Final Answer:
The focus is at (2, -4), the directrix is y = -7, and the equation of the parabola is (y + 4)² = 4(x - 2).
Step-by-step explanation:
In the given graph, the vertex of the parabola is at the point (2, -4), which serves as the focus. The parabola is oriented vertically, and the directrix is a horizontal line passing through the y-coordinate of the vertex, which is y = -4 - 3 = -7.
The standard form of the equation for a vertical parabola is (y - k)² = 4a(x - h), where (h, k) is the vertex. Plugging in the values, we get (y + 4)² = 4(x - 2), representing the equation of the parabola.
Understanding the focus and directrix helps visualize how the parabola is shaped. The focus is the point where all the parabola's reflecting rays converge, and the directrix is a line perpendicular to the axis of symmetry, creating a symmetrical relationship with the parabola. The equation of the parabola indicates how it is stretched or compressed and in which direction it opens.