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# Write the equation!

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Write the equation! Feb 26, 2019

#1
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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   Feb 26, 2019
#2
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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

so

(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 ! Feb 26, 2019
edited by ElectricPavlov  Feb 26, 2019