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All courses Math Foundations · MF.GR.8 47 of 60
Distinguish proportional relationships from linear but non-proportional relationships.

Decide whether a linear pattern is proportional

Problem
For \(x\): \(0,~1,~2,~3\) and \(y\): \(0,~3,~6,~9\), is the relationship proportional or linear but non-proportional? Does it pass through \((0,~0)\)?
Your answer
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Hint

Write the table as points and check the row where x = 0. Ask: does the relationship include the origin?

A proportional relationship has the form y = kx: it passes through (0,0) and has the same y/x ratio for all nonzero x-values.

Solution walkthrough

01

Pair the values

\[(x,y)=(0,0),(1,3),(2,6),(3,9)\]

The two aligned lists define four input-output points.

02

Check the constant multiplier

\[3/1=3;~6/2=3;~9/3=3\]

For every nonzero input, y divided by x equals the same constant three.

03

Check the origin

\[x=0~⇒~y=0~⇒~(0,0)\]

A proportional relationship must include the origin, and this one does.

04

Write the relationship

\[y=3x\]

A constant multiplier with no added starting term is proportional.

05

Report classification and origin check

\[\text{proportional};~\text{passes}~\text{through}~(0,0)~=~\text{yes}\]

Both the constant ratio and zero-input point establish proportionality.

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Another way

  1. Plot the points and observe a straight line through the origin with slope three.

!

Common mistake

Do not use y=3x+0 as evidence of a non-proportional start; adding zero leaves a proportional equation.

Solution walkthrough video