Final answer:
The vertex of the parabola is (4, 5), the focus is at (7, 5), and the directrix is x = 1.
Step-by-step explanation:
The given equation represents the parabola in the form (y - k)² = 4a(x - h), where (h, k) is the vertex and 4a is the focal length. Comparing the given equation with the standard form, we have (y - 5)² = 12(x - 4). So, the vertex of the parabola is (4, 5). The focal length is given by 4a = 12, so a = 3. Therefore, the focus is at a distance of 3 units to the right of the vertex, which gives us the coordinates of the focus as (4 + 3, 5) = (7, 5). Finally, the directrix is a vertical line at a distance of 3 units to the left of the vertex, which gives us the equation of the directrix as x = 4 - 3 = 1.