Question

Find the maximum value of the parabola: y = -x^2 + 8x - 7. Simplify your answer and write it as a proper fraction, improper fraction, or integer.

Answer 1

To find the maximum value of the parabola given by the equation \( y = -x^2 + 8x - 7 \), we first note that it is a downward-opening parabola because the coefficient of \( x^2 \) is negative.

The maximum value of a parabola in the form \( y = ax^2 + bx + c \) can be found using the vertex formula, which gives the \( x \)-coordinate of the vertex as:

\[
x = -\frac{b}{2a}
\]

Here, \( a = -1 \), \( b = 8 \), and \( c = -7 \).

Calculating the vertex:

\[
x = -\frac{8}{2 \cdot -1} = -\frac{8}{-2} = 4
\]

Next, we substitute \( x = 4 \) back into the equation to find the maximum \( y \)-value:

\[
y = - (4^2) + 8(4) - 7
\]
\[
y = -16 + 32 - 7
\]
\[
y = 16 - 7
\]
\[
y = 9
\]

Thus, the maximum value of the parabola is:

\[
\boxed{9}
\]