Final answer:
The standard form of the parabola is y = 4x^2 + 16x - 27.
Step-by-step explanation:
The standard form of a parabola is given by the equation y = ax^2 + bx + c. To find the values of a, b, and c, we need to use the focus-directrix definition of a parabola. The focus is given as (2,6), which means that the vertex of the parabola is halfway between the focus and the directrix. Since the directrix is y = 4, the vertex is at (2, 5). Using the vertex and the focus, we can find the value of a in the standard form equation. The distance between the vertex and the focus is equal to |a|/(4a). Hence, we have |a|/(4a) = 1. Solving this equation gives us a = 4. Now that we have the value of a, we can find the value of b. The distance from the vertex to the directrix is equal to |b|/(4a). Hence, we have |b|/(4a) = 1. Plugging in the value of a, we get |b|/16 = 1. Solving this equation gives us b = 16. Therefore, the standard form of the parabola is y = 4x^2 + 16x + c. To find the value of c, we can use any point on the parabola. Let's use the vertex. Plugging in the values of x and y for the vertex, we get 5 = 4(2)^2 + 16(2) + c. Solving this equation gives us c = -27. Therefore, the standard form of the parabola is y = 4x^2 + 16x - 27.