To determine the appropriate function model for the given dataset, we first analyze the growth of the y-values as the x-values increase.
Looking at the changes in the y-values:
- From 0 to 1: 5 to 35 (increases by 30)
- From 1 to 2: 35 to 245 (increases by 210)
- From 2 to 3: 245 to 1715 (increases by 1470)
- From 3 to 4: 1715 to 12005 (increases by 10290)
- From 4 to 5: 12005 to 84035 (increases by 72030)
The increases are not constant, which indicates that it is not a linear function.
Next, let's look at whether the function could be exponential or quadratic. In an exponential function, the ratio of successive y-values tends to be constant (or increases at a constant rate), while in a quadratic function, the second differences of the y-values tend to be constant.
Calculating the ratios:
- 35 / 5 = 7
- 245 / 35 = 7
- 1715 / 245 = 7
- 12005 / 1715 ≈ 7
- 84035 / 12005 ≈ 7
Since the ratio of successive y-values is consistent and appears to be approximately 7, this suggests that the function is indeed exponential.
Based on the options given, the function that fits the data well is:
f(x) = 5(7)^x and it represents an exponential function.
The correct choice would therefore be: f(x)=5(7)^x and exponential function.