Skill: Determine rate of change from a table

The rate of change tells how much the output changes for each 1-unit change in the input.

Use the formula

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

Suppose this table shows the distance a cyclist travels:

Time, $x$ (hours)	Distance, $y$ (kilometers)
$2$	$6$
$5$	$15$
$8$	$24$

Step 1: Choose two rows

Use the first two rows:

• Input changes from $2$ hours to $5$ hours.
• Output changes from $6$ kilometers to $15$ kilometers.

Step 2: Find each change

$$\Delta y=15-6=9\text{ km}$$ $$\Delta x=5-2=3\text{ hours}$$

Step 3: Divide the changes

$$\text{rate of change}=\frac{9\text{ km}}{3\text{ hours}} =3\text{ km per hour}$$

The cyclist travels $3$ kilometers for every 1 hour.

Step 4: Check another pair

Using the second and third rows:

$$\frac{24-15}{8-5} =\frac{9}{3} =3\text{ km per hour}$$

The rate is the same, so the table has a constant rate of change.

Remember to divide the change in $y$ by the change in $x$. Do not simply divide $y$ by $x$; the rate uses differences between two rows.