A way to graph #y = 1/2(x-2)^2 + 4# is to find the vertex and then create a table of values to graph. Since the equation is in vertex form (#y = a(x-h)^2 +k#), we know the vertex (h,k) is #(2, 4)#. This helps to focus on which x values to use, since the vertex must be in the middle.
Now to create a table: (choose an x value, plug it into the equation to find a y value)
(x,y)
#x = 0, y = 1/2(0-2)^2 + 4 => y = 1/2(-2)^2 + 4 => y = 1/2(4) + 4 => y = 2 + 4 => y = 6#
#(0,6)#
#x = 1, y = 1/2(1-2)^2 + 4 => y = 1/2(-1)^2 + 4 => y = 1/2(1) + 4 => y = 1/2 + 4 => y = 4.5 (4 1/2)#
#(1, 4.5)#
Vertex #(2,4)#
(the next values should match x=1 and x=0 since quadratics are symmetric)
#x = 3, y = 1/2(3-2)^2 + 4 => y = 1/2(1)^2 + 4 => y = 1/2(1) + 4 => y = 1/2 + 4 => y = 4.5 (4 1/2)#
#(3, 4.5)#
#x = 4, y = 1/2(4-2)^2 + 4 => y = 1/2(2)^2 + 4 => y = 1/2(4) + 4 => y = 2 + 4 => y = 6#
#(4,6)#
Now graph the following points:
#(0, 6), (1, 4.5), (2, 4), (3, 4.5), (4, 6)#
Connect the dots, graph should be shaped as "u". That is the graph of the quadratic equation #y = 1/2(x-2)^2 + 4#.
