Start with the graph of y=sqrtx. Remember that this means our domain is x>0, and we can graph the points (0,0),(1,1),(4,2),(9,3),(16,4), and so on.
graph{sqrtx [-4.73, 35.82, -8.76, 11.51]}
The next step is to graph y=sqrt(x-4). Notice that at x=4, this gives us sqrt0=0. At x=13, we have sqrt(13-4)=sqrt9=3. What adding the -4 within the function does is actually shift the function 4 units to the right. (Think about comparing the point on y=sqrtx of (9,3) with the point on y=sqrt(x-4) of (13,3)--that's the shift.)
graph{sqrt(x-4) [-4.73, 35.82, -8.76, 11.51]}
Next, we can graph y=1/2sqrt(x-4). This means you can take whatever y value previously existed, and halve it. So, the point at (20,4) will become (20,2) and the point at (5,1) will become (5,1/2).
graph{1/2sqrt(x-4) [-4.73, 35.82, -8.76, 11.51]}
For y=-1/2sqrt(x-4), just reflect all the points over the x-axis, which is the same as taking the negative versions of all the existing y-values.
graph{-1/2sqrt(x-4) [-4.73, 35.82, -8.76, 11.51]}
Finally, for y=-1/2sqrt(x-4)+1, take the graph and shift it upwards 1 point:
graph{-1/2sqrt(x-4)+1 [-4.73, 35.82, -8.76, 11.51]}
