The normal line to the graph of parabola

at the point

,

is

, or equivalently

. The number of real roots of a reduced cubic

depends on the sign of

. So the number of normal lines to

through

depends on the sign of

. If

, there is one normal line to

through

; if

, there are two normal lines; and if

, there are three. (For the cusp

, the only normal line is

.)