The graph of f(x) was vertically translated down by a value of k to get the function g(x) = 5x + k. what is the value of k?

See attached graphic.  what you should know for this case case is what is the cut point with the y axis.   let's suppose that the original graph is:   f (x) = 5x  the graph with the applied translation is:   g (x) = 5x -3  in this case we see the cut point with the y-axis in (0, -3).  we have then  k = -3.