Answer
$$y = - \frac{1}{2}{x^2}$$
Work Step by Step
$$\eqalign{
& r = - 2\sec \theta \tan \theta \cr
& r = - 2\left( {\frac{1}{{\cos \theta }}} \right)\left( {\frac{{\sin \theta }}{{\cos \theta }}} \right) \cr
& {\text{Multiply both sides of the equation by }}\cos \theta \cr
& r\cos \theta = - 2\cos \theta \left( {\frac{1}{{\cos \theta }}} \right)\left( {\frac{{\sin \theta }}{{\cos \theta }}} \right) \cr
& r\cos \theta = - 2\left( {\frac{{\sin \theta }}{{\cos \theta }}} \right) \cr
& {\text{Convert to rectangular form, using }}y = r\sin \theta ,{\text{ }}x = r\cos \theta \cr
& x = - 2\left( {\frac{{y/r}}{{x/r}}} \right) \cr
& x = - 2\left( {\frac{y}{x}} \right) \cr
& - \frac{1}{2}{x^2} = y \cr
& {\text{The equation represents a parabola}} \cr
& {\text{Graph}} \cr} $$
