Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find that value.f(x) = -x2 + 2x - 7

The given function is a quadratic function of the form f(x) = ax^2 + bx + c, where a = -1, b = 2, and c = -7.

The sign of the coefficient 'a' determines whether the function has a maximum or minimum value. If 'a' is positive, the function has a minimum value. If 'a' is negative, the function has a maximum value.

In this case, a = -1, which is negative. Therefore, the function has a maximum value.

The maximum or minimum value of a quadratic function f(x) = ax^2 + bx + c is given by the formula f(-b/2a).

Substituting the given values of a and b into this formula, we get:

f(-b/2a) = f(-(2)/(-2*1)) = f(1)

Substitute x = 1 into the function:

f(1) = -(1)^2 + 2*(1) - 7 = -1 + 2 - 7 = -6

Therefore, the maximum value of the function is -6.