Maths Club: From Circles to Approaching Squares
By: Barry Leung
Tags: Math, Function, Education, Mathematics, Graph

Photo by Roman Mager on Unsplash

Monday was the final meeting of maths club before Christmas, we explored the graphs of equations of circles, going into the realm of analyzing equations x⁴ +y⁴ = 1 and x³ +y³ = 1.

The Unit Circle

The unit circle is a circle of unit radius centered at the origin in the Cartesian plane. It is represented by the equation x² + y² = 1. Knowledge of transformations allow us to plot circle equations of the form (x – k)² + (y – h)² = r².

Photo by Matheo JBT on Unsplash

Sketching x⁴ + y⁴ = 1

If we know that the curve x² + y² = 1 is a unit circle, can we sketch the curve x⁴ +y⁴ = 1 using this information? And what happens as the powers become bigger even integers?

One skill for sketching unfamiliar functions is to consider the intercepts on both axes. Substituting x = 0 gives us y-intercepts at x = 1 and x = -1, and likewise, putting y = 0 gives us x-intercepts at y = 1 and y = -1. Looking at our unit circle, we know that our new curve intersects with the circle at both axes.

Rearranging equations into forms with y as the subject can help us further understand the behaviour of graphs. In this case, we can show that for our new equation, y = ±(1 - x⁴)^(1/4) and the unit circle can be rearranged into y = ±(1 - x²)^(1/2). Algebraically, we can start with the inequality x⁴ - x² ≤ 0 and prove that y = ±(1 - x⁴)^(1/4) is greater than or equal to y = ±(1 - x²)^(1/2) for -1 ≤ x ≤ 1, meaning the unit circle is bounded by our new equation.

Unit Circle bounded by our new equation

Try playing around with different even integer powers and observe the behavior of the graphs. What happens when the powers are 100 or even 1000? It looks like the curve is becoming a square as the powers increase to higher even integers. Interestingly, if you actually zoom in on the graphs and look at the corners, you will notice that it’s in fact a rounded shape and the corners are smooth. And that is true no matter how big the powers are, the graph will never become a square.

Image by ladyeleanor from Pixabay

Unfolding the Mystery

In general, the graph x^(2k) +y^(2k) = 1 for integers k are differentiable at every single point. In mathematics analysis, our equation is said to be smooth. And since a square has sharp corners, similar to the absolute value function f(x) = |x|, it’s not differentiable at point x = 0. Intuitively, you can’t draw a tangent at a corner like you do at a point on a curve. And because non-differentiable curves can only be generated using special mathematical operators like the modulus sign, our equation that consists of polynomials alone does not have sharp corners and therefore cannot become a square.

x¹⁰⁰ + y¹⁰⁰ =1

Sketching x³ + y³ = 1

What about the graph of x³ + y³ =1. Notice the equations discussed above all have even integer powers. And it should be noted that the main difference between x³ + y³ =1 and x² + y² = 1 is that the former is a function because for every for every y value, there is only one corresponding x value, whereas the latter is an equation since for every y value, there is more than one corresponding x value.

Rearranging the equation gives us y = (1 - x³)^(1/3) and we find that the curve crosses points (0,1) and (1,0) by evaluating the intercepts on both axes. And it putting 0≤ x ≤ 1 gives us a similar shape of a circle in the first quadrant. Now, to sketch other parts of the curve, let’s consider what happens when x approaches ±infinity. In our newly arranged form y = ±(1 — x³)^(1/3), as x approaches a bigger and bigger value, the effect of 1 becomes negligible and treat the equation as y ≈ (x³)^(1/3), which is simply y = -x. And we have successfully found the oblique asymptote of our function. On the graph, the curve will approach the line y = -x for both positive and negative x values.

curve approaching y = -x for extreme values of x

Asymptotes in a Nutshell

Photo by Michael Dziedzic on Unsplash

For most people the idea of asymptotes is introduced when learning about the reciprocal functions like f(x) = 1/x. The domain is all real numbers except for 0. Division by 0 is a common theme in functions involving asymptotes. And an oblique asymptote mentioned above can be found by putting extreme values of x to evaluate the behavior of the graph.

Oblique asymptotes are like your really good friends; you can cross them, but you’ll stay close in the end.