Answer:
22
Explanation:
You want the maximum value of f(x) = -x^2 +8x +6.
Graph
Perhaps the easiest way to find the maximum value is to let a graphing calculator show it to you. Type the function definition into the calculator input box. The attachment shows the maximum is 22 at x=4.
Vertex
The equation is of a parabola that opens downward (2nd-degree, leading coefficient negative). This means you can find the maximum value from the equation when it is written in vertex form.
f(x) = -x^2 +8x +6 . . . . . . given equation
f(x) = -(x^2 -8x) +6 . . . . . . leading coefficient factored out of x-terms
At this point, we can "complete the square" by adding the square of half the x-coefficient inside parentheses, and adding an equivalent amount outside parentheses.
f(x) = -(x^2 -8x +16) +6 +16
f(x) = -(x -4)^2 +22 . . . . . . . . . vertex form
Compare this to the vertex form equation ...
f(x) = a(x -h)^2 +k . . . . . . . . scale factor 'a', vertex (h, k)
We see that (h, k) is (4, 22), so the y-value is 22 at the most extreme point on the graph.