Paramaterizing a Parabola with $3$ points.

Question

Let $A, B, C$ be vectors in $\mathbb R^2$. I want to show that the set $\{A+tB+t^2C\mid t\in\mathbb R\}$ defines a parabola in $\mathbb R^2$, but I'm having a hard time doing so, since I can't solve for one coordinate in terms of the other.

Answer 1

Rotate the coordinate system such that $C$ is parallel to the $y$-axis.

Then, given $x$ you can solve for $t$ (uniquely) and find the corresponding $y$ as a second-degree polynomial in $x$.