Skill: Recognize constant rate of change

A relationship has a constant rate of change when the output changes by the same amount whenever the input changes by the same amount.

To check a table:

1. Find the change in the input values.
2. Find the change in the output values.
3. Divide the output change by the input change:

$$\text{rate of change}=\frac{\text{change in }y}{\text{change in }x}$$

4. If the rate is the same for every interval, the relationship has a constant rate of change.

Example

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

The input increases by $1$ each time. Now compare the output changes:

• From $5$ to $8$: $8-5=3$
• From $8$ to $11$: $11-8=3$
• From $11$ to $14$: $14-11=3$

The output increases by $3$ for every increase of $1$ in the input. The rate of change is

$$\frac{3}{1}=3.$$

Therefore, this relationship has a constant rate of change of $3$.

It can be written as

$$y=3x+5.$$

The $3$ shows the constant rate of change. The relationship does not need to pass through $(0,0)$; here, when $x=0$, $y=5$.