How to flip this equation on a graph

Question

$x^2+y^2=x^{(\sqrt\pi)}$ I was messing around on desmos, and graphed this and cannot figure out how to flip it over the x axis. Appreciate the help

Answer 1

Instead of the two main points being (0,0) and (1,0) I want it to be (0,0) and (0,-1).

Ah that's simple.

how to do that

If you exchange the x with the - you get the graph rotated $90^{\circ}$ counter-clockwise (this is always the case). This is because $+y$ is a $90^{\circ}$ counter-clockwise rotated $+x$.:

$ y^{2} + x^{2} = y^{\sqrt{\pi}} $ is $ y^{2} + x^{2} = x^{\sqrt{\pi}} $ but rotated $90^{\circ}$ counter-clockwise.

If you now multiply all $y$ by $-1$, the graph that you rotated $90^{\circ}$ counterclockwise now rotates $180^{\circ}$ counterclockwise (this is always the case too). This is because we use it to mirror the therm about the x-axis.

$ (-y)^{2} + x^{2} = y^{2} + x^{2} = (-y)^{\sqrt{\pi}} $ is $ y^{2} + x^{2} = x^{\sqrt{\pi}} $ but rotated $90^{\circ}$ clockwise.

soluton

Aka you're soluton is $ y^{2} + x^{2} = (-y)^{\sqrt{\pi}} $

The plot of this $ y^{2} + x^{2} = (-y)^{\sqrt{\pi}} $: