To find the vertex of the quadratic function \( p(x) = -2x^2 + 16x - 24 \), we first need to identify whether we need the x-coordinate or the y-coordinate of the vertex, since you've indicated that the answer is NOT a coordinate point.
The x-coordinate of the vertex of a quadratic function in the form \( ax^2 + bx + c \) can be calculated using the formula:
\[ x = -\frac{b}{2a} \]
In this case, \( a = -2 \) and \( b = 16 \). Plugging in these values:
\[ x = -\frac{16}{2 \cdot -2} = -\frac{16}{-4} = 4 \]
Now, we can substitute \( x = 4 \) back into the function to find the value of \( p(4) \):
\[ p(4) = -2(4)^2 + 16(4) - 24 \] \[ = -2(16) + 64 - 24 \] \[ = -32 + 64 - 24 \] \[ = 32 - 24 \] \[ = 8 \]
So, the value of \( p(x) \) at the vertex, which is also the maximum (since the parabola opens downwards), is \( 8 \) (thousand dollars).
Thus, the answer is:
8 (thousand dollars).
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