Answer:
To determine the type of function that best models the data in the table, we can observe the differences or ratios between successive values of y.
Using differences:
The first differences between successive values of y are:
1.5 - 0 = 1.5
6 - 1.5 = 4.5
13.5 - 6 = 7.5
24 - 13.5 = 10.5
The second differences between successive values of y are:
4.5 - 1.5 = 3
7.5 - 4.5 = 3
10.5 - 7.5 = 3
Since the second differences are constant, the function that best models the data is a quadratic function.
Using ratios:
The ratios between successive values of y are:
1.5/0 = undefined
6/1.5 = 4
13.5/6 = 2.25
24/13.5 ≈ 1.78
Since the ratios are not constant, the function that best models the data is not an exponential function.
Therefore, the type of function that best models the data in the table is a quadratic function. We can try to find a quadratic function of the form y = ax^2 + bx + c that passes through the given points. Using the first three points, we get the system of equations:
a(0)^2 + b(0) + c = 0
a(1)^2 + b(1) + c = 1.5
a(2)^2 + b(2) + c = 6
Solving this system of equations, we get:
a = 3/2
b = 0
c = 0
Therefore, the quadratic function that best models the data is y = (3/2)x^2.