Final answer:
The quadratic equation representing the parabolic arch of the doorway in standard and vertex form is y = -1/3x^2 + 12, with the vertex at (0, 12) and x-intercepts at (-6, 0) and (6, 0).
Step-by-step explanation:
The equation that represents the quadratic function of a parabolic arch of a doorway which is 12 feet high and 12 feet wide can be found using the fact that the vertex is at (0, 12) and it crosses the x-axis at (-6, 0) and (6, 0).
The general form of a quadratic equation in vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. For the given arch, h = 0 and k = 12.
To find the value of 'a', we use one of the x-intercepts, for example, (6, 0):
0 = a(6 - 0)^2 + 12
Solving for 'a' gives us a = -1/3. Therefore, the equation in vertex form is y = -1/3x^2 + 12.
To convert this to standard form, which is y = ax^2 + bx + c, we simply recognize that b = 0 and c = 12, resulting in the standard form of the equation being y = -1/3x^2 + 12.