In this explainer, we will learn how to interpret the slope of a straight line as the rate of change of two quantities.
To describe how a quantity changes in relation to a corresponding change in another quantity, we use the so-called rate of change.
Definition: Rate of Change
The rate of change of quantity with respect to quantity is the rate of change in to the change in :
It is expressed as a change in per change of one unit in .
In some cases, the rate of change is constant. Let us explore the implications of a constant rate of change with an example.
Imagine a cuboid-shaped swimming pool that is being filled. If the water flow used to fill the pool is constant, then the rate of change, which is the increase in the water level per minute (or per hour), is constant. The person in charge of checking the filling of the swimming pool made the measurements given in the table.
Now, let us look attentively at the changes in the height of the water for the given periods of time. We see that the rate of change in height to the change in time (i.e., the period of time considered) is constant .
This means that the change in height is proportional to the period of time considered: it is given by multiplying the period of time (which is given by the change in time) in hours by the rate of change, here 0.6 metres per hour.
Let us now plot the four ordered pairs (9:30, 0.2), (10:30, 0.8), (13:30, 2.6), and (14:00, 2.9) on a coordinate plane.
We see that the four points are aligned. And as the right triangles with small sides of lengths 1 and 0.6 show, any two points on the line corresponding to a difference in time of one hour describe a difference in water height of 0.6 m.
It follows that two points at a horizontal distance of two hours correspond to a difference in water height of 1.2 m (having two of our blue triangles between them), and those at a horizontal distance of three hours are 1.8 m apart along the -axis, and so on. We see that the graph of two quantities with a constant rate of change (here, 0.6 metres per hour) between them is a straight line.
From this, we can conclude that when the rate of change between the two quantities and is constant, then any change in is proportional to the related change in . When plotted on a coordinate plane, all the ordered pairs are on a straight line. This type of relationship between and is called a linear relationship. The function that assigns a -value to any -value is called a linear function.
Keep in mind that, in linear relationships, the change in one quantity is proportional to the change in the other, not the quantities themselves. Only in proportional relationships are the quantities proportional to each other; proportional relationships are a special case of linear relationships where the rate of change is also the unit rate (or coefficient of proportionality).
Let us summarize our main findings.
Definition: Linear Functions
A linear function is characterized by a constant rate of change between and .
The graph of a linear function is a straight line.
Inversely, any function that has a straight-line graph is a linear function.
Example 1: Finding the Rate of Change from a Graph
The graph below shows the relation between the cost of a party and the number of people attending. Determine the rate of change.
Answer
The graph gives us the cost of a party against the number of people attending. We see that there is a fixed cost of around $25 (given at the point where the number of people is zero), and then there is a variable cost that is proportional to the number of people attending the party. We know this because the graph is a straight line: whenever the number of people increases by one, the cost increases by the same amount. This cost increase when the number of people increases by one is the rate of change of the cost with respect to the number of people. This is the answer to the question “how will the cost change if one more person is attending?”
To determine the rate of change, we need to find two well-defined points on the graph, for instance, (4, 100) and (12, 250). These two points mean that a party for four people costs $100, while it costs $250 for 12 people. The rate of change is given by dividing the change in the cost by the change in the number of people. Hence, we have
We have found that the rate of change is per person (i.e., 18.75 dollars per person).
Example 2: Identifying Whether a Relation between Two Quantities Is Linear
The relation between the cost of wrapping paper and the number of rolls bought is shown in the table. Is the relationship between the two quantities linear? If so, find the constant rate of change.
| Number of Rolls | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Total Cost ($) | 3 | 7 | 9 | 12 |
Answer
The table gives the cost of different numbers of rolls. To determine whether the relationship between the two quantities (cost and number of rolls) is linear, we need to work out the rate of change between different pairs of data.
We see here that while the number of rolls increases from one column of the table to the other by one, the cost increases by 4, 2, and 3, giving a rate of change of 4, 2, and 3 dollars per roll respectively.
Hence, the rate of change of the cost with respect to the number of rolls is not constant. The relationship between these two quantities is therefore not linear.
Example 3: Finding and Interpreting the Rate of Change from a Table
Which of the following interprets the rate of change of the data in the table?
| Remaining Gas (gal) | 19 | 14 | 9 | 4 |
|---|---|---|---|---|
| Miles Driven | 0 | 100 | 200 | 300 |
- A decrease of 19 gallons per 100 miles
- A decrease of 5 gallons per 100 miles
- A decrease of 19 gallons per 50 miles
- A decrease of 5 gallons per 50 miles
Answer
The table gives the volume of the remaining gas in a vehicle (in gallons) and the miles driven since the volume was 19 gallons. We see that, for every 100 miles driven, the volume of the remaining gas decreased by five gallons. The rate of change of the remaining volume of gas with respect to the miles driven can therefore be interpreted as a decrease of 5 gallons per 100 miles.
Example 4: Finding the Rate of Change from a Table
Use the information in the given table to determine the rate of change in the number of seeds for every orange.
| Number of Oranges | 3 | 5 | 7 |
|---|---|---|---|
| Number of Seeds | 54 | 90 | 126 |
Answer
The table gives us different numbers of oranges and the total number of seeds in these oranges. We want to find the rate of change in the number of seeds for every orange; that is, if the number of oranges increases by one, by how much will the number of seeds increase? Let us analyze how the number of seeds increases when the number of oranges increases.
We find that when the number of oranges increases by two, the number of seeds increases by 36. Hence, we have
The rate of change in the number of seeds is 18 seeds for 1 orange.
In this last example, note that the number of seeds is proportional to the number of oranges with a proportionality factor of 18 seeds per orange. In this case, the rate of change is actually the unit rate (or coefficient of proportionality). Be careful, however, not to think the reverse is true. A relationship with a constant rate of change is not necessarily proportional, even if the same unit is used for both the unit rate and the rate of change (here seeds per orange).
Example 5: Finding the Rate of Change of a Function from Its Graph
What is the rate of change shown by this graph of a function?
Answer
The graph of against is a straight line. Therefore, the relationship between and is linear, which means that the rate of change of with respect to is constant. To find this rate of change from the graph, we need to take two well-identified points, such as and . The rate of change is the change in when increases by 1; it is therefore the change in divided by the change in between these two points. Hence, we have
The rate of change of with respect to is .
We can check our result by checking that when increases by 5 (from to 0), changes by 5 times the rate of change, that is, by . It is indeed the case since increases by 2 when it goes from to . Our answer is correct.
Key Points
- The rate of change of quantity with respect to quantity is the rate of change in to the corresponding change in :
- The rate of change is expressed as a change in per change of one unit in .
- A linear function is characterized by a constant rate of change between and . This means that any change in is proportional to the corresponding change in .
- The graph of a linear function is a straight line. Inversely, any function that has a straight-line graph is a linear function.
- Proportional relationships are a special case of linear relationships where the rate of change is also the unit rate (or coefficient of proportionality).
