AP®MathsCollege BoardPrecalculusRevision NotesExponential & Logarithmic FunctionsModeling with Exponential & Logarithmic FunctionsConstructing Models from a Data Set

Constructing Models from a Data Set (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Constructing models from a data set

Why might several model types fit the same data?

When a data set shows a rate of change that is slightly changing (neither perfectly constant nor dramatically varying)
- it can often be modeled reasonably well by more than one function type
In particular, linear, quadratic, and exponential models can all produce similar-looking curves over a limited range of data
Remember with these three function types
- A linear model assumes a constant rate of change
- A quadratic model assumes a rate of change that changes at a constant rate (constant 2nd differences)
- An exponential model assumes outputs that change by a constant ratio
Over a small portion of data, these can look very similar
- It is only over larger intervals that the differences between the models become more apparent

E.g. consider a data set showing a quantity that starts at 100 and grows slowly at first:

$x$	0	1	2	3	4
$y$	100	108	117	127	138

There are different possible modeling choices
- A linear model might use a constant increase of about 9.5 per unit
  - $y \approx 100 + 9.5 x$
- A quadratic model might capture the slight acceleration
  - $y \approx 100 + 8 x + 0.5 x^{2}$
- An exponential model might fit the proportional growth
  - $y \approx 100 (1.083)^{x}$
All three models produce values close to the actual data over this limited range

$x$	0	1	2	3	4
$y = 100 + 9.5 x$	100	109.5	119	128.5	138
$y \approx 100 + 8 x + 0.5 x^{2}$	100	108.5	118	128.5	140
$y \approx 100 (1.083)^{x}$	100	108.3	117.289	127.024	137.567

How do I decide which model is most appropriate?

When multiple models can fit the same data, you need additional information to choose the best one
Contextual clues can help determine the most appropriate model
- Does the context suggest a constant rate of change?
  - e.g. fixed salary increase per year
  - this would suggest a linear model
- Does the context suggest a constant percent change?
  - e.g. compound interest, population growth, radioactive decay
  - this would suggest an exponential model
- Does the context involve a quantity that increases then decreases (or vice versa)
  - e.g. projectile motion
  - this would suggest a quadratic model

Mathematical analysis of the data can also help
- Check 1st differences
  - if approximately constant, a linear model is appropriate
- Check 2nd differences
  - if approximately constant, a quadratic model is appropriate
- Check ratios of successive outputs
  - if approximately constant, an exponential model is appropriate
- Use technology to run regressions and compare

Applicability and limitations:
- Consider how the model behaves outside the range of the given data
  - A linear model predicts unbounded growth at a constant rate
    - Is that realistic?
  - A quadratic can change direction (from increase to decrease or vice versa)
    - Does that make sense?
  - An exponential model grows (or decays) without bound
    - Is that reasonable for the context?
The model that makes the most sense both within the data range and for reasonable predictions beyond it is typically the best choice

Examiner Tips and Tricks

When choosing between models, always consider the context first

Mathematical fit alone may not be enough if the model behaves unrealistically outside the data range

On the exam, if you are asked to justify a model choice, make sure your reasoning connects

the mathematical properties of the data (differences, ratios)
to the characteristics of the function type

Worked Example

A company tracks its monthly revenue (in thousands of dollars) over its first five months of operation. The data is shown in the table below.

Month ( $t$ )	1	2	3	4	5
Revenue ( $R$ )	10	16	25	40	65

A quadratic model and an exponential model are each constructed for this data. Which model is more appropriate? Give a reason based on the data.

Answer:

First, check the 2nd differences and ratios of the output values:

1st differences:
$16 - 10 = 6$ , $25 - 16 = 9$ , $40 - 25 = 15$ , $65 - 40 = 25$

2nd differences:
$9 - 6 = 3$ , $15 - 9 = 6$ , $25 - 15 = 10$

Ratios:
$\frac{16}{10} = 1.6$ , $\frac{25}{16} \approx 1.563$ , $\frac{40}{25} = 1.6$ , $\frac{65}{40} = 1.625$

The 2nd differences are not constant (they increase from 3 to 10), so a quadratic model is not ideal
The ratios are approximately constant (around 1.563 to 1.625), which suggests an exponential model.

An exponential model is more appropriate because the successive output values are approximately proportional over equal-length input-value intervals

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