Answer: The standard equation of a parabola is given by:
4p(x - h) = (y - k)^2
where (h, k) is the vertex of the parabola, p is the distance from the vertex to the focus and also the distance from the vertex to the directrix.
In this case, the focus is at (-2,1), so the vertex is halfway between the focus and the directrix, which is at (x, y) = (-2, (1+8)/2) = (-2, 4.5). The distance from the vertex to the focus (and to the directrix) is p = 3.5.
Substituting these values into the equation, we get:
4(3.5)(x + 2) = (y - 4.5)^2
Simplifying, we get:
14(x + 2) = (y - 4.5)^2
This is the standard equation of the parabola.
To sketch the graph, we can use the vertex (-2, 4.5) as a starting point, and use the distance from the vertex to the focus to plot some points on either side of the vertex. Since the focus is to the left of the vertex, we know that the parabola will open to the left.
Using the distance formula, we can find the coordinates of two points on the parabola that are equidistant from the focus and the directrix:
Point 1: (-5.5, 4.5)
Distance from focus: sqrt((-5.5 + 2)^2 + (4.5 - 1)^2) = sqrt(44.5) ≈ 6.67
Distance from directrix: |4.5 - 8| = 3.5
Point 2: (-5.5, 1.5)
Distance from focus: sqrt((-5.5 + 2)^2 + (1.5 - 1)^2) = sqrt(36.5) ≈ 6.04
Distance from directrix: |1.5 - 8| = 6.5
Parabola with a focus at (-2,1) and directrix y=8
The vertex is labeled as V, the focus as F, and the directrix as D. The distance from the vertex to the focus (and directrix) is labeled as p. The parabola opens to the left and is symmetric about the axis of symmetry, which is the vertical line passing through the vertex.