Answer the questions below about the quadratic function. \[ f(x)=-x^{2}-6 x-10 \] Does the function have a minimum or maximum value? Minimum Maximum What is the function's minimum or maximum value? \( \square \) Where does the minimum or maximum value occur? D
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Quadratic functions can be represented in the form \( f(x) = ax^2 + bx + c \). In this case, since the coefficient \( a \) is negative (\( a = -1 \)), your function opens downwards, which means it has a maximum value! To find this maximum value, we can use the vertex formula, \( x = -\frac{b}{2a} \), where \( a = -1 \) and \( b = -6 \). Calculating \( x \) gives you the x-coordinate of the vertex, which is the location of the maximum value. Plugging in \( b \) and \( a \), we find \( x = -\frac{-6}{2 \times -1} = 3\). Then substitute \( x = 3 \) back into the original function to find the maximum value, which turns out to be \( f(3) = -3^2 - 6(3) - 10 = -36 \). So, the maximum value is \( -36 \), occurring at \( x = 3 \).
