AP®MathsCollege BoardPrecalculusRevision NotesExponential & Logarithmic FunctionsSequences & Exponential FunctionsComparing Linear & Exponential Functions

Comparing Linear & Exponential Functions (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 30 June 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
For an exponential function, the constant ratio cannot equal 0 or 1
- If the ratio were 1, every output value would be the same, giving a constant function rather than exponential growth or decay
- Equivalently, the base of an exponential function must satisfy $b > 0$ and $b \neq 1$

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

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

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