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All courses Math II · F-LE.3 26 of 73
Compare growth of exponential, linear, quadratic, and other polynomial models using graphs/tables.

Compare linear and exponential table growth precisely

Problem
Analyze the table for n=0,1,2,3. Compute consecutive differences for L and consecutive ratios for E, record the leader or tie at each listed n, then extend both patterns to n=4 and identify the first overtake after the table.
Source table for n=0,1,2,3: L values 10,20,30,40 and E values 5,10,20,40. Open full size
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Hint

Compute consecutive differences for L and ratios for E.

Linear tables have a constant difference; exponential tables have a constant nonunit factor.

Solution walkthrough

01

Identify the linear pattern

\[L:~10,~20,~30,~40\]

The linear values increase by \(+10\) each step, which is constant additive growth.

02

Identify the exponential pattern

\[E:~5,~10,~20,~40\]

The exponential values double each step, which is multiplicative growth.

03

Compare their values over the table

\[\text{at}~n~=~3,~\text{both}~\text{are}~40\]

The exponential starts lower, but its doubling pattern lets it catch up to the linear table by \(n=3\).

04

State the comparison

\[L~\text{model}=\text{linear};~L~\text{constant}~\text{difference}=+10;~E~\text{model}=\text{exponential};~E~\text{constant}~\text{factor}=×2;~\text{listed}~\text{leader}~\text{sequence}=L,L,L,\text{tie};~\text{first}~\text{listed}~\text{equality}=n=3;~\text{first}~\text{listed}~\text{overtake}=\text{none};~\text{next}-\text{row}~\text{check}~n=4:L=50,E=80;~\text{first}~\text{overtake}~\text{after}~\text{table}=E~\text{at}~n=4;~\text{eventual}~\text{dominance}=E\]

This compares both the matching value and the reason the exponential catches up.

+

Another way

  1. Compute differences for the linear table and ratios for the exponential table to see the two growth rules side by side.

!

Common mistake

Saying the exponential already exceeds the linear at \(n=3\), even though both tables show \(40\) there.

Answer comparison: L has constant difference +10; E has factor 2. Leaders are L,L,L,tie for n=0–3; at n=4, L=50 and E=80, so E first overtakes after the table.
L model=linear; L constant difference=+10; E model=exponential; E constant factor=×2; listed leader sequence=L,L,L,tie; first listed equality=n=3; first listed overtake=none; next-row check n=4:L=50,E=80; first overtake after table=E at n=4; eventual dominance=E