Selecting a model (linear vs quadratic vs exponential)
Selecting a Model
Selecting a Model is the opening topic of Modelling with Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.8(A), which requires students to analyze data to select the appropriate model from among linear, quadratic, and exponential models.
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.