SOLUTION: State the minimum or maximum value for the quadratic function and name the x value at which this occurs: f(x)-2^2-6+5

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Question 111497: State the minimum or maximum value for the quadratic function and name the x value at which this occurs: f(x)-2^2-6+5
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
I'm going to assume that you meant f%28x%29=-2x%5E2-6x%2B5.
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For any quadratic function f%28x%29=ax%5E2%2Bbx%2Bc the maximum or minimum occurs at the vertex of the parabola described by the function. The the x-coordinate of the vertex (h, k) of the parabola ax%5E2%2Bbx%2Bc is given by h=%28-b%29%2F2a. In this case, -%286%2F4%29. This is the x value at which the maximum or minimum occurs.
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To find the maximum or minimum value, you need to evaluate f%28h%29.
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f%28-6%2F4%29=-2%28-6%2F4%29%5E2-6%28-6%2F4%29%2B5
f%28-6%2F4%29=-2%2836%2F16%29%2B%2836%2F4%29%2B5
f%28-6%2F4%29=-72%2F16%2B144%2F16%2B80%2F16
f%28-6%2F4%29=152%2F16=38%2F4
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You didn't say that you needed to identify the value as either a minimum or maximum. There are a couple of ways to do that. The easiest is to pick another value for x and evaluate the function at the new value. If this new value for the function is smaller than the vertex y-coordinate, you know you have found a maximum, and vice versa. Let's evaluate f%280%29=5. Since 5 (or 20/4) is smaller than 38/4, the point must be a maximum.
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You could also graph the function, and determine whether the vertex was a minimum or maximum by inspection.
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graph%28400%2C400%2C-10%2C10%2C-10%2C10%2C-2x%5E2-6x%2B5%29
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There is yet another way, but you need Calculus. Write back if you know how to take the first and second derivatives of a function.