Final answer:
The standard equation of the parabola with the vertex at (0,2) and the focus at (-4,2) is (y - 2)^2 = -16x.
Step-by-step explanation:
The student's question is about determining the standard equation of a parabola with a given vertex and focus. Given the information that the vertex is at (0,2) and the focus is at (-4,2), we can infer that the parabola opens to the left since the focus is to the left of the vertex, and both are on the same horizontal line. The standard form equation for a horizontally oriented parabola is (y - k)^2 = 4p(x - h), where (h, k) is the vertex of the parabola and p is the distance from the vertex to the focus (which is positive if the parabola opens to the right and negative if it opens to the left).
Here, since our vertex (h, k) is (0, 2), and the focus is 4 units to the left, p would be -4. Plugging in these values yields the equation (y - 2)^2 = 4*(-4)(x - 0), which simplifies to (y - 2)^2 = -16x. Hence, the standard equation of the parabola is (y - 2)^2 = -16x.