By the geometric definition of a parabola, which relationship is always true for any point P on the curve?

Convert the equation $y=2x^2+8x+5$ into vertex form by completing the square, and identify the vertex.

Given the equation of the vertical parabola $y=0.5(x-1{)}^2+3$, determine the coordinates of the focus and the equation of the directrix.

A horizontal parabola has its vertex at the origin and its focus at $(4,0)$. Which equation represents this parabola?

For the horizontal parabola $x=-\frac{1}{4}(y-2{)}^2+1$, determine the coordinates of the focus and the equation of the directrix.