Final answer:
To sketch the parabola with the given directrix and focus, we first determine the vertex and then use the standard form of a parabolic equation. The vertex is found by halving the distance between the directrix and focus, and the equation is derived from the standard form by substituting the vertex and distance value.
Step-by-step explanation:
To sketch the parabola with a directrix of y = 4 and a focus of (2, 6), we need to understand the basic properties of a parabola. The parabola is a U-shaped curve that is symmetric both horizontally and vertically. The directrix is a horizontal line, so the parabola opens vertically. The focus is a point located on the axis of symmetry of the parabola, which is equidistant from the vertex and the directrix.
Given that the directrix is y = 4 and the focus is at (2, 6), the vertex is halfway between the directrix and the focus at a distance of 2 units below the focus. Therefore, the vertex is at (2, 8).
Now, we can draw a rough sketch of the parabola, knowing that it opens upwards, has a vertex at (2, 8), and passes through the focus at (2, 6). (Refer to the graph provided.)
To write the equation of the parabola, we use the standard form for a parabolic equation: (y - k) = 4p(x - h)^2. In this equation, (h, k) are the coordinates of the vertex, and p is the distance between the vertex and the focus.
Substituting the values (h, k) = (2, 8) and p = 2 into the standard form equation, we obtain the equation of the parabola: (y - 8) = 4(2)(x - 2)^2.