Transforming Polynomials

First consider that you have the basic 2nd degree polynomial:

That can be written as a parametric equation vector:

And you can also consider a linear transformation matrix like so:

Multiplying the vector by the matrix gives a new parametric vector:

Which after solving for t and plugging into the y component is transformed back into a standard polynomial:

So for example if you have a target polynomial (random):

By setting the coefficients equal to the last equation, hat creates 3 equations with 3 unkowns:

These can be solved for a, b, c:

So the transformation matrix is:

This is the linear transformation matrix to convert the parabola x^2+x+1 to 5x^2-2x+3

So it is pretty easy to see that every possible parabola is the basic parabola times a linear transformation matrix that it is possible to solve for. 

I believe but I'm not sure that it can apply to polynomials of any degree but you have to add more degrees of  freedom to the transformation matrix; like for cubics:

A transformation matrix like the above should give 4 equations with four unknowns, etc.