The problem in the image asks for the average rate of change of the function $f(x) = x^2 + 4x + 1$ over the interval $-8 \leq x \leq 4$.

Solution:

The average rate of change of a function over an interval is given by the formula:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

Where:

• $a$ and $b$ are the endpoints of the interval ($a = -8, b = 4$).
• $f(a)$ and $f(b)$ are the values of the function at those points.

1. Find $f(a)$, where $a = -8$: $f(-8) = (-8)^2 + 4(-8) + 1$ $f(-8) = 64 - 32 + 1 = 33$

2. Find $f(b)$, where $b = 4$: $f(4) = (4)^2 + 4(4) + 1$ $f(4) = 16 + 16 + 1 = 33$

3. Substitute into the formula: $\text{Average Rate of Change} = \frac{f(4) - f(-8)}{4 - (-8)}$ $\text{Average Rate of Change} = \frac{33 - 33}{4 + 8}$ $\text{Average Rate of Change} = \frac{0}{12} = 0$

Final Answer:

$\boxed{0}$

Would you like further clarification or details about this process?

Related Questions:

1. What does the average rate of change tell us about the behavior of a function?
2. How would the result change if the interval was different, for example $-8 \leq x \leq 0$?
3. Can the average rate of change be negative? Why or why not?
4. How does the formula for the average rate of change relate to the slope of a secant line?
5. How would the average rate of change differ if the function were linear?

Tip:

When calculating the average rate of change, always simplify the interval difference first to avoid sign mistakes.