How to write an equation that creates a curve that touches X and Y axis, while the apex of the curve approaches the origin.

Question

Looking for an equation that looks similar to y=1/x but will actually touch the x/y axis at some point. Lets say 1. Then have some variable that when set higher/lower the curve will become sharper as it gets closer to the origin.

The image of the graph is an example of what I'm looking to do. The change of the blue line to the red line is what I'm trying to accomplish.

graph image

Answer 1

Perhaps you're interested in one quadrant of a super-ellipse a.k.a. squircle that arises from generalizing the defining equation of a circle $$ x^2 + y^2 = 1 $$ to $$ x^a + y^a = 1 $$ where $a \geq 2$.

Of course, for these arbitrary powers to be defined, we have to have a non-negative base (this is redundant in the $a=2$ case), so the equation is actually $$ |x|^a + |y|^a = 1. $$

If you want the curve to match your specifications exactly, then you have to shift the center over to $(x, y) = (1, 1)$ and restrict to just one quadrant, so $$ |x-1|^a + |y-a|^a = 1, $$ where $x \leq 1$ and $y \leq 1$.

Here's what a few of these curves look like for various values of $a$.

You drag a slider in this interactive graph to mess around with these curves.

Answer 2

A simple one is given in an implicit form ($f(x,y)=c$) by

$$|x|^{1/n}+|y|^{1/n}=1$$

whose upper part is given by equation :

$$y=(1-|x|^{1/n})^n$$

This curve is tangent to the coordinate axes in $(1,0),(-1,0)$ and $(0,1).$

The higher the value of $n$, the closer the curve is to the origin.