To find the equation of a quadratic graph (a parabola) given its focus and directrix, we start by identifying the properties of the parabola.
The focus of the parabola is the point where all the light rays reflected off the curve converge. In this case, the focus is located at (5, 6). The directrix is a line where, for any point on the parabola, the distance to the focus is equal to the perpendicular distance to the directrix. The directrix provided is y = 2.
First, we calculate the vertex of the parabola, which lies midway between the focus and the directrix. The y-coordinate of the vertex can be found by averaging the y-coordinate of the focus (6) and the y-coordinate of the directrix (2):
Vertex y-coordinate = (6 + 2) / 2 = 4.
The vertex x-coordinate will be the same as the focus’s x-coordinate, which is 5. Hence, the vertex of the parabola is located at (5, 4).
The distance from the vertex to the focus (p) is calculated as:
p = 6 – 4 = 2.
Since the focus is above the directrix, this parabola opens upwards. The standard form of a parabola that opens upwards with vertex at (h, k) is:
(x – h)2 = 4p(y – k)
Substituting h = 5, k = 4, and p = 2 into the equation gives:
(x – 5)2 = 8(y – 4)
Thus, the equation of the quadratic graph with a focus of (5, 6) and a directrix of y = 2 is:
(x – 5)2 = 8(y – 4)