Selecting a model (linear vs quadratic vs exponential)
Selecting a Model
Selecting a Model is the opening topic of Modelling with Functions in the California Common Core State Standards. It is aligned to Standard S-ID.6a, which requires students to fit a function to data and informally assess how well the model fits.
Choosing a model matches the data pattern: a constant difference is linear, a constant second difference quadratic, and a constant ratio exponential.
Theory
Match the pattern in the data to a model:
- Constant difference \(\Rightarrow\) linear.
- Constant second difference \(\Rightarrow\) quadratic.
- Constant ratio \(\Rightarrow\) exponential.
The tests:
How to choose a model
- Compute the first differences.
- If not constant, compute second differences.
- Also check the ratios of consecutive terms.
- Match the constant pattern to the model.
Each term is \(\times2\) the last — a constant ratio.
| \(\dfrac{4}{2}=\dfrac{8}{4}\) | \(=\) | \(2\) |
| \(\Rightarrow\) | \(\text{exponential}\) |
Each term is \(+2\) — a constant difference.
| \(5-3=7-5\) | \(=\) | \(2\) |
| \(\Rightarrow\) | \(\text{linear}\) |
First differences \(3,5,7\); second differences constant \(2\).
| \(\text{2nd differences}\) | \(=\) | \(2\) |
| \(\Rightarrow\) | \(\text{quadratic}\) |
A constant percent change is a constant ratio.
| \(\text{constant } \%\) | \(\Rightarrow\) | \(\text{exponential}\) |
Common pitfalls
Frequently asked questions
How do you know a model is linear?
The first differences are constant.
What indicates a quadratic model?
The second differences are constant.
What indicates an exponential model?
The ratios of consecutive values are constant.
What does a constant percent change mean?
An exponential model.