Question
Which type of function best models the data in the table?
Time and Growth Data
Time (days) Growth (inches)
1
0
3
12
5
32
7
60
(1 point)
Responses
exponential
exponential
quadratic
quadratic
linear
linear
cubic
Time and Growth Data
Time (days) Growth (inches)
1
0
3
12
5
32
7
60
(1 point)
Responses
exponential
exponential
quadratic
quadratic
linear
linear
cubic
Answers
GPT-4o mini
To determine the best type of function that models the given data (Time and Growth), let's analyze the provided points:
- At 1 day, growth is 0 inches.
- At 3 days, growth is 12 inches.
- At 5 days, growth is 32 inches.
- At 7 days, growth is 60 inches.
We can consider the nature of the growth:
1. **Linear**: This would mean a constant rate of growth, which does not seem to fit the data as the growth rate increases significantly between days.
2. **Quadratic**: A quadratic function suggests a parabolic growth, which could fit some data sets, but we need to check if the growth changes non-linearly across intervals.
3. **Cubic**: A cubic function would fit a more complex data set with inflection points, but from the data, we would first check if quadratic fits better before considering cubic.
4. **Exponential**: This suggests that growth accelerates over time, which is possible given the data increments.
By calculating the growth differences:
- From Day 1 to Day 3: Growth changes from 0 to 12 (12 units over 2 days)
- From Day 3 to Day 5: Growth changes from 12 to 32 (20 units over 2 days)
- From Day 5 to Day 7: Growth changes from 32 to 60 (28 units over 2 days)
As we see, the amount of growth itself is increasing by larger amounts, suggesting this could potentially be exponential or quadratic, but the rate of increase is escalating, likely indicating exponential growth more than linear or quadratic.
Considering the rapid increase, **exponential** is the best fit for modeling this data.
So the answer is **exponential**.
- At 1 day, growth is 0 inches.
- At 3 days, growth is 12 inches.
- At 5 days, growth is 32 inches.
- At 7 days, growth is 60 inches.
We can consider the nature of the growth:
1. **Linear**: This would mean a constant rate of growth, which does not seem to fit the data as the growth rate increases significantly between days.
2. **Quadratic**: A quadratic function suggests a parabolic growth, which could fit some data sets, but we need to check if the growth changes non-linearly across intervals.
3. **Cubic**: A cubic function would fit a more complex data set with inflection points, but from the data, we would first check if quadratic fits better before considering cubic.
4. **Exponential**: This suggests that growth accelerates over time, which is possible given the data increments.
By calculating the growth differences:
- From Day 1 to Day 3: Growth changes from 0 to 12 (12 units over 2 days)
- From Day 3 to Day 5: Growth changes from 12 to 32 (20 units over 2 days)
- From Day 5 to Day 7: Growth changes from 32 to 60 (28 units over 2 days)
As we see, the amount of growth itself is increasing by larger amounts, suggesting this could potentially be exponential or quadratic, but the rate of increase is escalating, likely indicating exponential growth more than linear or quadratic.
Considering the rapid increase, **exponential** is the best fit for modeling this data.
So the answer is **exponential**.