The function f(x) = –x^2 + 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?
part b: determine the x-intercepts. what do these values mean in the context of the problem?

we can write  by completing the square:





thus,



The form  of writing a quadratic function is called the Vertex form, and (h, k) is the vertex.

So in , h=14, k=4, 

thus the vertex is (14, 4)

The coefficient of the largest term is negative, this means that the parabola opens downwards and the vertex is the highest point.

In the context of the problem, the vertex (14, 4) means that the largest

profit is  $4, made for selling 14 sodas.

The x-intercepts are the roots of f(x), that is the values of x for which f(x)=0, 

so we need to solve f(x)=0





x-14= 2,     or  x-14=-2

thus the roots are x=16 and x=12

when selling 12 and 16 sodas, the profit is 0.

Part A:

the vertex is (14, 4)

the vertex (14, 4) means that the largest profit is  $4, made for selling 14 sodas.

Part B:

the x-intercepts are 12 and 16.

when selling 12 and 16 sodas, the profit is 0.