AP®MathsCollege BoardPrecalculusRevision NotesExponential & Logarithmic FunctionsModeling with Exponential & Logarithmic FunctionsSelecting & Constructing Exponential Models

Selecting & Constructing Exponential Models (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Selecting & constructing exponential models

When is an exponential model appropriate?

An exponential function models situations where successive output values over equal-length input-value intervals are proportional
- i.e. they have a constant ratio
When the input values are whole numbers
- this corresponds to repeated multiplication of a constant factor applied to an initial value
To determine whether data should be modeled by an exponential function
- check whether the ratios of consecutive output values
- for equally spaced input values
- are approximately constant
E.g. given the data $t = 0, 1, 2, 3$ with $f (t) = 20, 30, 45, 67.5$ :
- $\frac{f (1)}{f (0)} = \frac{30}{20} = 1.5$ , $\frac{f (2)}{f (1)} = \frac{45}{30} = 1.5$ , $\frac{f (3)}{f (2)} = \frac{67.5}{45} = 1.5$
- The ratio is constant at $1.5$ , so an exponential model is appropriate
Compare this with the tests for other function types
- Linear: successive output values have a constant difference (1st differences are constant)
- Quadratic: the 2nd differences in output values are constant
- Exponential: successive output values have a constant ratio

What if the ratios aren't quite constant?

Sometimes a constant needs to be added to (or subtracted from) the output values before the proportional pattern becomes visible
E.g. the data $(1, 7), (2, 9), (3, 13), (4, 21)$ does not have constant ratios as is
- But subtracting 5 from each output gives $(1, 2), (2, 4), (3, 8), (4, 16)$
- for which the output values have a constant ratio of 2
  - I.e. $\frac{4}{2} = \frac{8}{4} = \frac{16}{8} = 2$
This means the original data can be modeled by a vertically shifted exponential function
- I.e. an additive transformation of an exponential
In general, if the output values of an additive transformation of a function are proportional over equal-length input-value intervals
- then the function can be modeled by a transformation of an exponential function

Examiner Tips and Tricks

When checking ratios to identify an exponential model, make sure the input-value intervals are equal in length. The proportional ratio test only works for equally spaced inputs

How do I construct an exponential model from data?

Method 1: From the initial value and ratio
- If you can identify the initial value $a = f (0)$
  - and the constant ratio $b$ between consecutive outputs (for unit input intervals)
- then the model is $f (x) = a b^{x}$
- E.g. for the data above with $f (0) = 20$ and ratio $b = 1.5$
  - $f (t) = 20 {(\frac{3}{2})}^{t}$

Method 2: From two input-output pairs
- If you have two data points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ ,
  - you can construct the model $y = a b^{x}$ by solving a system of two equations
- Substituting the two points gives
  - $y_{1} = a b^{x_{1}}$
  - $y_{2} = a b^{x_{2}}$
- Dividing the second equation by the first eliminates $a$
  - $\frac{y_{2}}{y_{1}} = \frac{a b^{x_{2}}}{a b^{x_{1}}} = \frac{b^{x_{2}}}{b^{x_{1}}} = b^{(x_{2} - x_{1})}$
- You can then solve for $b$ , and substitute back into $y = a b^{x}$ to find $a$
- E.g. given the data points $(0, 5)$ and $(3, 40)$ :
  - $\frac{40}{5} = \frac{a b^{3}}{a b^{0}} = \frac{b^{3}}{b^{0}} = b^{3}$
    - so $b^{3} = \frac{40}{5} = 8 \Rightarrow b = 2$
  - Using $b = 2$ and the point $(0, 5)$ in $y = a b^{x}$
  - $5 = a \cdot b^{0} \Rightarrow a = 5$
  - Model: $f (x) = 5 \cdot 2^{x}$

Examiner Tips and Tricks

Constructing a model from two data points tends to appear in a free response question on every exam, so practice this skill thoroughly.

Non-exact values of $a$ and $b$ must be given as decimal approximations correct to three decimal places
- The scoring guidelines are strict about this
You can solve for $a$ and $b$ either algebraically or by using your graphing calculator
- The scoring guidelines note that "supporting work is not required", so using a calculator is perfectly acceptable
Store intermediate values in your calculator rather than rounding them
- Rounding too early can produce final answers that are not accurate to three decimal places

Method 3: Applying transformations to $f (x) = a b^{x}$
- An exponential model can be built by applying transformations based on the characteristics of the data or context
- E.g. if data shows exponential growth starting from a value of 10 at $t = 2$ with a growth factor of 3
  - the model might be $f (t) = 10 \cdot 3^{(t - 2)}$
  - which is a horizontal shift of $y = 10 \cdot 3^{t}$

Method 4: Using technology (exponential regression)
- When data is not perfectly exponential
  - a graphing calculator can fit an exponential regression model of the form $y = a b^{x}$ to the data
- This is done using the ExpReg function on most graphing calculators
  - The calculator determines the values of $a$ and $b$ that best fit the data using a least-squares method
  - The resulting model can then be used to predict output values at input values not in the original data set (including non-integer values)
- See the first Worked Example for an example of this

Examiner Tips and Tricks

When using your calculator for exponential regression, remember to give final predicted values correct to three decimal places.

What role does the natural base e play?

The natural base $e \approx 2.718$ is commonly used in exponential models for real-world situations
Any exponential function $f (x) = a b^{x}$ can be rewritten using base $e$
- $f (x) = a e^{(\ln b) x}$
Models using base $e$ appear frequently in science, economics, and engineering
- e.g. continuous growth/decay, compound interest

How do I identify the correct model type on the exam?

In exam questions, you are typically given a table of values and asked which function type best models the data
- e.g. linear, quadratic, exponential, or logarithmic
Check the 1st differences of the output values (for equally spaced inputs)
- If these are constant, the model should be linear
Check the 2nd differences of the output values (for equally spaced inputs)
- If these are constant, the model should be quadratic
Check the ratios of successive output values (for equally spaced inputs)
- If these are constant, the model should be exponential
If none of the above are constant, consider a logarithmic model
- Here the input values change proportionally
  - while output values change additively
To earn full credit you must give a reason that references specific values from the table
- E.g. "The ratios of successive output values are constant at 0.5 over equal-length input-value intervals of 1, so an exponential model is best."

Examiner Tips and Tricks

When identifying model types in a free response question, simply stating "exponential" is not enough to earn full credit.

You must show the constant ratio using values from the table
and demonstrate that it applies to more than one pair of successive outputs

Saying "I used exponential regression" or citing $r$ -values or $r^{2}$ -values found with your calculator is not sufficient reasoning for these questions.

The exam expects reasoning based on the proportional relationship of output values

Worked Example

The table presents values for a function $f$ at selected values of $x$ .

$x$	0	1	2	3
$f (x)$	10	17	28	48

An exponential regression $y = a b^{x}$ is used to model these data. What is the value of $f (1.5)$ predicted by the exponential function model?

(A) 21.863

(B) 22.166

(D) 22.500

Answer:

Using a graphing calculator, enter the data and run an exponential regression (ExpReg) of the form $a b^{x}$

The constants $a$ and $b$ are calculated as

$a = 10.0149116$ $b = 1.68284176$

(values may vary slightly depending on the calculator)

This gives the following model

$y = 10.0149116 \cdot (1.68284176)^{x}$

Substitute in $x = 1.5$

$f (1.5) = 10.015 \cdot (1.683)^{1.5} = 21.8631150 . . .$

Rounded to 3 decimal places, that is answer (A)

It is worth considering the incorrect answer options and where they came from:

(B) is the result of using only the first ratio ( $17 / 10 = 1.7$ ) and then computing $10 \cdot (1.7)^{1.5}$
- This uses a less accurate model based on only two data points rather than the full regression
(C) is the geometric mean of $f (1)$ and $f (2)$ , i.e. $\sqrt{17 \times 28}$
- This would only be exact for a perfectly exponential data set
(D) is the linear interpolation $(17 + 28) / 2$
- This assumes a linear model between the two points, which is inappropriate for exponential data

(A) 21.863

Worked Example

The function $f$ is decreasing and is defined for all real numbers. The table gives values for $f (x)$ at selected values of $x$ .

$x$	0	2	4	6	8
$f (x)$	96	48	24	12	6

(i) Based on the table, which of the following function types best models function $f$ : linear, quadratic, exponential, or logarithmic?

(ii) Give a reason for your answer based on the relationship between the change in the output values of $f$ and the change in the input values of $f$ . Refer to the values in the table in your reasoning.

Answer:

The input value intervals are all equal to 2, and successive output values have a ratio of 0.5

So an exponential model is appropriate
But be sure to explain your answer correctly to gain full credit

(i)

An exponential function best models $f$

(ii)

The input-value intervals all have equal length 2

The ratios of successive output values are

$\frac{f (2)}{f (0)} = \frac{48}{96} = 0.5$ , $\frac{f (4)}{f (2)} = \frac{24}{48} = 0.5$ ,

$\frac{f (6)}{f (4)} = \frac{12}{24} = 0.5$ , $\frac{f (8)}{f (6)} = \frac{6}{12} = 0.5$

Because the successive output values over equal-length input-value intervals are proportional (with a constant ratio of $0.5$ ), an exponential model is the best fit

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Selecting & Constructing Exponential Models (College Board AP® Precalculus): Revision Note

Selecting & constructing exponential models

When is an exponential model appropriate?

What if the ratios aren't quite constant?

Examiner Tips and Tricks

How do I construct an exponential model from data?

Examiner Tips and Tricks

Examiner Tips and Tricks

What role does the natural base e play?

How do I identify the correct model type on the exam?

Examiner Tips and Tricks

Worked Example

Worked Example

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Author: Roger B

Reviewer: Mark Curtis