Skill: Determine rate of change from a table

To find the rate of change from a table, compare how much the output $y$ changes to how much the input $x$ changes:

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

Use two rows from the table.

$x$	$y$
$2$	$7$
$5$	$16$
$9$	$28$

Step 1: Choose two points.
Use the first two points: $(2,7)$ and $(5,16)$.

Step 2: Find each change.

$$\Delta y=16-7=9$$ $$\Delta x=5-2=3$$

Step 3: Divide the changes.

$$\text{rate of change}=\frac{\Delta y}{\Delta x} =\frac{9}{3}=3$$

The rate of change is $3$. This means that for every increase of $1$ in $x$, $y$ increases by $3$.

Check the next pair of points. The input changes by $4$, and the output changes by $12$:

$$\frac{28-16}{9-5}=\frac{12}{4}=3$$

The rate is still $3$, so the table shows a linear relationship with a slope of $3$.

Do not usually divide $y$ by $x$. Instead, divide the change in $y$ by the corresponding change in $x$.