You can find the equation of a vertical parabola with vertex at the origin by using the geometric definition. If you denote the focus by (0, c), the directrix is the line with equation y = − c . y equals negative c .

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Here′s Why It Works Any point (x, y) on the parabola must be equidistant from the focus and the directrix. Use the Distance Formula.

( x − 0 ) 2 + ( y − c ) 2 = ( x − x ) 2 + ( y − ( − c ) ) 2 Distance Formula x 2 + ( y − c ) 2 = 0 2 + ( y + c ) 2 Square each side . x 2 + y 2 − 2 c y + c 2 = y 2 + 2 c y + c 2 Expand . x 2 − 2 c y = 2 c y Subtract  y 2  and  c 2  from each side . x 2 = 4 c y Add  2 c y  to each side . y = 1 4 c x 2 Standard quadratic form table with 6 rows and 3 columns , row1 column 1 , square root of open , x minus 0 , close squared . plus . open , y minus c , close squared end root , column 2 equals . square root of open , x minus x , close squared . plus . open . y minus . open , negative c , close . close squared end root , column 3 cap distancecap formula , row2 column 1 , x squared , plus . open , y minus c , close squared , column 2 equals , 0 squared , plus . open , y plus c , close squared , column 3 cap squareeachside . . , row3 column 1 , x squared , plus , y squared , minus 2 c y plus , c squared , column 2 equals , y squared , plus 2 c y plus , c squared , column 3 cap expand , . , row4 column 1 , x squared , minus 2 c y , column 2 equals 2 c y , column 3 cap subtract . y squared , and , c squared . fromeachside . . , row5 column 1 , x squared , column 2 equals 4 c y , column 3 cap add , 2 c y . toeachside . . , row6 column 1 , y , column 2 equals , fraction 1 , over 4 c end fraction . x squared , column 3 cap standardquadraticform , end table

Note that the equation has the expected quadratic form y = a x 2 y equals , eh x squared  for a vertical parabola with vertex at (0, 0). The coefficient a = 1 4 c eh equals , fraction 1 , over 4 c end fraction  determines both the focus (0, c) and the directrix y = − c . y equals negative c .  This is the key to shifting between the algebraic and geometric representations of a parabola.

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