To calculate the rate of change for the given data points, we will use the concept of slope between two points, which is defined as:
[tex]\[ \text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let’s calculate the rate of change between each pair of consecutive points.
1. Between the points [tex]\((-1, 5)\)[/tex] and [tex]\((2, -4)\)[/tex]:
[tex]\[
\text{Rate of Change 1} = \frac{-4 - 5}{2 - (-1)} = \frac{-4 - 5}{2 + 1} = \frac{-9}{3} = -3
\][/tex]
2. Between the points [tex]\((2, -4)\)[/tex] and [tex]\((7, -19)\)[/tex]:
[tex]\[
\text{Rate of Change 2} = \frac{-19 - (-4)}{7 - 2} = \frac{-19 + 4}{7 - 2} = \frac{-15}{5} = -3
\][/tex]
3. Between the points [tex]\((7, -19)\)[/tex] and [tex]\((10, -28)\)[/tex]:
[tex]\[
\text{Rate of Change 3} = \frac{-28 - (-19)}{10 - 7} = \frac{-28 + 19}{10 - 7} = \frac{-9}{3} = -3
\][/tex]
Thus, the rate of change between each pair of points is:
[tex]\[ -3, -3, -3 \][/tex]
This indicates that the rate of change is consistent and equal to [tex]\(-3\)[/tex] throughout the intervals given in the data.
