To determine the best type of function that models the data given, let's examine the growth values for even increments of time and see how they change.
- Time: 1 day → Growth: 0 inches
- Time: 3 days → Growth: 12 inches
- Time: 5 days → Growth: 32 inches
- Time: 7 days → Growth: 60 inches
Now, we can analyze the growth between the given time intervals:
- From 1 to 3 days:
- Growth change: \(12 - 0 = 12\)
- Time change: \(3 - 1 = 2\)
- From 3 to 5 days:
- Growth change: \(32 - 12 = 20\)
- Time change: \(5 - 3 = 2\)
- From 5 to 7 days:
- Growth change: \(60 - 32 = 28\)
- Time change: \(7 - 5 = 2\)
Next, calculate the average rates of growth:
- From 1 to 3 days: \( \frac{12}{2} = 6 \)
- From 3 to 5 days: \( \frac{20}{2} = 10 \)
- From 5 to 7 days: \( \frac{28}{2} = 14 \)
These rates are increasing, which suggests that the growth is accelerating.
Next, let’s check the second differences which might indicate a quadratic pattern (since linear would have constant first differences):
- First differences of growth:
- \(12, 20, 28\)
- Second differences (changes in first differences):
- From 12 to 20: \(20 - 12 = 8\)
- From 20 to 28: \(28 - 20 = 8\)
The second differences are constant (all 8), which indicates a quadratic relationship.
Given this analysis, the best model for the growth data over time is a quadratic function.