2) 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 (4,3) 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 ax^2+bx+c, then what is a+b+c?
The vertex of this parabola will lie at (4,2)
And we have this form
y = a(x - h)^2 + k and (h,k) = the vertex = (4,2) and since (8,6) is on the graph, we can solve for "a" thusly
6 = a(8 - 4)^2 + 2
6 = a(4)^2 + 2
4 = 16a
a = 1/4
So....our equation is
y = (1/4)(x - 4)^2 + 2 expand and simplify
y = (1/4) [x^2 - 8x + 16] + 2
y = (1/4)x^2 -2x + 4 + 2
y = (1/4)x^2 - 2x + 6 and a = (1/4) , b = -2 and c = 6 ...so ... a + b + c = (1/4) - 2 + 6 = 4 + 1/4 = 17/4
Here's a graph : https://www.desmos.com/calculator/htcmqubuwp