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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
Show answer choicesAnswer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
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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

List the ordered pairs from the table

\[(0,0),~(1,3),~(2,6),~(3,9)\]

Reading the table as ordered pairs makes it easier to check whether the relationship is proportional. We want to see whether the pattern matches the form y = kx.

02

Check whether the table includes the origin

\[x~=~0~\Rightarrow~y~=~0\]

A proportional relationship must pass through the origin, which is the point (0,0). This table does include (0,0), so that condition is met.

03

Check for a constant of proportionality

\[\frac{3}{1}=3,\;~\frac{6}{2}=3,\;~\frac{9}{3}=3\]

For each nonzero x-value, the ratio y/x is 3. That means y is always 3 times x, so the rule is y = 3x.

04

Classify the relationship

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

This relationship is proportional because it has the form y = kx and the table includes the origin. So the answer is: proportional.

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Another way

  1. You can look at how the values change: when x increases by 1, y increases by 3, and the table starts at (0,0). A constant rate plus a start at 0 means the relationship is proportional.

  2. You can graph the points (0,0), (1,3), (2,6), and (3,9). They lie on a straight line through the origin, which confirms the relationship is proportional.

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Common mistake

A common mistake is to say a relationship is proportional just because y goes up by 3 each time x goes up by 1. That only shows the relationship is linear. To be proportional, it must also pass through (0,0). This table does, so it is proportional.