also useful to have a mathematical justification. But this justification only comes after an observation is made, which is unlikely to be made without the assistance of technology.

Students can be expected to learn quite a bit about functions, and understand that a function doesn't have to be smooth or "normal" in order to be a valid function. Understanding a function in depth requires some extensive work with each part of the equation. This is extremely likely to happen if students are asked to explore such an interesting family of functions.