Complete the square and explain how to transform the graph of y=x^2 into the graph of the given function.
f(x)=x^2+4x+6
To complete the square on ax^2 + bx + c.
Firstly take out the coefficient of x^2 as a factor: a(x^2 + (b/a)x + (c/a)).
Next write the square of x plus half the coefficient of the x term: (x+b/2a)^2
Then add and subtract the constant term of that: b^2/4a^2: a((x+b/2a)^2 + c/a-b^2/4a^2)
Then simplify: a((x+b/2a)^2 - (b^2-4ac)/4a).
In your case: a=1
x^2+4x+6 = (x+2)^2 + (6-4) = (x+2)^2 + 2.
The relation between the graph of y = x^2 and the graph of y = (x+2)^2 + 2
is a left shift of 2 (x -> x+2) and an upward shift of 2.