Distinguish linear and non-linear relationships

A linear relationship has a constant rate of change: for equal changes in one quantity, the other changes by equal amounts, producing a straight-line graph and an equation of the form $y=mx+b$ when appropriate. A nonlinear relationship has a changing rate of change, so its table, graph, or equation does not maintain this pattern; distinguishing the two prevents confusing any increasing relationship with a linear one and supports later algebraic modeling.

Detailed Explanation: Distinguish linear and non-linear relationships

To decide whether a relationship is linear, check whether it has a constant rate of change.

Look at how the $x$ -values change.
Find how the $y$ -values change.
If equal changes in $x$ always produce equal changes in $y$ , the relationship is linear.
If the changes in $y$ are different, the relationship is nonlinear.

Example

Determine whether the table shows a linear or nonlinear relationship.

$x$	$y$
$1$	$2$
$2$	$4$
$3$	$8$
$4$	$(16)$

Step 1: Check the changes in $x$ .

The $x$ -values increase by $1$ each time:

1 \to 2 \to 3 \to 4

Step 2: Check the changes in $y$ .

2 \to 4 \quad \text{increases by } 2

4 \to 8 \quad \text{increases by } 4

8 \to 16 \quad \text{increases by } 8

Step 3: Compare the changes.

The $x$ -values change by the same amount each time, but the $y$ -values change by different amounts: $2$ , $4$ , and $8$ .

Therefore, the rate of change is not constant, so the relationship is nonlinear.

A linear table would have the same $y$ -change each time, such as $(+3,+3,+3)$ .

Learn by doing: Distinguish linear and non-linear relationships

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