Linear Rates of Change
Consider these scenarios:
- The horizontal speed of a projectile
- The reproduction rate of a colony of bacteria
- The revenue growth of a business over a year
- The flow in and out of a body of water
In each of these cases, we are interested in how one measurement varies as another one does. That is, there is a dependent variable that varies with respect to an independent variable. However, we’re not interested in the measurements themselves but in how they vary. We call this the rate of change.
Exploration
Let’s focus on the first scenario. We can’t directly measure speed, but we can measure distance and time. Notice that speed is the amount that distance changes per unit of time. That is, speed is the rate of change of distance with respect to time. So we can use our measurements of distance and time to figure out the speed.
Suppose that the distance ($x$x, in metres) is related to the time ($t$t, in seconds) by the relationship $x=3t$x=3t. Let’s plot this relationship first:
Notice that this is a linear function. Since speed is the rate of change of distance over time, we want to find out how the distance changes over any amount of time. Let’s start by picking two points on the line. We’ll use $\left(1,3\right)$(1,3) and $\left(3,9\right)$(3,9):
First we find the change in the independent and dependent variables:
| Change in time | $=$= | $3-1$3−1 m |
| $=$= | $2$2 m | |
| Change in distance | $=$= | $9-3$9−3 s |
| $=$= | $6$6 s |
And then we divide the change in distance by the change of time to get the rate of change:
| Rate of change | $=$= | $\frac{6}{2}$62 m/s |
| $=$= | $3$3 m/s |
So the speed is $3$3 m/s. We will get the same result no matter which two points we choose. Notice that this is the same as the gradient of a linear function. In fact, this is always the case when the function is linear.
Using this we can come up with a general formula for the rate of change in a linear function. If our two points are $\left(x_1,y_1\right)$(x1,y1) and $\left(x_2,y_2\right)$(x2,y2), then the rate of change is:
| Rate of change | $=$= | $\frac{\text{change in dependent variable}}{\text{change in independent variable}}$change in dependent variablechange in independent variable |
| $=$= | $\frac{x_2-x_1}{y_2-y_1}$x2−x1y2−y1 | |
| $=$= | gradient of the linear function |
The rate of change of a dependent variable with respect to an independent variable is how much the dependent variable changes as the independent variable changes.
In the case of a linear function, the rate of change is the gradient.
Note that the independent variable is most often time, but can be anything else.
Practice questions
Question 1
Sarah is planning to donate some scarves to the local charity for their winter drive next month. She already has $8$8 old scarves that she is willing to give away and plans on adding to this amount by knitting $2$2 each week.
Complete the table.
Week Number of scarves $0$0 $\editable{}$ $1$1 $\editable{}$ $2$2 $\editable{}$ $3$3 $\editable{}$ $4$4 $\editable{}$ What amount is added to the number of scarves in each row?
What does this amount represent?
the number of scarves that Sarah will buy each week
Athe number of scarves that Sarah will knit each week
Bthe difference between the number of scarves that Sarah will have knitted and the number of scarves she will have bought
Cthe number of scarves that Sarah will have knitted after $20$20 weeks
D
Question 2
The graph shows the progress of two competitors in a cycling race.
Who is travelling faster?
Justin
AOliver
BHow much faster is Oliver travelling?