How do you find transformations?

I think this question meant to ask how the graphs of functions
are transformed or shift when the functions change from
y = x^2 to y = 3(x + 2)^2 + 1
As well as giving an answer, I have tried to explain why.
By the way, all this is easier to understand if you're lucky enough
to have a graphic calculator to demonstrate the changes.
 
1) Changing from y = x^2 to y = 3x^2
In both cases when x = 0 y = 0,
so the graph of the second function still has its vertex at 0, 0
but all y coordinates are multiplied by 3
so the second graph is stretched upwards by a factor of three.

2) Changing from y = 3x^2 to y = 3(x + 2)^2
The line y = 3 cuts y = 3x^2 at x = 1 and x = -1
The line y = 3 cuts y = 3(x + 2)^2 when x + 2 = 1 and x + 2 = -1
These are the points x = - 3 and x = -1
This tells us that replacing x by x + 2 has made the
whole graph shift two units in the negative x direction.

3) Changing from y = 3(x + 2)^2 to y = 3(x + 2)^2 + 1
The least value for y in both functions occurs when x = -2
The first function has a vertex at -2, 0
while the second one has its vertex at -2, 1
Thus the effect of increasing the function by the constant 1
is to shift the whole graph upwards by one unit. 

I hope these are the transformations you meant
and that you understand why they occur.

Regards - Ian