12in an experiment, a petri dish with a colony of bacteria is exposed to cold temperatures and then warmed again.
find a quadratic model for the data in the table. type your answer below. show your work.

time (hours)
0
1
2
3
4
5
6
population (1000s)
5.1
3.03
1.72
1.17
1.38
2.35
4.08

Let the equation be : at^2 + bt + c = P,
where t = time (hrs),
P = population (1000's).
 When t = 1, P = 3.03.
When t = 2, P = 1.72.
When t = 3, P = 1.17.
 Substitute these into the equation to obtain these 3 simultaneous equations : a + b + c = 3.03
4a + 2b + c = 1.72
9a + 3b = c = 1.17
 
Solving gives :
a = 0.38,
b = -2.45,
c = 5.1.
 
The equation is therefore,
P = 0.38t^2 - 2.45t + 5.1
 Testing with t = 0 to 6 gives the population values as provided, so it seems to be a valid model.
 At t = 9 hrs,
P = 0.38*9^2 - 2.45*9 + 5.1
   = 13.83.