Constructing Linear &
Exponential Functions

Learn to build linear and exponential functions from tables, graphs, descriptions, and input-output pairs: including arithmetic and geometric sequences.

Algebra 2

Two Fundamental Growth Models

Every relationship that changes at a predictable pattern can usually be modeled by one of two function families. The key question is: does the output change by a constant amount or by a constant factor?

Linear Functions

Output changes by a constant difference for each unit increase in input. Form: $ f(x) = mx + b $

Exponential Functions

Output changes by a constant ratio for each unit increase in input. Form: $ f(x) = a \cdot r^x $

Sequences: Arithmetic vs Geometric

Functions and sequences are deeply connected. An arithmetic sequence is the discrete version of a linear function, and a geometric sequence is the discrete version of an exponential function.

Arithmetic

Add a common difference $ d $ each step.
Example: $ 3, 7, 11, 15, \ldots $ with $ d = 4 $

Geometric

Multiply by a common ratio $ r $ each step.
Example: $ 2, 6, 18, 54, \ldots $ with $ r = 3 $

Seeing the Difference on a Graph

The Core Formulas

Linear Function

$$ f(x) = mx + b $$ $ m $ : slope (constant rate of change)
$ b $ : y-intercept (initial value)

Exponential Function

$$ f(x) = a \cdot r^x $$ $ a $ : initial value (when $ x = 0 $)
$ r $ : common ratio (growth/decay factor)

Building from Two Points

Given two input-output pairs $ (x_1, y_1) $ and $ (x_2, y_2) $, you can construct either function type:

Linear

Find slope: $ m = \dfrac{y_2 - y_1}{x_2 - x_1} $

Then: $ b = y_1 - m \cdot x_1 $

Exponential

Find ratio: $ r = \left(\dfrac{y_2}{y_1}\right)^{\!1/(x_2 - x_1)} $

Then: $ a = \dfrac{y_1}{r^{\,x_1}} $

Constant Difference vs Constant Ratio

	Linear	Exponential
Test from table	Subtract consecutive outputs: $ y_2 - y_1 $ is constant	Divide consecutive outputs: $ y_2 / y_1 $ is constant
Parameter found	Common difference $ d = m $	Common ratio $ r $
Graph shape	Straight line	Curved (concave up or down)
Long-term behavior	Grows without bound (linearly)	Grows/decays much faster

Step-by-Step Construction Method

1

Identify the data. Extract at least two input-output pairs from the table, graph, or description.

2

Test for type. Compute consecutive differences and consecutive ratios to determine if the relationship is linear or exponential.

3

Find the parameters. For linear: calculate $ m $ and $ b $. For exponential: calculate $ r $ and $ a $.

4

Write the function. Plug the parameters into $ f(x) = mx + b $ or $ f(x) = a \cdot r^x $.

5

Verify. Substitute a known point back into your function to confirm correctness.

Example 1: Linear from a Table

$ x $	0	1	2	3	4
$ f(x) $	5	8	11	14	17

Example 2: Exponential from a Table

$ x $	0	1	2	3	4
$ g(x) $	4	12	36	108	324

Example 3: Exponential from Two Points

Construct an exponential function through the points $ (1, 6) $ and $ (3, 54) $.

Example 4: Linear from a Description

A pool is being filled at a constant rate. After 2 hours, it holds 500 gallons. After 5 hours, it holds 1,100 gallons. Write a function for gallons $ G $ as a function of time $ t $ (hours).

Reading Functions from a Graph

Example 5: Geometric Sequence

Write an explicit formula for the geometric sequence: $ 6, 15, 37.5, 93.75, \ldots $

Common Mistakes to Avoid

✕ Confusing $ r $ and $ d $

Using subtraction to find the ratio, or division to find the difference. Always match the operation to the function type.

✕ Forgetting $ x = 0 $ gives $ a $

In $ f(x) = a \cdot r^x $, when $ x = 0 $, $ r^0 = 1 $, so $ f(0) = a $. Don't confuse $ a $ with the first term of a sequence indexed from $ n = 1 $.

✕ Non-uniform $ x $-spacing

When checking ratios/differences in a table, ensure $ x $-values increase by 1 each time. If $ x $ jumps by $ \Delta x $, the ratio will be $ r^{\Delta x} $ and the difference will be $ m \cdot \Delta x $.

Summary Comparison

Feature	Linear	Exponential
General form	$ f(x) = mx + b $	$ f(x) = a \cdot r^x $
Sequence form	$ a_n = a_1 + (n-1)d $	$ a_n = a_1 \cdot r^{n-1} $
Identifying pattern	Constant difference	Constant ratio
Graph	Straight line	Curve
Key parameter	Slope $ m $	Ratio $ r $
Initial value	y-intercept $ b $	$ a = f(0) $

Question 1 of 5

A table shows $ f(0) = 3 $, $ f(1) = 9 $, $ f(2) = 27 $, $ f(3) = 81 $. Which function models this data?

The consecutive ratios are $ 9/3 = 3 $, $ 27/9 = 3 $, $ 81/27 = 3 $. So $ r = 3 $ and $ a = f(0) = 3 $. The function is $ f(x) = 3 \cdot 3^x $. Note that $ f(x) = 3^x $ gives $ f(0) = 1 $, not 3.

Question 2 of 5

An arithmetic sequence has $ a_1 = 7 $ and $ a_5 = 23 $. What is the explicit formula for $ a_n $?

The common difference is found by $ d = \frac{a_5 - a_1}{5 - 1} = \frac{23 - 7}{4} = 4 $. The standard explicit formula is $ a_n = a_1 + (n-1)d $, which gives $ a_n = 7 + 4(n-1) $. Choice (a) incorrectly uses $n$ instead of $n-1$, giving $ a_1 = 7 + 4(1) = 11 $ instead of 7. Choice (d) confuses the first term and the common difference.

Question 3 of 5

An exponential function passes through $ (2, 12) $ and $ (4, 48) $. What is the value of $ r $?

Using $ f(x) = a \cdot r^x $: divide $ f(4)/f(2) = 48/12 = 4 = r^{4-2} = r^2 $. So $ r^2 = 4 $, giving $ r = 2 $.

Question 4 of 5

A population of bacteria doubles every hour. If there are 500 bacteria at $ t = 0 $, which function models the population $ P(t) $?

"Doubles every hour" means constant ratio $ r = 2 $, making this exponential. The initial value is $ a = 500 $, so $ P(t) = 500 \cdot 2^t $. Choice (a) is linear and only adds 2 per hour, not doubling.

Question 5 of 5

Which test determines whether a table of values represents a linear function (assuming equally spaced $ x $-values)?

A linear function has a constant rate of change, meaning consecutive output values differ by the same amount (constant difference). A constant ratio indicates an exponential function. Outputs being positive or increasing does not determine function type.

	Linear	Exponential
Test from table	Subtract consecutive outputs: \( y_2 - y_1 \) is constant	Divide consecutive outputs: \( y_2 / y_1 \) is constant
Parameter found	Common difference \( d = m \)	Common ratio \( r \)
Graph shape	Straight line	Curved (concave up or down)
Long-term behavior	Grows without bound (linearly)	Grows/decays much faster

Feature	Linear	Exponential
General form	\( f(x) = mx + b \)	\( f(x) = a \cdot r^x \)
Sequence form	\( a_n = a_1 + (n-1)d \)	\( a_n = a_1 \cdot r^{n-1} \)
Identifying pattern	Constant difference	Constant ratio
Graph	Straight line	Curve
Key parameter	Slope \( m \)	Ratio \( r \)
Initial value	y-intercept \( b \)	\( a = f(0) \)

Constructing Linear &Exponential Functions

Two Fundamental Growth Models

Sequences: Arithmetic vs Geometric

Seeing the Difference on a Graph

The Core Formulas

Building from Two Points

Constant Difference vs Constant Ratio

Step-by-Step Construction Method

Example 1: Linear from a Table

Example 2: Exponential from a Table

Example 3: Exponential from Two Points

Example 4: Linear from a Description

Reading Functions from a Graph

Example 5: Geometric Sequence

Common Mistakes to Avoid

Summary Comparison

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Constructing Linear &
Exponential Functions