To determine the average rate of change of the function $f(x) = x^2 + 4x - 2$ over the interval $-8 \leq x \leq 3$, we use the formula for the average rate of change:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

where $a = -8$ and $b = 3$ are the endpoints of the interval.

Step 1: Find $f(-8)$ and $f(3)$

1. Evaluate $f(-8)$: $f(-8) = (-8)^2 + 4(-8) - 2$ $f(-8) = 64 - 32 - 2 = 30$

2. Evaluate $f(3)$: $f(3) = (3)^2 + 4(3) - 2$ $f(3) = 9 + 12 - 2 = 19$

Step 2: Calculate the average rate of change

Now that we know $f(-8) = 30$ and $f(3) = 19$, we can calculate the average rate of change:

$\text{Average rate of change} = \frac{f(3) - f(-8)}{3 - (-8)}$ $\text{Average rate of change} = \frac{19 - 30}{3 - (-8)} = \frac{-11}{3 + 8} = \frac{-11}{11} = -1$

Final Answer:

The average rate of change of the function $f(x) = x^2 + 4x - 2$ over the interval $-8 \leq x \leq 3$ is -1.

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Would you like further details or have any other questions?

Here are five related questions to explore:

1. How would you find the instantaneous rate of change at a particular point?
2. How do the concepts of average rate of change and instantaneous rate of change differ?
3. How can you apply the average rate of change formula in real-life scenarios?
4. What is the significance of the derivative in the context of rate of change?
5. How would you calculate the average rate of change for a piecewise function?

Tip: To better understand rate of change, try visualizing it as the slope of a line connecting two points on the graph of the function.