Answer: min value is y = 0
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Step-by-step explanation:
You can factor x^2+6x+9 into (x+3)^2, which turns into 1(x - (-3) )^2 + 0
So y = x^2+6x+9 becomes y = 1(x - (-3) )^2 + 0
That matches with vertex form y = a(x-h)^2 + k
The vertex is (-3,0). The y coordinate of the vertex directly tells us the minimum value. We know we have a minimum because a = 1 is positive. If a < 0, then we'd have a max y value instead.
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Alternatively, compare y = x^2+6x+9 to the form y = ax^2+bx+c
We see that a = 1, b = 6, c = 9
The x coordinate of the vertex is
h = -b/(2a)
h = -6/(2*1)
h = -3
The x coordinate of the vertex is x = -3. Plug this into the original equation to find the y coordinate of the vertex
y = x^2 + 6x + 9
y = (-3)^2 + 6(-3) + 9
y = 9 - 18 + 9
y = -9 + 9
y = 0
The y coordinate of the vertex is y = 0. This is the smallest y output possible.