Question 117704: How do you find the vertex of this quadratic equation?
y=x^2+6x+8
Answer by josmiceli(19441)   (Show Source):
You can put this solution on YOUR website! 
First, it helps if you can picture it. Does it look like a pointy
cup that holds water, or is it a pointy cup that poured all its
water out?
If the number in front of  is positive, it's a pointy cup
that holds water. In this case, The number in front of
is  , since  .
Now you can picture it.
(1) If it touches the x-axis it has 1 real solution for x.
That means you can somehow get  into the form
 . Then  is the only solution.
The other possibilities are:
(2) It floats above the x-axis without touching it.
(3) It crosses the x-axis in 2 places
That's almost all you need to know. Now picture it crossing the x-axis
in 2 places. Where is the vertex? It's exactly in the middle between
the 2 crossing points. The crossing points are the roots
So, if the roots were at  and  , then the vertex
would be at  .
Now all you need to know are the roots of
They occur at  , so find  when
Right away I see that  and  .
That leads me to  . What makes this true?
Either  or  These are the roots. What's
exactly in the middle?  is right between them.
So, that's where the vertex is. A point needs an x and a y, so what
is the y? Plug into the original equation
So, the vertex is at (-3,-1)
Here's the graph
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