Given a straight line (a directrix) and a point (the focus) which is not on that line, a parabola is locus of all points whose distance form the directrix is the same as its distance from the focus.
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Locus
The given terms can't be an equation without an equality sign but a negative parabola opens down wards whereas a positive parabola opens up wards.
If the equation of the parabola isy = ax^2 + bx + c, then it opens above when a>0 and opens below when a<0. [If a = 0 then the equation describes a straight line, and not a parabola!].
It is the apex of the parabola.
No, a parabola does not have to have an x-intercept. ex. -2(x-2)^2 - 4 is a parabola that has no x-intercept.