How is the graph of y= (x-1)^2-3 transformed to produce the graph of y= 1/2(x+4)^2?

Answer:The transformation that follows are:Shift 2 units up.Shift 5 units to the left.Vertical shrink by a factor of 2.Step-by-step explanation:We are given a graph as:      [tex]y= (x-1)^2-3[/tex]Now, this graph is translated to get a new graph as:        [tex]y=\dfrac{1}{2}(x+4)^2[/tex]First the graph of the original function is shifted 3 units up.Since, the translation of the type: f(x) to f(x)+k is a shift k units up or k units down depending on whether k is positive or negative respectively.[tex]y=(x-1)^2-3+3\\\\i.e.\\\\y=(x-1)^2[/tex]Now, this graph is then shifted 5 units left.since the translation of the type f(x) to f(x+k) is a shift of  the function f(x) k units to the left or k units to the right depending on k whether k is positive or negative respectively.Here,[tex]y=(x-1+5)^2\\\\i.e.\\\\y=(x+4)^2[/tex]Now, this graph is then vertically shrink by a factor of 2.Because the transformation of the type:          f(x) to kf(x) is a vertical stretch if k>1and vertical shrink if 0<k<1.i.e.[tex]y=\dfrac{1}{2}(x+4)^2\\\\i.e.\\\\k=\dfrac{1}{2}<1[/tex]