To calculate the average rate of change of the function [tex]\( f(x) = x^2 + 5x - 12 \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 1 \)[/tex], let's go through the steps:
1. First, we find the values of the function at [tex]\( x = -2 \)[/tex] and [tex]\( x = 1 \)[/tex].
- Evaluating the function at [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = (-2)^2 + 5(-2) - 12
= 4 - 10 - 12
= -18
\][/tex]
Therefore, [tex]\( f(-2) = -18 \)[/tex].
- Evaluating the function at [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 1^2 + 5(1) - 12
= 1 + 5 - 12
= -6
\][/tex]
Therefore, [tex]\( f(1) = -6 \)[/tex].
2. Next, we calculate the change in [tex]\( y \)[/tex] (i.e., the difference between the function values at these points):
[tex]\[
\Delta y = f(1) - f(-2) = -6 - (-18) = -6 + 18 = 12
\][/tex]
3. The change in [tex]\( x \)[/tex] is given by the difference in the [tex]\( x \)[/tex]-values. Here, [tex]\( x \)[/tex] changes from [tex]\( -2 \)[/tex] to [tex]\( 1 \)[/tex]:
[tex]\[
\Delta x = 1 - (-2) = 1 + 2 = 3
\][/tex]
4. Finally, we find the average rate of change by dividing the change in [tex]\( y \)[/tex] by the change in [tex]\( x \)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{\Delta y}{\Delta x} = \frac{12}{3} = 4
\][/tex]
Summarizing the steps:
- [tex]\( f(-2) = -18 \)[/tex]
- [tex]\( f(1) = -6 \)[/tex]
- Change in [tex]\( y = 12 \)[/tex]
- Change in [tex]\( x = 3 \)[/tex]
- Average rate of change = 4
Therefore, the average rate of change of the function [tex]\( f(x) \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 1 \)[/tex] is [tex]\( 4 \)[/tex].
