To determine the average rate of change of the quadratic function for the interval from [tex]\( x = 9 \)[/tex] to [tex]\( x = 10 \)[/tex], follow these steps:
1. Identify the [tex]$y$[/tex]-values for the given [tex]$x$[/tex]-values:
- Given: [tex]\( x = 9 \)[/tex] and [tex]\( x = 10 \)[/tex].
- From the table, the [tex]$y$[/tex]-values corresponding to the given [tex]$x$[/tex]-values:
[tex]\[
(9, -82) \quad \text{and} \quad (10, -101)
\][/tex]
2. Calculate the change in [tex]$y$[/tex] ([tex]\(\Delta y\)[/tex]) and the change in [tex]$x$[/tex] ([tex]\(\Delta x\)[/tex]):
[tex]\[
\Delta y = y_{final} - y_{initial} = -101 - (-82)
\][/tex]
Simplifying:
[tex]\[
\Delta y = -101 + 82 = -19
\][/tex]
[tex]\[
\Delta x = x_{final} - x_{initial} = 10 - 9 = 1
\][/tex]
3. Calculate the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{\Delta y}{\Delta x} = \frac{-19}{1} = -19
\][/tex]
So, the average rate of change for the interval from [tex]\( x = 9 \)[/tex] to [tex]\( x = 10 \)[/tex] is [tex]\(\boxed{-19}\)[/tex].
