(4pt) Petermine an equation for the parabola with focus \( (-2,4) \) and directrix \( y=10 \). The of the grid below is optional.)
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To find the equation of a parabola given its focus and directrix, we start by identifying the vertex. The vertex is located halfway between the focus and the directrix. Hence, the y-coordinate of the vertex can be found by averaging the y-coordinates of the focus (4) and the directrix (10). Therefore, the vertex is at \( (-2, \frac{4 + 10}{2}) = (-2, 7) \). The distance \( p \) from the vertex to the focus or directrix is \( 10 - 7 = 3 \) (since the parabola opens downwards, \( p \) is negative). Parabolas with vertical axes of symmetry can be given by the equation \( (x - h)^2 = 4p(y - k) \), where \( (h,k) \) is the vertex. Plugging in values, we have: \[ (x + 2)^2 = -12(y - 7) \] This is the equation of the parabola with a focus at \( (-2, 4) \) and directrix \( y=10 \). And voilà! You’ve got the equation! Just remember, if you ever need to sketch this parabola, plotting the focus and the directrix can help you see how the curve behaves. The distance from any point on the parabola to the focus equals the distance from that point to the directrix, making it a perfect geometric shape. Happy graphing!
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