Final answer:
To write a quadratic function with a focus of (0,2) and a directrix of y=6, use the equation of a parabola and plug in the given values to get x^2 = 16y-32.
Step-by-step explanation:
To write a quadratic function with a focus of (0,2) and a directrix of y=6, we can use the formula for the equation of a parabola.
The general equation of a parabola with a vertical axis of symmetry is (x-h)^2 = 4p(y-k), where (h,k) is the vertex and p is the distance from the vertex to the focus or directrix.
In this case, the vertex is (0,2) and the directrix is y=6. The distance from the vertex to the directrix is 6-2 = 4. Therefore, p=4.
Plugging the values into the equation, we get x^2 = 4(4)(y-2). Simplifying further, x^2 = 16y-32 is the quadratic function with the given focus and directrix.