now find the change in x. the change in x is the average rate of change of f(x)=x² + 5x - 12 from x=-2 to…

Explanation:

Step1: Calculate change in x

The change in (x), denoted as (\Delta x), is found by subtracting the initial (x) - value from the final (x) - value. Given (x_1=-2) and (x_2 = 1), then (\Delta x=x_2 - x_1). (\Delta x=1-(-2)=3)

Step2: Calculate (f(x_1)) and (f(x_2))

First, find (f(x_1)) when (x_1=-2). [ \begin{align*} f(-2)&=(-2)^2+5\times(-2)-12\ &=4 - 10-12\ &=-18 \end{align*} ] Then, find (f(x_2)) when (x_2 = 1). [ \begin{align*} f(1)&=1^2+5\times1-12\ &=1 + 5-12\ &=-6 \end{align*} ]

Step3: Calculate average rate of change

The average rate of change of a function (y = f(x)) from (x = x_1) to (x = x_2) is given by (\frac{\Delta y}{\Delta x}=\frac{f(x_2)-f(x_1)}{x_2 - x_1}). We know (f(x_2)-f(x_1)=-6-(-18)=12) and (\Delta x = 3). So the average rate of change is (\frac{-6-(-18)}{1-(-2)}=\frac{12}{3}=4)

Answer:

The change in (x) is (3). The average rate of change of (f(x)=x^{2}+5x - 12) from (x=-2) to (x = 1) is (4).