SOLUTION: What is the optimal value of this function: y=-0.5x^2+2x+3

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Question 253267: What is the optimal value of this function: y=-0.5x^2+2x+3
Found 2 solutions by jim_thompson5910, Alan3354:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Take note that y=-0.5x%5E2%2B2x%2B3 is equivalent to y=-%281%2F2%29x%5E2%2B2x%2B3 since -1%2F2=-0.5


To find the optimal value, we need to find the vertex.


In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula: x=%28-b%29%2F%282a%29.


x=%28-b%29%2F%282a%29 Start with the given formula.


From y=-%281%2F2%29x%5E2%2B2x%2B3, we can see that a=-1%2F2, b=2, and c=3.


x=%28-%282%29%29%2F%282%28-1%2F2%29%29 Plug in a=-1%2F2 and b=2.


x=%28-2%29%2F%28-1%29 Multiply 2 and -1%2F2 to get -1.


x=2 Divide.


So the x-coordinate of the vertex is x=2. Note: this means that the axis of symmetry is also x=2.


Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.


y=-%281%2F2%29x%5E2%2B2x%2B3 Start with the given equation.


y=-%281%2F2%29%282%29%5E2%2B2%282%29%2B3 Plug in x=-2.


y=-%281%2F2%29%284%29%2B2%282%29%2B3 Square -2 to get 4.


y=-2%2B2%282%29%2B3 Multiply -1%2F2 and 4 to get -2.


y=-2%2B4%2B3 Multiply 2 and 2 to get 4.


y=5 Combine like terms.


So the y-coordinate of the vertex is y=5.


So the vertex is .


Since the max/min occurs at the vertex as the 'y' value, this means that the max is y=5. So the optimal value is is y=5.


Here's a graph to visually confirm the answer:




Graph of y=-0.5x%5E2%2B2x%2B3 with the vertex of (2,5). Here we can see that the largest y value is y=5.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
From dictionary.com:
op·ti·mal (ŏp'tə-məl)
adj. Most favorable or desirable; optimum
Sounds like a matter of opinion.
I like 377.
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If you're told in class that this is "wrong", object to it.