Q.Find the equation of the parabola with vertex at the origin, axis along the $x$-axis, and passing through the point $(1,-2)$.

A parabola is the locus of a point equidistant from a fixed point, the focus, and a fixed line not through the focus, the directrix -- the defining focus-directrix property that distinguishes it, and the ellipse and hyperbola after it, from the circle's simpler single-distance definition. Applying this property with focus $(a,0)$ and directrix $x=-a$, and equating the two distances via the distance formula, derives the standard equation $y^2=4ax$ directly; three further orientations ($y^2=-4ax$, $x^2=4ay$, $x^2=-4ay$) cover every other choice of axis and opening direction. The vertex is the origin, the axis is the line of symmetry through the vertex and focus, and …