How to change from quadratic to linear equation

Question

Equation of curve

$ y = 4x^2 -2 $.

The following equation is obtained when this curve equation is expressed in linear form

$Y =-2X+4 $.

How do we change the curve equation to the linear equation? I tried the following

$ y = 4x^2 -2$

$y + 2 = 4x^2$

$y + 2 = 4(x)^2$

$y + 2 = 4(x-0)^2$

$y = 4(x-0)^2 - 2$

I need to express $X$ and $Y$ in terms of $x$ and $y$

Answer 1

$$y=4x^2-2 \\ y+6=4x^2+4$$ Compare with $Y=-2X+4$, this implies, $Y=y+6$, $X=-2x^2$.

Answer 2

There is no such unique change of variables to convert between the two, at least not without additional constraints. For example $\;X=f(x,y),\, Y=y-4x^2-2f(x,y)+6\;$ means that $\,Y=-2X+4 \iff y=4x^2-2\,$ holds for any arbitrary function $\,f(x,y)\,$.