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​ ($)
88
99
1010
1111
1212
Revenue​ ($)
29912991
37313731
41754175
43234323
41754175

Answer 1

Interpret the prices as $8, $9, $10, $11, $12 and the revenues as 29,912; 37,313; 41,754; 43,234; 41,754. The vertex is at price h = 11 with revenue k = 43,234, so use vertex form R(p) = a(p − 11)^2 + 43,234.

Use p = 10, R = 41,754 to find a:
41,754 = a(10 − 11)^2 + 43,234 ⇒ a = 41,754 − 43,234 = −1,480.

Thus the quadratic model is
R(p) = −1,480(p − 11)^2 + 43,234 (dollars).

Expanded (standard) form:
R(p) = −1,480p^2 + 32,560p − 135,846.

This model predicts a maximum revenue of $43,234 at ticket price $11.