Modeling quadratic functions

Question

A movie theater recorded its monthly revenue for the past 5 months. The table below 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 10, 11, 12, 13, 14

Revenue 3072, 3792, 4224, 4368, 4224

Answer 1

3072, 3792, 4224, 4368, 4224

3792-3072=720 f(x₂)-f(x₁)

4224-3792=432 f(x₃)-f(x₂)

432-720=-288,,,,f(x₃)-2f(x₂)+f(x₁)

4224-3792=432 f(x₄)-f(x₃)

4368-4224=144 f(x₃)-f(x₂)

144-432=-288 f(x₄)-2f(x₃)+f(x₂)

yay, when independent variable increases equal amount for data points 10,11,12,13

then f(x₄)-2f(x₃)+f(x₂)=f(x₃)-2f(x₂)+f(x₁)

is a requirement for the function to be quadratic.

the result (-288) will always be 2a where f(x)=ax²+bx+c,,, we have a=-144

note: result is 2a if increments in variable a one unit.

for b and c,

-144(10²)+b(10)+c=3072

-144(11²)+b(11)+c=3792

subtracting

b=720+3024=3744

-14400+37440+c=3072

c=-19968

f(x)=-144x²+3744x-19968

[f(x)/-144]=x²-26x+414/3=(x-13)²-169+414/3=(x-13)²-91/3

f(x)=-144(x-13)²+91(144)/3