The function f(x) = –x2 + 18x – 72 models the daily profit, in dollars, a gym makes for selling memberships, where x is the number of memberships sold, and f(x) is the amount of profit. part a: determine the vertex. what does this calculation mean in the context of the problem? (5 points) part b: determine the x-intercepts. what do these values mean in the context of the problem? (5 points)

A:
Complete the square for the vertex
f(x) = -(x - 18x) - 72
f(x) = -(x - 18x + 81) - 72 + 81
f(x) = -(x - 9)^2 + 9
so the vertex is (9, 9).  The vertex is the point where the maximum amount of profit is made.

B:
f(x) = x^2 - 18x + 72 (this makes the factoring easier and doesn't change the x-intercepts)
f(x) = (x - 12)(x - 6) = 0
set both factors equal to 0
x - 12 = 0                        x - 6 = 0
x = 12                             x = 6
These are the number of memberships where the gym will break even meaning neither make a profit nor take a loss.