AP®MathsCollege BoardPrecalculusStudy GuidesExponential & Logarithmic FunctionsSequences & Exponential FunctionsComparing Linear & Exponential Functions

Comparing Linear & Exponential Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 10 April 2026

Comparing linear & exponential functions

How can you tell if a function is linear or exponential from its data?

The key distinction is in how the output values change over equal-length input-value intervals:
- If the output values change at a constant rate (i.e. the differences between successive outputs are constant)
  - then the function is linear
- If the output values change proportionally (i.e. if the ratios between successive outputs are constant)
  - then the function is exponential
E.g. consider these two functions, $f$ and $g$
- The table shows output values for different values of the input
- The inputs increase by 1 each time

$x$	0	1	2	3	4
$f (x)$	4	10	16	22	28
$g (x)$	4	8	16	32	64

For $f$
- the differences are $10 - 4 = 6$ , $16 - 10 = 6$ , $22 - 16 = 6$ , $28 - 22 = 6$
- I.e. the differences are constant
  - so $f$ is linear
For $g$
- the ratios are $\frac{8}{4} = 2$ , $\frac{16}{8} = 2$ , $\frac{32}{16} = 2$ , $\frac{64}{32} = 2$
- I.e. the ratios are constant
  - so $g$ is exponential

What are the structural similarities between linear and exponential functions?

Both linear and exponential functions can be expressed in terms of
- an initial value
- and a constant involved with change

	Linear function	Exponential function
General form	$f (x) = b + m x$	$f (x) = a \cdot b^{x}$
Initial value	$b$ (value when $x = 0$ )	$a$ (value when $x = 0$ )
Constant involved with change	$m$ (slope)	$b$ (base)
How change works	Addition: add $m$ for each unit increase in $x$	Multiplication: multiply by $b$ for each unit increase in $x$

The only structural difference is the operation
- Linear functions are built on repeated addition
- Exponential functions are built on repeated multiplication
This parallels the relationship between arithmetic and geometric sequences (common difference vs common ratio)

How many values do you need to determine a linear or exponential function?

Arithmetic sequences, linear functions, geometric sequences, and exponential functions all share one important property
- They can be determined by two distinct values
For a linear function two points determine the slope $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ ,
- and from there the full function can be written
For an exponential function two points determine the base $b$ (by dividing the outputs and accounting for the difference in inputs)
- and from there the full function can be written
This is a useful property
- It means that if you know a function is linear or exponential, you only need two data points to write the function completely

Examiner Tips and Tricks

Exam questions often give a table of data and ask you to identify the function type (linear, quadratic, or exponential) and justify your answer. For this:

Check differences first
- If first differences are constant, the function is linear
- If second differences are constant, the function is quadratic
Check ratios next
- If they are constant, the function is exponential

To earn all the points in a free response question, make sure to reference specific values from the table in your reasoning. Don't just state the general rule.

Worked Example

In a certain simulation, the population of a virus can be modeled using an exponential function, where time $t$ is measured in hours and $t = 0$ is the start of the simulation. At time $t = 4$ the population was 128,000, and at time $t = 10$ the population was 18,000. What was the population at time $t = 7$ based on the simulation?

(A) 505,320

(B) 73,000

(D) 32,423

Answer:

For an exponential function, the output values over equal-length input-value intervals change by a constant proportion

I.e. the output values will form a geometric sequence

You can divide the interval between $t = 4$ and $t = 10$ into 6 equal-length intervals of length 1 hour

If $r$ is the common ratio of the corresponding geometric sequence, then

$128000 \cdot r^{6} = 18000$

Solve that for $r$

$r^{6} = \frac{18000}{128000} = \frac{9}{64}$

$r = {(\frac{9}{64})}^{\frac{1}{6}}$

Between $t = 4$ and $t = 7$ there are 3 intervals of length 1 hour

So at $t = 7$ the population will be

$\begin{array}{rcl} 128000 \cdot r^{3} & = & 128000 \cdot {({(\frac{9}{64})}^{\frac{1}{6}})}^{3} \\ = & 128000 \cdot {(\frac{9}{64})}^{\frac{1}{2}} \\ = & 128000 \cdot \frac{3}{8} \\ = & 48000 \end{array}$

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent