The vertex is ( 5, [13 + 2]/2 ) = (5, 7.5)
p is the distance from the vertex to the focus = 7.5 - 2 = 5.5
The parabola turns downward because the directrix is above the focus
We have the form :
4p ( y - k)^2 = - (x - h)^2 wher (h, k) = the vertex....so we have
4(5.5) (y - 7.5) = - (x - 5)^2 simplify
22(y - 7.5) = - ( x - 5)^2 divide both sides by 22
y - 7.5 = -(1/22)(x - 5)^2 add 7.5 to both sides
y = -(1/22)(x - 5)^2 + 7.5
Here is a graph : https://www.desmos.com/calculator/a8n6hmiw4w
CPhill is working on this...I know he does this differently than what I am going to show ya....
Definition of parabola is the set of points that are equidistant from a point called the focus and a line called the directrix.
This parabola has the focus BELOW the directrix, so it opens downward....it looks like a dome.
Point x,y on the parabola has the same distance to the focus and the line y= 13
Distance to the focus is (x-5)^2 +(y-2)^2 = d^2
Distance to the directrix is simply d = 13-y d^2 = (13-y)^2 these two d^2 are equal
(x-5)^2 + (y-2)^2 = (13-y)^2 Expand and simplify to find your parabola equation
x^2 - 10x + 25 + y^2 - 4y + = y^2 -26y + 169 gather like terms and simplify and you will get
y = -1/22 x^2 + 10/22 x + 140/22 for your parabola
...here is a graph..... Well...you can look at CPHill's graph !