Find the vertex, focus, directrix, and axis of symmetry of each parabola (without completing the square ), and determine whether the parabola opens upward of downward. y = − 2 x 2 − 6

Question

Find the vertex, focus, directrix, and axis of symmetry of each parabola (without completing the square ), and determine whether the parabola opens upward of downward.  y  =  −  2      x          2        −  6

Gillian Howell · Accepted Answer

First of all, the negative on the       x          2       term tells youthat it opens downward;To find the vertex, you can add 6 to each side to see what hand k are (or whatever variables your book uses for vertex form of the equation).  y  +  6  =  −  2      x          2              Compare with    y  −  k  =  a  (  x  −  h      )          2      The y + 6 is y - (-6), so k is-6, and there is no h, so h = 0.So vertex is (0, -6).(You can also find the xcoordinate of the vertex by: x = - b /2a= 0 / -4 = 0 for     y  =  −  2      x          2        +  0  x  −  6  −  2  =  a,   0 = b, and -6 = c Then plug in zero for xto get the y value of the vertex:   y  =  −  2  (  0      )          2        −  6  =  −  6 So, (0, -6) is the vertex.)The axis of symmetry goes through the vertex, along thedirection that the parabola opens - in this case, along the yaxis.The equation for that axis is x = 0.Focus and directrix are found by : c =1 / 4a = 1 / (4 * 2) = 1/8.So the focus is 1/8 of a unit down from thevertex, at ( 0, -6 1/8), and the directrixis 1/4 unit up from the vertex, aty = - 5 7/8

Find the vertex, focus, directrix, and axis of symmetry of each parabola (without completing the square ), and determine whether the parabola opens upward of downward. y = -2x^2 - 6

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