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Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. "f(x) \( =-2 x^{2}+4 x-7 \) Does the quadratic function f have a minimum value or a maximum value? The function \( f \) has a minimum value. The function \( f \) has a maximum value. What is this minimum or maximum value? Simplify your answer.)

Ask by Sanders Gibson. in the United States
Mar 15,2025

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The quadratic function \( f(x) = -2x^2 + 4x - 7 \) has a maximum value of \(-5\).

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The given quadratic function \( f(x) = -2x^2 + 4x - 7 \) has a leading coefficient of -2, which is negative. This indicates that the parabola opens downward, meaning the function has a maximum value. To find the maximum value, you can use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = 4 \). Calculating the vertex gives: \[ x = -\frac{4}{2(-2)} = 1 \] Next, substitute \( x = 1 \) back into the function to find the maximum value: \[ f(1) = -2(1)^2 + 4(1) - 7 = -2 + 4 - 7 = -5 \] Thus, the function \( f \) has a maximum value of -5.

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