Determine the rate of change from a table

Given a table of paired values, the learner determines rate of change by dividing the change in the dependent variable by the corresponding change in the independent variable, including when input intervals are unequal. The resulting slope is interpreted with its units and sign: a constant positive, negative, or zero rate describes a linear relationship, whereas varying rates indicate that the table does not represent a single linear rate. This treatment excludes instantaneous rates and calculus-based analysis.

Detailed Explanation: Determine the rate of change from a table

The rate of change tells how much the dependent variable changes for each 1 unit of change in the independent variable.

Use the formula

\text{rate of change}=\frac{\text{change in dependent variable}}{\text{change in independent variable}} =\frac{\Delta y}{\Delta x}

Consider this table showing the distance traveled over time:

Time, $x$ (hours)	Distance, $y$ (miles)
$0$	$0$
$2$	$6$
$5$	$15$
$9$	$27$

Here, time is the independent variable, and distance is the dependent variable.

Step 1: Compare two consecutive rows

From $x=0$ to $x=2$ , the changes are

\Delta x=2-0=2

and

\Delta y=6-0=6

So the rate of change is

\frac{\Delta y}{\Delta x}=\frac{6}{2}=3

The rate is $3$ miles per hour.

Step 2: Check the next interval

From $x=2$ to $x=5$ ,

\Delta x=5-2=3

and

\Delta y=15-6=9

Thus,

\frac{\Delta y}{\Delta x}=\frac{9}{3}=3

The input interval is different, but the rate is still $3$ miles per hour.

Step 3: Check the last interval

From $x=5$ to $x=9$ ,

\Delta x=9-5=4

and

\Delta y=27-15=12

So,

\frac{\Delta y}{\Delta x}=\frac{12}{4}=3

Conclusion

Every interval has the same rate of change:

\boxed{3\text{ miles per hour}}

Because the rate is constant and positive, the table represents a linear relationship. The positive sign means the distance increases as time increases.

If the rates from the different intervals were not equal, the table would not represent one constant linear rate.

Learn by doing: Determine the rate of change from a table

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