Answer
$$y = \frac{3}{2}x - 2$$
Work Step by Step
$$\eqalign{
& {y^2} = - 12x;{\text{ }}\left( { - \frac{4}{3}, - 4} \right) \cr
& {\text{Find }}\frac{{dy}}{{dx}}{\text{ using the implicit differentiation}} \cr
& \frac{d}{{dx}}\left[ {{y^2}} \right] = \frac{d}{{dx}}\left[ { - 12x} \right] \cr
& 2y\frac{{dy}}{{dx}} = - 12 \cr
& \frac{{dy}}{{dx}} = - \frac{{12}}{{2y}} \cr
& \frac{{dy}}{{dx}} = - \frac{6}{y} \cr
& {\text{Calculate the slope at the point }}\left( { - \frac{4}{3}, - 4} \right) \cr
& m = - \frac{6}{{ - 4}} \cr
& m = \frac{3}{2} \cr
& {\text{The equation of the tangent line at the given point is}} \cr
& y - {y_1} = m\left( {x - {x_1}} \right) \cr
& y + 4 = \frac{3}{2}\left( {x + \frac{4}{3}} \right) \cr
& y + 4 = \frac{3}{2}x + 2 \cr
& y = \frac{3}{2}x - 2 \cr
& \cr
& {\text{Graph}} \cr} $$
