Part A details a quadratic equation in Vertex form, or f(x) = (x-h)^2 + k. In this form, h represents the x-value at which the graph is shifted, and k represents the y-value at which the graph intersects the y-axis, both in “units”. In this case, the f(x) is translated down five units to become g(x).
Part B also details a quadratic equation in Vertex form, instead a constant is added to the squared quantity. Using f(x)=(x-h)^2 + k, we find that h=-1 (if you plug -1 in for h, you’ll get (x+1)^2). Therefore, we have a translation from 0 to -1 along the x-axis, or a translation left by one unit.