Finding the equation for image of a map Consider the morphism $\mathbb A^1 \to \mathbb A^2$ given by $f(t) = (t^n,(1-t)^n)$.  What is the defining equation of the image?  Is there a systematic way to do this?

Too long for a comment.
In M2
R=QQ[t,x,y,MonomialOrder=>Eliminate 1]
I1=ideal(x-t,y-(1-t))
toString (gens gb I1)_0_0
-- x+y-1

I2=ideal(x-t^2,y-(1-t)^2)
toString (gens gb I2)_0_0
-- x^2-2*x*y+y^2-2*x-2*y+1

I3=ideal(x-t^3,y-(1-t)^3)
toString (gens gb I3)_0_0
-- x^3+3*x^2*y+3*x*y^2+y^3-3*x^2+21*x*y-3*y^2+3*x+3*y-1

I4=ideal(x-t^4,y-(1-t)^4)
toString (gens gb I4)_0_0
-- x^4-4*x^3*y+6*x^2*y^2-4*x*y^3+y^4-4*x^3-124*x^2*y-124*x*y^2-4*y^3+6*x^2-124*x*y+6*y^2-4*x-4*y+1

The leading terms seem to be $(x+y)^n$ for $n$ odd and $(x-y)^n$ for $n$ even.
In geogebra

