Question

*Hint: What is the vertex? The answer is NOT a coordinate point (x, y).
p(x)=-2x2+16x-24
16 (thousand dollars)
8 (thousand dollars)
10 (thousand dollars)
12 (thousand dollars)

Answer 1

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)**.