Question

The function f(x) = -x²+ 36x-224 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. Determine the vertex. What does this calculation mean in the context of the problem?

Answer 1

The vertex of the profit function f(x) = -x² + 36x - 224 is (18, 100), indicating that the maximum profit of $100 occurs when 18 games are sold.

Step-by-step explanation:

To determine the vertex of the quadratic function f(x) = -x² + 36x - 224, which models the profit from selling games, we can use the formula for the vertex of a parabola, h = -b/2a when the function is in the form ax² + bx + c. In this case, a is -1, and b is 36, so the x-coordinate of the vertex is h = -36/(2*(-1)) = 18. To find the y-coordinate of the vertex, we substitute x with 18 into the function: f(18) = -(18)² + 36*18 - 224 = -324 + 648 - 224 = 100. Thus, the vertex is at the point (18, 100).

In the context of the problem, the vertex represents the maximum profit that can be achieved. This occurs when 18 games are sold, resulting in a profit of $100. It is the highest point on the graph of the profit function, indicating the optimal sales quantity for maximum profit.