D. Interpreting the Slope of a Line as a Rate

Slope represents the relationship between the change in vertical values (rise) and the corresponding change in horizontal values (run). Slope is also commonly represented by m.

Slope can be represented in the following ways:

Because slope relates the independent and dependent variables, slope can also be described as a rate of change.

A distance vs. time graph shows the distance travelled in a certain time interval. Kilometres per hour (distance/time) is a rate called speed or velocity.

Example 1

Using the graph above, determine the unit rate for the speed of the motorized scooter.

The graph shows distance, in km, versus time, in hours. By picking two known points on the straight-line graph, you can determine the change in both time and distance and use those values to calculate the rate (speed).

One clearly marked point on the graph is the distance of 0 km when the time is 0 hours. Another clearly marked point on the graph is the distance of 60 km when the time is 2 hours.

Therefore, a distance of 60 km - 0 km = 60 km was travelled in a time of 2 h - 0 h = 2 h.

The scooter's rate of speed was 30 km/h.

The fuel consumption of a certain vehicle can be expressed as the number of miles travelled per gallon of liquid fuel.

Example 2

Given that a 2-door vehicle averages 37 kilometres per gallon in the graph above, how many kilometres could be driven using 4.5 gallons of gasoline?

Set up a proportion relating the average fuel consumption, as a unit rate, to the number of kilometres, n, travelled on 4.5 gallons.

A 2-door vehicle can travel 166.5 kilometres using 4.5 gallons of fuel.