To determine the rate of change of the function represented by the given table, we need to follow these steps:
1. Identify the points on the table:
- Point 1: [tex]\( (1, -8.5) \)[/tex]
- Point 2: [tex]\( (2, -6) \)[/tex]
- Point 3: [tex]\( (3, -3.5) \)[/tex]
- Point 4: [tex]\( (4, -1) \)[/tex]
2. Calculate the change in [tex]\( y \)[/tex] ([tex]\(\Delta y\)[/tex]) over the range of [tex]\( y \)[/tex] values:
[tex]\[
\Delta y = y_{\text{final}} - y_{\text{initial}} = -1 - (-8.5)
\][/tex]
[tex]\[
\Delta y = -1 + 8.5 = 7.5
\][/tex]
3. Calculate the change in [tex]\( x \)[/tex] ([tex]\(\Delta x\)[/tex]) over the range of [tex]\( x \)[/tex] values:
[tex]\[
\Delta x = x_{\text{final}} - x_{\text{initial}} = 4 - 1
\][/tex]
[tex]\[
\Delta x = 3
\][/tex]
4. Finally, calculate the rate of change ([tex]\(\frac{\Delta y}{\Delta x}\)[/tex]):
[tex]\[
\text{Rate of change} = \frac{\Delta y}{\Delta x} = \frac{7.5}{3}
\][/tex]
[tex]\[
\text{Rate of change} = 2.5
\][/tex]
Therefore, the rate of change of the function represented by the given table is [tex]\( 2.5 \)[/tex].
