Answer
$$y = - \frac{1}{2}x - 4$$
Work Step by Step
$$\eqalign{
& {y^2} = 8x,{\text{ point }}\left( {8, - 8} \right) \cr
& {\text{By the implicit differentiation}} \cr
& \frac{d}{{dx}}\left[ {{y^2}} \right] = \frac{d}{{dx}}\left[ {8x} \right] \cr
& 2y\frac{{dy}}{{dx}} = 8 \cr
& {\text{Solve for }}\frac{{dy}}{{dx}} \cr
& \frac{{dy}}{{dx}} = \frac{8}{{2y}} \cr
& \frac{{dy}}{{dx}} = \frac{4}{y} \cr
& {\text{Calculate the slope at the point }}\left( {8, - 8} \right) \cr
& m = {\left. {\frac{{dy}}{{dx}}} \right|_{\left( {8, - 8} \right)}} = \frac{4}{{ - 8}} \cr
& m = - \frac{1}{2} \cr
& {\text{Using the point - slope form }}y - {y_1} = m\left( {x - {x_1}} \right),{\text{ with }}\underbrace {\left( {8, - 8} \right)}_{\left( {{x_1},{y_1}} \right)} \cr
& y - {y_1} = m\left( {x - {x_1}} \right) \cr
& y - \left( { - 8} \right) = - \frac{1}{2}\left( {x - 8} \right) \cr
& y + 8 = - \frac{1}{2}x + 4 \cr
& y = - \frac{1}{2}x + 4 - 8 \cr
& y = - \frac{1}{2}x - 4 \cr} $$
