y = 
x² + 2x + 3
Clear of fractions:
3y = x² + 6x + 9
Factor 1 out of first two terms on the right:
3y = 1(x² + 6x) + 9
Change the parentheses to brackets so it can contain parentheses:
3y = 1[x² + 6x] + 9
To complete the square inside the bracket:
1. Take one-half of the coefficient of x. 
·(6) = 3
2. Square the result. (3)² = 9
3. Add it then subtract it in the bracket: Add + 9 - 9
3y = 1[x² + 6x + 9 - 9] + 9
Factor the first three terms inside the bracket as a perfect square:
3y = 1[(x+3)² - 9] + 9
Remove the bracket by distributing the 1 leaving the parentheses intact:
3y = 1(x+3)² - 9 + 9
The -9 and the +9 gives 0
3y = 1(x+3)² + 0
Solve for y by dividing through by 3
y = (x+3)² + 0
Compare that to the standard vertex form:
y = a(x-h)² + k
We see that the vertex (h,k) is (-3,0) and from the original
equation y = x² + 2x + 3 tells us that the y-intercept
is (0,3)
The axis of symmetry is the vertical line through the vertex, and
is therefore the vertical line whose equation is x=-3:
The point that matches the y-intercept on the other side of the
axis of symmetry is (-6,3), so we plot that point:
and we sketch in the parabola:
Edwin
