SOLUTION: i need to find the vertex of y=x^2+12x+19

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Question 586052: i need to find the vertex of y=x^2+12x+19
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E2%2B12x%2B19 Start with the given expression.


Take half of the x coefficient 12 to get 6. In other words, %281%2F2%29%2812%29=6.


Now square 6 to get 36. In other words, %286%29%5E2=%286%29%286%29=36


x%5E2%2B12x%2Bhighlight%2836-36%29%2B19 Now add and subtract 36. Make sure to place this after the "x" term. Notice how 36-36=0. So the expression is not changed.


%28x%5E2%2B12x%2B36%29-36%2B19 Group the first three terms.


%28x%2B6%29%5E2-36%2B19 Factor x%5E2%2B12x%2B36 to get %28x%2B6%29%5E2.


%28x%2B6%29%5E2-17 Combine like terms.


So after completing the square, x%5E2%2B12x%2B19 transforms to %28x%2B6%29%5E2-17. So x%5E2%2B12x%2B19=%28x%2B6%29%5E2-17.


So y=x%5E2%2B12x%2B19 is equivalent to y=%28x%2B6%29%5E2-17.


So the equation y=%28x%2B6%29%5E2-17 is now in vertex form y=a%28x-h%29%5E2%2Bk where a=1, h=-6, and k=-17


Remember, the vertex of y=a%28x-h%29%5E2%2Bk is (h,k).


So the vertex of y=%28x%2B6%29%5E2-17 is (-6,-17) since h = -6 and k = -17


A graph will confirm this



Graph of y=x%5E2%2B12x%2B19 with vertex (-6,-17)
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