Formula for minimum of a changing parabola

Question

So what I want to find out is the path drawn by the parabola $y=x^2+bx$ minimum as b changes. Im pretty sure it draws an upside down parabola because if I put in Desmos and slide the b it goes in the pattern if you look at it. How can find this formula and what is it called?

Answer 1

The minimum of the parabola $x^2+bx = (x-\frac{b}{2})^2 - \frac{b^2}{4}$ is achieved at $(\frac{b}{2},-\frac{b^2}{4})$, which is precisely the parabola $y=-x^2$.