Learning Objectives
- Determine the rate of change between two data points.
- Determine if the rate of change is linear.
Key words
- Rate of change: the ratio of the change in one variable with respect to the change in another variable
- Linear rate of change: a constant rate of change between any two data points
Linear Rate of Change
Rate of change is defined as the ratio of the change in one variable with respect to the change in another variable. The following table shows values of two variables, [latex]x[/latex] and [latex]y[/latex].
| [latex]x[/latex] | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| [latex]y[/latex] | -7 | -5 | -3 | -1 | 1 | 3 | 5 |
Notice that as the values of [latex]x[/latex] increase so do the values of [latex]y[/latex]. As the value of [latex]x[/latex] increases by 1 unit, the value of [latex]y[/latex] increases by 2 units. The ratio of change in [latex]y[/latex] per change in [latex]x =\frac{\text{change in }y}{\text{change in } x} = \frac{2}{1} = 2[/latex]. This rate of change is constant. That is what makes it linear.
Linear rate of change means the rate of change ratio for any two data points is always the same. To illustrate, let’s use the values in the following table to find the rate of change for any two points. For example, for the two data points (-2, -7) and (1, -1), the rate of change ratio of change in [latex]y[/latex] over change in [latex]x[/latex] is:
[latex] \frac{\text{change in }y}{\text{change in }x} = \frac{-1-(-7)}{1-(-2)} = \frac{6}{3} = 2[/latex].
For another two data points (12, 21) and (25, 47), the rate of change ratio of change in [latex]y[/latex] over change in [latex]x[/latex] is:
[latex]\frac{\text{change in }y}{\text{change in }x} = \frac{47-21}{25-12} = \frac{26}{13} = 2[/latex]
| [latex]x[/latex] | -2 | 1 | 4 | 6 | 12 | 25 | 35 |
|---|---|---|---|---|---|---|---|
| [latex]y[/latex] | -7 | -1 | 5 | 9 | 21 | 47 | 67 |
The table has a linear rate of change ratio because the ratio is always 2 between any two data points.
Rate of change
Rate of Change = [latex] \frac{y_{2}-y_{1}}{x_{2}-x_{1}} [/latex], where [latex]\left ( x_1, y_1\right )[/latex] and [latex]\left (x_2, y_2\right )[/latex] are two data points.
The rate of change is linear when it is constant for any two data points.
Example
Determine if the data in the table represents a linear rate of change.
| [latex]x[/latex] | -3 | -1 | 1 | 3 |
|---|---|---|---|---|
| [latex]y[/latex] | -4 | 0 | 4 | 8 |
Solution
Moving from left to right, [latex]x[/latex] increases by 2 units between cells, and [latex]y[/latex] increases by 4 units between cells.
The rate of change between points is constant at [latex]\frac{2}{4}=\frac{1}{2}[/latex], which is a linear rate of change.
Example
Determine if the data in the table represents a linear rate of change.
| [latex]x[/latex] | -5 | -1 | 3 | 7 |
|---|---|---|---|---|
| [latex]y[/latex] | 2 | 0 | -4 | -8 |
Solution
Moving from left to right, [latex]x[/latex] increases by 4 units between cells, but [latex]y[/latex] decreases by 2 units, then 4 units, then 4 units between cells.
The rate of change between points is not constant so this is not a linear rate of change.
Try It
Determine if the data in the table represents a linear rate of change.
| [latex]x[/latex] | 1 | 5 | 9 | 11 |
|---|---|---|---|---|
| [latex]y[/latex] | 2 | 4 | 6 | 7 |
Example
Find the rate of change between the points [latex]\left (4, -7\right )[/latex] and [latex]\left (-3, 4\right )[/latex].
Solution
If we let [latex]\left (4, -7\right )=\left ( x_1, y_1\right )[/latex] and [latex]\left (-3, 4\right )=\left ( x_2, y_2\right )[/latex],
[latex]\begin{equation}\begin{aligned}\text{Rate of Change} & =\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ & = \frac{4-(-7)}{-3-4} \\ & = \frac{11}{-7} \\ & = -\frac{11}{7} \end{aligned}\end{equation}[/latex]
On the other hand, if we let [latex]\left (-3, 4\right )=\left ( x_1, y_1\right )[/latex] and [latex]\left (4, -7\right )=\left ( x_2, y_2\right )[/latex],
[latex]\begin{equation}\begin{aligned}\text{Rate of Change} & =\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ & = \frac{-7-4}{4-(-3)} \\ & = \frac{-11}{7} \\ & = -\frac{11}{7} \end{aligned}\end{equation}[/latex]
Notice that we get the same rate of change irrespective of the designation of the points.
Try It
Find the rate of change between the points [latex]\left (0, -3\right )[/latex] and [latex]\left (5, -4\right )[/latex].
