Defining a region in $\mathbb{R}^2$

Question

I was trying to do this exercise but my answer doesn't match with the solution and I'm wondering why:

Consider the coordinates transformation defined by $x=2u+v$ and $y=u^2-v$. Being $T$ the triangle with vertices $(0,0)$, $(1,0)$, $(0,2)$ on plan $uv$ determine the image of $T$ on plan $xy$ by the coordinates transformation.

So what I did was take the vertices and apply the the transformation obtaining $(0,0)$ $(2,1)$ and $(2,-2)$.

Then I represented then in the plane $xy$ and I formed the triangle.

Then I defined my conditions:

• $0<x<2$

• $-x<y<\frac{x}{2}$

However the solutions say the proper answer for $y$ would be:

• $-x<y<\frac{x^2}{4}$

And I wonder why since it's supposed to be a triangle. So is it possible to be limited by a parabola?

Answer 1

You found the images of the vertices of the given triangle. But the images of the straight lines making up the sides of the given triangle are not mapped to straight lines.