The vertex form equation of a parabola is:
\[ y=\frac{(x-h)^2}{4p} + k\]
Expanding this:
\[\begin{gather} y=\frac{x^2-2hx+h^2}{4p} + k \\ y=\frac{x^2}{4p} - \frac{hx}{2p} + \frac{h^2}{4p} + k \end{gather}\]
We can write this as:
\[ y= \left( \frac{1}{4p} \right) x^2 + \left(- \frac{h}{2p} \right) x + \left( \frac{h^2}{4p} + k \right)\]
We can write the coeffecient as \(a\), \(b\) and \(c\):
\[ y= ax^2 + b x + c\]
This is the standard form equation of the parabola.