To find the quadratic model that fits the given data, we need to find a quadratic equation of the form ŷ = ax^2 + bx + c, where ŷ represents the predicted profit and x represents the number of weeks.
We can use the given data points to create a system of equations. Let's plug in the values from the table:
Week 0: ŷ = -7
Week 1: ŷ = 4
Week 2: ŷ = 14
Week 3: ŷ = 23
Week 4: ŷ = 29
Week 5: ŷ = 35
Week 6: ŷ = 38
Week 7: ŷ = 40
Week 8: ŷ = 41
Week 9: ŷ = 40
Week 10: ŷ = 38
Week 11: ŷ = 34
Week 12: ŷ = 29
Now we can set up the system of equations:
-7 = a(0)^2 + b(0) + c ---> c = -7
4 = a(1)^2 + b(1) + c ---> a + b + c = 4
14 = a(2)^2 + b(2) + c ---> 4a + 2b + c = 14
23 = a(3)^2 + b(3) + c ---> 9a + 3b + c = 23
29 = a(4)^2 + b(4) + c ---> 16a + 4b + c = 29
35 = a(5)^2 + b(5) + c ---> 25a + 5b + c = 35
38 = a(6)^2 + b(6) + c ---> 36a + 6b + c = 38
40 = a(7)^2 + b(7) + c ---> 49a + 7b + c = 40
41 = a(8)^2 + b(8) + c ---> 64a + 8b + c = 41
40 = a(9)^2 + b(9) + c ---> 81a + 9b + c = 40
38 = a(10)^2 + b(10) + c ---> 100a + 10b + c = 38
34 = a(11)^2 + b(11) + c ---> 121a + 11b + c = 34
29 = a(12)^2 + b(12) + c ---> 144a + 12b + c = 29
We can solve this system of equations to find the values of a, b, and c.
After solving the system, we get the following values:
a ≈ -0.76
b ≈ 12.07
c ≈ -6.99
Therefore, the quadratic model that fits the given data is:
ŷ = -0.76x^2 + 12.07x - 6.99
Hence, the correct answer is ŷ = -0.76x^2 + 12.07x - 6.99.