A parabola can be written in its vertex form, which is shown below

Suppose that we know the vertex (h,k) and one other point on the parabola (b,c); then,
![\begin{gathered} y=a(x-h)^2+k \\ a\to to\text{ be defined} \end{gathered}]()
We only need to know the value of a to completely identify the equation of the parabola, which can be done as shown below

Then, we can indeed determine the equation given the vertex and one other point on the parabola.