Question

What is the vertex of the graph of y + 2x + 3 = –(x + 2)2 + 1?

Answer 1

There are a lot of steps involved in this, so pay attention. First step is to expand the squared quantity and FOIL it out, like this:
$y+2x+3=-(x+2)(x+2)+1$ and
$y+2x+3=- x^{2} -4x-4+1$
We are going to combine all the like terms now and get them all on one side of the equation:
$- x^{2} -6x-6=y$
Now we are going to complete the square on the polynomial in order to find the vertex.  Do this by first setting the equation equal to 0 and then moving the constant over to the other side of the equals sign, like this:
$- x^{2} -6x=6$ and now factor out the negative sign (cuz negative signs are a pain):
$-( x^{2} +6x)=6$.  To complete the square, you take half the linear term, square it, and add it in to both sides.  Our linear term is 6x.  Half of 6 is 3, and 3 squared is 9.  That's easy to add in on the left side, but we cannot forget that fact that we factored out a negative 1, and that the negative 1 is still there and has to be taken into consideration when we "add" in a 9 to the other side.  We actually multiply the negative 1 times the 9 and that's what's added in to the right:
$-( x^{2} +6x+9)=6-9$
What you do when you complete the square is create a perfect square binomial on the left, which we have and which looks like this:
$-(x+3) ^{2} = -3$
When we move the 3 back over to be with its mates (the 3 is the y coordinate for the vertex), we have the actual sign of the y coordinate.  The number inside the parenthesis with the x is the x coordiante of the vertex in the form $(x-h) ^{2}$.  So the vertex of your problem is (-3, 3).  The negative outside the parenthesis just indicates to us that the parabola is an upside down one, like a mountain instead of a valley.