Geometrically speaking, a parabola is defined as the set of points that are the same distance from a given point and a given line. The point is called the focus of the parabola and the line is called the directrix of the parabola.
Suppose P is a parabola with focus (4,2) and directrix y=1. The point (8,6) is on P because (8,6) is 5 units away from both the focus and the directrix.
If we write the equation whose graph is P in the form y=ax^2+bx+c then what is a+b+c?
The vertex of the parabola will be found half-ways between the focus and the directrix.
The vertex will be the point (4, 1.5).
A formula for the parabola is: y - k = a(x - h)2 where (h, k) are the coordinates of the vertex.
For this situation: y - 1.5 = a(x - 4)2.
Since the point (8, 6) is on the parabola: 6 - 1.5 = a(8 - 4)2
---> 4.5 = a(4)2 ---> 4.5 = 16·a ---> a = 0.28125
The equation is: y - 1.5 = 0.28125(x - 4)2.
Multiplying out: y - 1.5 = 0.28125(x2 - 8x + 16)
y - 1.5 = 0.28125x2 - 2.25x + 4.5
y = 0.28125x2 - 2.25x + 6