The equation of a quadratic graph can be derived from its focus and directrix. In this case, the focus is at the point (4, 3), and the directrix is the line y = 13.
To find the equation, we first recognize that the vertex of the parabola lies midway between the focus and the directrix. The distance from the focus to the directrix can be calculated as follows:
- Focus: (4, 3)
- Directrix: y = 13, which can be represented as the point (x, 13) for any x.
The distance from the focus to the directrix in the y-direction is:
|3 – 13| = 10 units.
Since the focus is below the directrix, we can find the vertex by going down 5 units from the directrix (half of 10) to find the vertex’s y-coordinate:
Vertex y-coordinate = 13 – 5 = 8.
Thus the vertex is at (4, 8). The distance ‘p’ from the vertex to the focus (or the vertex to the directrix) is 5. Since the parabola opens downward (because the focus is below the directrix), the equation of a parabola in vertex form is given by:
(x – h)² = 4p(y – k)
Where (h, k) is the vertex and p is the directed distance. Plugging in these values (h = 4, k = 8, p = -5, since it opens downwards):
(x – 4)² = -20(y – 8)
We can expand this to standard form if necessary:
(x – 4)² = -20y + 160
x² – 8x + 16 = -20y + 160
20y = -x² + 8x – 144
Thus, the equation of the quadratic graph is:
y = -(1/20)x² + (2/5)x – 7.2.