They may be defined in a number of different ways. One possible description is that they are conic sections.
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No, a conic section does not have vertices. If it is a circle, it has a center; if it is a parabola or hyperbola, it has a focus; and if it is an ellipse, it has foci.
A point, a straight line, a circle, an ellipse, a parabola and half a hyperbola.
Circle. Ellipse. Rectangle. Square. Triangle. Parallelogram. Trapezoid. Losenge. Deltoid. Pentagon. Hexagon. Parabola. That's twelve.
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.