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Determine whether the quadratic function shown below has a minimum or maximum, then determine the minimum or maximum value of the function.
\[
f(x)=-x^{2}+7 x-18
\]

Answer Attempt 2 out of 2

The maximum $\vee$ value is
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Step 1 ：The given function is a quadratic function in the form of \(f(x) = ax^2 + bx + c\), where \(a = -1\), \(b = 7\), and \(c = -18\).

Step 2 ：A quadratic function has a maximum value when the coefficient of \(x^2\) is negative and a minimum value when the coefficient of \(x^2\) is positive. In this case, the coefficient of \(x^2\) is -1, which is negative. Therefore, the function has a maximum value.

Step 3 ：The maximum or minimum value of a quadratic function \(f(x) = ax^2 + bx + c\) is given by the formula \(-\frac{b^2}{4a} + c\).

Step 4 ：So, we can substitute \(a = -1\), \(b = 7\), and \(c = -18\) into the formula to find the maximum value of the function.

Step 5 ：\(a = -1\)

Step 6 ：\(b = 7\)

Step 7 ：\(c = -18\)

Step 8 ：max_value = -5.75

Step 9 ：The maximum value of the function is -5.75. This is the final answer.

Step 10 ：Final Answer: The maximum value of the function is \(\boxed{-5.75}\).