Skill: Distinguish linear and non-linear relationships

To distinguish a linear relationship from a nonlinear one, look for a constant rate of change.

• In a table, check whether the first differences are the same.
• On a graph, a linear relationship makes a straight line.
• In an equation, a linear relationship can be written as $y=mx+b$, where $m$ is the constant rate of change.

Example: Is the relationship in the table linear or nonlinear?

$x$	$y$
$0$	$5$
$1$	$8$
$2$	$11$
$3$	$14$

Step 1: Find the changes in $y$.

As $x$ increases by $1$ each time:

$$8-5=3$$ $$11-8=3$$ $$14-11=3$$

The first differences are all $3$.

Step 2: Decide.

Because the change in $y$ is constant, the relationship is linear. Its rate of change is $3$.

Step 3: Write an equation.

When $x=0$, $y=5$, so the starting value is $b=5$. The rate of change is $m=3$.

Therefore, the equation is

$$y=3x+5$$

The graph of this relationship would be a straight line. If the differences had changed instead of staying at $3$, the relationship would be nonlinear.