Sure! To find the average rate of change between [tex]\( x = 1 \)[/tex] and [tex]\( x = 6 \)[/tex], we follow these steps:
1. Identify the values of [tex]\( y \)[/tex] corresponding to [tex]\( x = 1 \)[/tex] and [tex]\( x = 6 \)[/tex]:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 12 \)[/tex]
- When [tex]\( x = 6 \)[/tex], [tex]\( y = 27 \)[/tex]
2. Calculate the change in [tex]\( y \)[/tex] ([tex]\( \Delta y \)[/tex]):
[tex]\[
\Delta y = y(6) - y(1) = 27 - 12 = 15
\][/tex]
3. Calculate the change in [tex]\( x \)[/tex] ([tex]\( \Delta x \)[/tex]):
[tex]\[
\Delta x = 6 - 1 = 5
\][/tex]
4. Find the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{\Delta y}{\Delta x} = \frac{15}{5} = 3.0
\][/tex]
Therefore, the average rate of change between [tex]\( x = 1 \)[/tex] and [tex]\( x = 6 \)[/tex] is [tex]\( 3.0 \)[/tex].
