Final answer:
The equation of the parabola with a given focus and directrix in vertex form is y = 5.
Step-by-step explanation:
The equation of a parabola in vertex form is given by y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
The vertex form of the equation allows us to easily identify the vertex and the direction of opening of the parabola.
In this case, the given focus is (4, 5), which is the vertex of the parabola.
The directrix is the line y = -3, which is a horizontal line.
Since the vertex form of the equation is in the form y = a(x - h)^2 + k, we can substitute the vertex coordinates into the equation and solve for a to find the equation of the parabola.
Substituting the vertex coordinates (4, 5) into the equation, we get:
5 = a(4 - 4)^2 + 5
5 = a(0)^2 + 5
5 = 5
Since the equation is true for all values of a, we can conclude that the equation of the parabola is y = 5.