Skill: Distinguish linear and non-linear relationships

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

1. Look at how the $x$-values change.
2. Find how the $y$-values change.
3. If equal changes in $x$ always produce equal changes in $y$, the relationship is linear.
4. 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 $(+3,+3,+3)$.