Question

now find the change in x. the change in x is the average rate change of $f(x)=x^{2}+5x - 12$ for $x = - 2$ to $x = 1$ is

Explanation:

Step1: Calculate the change in \(x\)

The change in \(x\), denoted as \(\Delta x\), is calculated by subtracting the initial \(x\)-value from the final \(x\)-value.
If the initial \(x\)-value is \(x_1=-2\) and the final \(x\)-value is \(x_2 = 1\), then \(\Delta x=x_2 - x_1\).
Substitute the values: \(\Delta x=1-(-2)=1 + 2=3\).

Step2: Calculate the average rate of change

The formula for the average rate of change of a function \(y = f(x)\) from \(x=a\) to \(x = b\) is \(\frac{f(b)-f(a)}{b - a}\).
First, find \(f(-2)\):

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Then, find \(f(1)\):

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The average rate of change \(\frac{f(1)-f(-2)}{1-(-2)}=\frac{-6-(-18)}{3}=\frac{-6 + 18}{3}=\frac{12}{3}=4\).

Answer:

The change in \(x\) is \(3\). The average rate of change is \(4\).