Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then
find that value.
-$$f ( x ) = - x ^ { 2 } - 2 x - 8$$
A) maximum; - 1
B) minimum; - 7
C) maximum; - 7
D) minimum; - 1

Question

Quizplus · Accepted Answer

The quadratic function is in the form $f(x) = ax^2 + bx + c$, where $a = -1$, $b = -2$, and $c = -8$. Since $a < 0$, the parabola opens downwards, indicating it has a maximum value. To find the maximum value, use the vertex formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex. Substituting $a = -1$ and $b = -2$ gives $x = -\frac{-2}{2(-1)} = 1$. Then, substitute $x = 1$ into the original equation to find the y-coordinate (maximum value): $f(1) = -(1)^2 - 2(1) - 8 = -1 - 2 - 8 = -11$. However, there seems to be a mistake in my calculation of the y-coordinate of the vertex. Let's correct that: The correct x-coordinate of the vertex is actually $x = -\frac{-2}{2(-1)} = -\frac{-2}{-2} = 1$, which was incorrectly calculated. The correct calculation for the x-coordinate of the vertex should be $x = -\frac{-2}{2(-1)} = -\frac{-2}{-2} = 1$. To find the y-coordinate (maximum value), we should correctly substitute $x = -1$ (the actual x-coordinate of the vertex) into the original equation: $f(-1) = -(-1)^2 - 2(-1) - 8 = -1 + 2 - 8 = -7$. Therefore, the function has a maximum value at $-7$, indicating a correction in the explanation towards the correct answer.

Determine, Without Graphing, Whether the Given Quadratic Function Has a Maximum