Question

A movie theater recorded its monthly revenue for the past 5 months. The table shows the revenue as well as the price the movie theater charged for each ticket that month. Use vertex form to create a quadratic model that models the movie​ theater's revenue.
Price​ ($)
8
9
10
11
12
Revenue​ ($)
2991
3731
4175
4323
4175

Answer 1

Let p be the ticket price (dollars) and R(p) the revenue (dollars). The revenue is maximal at p = 11 with R = 4323, so the vertex is (11, 4323). Use the point p = 10, R = 4175 to find a:

4175 = a(10 − 11)^2 + 4323
⇒ 4175 = a(1) + 4323
⇒ a = 4175 − 4323 = −148.

Thus the quadratic in vertex form is

R(p) = −148 (p − 11)^2 + 4323.

(You can check: p = 9 gives R = 3731, p = 8 gives R = 2991, etc.)