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 \(\mathcal{P}\) is a parabola with focus \((4,3)\) and directrix \(y=1\). The point \((8,6)\) is on \(\mathcal{P}\) because \((8,6)\) is 5 units away from both the focus and the directrix.
If we write the equation whose graph is \(\mathcal{P}\) in the form \(y=ax^2+bx+c\), then what is \(a+b+c\)?
+23245 A graphing form for a parabola is: y - k = a(x - h)2 where (h, k) is the vertex.
The vertex is midway between the focus and the directrix.
Since the focus is (4, 3) and the directrix is y = 1, the vertex is at (4, 2).
Therefore, the graphing form is: y - 2 = a(x - 4)2.
To find a, let's put the value of (8, 6) into this formula: 6 - 2 = a(8 - 4)2
4 = a(4)2
4 = a·16
a = 4/16 ---> a = ¼
So, the equation is: y - 2 = ¼(x - 4)2
Multiplying out: y - 2 = ¼(x2 - 8x + 16)
y - 2 = ¼x2 - 2x + 4
y = ¼x2 - 2x + 6
a = ¼ b = -2 c = 6
