Skill: Recognize constant first differences in a table

To recognize a constant first difference, compare each pair of neighboring input values and each pair of neighboring output values.

Example

Input $x$	Output $y$
$1$	$4$
$3$	$(10)$
$5$	$(16)$
$7$	$(22)$

Step 1: Check how the inputs change.

The inputs increase by equal amounts:

$$3-1=2,\qquad 5-3=2,\qquad 7-5=2$$

So, the input increases by $2$ each time.

Step 2: Find the successive output differences.

$$10-4=6,\qquad 16-10=6,\qquad 22-16=6$$

The outputs increase by $6$ each time. Since the output difference is constant, the table represents a linear relationship.

Step 3: Distinguish the output change from the input change.

The output changes by $6$ while the input changes by $2$. The rate of change is

$$\frac{\text{change in output}}{\text{change in input}} = \frac{6}{2}=3$$

This means the output increases by $3$ for every increase of $1$ in the input.

Step 4: Connect the pattern to an equation.

The rule is

$$y=3x+1$$

For example, when $(x=1)$,

$$y=3(1)+1=4$$

and when $(x=3)$,

$$y=3(3)+1=10$$

The points from the table would lie on a straight line when graphed. Therefore, equal input increases and constant output differences show a linear relationship.