Determine rate of change from a table

Rate of change describes how much one quantity changes for each unit change in another, found from a table by dividing the change in the output by the corresponding change in the input, $\Delta y/\Delta x$ . The rate may be positive, negative, or zero and should remain constant for a linear relationship; it is expressed with appropriate units, rather than confused with the ratio $y/x$ . The focus is on constant rates in tables, not instantaneous rates or advanced slope theory.

Detailed Explanation: 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.

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