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All courses Algebra I · F-LE.1.a 28 of 76
Distinguish linear from exponential situations by equal differences versus equal factors over equal intervals.

Use first differences to identify a linear pattern

Problem
Calculate the first differences in the table. Are they constant? Use the differences to decide whether the pattern is linear.
Raw input/output table without computed differences, ratios, or classification cues. Open full size
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Hint

Subtract consecutive outputs to find the first differences: \(5-2\), \(8-5\), and \(11-8\).

A pattern is linear when the first differences are constant over equal input intervals.

Solution walkthrough

01

Read the table in input order

\[\text{inputs}:~0,1,2,3;~\text{outputs}:~2,5,8,11\]

The inspected table uses equal one-unit input intervals, so consecutive output differences can test whether the pattern is linear.

02

Compute every first difference

\[5-2=3;~8-5=3;~11-8=3\]

Each subtraction uses the later output minus the preceding output. Every first difference equals positive 3.

03

Apply the linear test

\[\text{equal}~\text{input}~\text{steps}~+~\text{constant}~\text{first}~\text{difference}~3~->~\text{linear}\]

A constant output difference over equal input intervals is decisive evidence of a linear pattern. The positive difference also shows the outputs increase.

04

Report the requested evidence

\[\text{linear};~\text{differences}:~3,~3,~3\]

The classification and the three independently computed differences exactly answer the prompt.

+

Another way

  1. The outputs fit y=3x+2 for every displayed input, another confirmation that the pattern is linear.

!

Common mistake

Do not reverse the subtraction order. Reading downward in the table requires later output minus earlier output, giving 5-2=3 rather than -3.

First-differences pattern card showing outputs 2, 5, 8, 11 with cues to subtract consecutive outputs and check whether the differences match.
classification: linear; differences: 3, 3, 3

Solution walkthrough video