Writing Exponential Functions

Q: How do I know a table is exponential and not linear?

Check the outputs as $x$ increases by $1$. A constant difference (add the same amount) means linear. A constant ratio (multiply by the same amount) means exponential. Divide consecutive outputs — if you get the same number every time, it is exponential.

Q: How do I check my function?

Plug in an $x$-value from a row you did not use to build the rule. If the output matches the table, your $a$ and $b$ are correct.

Writing an exponential function means filling in two blanks: $y = a \cdot b^x$ . The number $a$ is the starting value — the output when $x = 0$ — and $b$ is the constant ratio, the factor every output gets multiplied by when $x$ goes up by $1$ .

Most of these problems hand you a table. The whole job is two reads: pull $a$ from the row where $x = 0$ , then divide any output by the output before it to get $b$ . Once you have both numbers, write them into the form and you are done.

Step 1: find $a$ , the initial value

Look for the row where $x = 0$ . Whatever $y$ is there, that is $a$ , because $b^0 = 1$ leaves $y = a$ . If the table starts at $x = 0$ with $y = 4$ , then $a = 4$ — no work required.

If the table does not include $x = 0$ , work backwards: divide by $b$ to step left one column at a time until you reach $x = 0$ .

Step 2: find $b$ , the common ratio

Divide any output by the previous output. If the table reads $4, 12, 36, 108$ , then $\dfrac{12}{4} = 3$ , and checking another pair, $\dfrac{36}{12} = 3$ . The ratio is constant, so $b = 3$ .

That constant-ratio check is also how you confirm the data is exponential at all. A constant difference between outputs means linear; a constant ratio means exponential. If neither is constant, it is neither type.

In the table below, each output is the previous one times $2$ : $\dfrac{6}{3} = 2$ , $\dfrac{12}{6} = 2$ , $\dfrac{24}{12} = 2$ . The constant ratio is $b = 2$ , and the value at $x = 0$ is $a = 3$ .

$x$	$y$
$0$	$3$
$1$	$6$
$2$	$12$
$3$	$24$

Step 3: write the rule

Substitute your two numbers into $y = a \cdot b^x$ . With $a = 4$ and $b = 3$ , the function is $y = 4 \cdot 3^x$ . Test it against a table row you did not use: $x = 2$ gives $4 \cdot 9 = 36$ , which matches — so the rule is right.

Worked examples

Example 1: a growth table

Write the exponential function for the table $x: 0, 1, 2, 3$ and $y: 4, 12, 36, 108$ .

Read the value at

x = 0

a = 4

Divide consecutive outputs

\dfrac{12}{4} = 3

Confirm the ratio holds

\dfrac{36}{12} = 3

Write the rule

y = 4 \cdot 3^x

Answer: $y = 4 \cdot 3^x$

Example 2: a decay table

Write the exponential function for the table $x: 0, 1, 2, 3$ and $y: 80, 40, 20, 10$ .

Read the value at

x = 0

a = 80

Divide consecutive outputs

\dfrac{40}{80} = \dfrac{1}{2}

A ratio between

0

and

1

means decay

b = \dfrac{1}{2}

Write the rule

y = 80 \cdot \left(\dfrac{1}{2}\right)^x

Answer: $y = 80 \cdot \left(\dfrac{1}{2}\right)^x$

Example 3: from $a$ and $b$ directly

A function has initial value $a = 2$ and common ratio $b = 6$ . Write it.

Start with the form

y = a \cdot b^x

Substitute both values

y = 2 \cdot 6^x

Answer: $y = 2 \cdot 6^x$

Try one yourself

Common questions

How do I know a table is exponential and not linear?

Check the outputs as $x$ increases by $1$ . A constant difference (add the same amount) means linear. A constant ratio (multiply by the same amount) means exponential. Divide consecutive outputs — if you get the same number every time, it is exponential.

What if the table skips

x = 0

Find $b$ first from any consecutive pair, then divide backwards. If $y = 18$ at $x = 1$ and $b = 3$ , then at $x = 0$ the value was $18 \div 3 = 6$ , so $a = 6$ .

Can

b

be a fraction?

Yes — that is exactly what decay looks like. A table like $80, 40, 20, 10$ has $b = \dfrac{1}{2}$ . The base just has to be positive and not equal to $1$ .

How do I check my function?

Plug in an $x$ -value from a row you did not use to build the rule. If the output matches the table, your $a$ and $b$ are correct.

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Writing Exponential Functions

Step 1: find $a$ , the initial value

Step 2: find $b$ , the common ratio

Step 3: write the rule

Worked examples

Example 1: a growth table

Example 2: a decay table

Example 3: from $a$ and $b$ directly

Try one yourself

Common questions

Keep learning

Introduction to Exponential Functions

Graphing Exponential Functions

Interpreting Parameters in Context

Geometric Sequences

Writing Exponential Functions

Step 1: find aaa, the initial value

Step 2: find bbb, the common ratio

Step 3: write the rule

Worked examples

Example 1: a growth table

Example 2: a decay table

Example 3: from aaa and bbb directly

Try one yourself

Common questions

Keep learning

Introduction to Exponential Functions

Graphing Exponential Functions

Interpreting Parameters in Context

Geometric Sequences

Step 1: find $a$ , the initial value

Step 2: find $b$ , the common ratio

Example 3: from $a$ and $b$ directly