Equation of the horizontal parabola with generic vertex

Let's consider the horizontal parabola with vertex at a generic point $A(x_0,y_0)$.

In this case the focus is on $F(x_0+\dfrac{p}{2},y_0)$ and the generator line is $x=x_0-\dfrac{p}{2}$.

The equation of the parabola under these conditions is $$(y-y_0)^2=2p(x-x_0)$$

The equation of the parabola which focus point is at $F(-2,4)$ and the vertex point at $A(3,4)$.

Identify $A(x_0,y_0)$ with $A(3,4)$ and $F(x_0+\dfrac{p}{2},y_0)$ with $F(-2,4)$. We obtain $x_0=3$ and $y_0=4$.

By analyzing the focus and the generic equation we know that $$x_0+\dfrac{p}{2}=3+\dfrac{p}{2}=-2$$, then $\dfrac{p}{2}=5$ and we obtain the parameter value $p=10$.

Substituting into the equation $(y-y_0)^2=2p(x-x_0)$ we obtain $$(y-4)^2=20(x-3)$$