# 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 $$(y-4)^2=20(x-3)$$\$