Sure, let's break down the process step-by-step to find 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].
1. Evaluate the function at [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = (-2)^2 + 5(-2) - 12 = 4 - 10 - 12 = -18
\][/tex]
2. Evaluate the function at [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 1^2 + 5(1) - 12 = 1 + 5 - 12 = -6
\][/tex]
3. Find the change in [tex]\( y \)[/tex] by subtracting the value of the function at [tex]\( x = -2 \)[/tex] from the value of the function at [tex]\( x = 1 \)[/tex]:
[tex]\[
\Delta y = f(1) - f(-2) = -6 - (-18) = -6 + 18 = 12
\][/tex]
4. Find the change in [tex]\( x \)[/tex] by subtracting [tex]\(-2\)[/tex] from [tex]\( 1 \)[/tex]:
[tex]\[
\Delta x = 1 - (-2) = 1 + 2 = 3
\][/tex]
5. Calculate the average rate of change by dividing the change in [tex]\( y \)[/tex] by the change in [tex]\( x \)[/tex]:
[tex]\[
\text{Average rate of change} = \frac{\Delta y}{\Delta x} = \frac{12}{3} = 4.0
\][/tex]
Summary of the calculations:
- The value of the function at [tex]\( x = -2 \)[/tex] is [tex]\( f(-2) = -18 \)[/tex].
- The value of the function at [tex]\( x = 1 \)[/tex] is [tex]\( f(1) = -6 \)[/tex].
- The change in [tex]\( y \)[/tex] is [tex]\( 12 \)[/tex].
- The change in [tex]\( x \)[/tex] is [tex]\( 3 \)[/tex].
- The average rate of change of [tex]\( f(x) = x^2 + 5x - 12 \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 1 \)[/tex] is [tex]\( 4.0 \)[/tex].
And hence, the average rate of change is 4.0.
