Sure, let's 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].
First, let's evaluate the function at [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = (-2)^2 + 5(-2) - 12 \][/tex]
[tex]\[ f(-2) = 4 - 10 - 12 \][/tex]
[tex]\[ f(-2) = -18 \][/tex]
So, [tex]\( f(-2) = -18 \)[/tex].
Next, let's evaluate the function at [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1^2 + 5(1) - 12 \][/tex]
[tex]\[ f(1) = 1 + 5 - 12 \][/tex]
[tex]\[ f(1) = -6 \][/tex]
So, [tex]\( f(1) = -6 \)[/tex].
Now, the change in [tex]\( y \)[/tex], which is [tex]\(\Delta y\)[/tex], is:
[tex]\[ \Delta y = f(1) - f(-2) \][/tex]
[tex]\[ \Delta y = -6 - (-18) \][/tex]
[tex]\[ \Delta y = -6 + 18 \][/tex]
[tex]\[ \Delta y = 12 \][/tex]
The change in [tex]\( y \)[/tex] is 12.
Finally, we calculate the average rate of change:
The change in [tex]\( x \)[/tex], which is [tex]\(\Delta x\)[/tex], is:
[tex]\[ \Delta x = 1 - (-2) \][/tex]
[tex]\[ \Delta x = 1 + 2 \][/tex]
[tex]\[ \Delta x = 3 \][/tex]
The average rate of change is:
[tex]\[ \text{Average rate of change} = \frac{\Delta y}{\Delta x} \][/tex]
[tex]\[ \text{Average rate of change} = \frac{12}{3} \][/tex]
[tex]\[ \text{Average rate of change} = 4.0 \][/tex]
Thus, the average rate of change from [tex]\( x = -2 \)[/tex] to [tex]\( x = 1 \)[/tex] of the function [tex]\( f(x) = x^2 + 5x - 12 \)[/tex] is:
[tex]\[ 4.0 \][/tex]
