The focus must be the point (0, 4), because it is equidistant from the vertex (0, 2) as the focus is from the directrix, y = 0, which indicates that the focus is twice the distance from the vertex to the directrix or (0, 4)
The evaluation that shows the reasons the focus must be the point (0, 4) are as follows;
The location of the vertex of the parabola at the point (0, 2), and the location of the directrix on the x-axis, we get;
The location of the focus is on the line passing through the vertex, which is the line x = 0
The definition of a parabola is the path of a point that moves such that the distance from the focus and the directrix are the same
The equation of the directrix is; y = 0
The shortest distance of the vertex from the directrix is 2 - 0 = 2 units
The distance from the focus to the vertex is therefore 2 units
Whereby the focus is 2 units above the x-axis, the focus, which is 2 units from the vertex on the remote side of the directrix is 2 + 2 = 4 units above the x-axis and the coordinates of the vertex must be (0, 4)
The definition of a parabola indicates that the location of the focus should be 2 units from the
The complete question found through search can be presented as follows;
A parabola is shown graphed on the grid below. Its directrix is the x-axis
(a) Explain why the focus must be the point (0, 4)
The coordinates of the vertex of the parabola is (0, 2)
The coordinates of other points on the parabola are (-8, 8), (8, 8)