SOLUTION: find the vertex, the line of symmetry and the maximum or minimum value of f(x). Graph the function f(x)=1/2(x+6)^2+3

Algebra ->  Algebra  -> Quadratic Equations and Parabolas -> SOLUTION: find the vertex, the line of symmetry and the maximum or minimum value of f(x). Graph the function f(x)=1/2(x+6)^2+3      Log On

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Question 236217: find the vertex, the line of symmetry and the maximum or minimum value of f(x). Graph the function f(x)=1/2(x+6)^2+3
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(48510) About Me  (Show Source):
You can put this solution on YOUR website!
find the vertex, the line of symmetry and the maximum or minimum value of f(x). Graph the function f(x)=1/2(x+6)^2+3
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Vertex (-6,3)
Line of symmetry: x = -6
Minimum at the vertex is f(x)=3
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graph%28400%2C300%2C-10%2C10%2C-10%2C10%2C%281%2F2%29%28x%2B6%29%5E2%2B3%29
=====================================================
Cheers,
Stan H.

Answer by Edwin McCravy(6931) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29=1%2F2%28x%2B6%29%5E2%2B3
The equation
f%28x%29=a%28x-h%29%5E2%2Bk
is
a parabola with a vertex at (h,k) and which passes
through the points (h+1,k+a) and (h-1,k+a), and its
axis of symmetry is the vertical line whose equation
is x=h
So in the problem:
f%28x%29=1%2F2%28x%2B6%29%5E2%2B3
a=1%2F2, h=-6, k=3
[Notice that the sign is changed for h but kept for k]
So the vertex is (h,k) or (-6,3),
The parabola passes through (h-1,k+a) and (h+1,k+a), or
(-6-1,3+1%2F2) or (-7,3%261%2F2) and
(-6+1,3+1%2F2) or (-5,3%261%2F2)
Its axis of symmetry is the vertical line whose equation
is x=h or x=-6
We plot the vertex (-6,3)
drawing%28400%2C400%2C-10%2C2%2C-3%2C9%2C+graph%28400%2C400%2C-10%2C2%2C-3%2C9%29%2C%0D%0A%0D%0Aline%28-6%2B.1%2C3%2C-6-.1%2C3%29%2C+line%28-6%2C3%2B.1%2C-6%2C3-.1%29%2C+line%28-6%2B.1%2C3%2B.1%2C-6-.1%2C3-.1%29%2C+line%28-6%2B.1%2C3-.1%2C-6-.1%2C3%2B.1%29%2C+locate%28-6%2C3%2C%27%286%2C3%29%27%29%0D%0A%0D%0A++%29
We draw the axis of symmetry which is a vertical line
through the vertex (the green line below:
drawing%28400%2C400%2C-10%2C2%2C-3%2C9%2C+graph%28400%2C400%2C-10%2C2%2C-3%2C9%29%2C%0D%0Agreen%28line%28-6%2C-4%2C-6%2C10%29%29%2C%0D%0Aline%28-6%2B.1%2C3%2C-6-.1%2C3%29%2C+line%28-6%2C3%2B.1%2C-6%2C3-.1%29%2C+line%28-6%2B.1%2C3%2B.1%2C-6-.1%2C3-.1%29%2C+line%28-6%2B.1%2C3-.1%2C-6-.1%2C3%2B.1%29%2C+locate%28-6%2C3%2C%27%286%2C3%29%27%29%0D%0A%0D%0A++%29
Then we plot the two points,
one on each side of the vertex
(-7,3%261%2F2) and (-5,3%261%2F2)


drawing%28400%2C400%2C-10%2C2%2C-3%2C9%2C+graph%28400%2C400%2C-10%2C2%2C-3%2C9%29%2C%0D%0Agreen%28line%28-6%2C-4%2C-6%2C10%29%29%2C%0D%0Aline%28-6%2B.1%2C3%2C-6-.1%2C3%29%2C+line%28-6%2C3%2B.1%2C-6%2C3-.1%29%2C+line%28-6%2B.1%2C3%2B.1%2C-6-.1%2C3-.1%29%2C+line%28-6%2B.1%2C3-.1%2C-6-.1%2C3%2B.1%29%2C+locate%28-6%2C3%2C%27%286%2C3%29%27%29%2C%0D%0A%0D%0Aline%28-7%2B.1%2C7%2F2%2C-7-.1%2C7%2F2%29%2C+line%28-7%2C7%2F2%2B.1%2C-7%2C7%2F2-.1%29%2C+line%28-7%2B.1%2C7%2F2%2B.1%2C-7-.1%2C7%2F2-.1%29%2C+line%28-7%2B.1%2C7%2F2-.1%2C-7-.1%2C7%2F2%2B.1%29%2C+%0D%0A%0D%0Aline%28-5%2B.1%2C7%2F2%2C-5-.1%2C7%2F2%29%2C+line%28-5%2C7%2F2%2B.1%2C-5%2C7%2F2-.1%29%2C+line%28-5%2B.1%2C7%2F2%2B.1%2C-5-.1%2C7%2F2-.1%29%2C+line%28-5%2B.1%2C7%2F2-.1%2C-5-.1%2C7%2F2%2B.1%29++%29
Finally we sketch in the parabola:
drawing%28400%2C400%2C-10%2C2%2C-3%2C9%2C+graph%28400%2C400%2C-10%2C2%2C-3%2C9%2C%0D%0A%281%2F2%29%28x%2B6%29%5E2%2B3%29%2C%0D%0Agreen%28line%28-6%2C-4%2C-6%2C10%29%29%2C%0D%0Aline%28-6%2B.1%2C3%2C-6-.1%2C3%29%2C+line%28-6%2C3%2B.1%2C-6%2C3-.1%29%2C+line%28-6%2B.1%2C3%2B.1%2C-6-.1%2C3-.1%29%2C+line%28-6%2B.1%2C3-.1%2C-6-.1%2C3%2B.1%29%2C+locate%28-6%2C3%2C%27%286%2C3%29%27%29%2C%0D%0A%0D%0Aline%28-7%2B.1%2C7%2F2%2C-7-.1%2C7%2F2%29%2C+line%28-7%2C7%2F2%2B.1%2C-7%2C7%2F2-.1%29%2C+line%28-7%2B.1%2C7%2F2%2B.1%2C-7-.1%2C7%2F2-.1%29%2C+line%28-7%2B.1%2C7%2F2-.1%2C-7-.1%2C7%2F2%2B.1%29%2C+%0D%0A%0D%0Aline%28-5%2B.1%2C7%2F2%2C-5-.1%2C7%2F2%29%2C+line%28-5%2C7%2F2%2B.1%2C-5%2C7%2F2-.1%29%2C+line%28-5%2B.1%2C7%2F2%2B.1%2C-5-.1%2C7%2F2-.1%29%2C+line%28-5%2B.1%2C7%2F2-.1%2C-5-.1%2C7%2F2%2B.1%29++%29
Edwin