Question

The function f(x) = −x^2 + 48x − 432 models the profit, in dollars, a game developer makes for selling a new game, where x is the number of games 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?

Answer 1

f(x) = −x^2 + 48x − 432

Part A

Find the zeros:

−x^2 + 48x − 432 = 0

x^2 - 48x + 432 = 0

Factor

(x - 36 ) (x - 12 )=0

Two solutions: x = 36 and x = 12

The vertex is in x = (36 +12 )/ 2 = 24

f(24) = -(24)^2 + 48(24) − 432 = 144

The vertex is (24, 144).

Given that the coefficient of x^2 is negative, the parabole opens downward, that mean that the vertex is a maximum value.

In the context of the problem that means that the maximum profit is 144 and is obtainded when the number of games sold is 24.

Part B.

The x-intercepts are the values of x when y=0. Which we calculated above.

They are x = 24 and x = 36.

Those values are the number of games for which the profit is zero.