The function f(x) = –x2 + 28x – 192 models the hourly profit, in dollars, a shop makes for selling sodas, where x is the number of sodas 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)

Part A:
The vertex :  x = - b / 2 a = - 28 : ( - 2 ) = 14
y = f ( 14 ) = - 14² + 28 · 14 - 192 = 196 + 392 - 192 = 4
The vertex is ( 14, 4 ). It means that the maximum hourly profit is $4, and that the shop should sell 14 sodas to reach the maximum.
Part B :
We will solve the quadratic equation:
x 1/2 = ( - 28 +/- sqrt ( 784 - 768 ) ) / ( - 2 )
x 1 = 12,  x 2 = 16.
It means that if the shop must sell more than 12 and less then 16 sodas if they want to make a profit ( 13, 14, or 15 ). If they sell 12, or 16 they will brake even. 

Main idea on left, details on right