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17 September, 15:01

What is the equation of the quadratic graph with a focus of (5, - 1) and a directrix of y = 1?

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  1. 17 September, 16:01
    0
    The answer is directrix is horizontal, so the parabola is vertical

    focus lies below the directrix, so the parabola opens downwards.

    General equation for down-opening parabola:

    y = a (x-h) ²+k

    with

    a<0

    vertex (h, k)

    focal length p = 1/|4a|

    focus (h, k-p)

    directrix (h, k+p)

    Apply your data and solve for h, k, and a.

    vertex is halfway between focus and directrix: (3,3)

    h = 3

    k = 3

    distance between focus and vertex is 2, so p = 2.

    a = - 1 / (4p) = - ⅛

    y = - ⅛ (x-3) ²+3
  2. 17 September, 18:20
    0
    (x-h) ^2=4p (y-k)

    vertex is (h, k)

    p is the distance from focus to vertex and distance from vertex to directix (vertex is in middle of directix and focus)

    if p is positive, the parabola opens up and the focus is above the directix

    if p is negative, the parabola opens down and the focus is below the directix

    we see the directix is over the focus (1>-1) so the parabola opens down and p is negative

    distance from (5,-1) to y=1 is 2 units

    2/2=1

    p=-1 since p is negative

    1 up from (5,-1) is (5,0)

    veretx is (5,0)

    (x-5) ^2=4 (-1) (y-0)

    (x-5) ^2=-4y is the equation
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