Distinguish linear and non-linear relationships

A linear relationship changes at a constant rate: its table has constant first differences, its graph forms a straight line, and it can be represented by an equation such as $y=mx+b$ , whether or not it passes through the origin. A nonlinear relationship has a changing rate of change and does not produce a straight-line graph; distinguishing these forms helps connect verbal situations, tables, graphs, and equations, without requiring formal function notation or advanced nonlinear models.

Detailed Explanation: 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.

Learn by doing: Distinguish linear and non-linear relationships

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First and Second Differences - Completed Table to Function Type

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Distinguish linear and non-linear relationships

Detailed Explanation: Distinguish linear and non-linear relationships

Learn by doing: Distinguish linear and non-linear relationships

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First and Second Differences - Completed Table to Function Type