Name Cumulative Test Due Monday \( 3 / 17 / 25 \) 10. (4pt) Determine an equation for the parabola with focus. \( (-2,4) \) and directrix \( y=10 \). (The use of the grid below is optional.)

To find the equation of the parabola, we first note that the focus \((-2, 4)\) and the directrix \(y = 10\) indicate that this is a vertical parabola that opens downward, since the focus is below the directrix. The vertex of the parabola is located halfway between the focus and the directrix. The vertex thus lies at the point \((-2, \frac{4 + 10}{2}) = (-2, 7)\). Using the standard form of the equation for a vertical parabola \((x - h)^2 = 4p(y - k)\), where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus (or directrix), we find \(p = 4 - 7 = -3\). Substituting \(h = -2\), \(k = 7\), and \(p = -3\) into the equation gives us: \[ (x + 2)^2 = -12(y - 7) \] So, the equation of the parabola is \((x + 2)^2 = -12(y - 7)\). Happy graphing!