The function f (x) = - x2 + 24x - 80 models the hourly profit, in dollars, a shop makes for selling coffee, where x is the number of cups of coffee 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? Part B: Determine the x-intercepts. What do these values mean in the context of the problem?

If we are given the function f (x) = - x² + 24x - 80 that models the hourly profit, and x is the number of cups of coffee sold, we can conclude that the graph would look like a parabola and would face downward. To solve this, we have to complete the square and then factor it;

f (x) = - x² + 24x - 80 = y

y + 144 = - (x² - 24x + 144 + 80)

y + 224 = - (x - 12) ²

The vertex is (12, - 224)