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 y=ax^2 + bx + c, then what is a+b+c?
I've answered the first one elsewhere today........
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 y=ax^2 + bx + c, then what is a*b*c ?
The vertex will be found at (4,2) and we can write that :
(y - 2) = (a)(x - 4)^2 and since (8,6) is on the curve, we can solve for a
(6 -2) = (a) (8 - 4)^2 simplify
4 = (a)(4)^2
4 = (a)16 → a = 4/16 = 1/4
Since the x coordinate of the vertex is given by -b/ (2a) we have that -b/[2 (1/4)] = 4 → -b / (1/2) = 4 → -b = 2 → b = -2
And using the fact that (4,2) is on the graph, we can find c, thusly :
y = ax^2 + bx + c ......so......
2 = (1/4)(4)^2 -2(4) + c
2 = 4 - 8 + c
2 = -4 + c
c = 6
Then a*b*c = (1/4) (-2) (6) = (1/4)(-12) = -3
+36916 You are given the directrix and the focus you should be able to deduce that the vertex (between the two)
is at 4,2 this is h,k
Parabola in vetex form y = a(x-h)^2 + k
y = a(x-4)^2 + 2 Sub in th epoint given to calc 'a'
6 = a(8-4)^2 + 2
a = 1/4
now your equation becomes y = 1/4 (x-4)^2 + 2 expand
= 1/4 x^2 -2x+6 now you can see what a b c are to add them.....
