Question 32675
This is problem requires a longer explanation so I will only answer part A and then you can apply these same principles to the remaining 3 parts. 
You have the problem y=x^2 - 2x + 3 and you want to manipulate it to the form y = a(x - h)^2 + k.  

y=x^2 - 2x + 3  First, a will always be the coefficient on x^2- in this case, it is 1.  To find the (x-h)^2 part, you will need to "complete the square".
(x - ?)(x - ?) will give you (x^2 - 2x + __ ) + 3.  Forget about the three while caculate what should go in the parenthesis to give you (x^2 - 2x + __ ) when you factor it.  You come up with (x - 1)(x - 1). WHen you foil this, you get (x^2 -2x + 1) and now, to keep the equation balanced, you need to subtract one.  (x-1)^2 + 3 - 1. Combine those, and you get y = 1(x-1)^2 + 2

y = x^2 - 2x + 3
y = 1x^2 - 2x + 3
y = 1(x^2 - 2x + ?) + 3
y = 1(x - ?)(x - ?) + 3 
y = 1(x^2 - 2x + 1) + 3 -1
y = 1(x - 1)(x - 1) + 3 -1 
y = 1(x-1)^2 + 2
Now, we have it in the form y = a(x - h)^2 + k.
The VERTEX of the equation is at the point (h,k).  Notice that the h is subtracted from the x so watch the sign carefully.  In this case, the vertex would be (1,2).  If a is positive, the vertex is a minimum and the parabola points upward (positive=happy face) and if a is negative, the vertex is a maximum (negative=frowny face).  In this case, a=1 and is positive so the vertex is a minimum and the parabola points upward like a "U".  To find the x interecepts, you substitute x=0 and solve for y.  
This should be enough information for a basic sketch.  Good luck with the remaining problems.