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thhe geometric definition of a parabola:
a set of points where each point has its distance from a fixed point equal to its distance for a fixed line. that fixed point is called the focus and that fixed line is called the directrix.
in other words, the points on a parabola are equidistant from a focus point and a directrix line.
the key to deriving this is using the distance formula.
the distance formula gives the distance between two points: and . the distance is
if we have a point on this parabola, the distance between and the focus point is
now, the distance between that same point and the directrix line at . when talking about distances from a point to a line, we want the shortest possible distance.
the directrix is a horizontal line, meaning that all the points on the line have the same y-coordinate. the points on the horizontal line can have any x-coordinate.
if we choose an x-coordinate that is the same as the x-coord for , then there is a point on the directrix . this will get us the shortest distance.
so the distance between on the parabola and the point on the directrix line (they share the same x-coordinate)
the definition of a parabola says that these two distances are equal.
now we can simplify to get a standard form equation of a parabola, which is . the amount of simplification depends on your teacher.
get rid of the square roots by squaring both sides of the equation.
i will do all expansions separately.
there is are formulas to with expanding binomials that are squared:
if we expand then
if we expand then
if we expand then
going back to our original equation, let us put in all the expansions involving y first.
combine like terms. i first subtract y² from both sides, then subtract 9/4 from both sides.
expanding out (also, 112/4 = 28)
so your parabola has the equation
if you want vertex form
then do not expand out . leave it alone at this step (i also simplify 112/4 to 28)
solve for y to get vertex form
so vertex form version is