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?

Part A.  To find the vertex, [tex](h,k)[/tex], of a function of the form [tex]f(x)=ax^2+bx+c[/tex], we vertex formula: [tex]h= \frac{-b}{2a} [/tex], and then we evaluate [tex]f(x)[/tex] at [tex]h[/tex] to find [tex]k[/tex].
We know from problem that [tex]a=-1[/tex] and [tex]b=24[/tex]. Lets replace those values in our formula:
[tex]h= \frac{-b}{2a} [/tex]
[tex]h= \frac{-24}{2(-1)} [/tex]
[tex]h= \frac{-24}{-2} [/tex]
[tex]h=12[/tex]
[tex]k=f(12)=-12^2+24(12)-80[/tex]
[tex]k=f(12)=-144+288-80[/tex]
[tex]k=64[/tex]

We can conclude that the coordinates of the vertex of our function are (12,64). In this situation the vertex represents the maximum hourly profit the shop can achieve.

Part B. To find the x-intercept of our function, we are going to set the function equal to zero and solve for [tex]x[/tex]:
[tex]-x^2+24x-80=0[/tex]
We can multiply both sides of our equation by -1 to change [tex]x^2[/tex] to positive:
[tex]x^2-24x+80=0[/tex]
Now we can easily factor our expression:
[tex](x-4)(x-20)=0[/tex]
[tex]x-4=0[/tex] and [tex]x-20=0[/tex]
[tex]x=4[/tex] and [tex]x=20[/tex]

We can conclude that the x-intercepts of our function are x=4 and x=20. They represent how many cups of coffee the shop will need to sell per hour to have no profit.