Questions: Use the formula (f(b)-f(a))/(b-a) to calculate the average rate of change over the interval [-2,0] given the function table for f(x)=x^2+6x+8. x f(x) -3 -1 -2 0 -1 3 0 8

Transcript text: Use the formula $\frac{f(b)-f(a)}{b-a}$ to calculate the average rate of change over the interval $[-2,0]$ given the function table for $f(x)=x^{2}+6 x+8$. \begin{tabular}{|c|c|} \hline $\mathbf{x}$ & $\mathrm{f}(\mathbf{x})$ \\ \hline-3 & -1 \\ \hline-2 & 0 \\ \hline-1 & 3 \\ \hline 0 & 8 \\ \hline \end{tabular} (1 point)

Solution

Solution Steps

To find the average rate of change of the function $ f(x) = x^2 + 6x + 8 $ over the interval $[-2, 0]$, we use the formula $\frac{f(b) - f(a)}{b - a}$. Here, $a = -2$ and $b = 0$. We need to find $f(-2)$ and $f(0)$ using the given function table, and then substitute these values into the formula to calculate the average rate of change.

Step 1: Identify the Function Values

From the function table, we have:

$ f(-2) = 0 $
$ f(0) = 8 $

Step 2: Apply the Average Rate of Change Formula

We use the formula for the average rate of change over the interval $[-2, 0]$: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] Substituting the values: \[ \text{Average Rate of Change} = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{8 - 0}{0 + 2} = \frac{8}{2} = 4 \]