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The problem asks to find the average rate of change of the function \( f(x) = x^2 + 8x \) from \( x_1 = 1 \) to \( x_2 = 4 \).

The formula for the average rate of change of a function between two points \( x_1 \) and \( x_2 \) is given by:

\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Let's break it down step by step:

1. First, find \( f(1) \) and \( f(4) \).

- \( f(1) = (1)^2 + 8(1) = 1 + 8 = 9 \)
- \( f(4) = (4)^2 + 8(4) = 16 + 32 = 48 \)

2. Now, use the formula for the average rate of change:

\[
\text{Average rate of change} = \frac{f(4) - f(1)}{4 - 1} = \frac{48 - 9}{4 - 1} = \frac{39}{3} = 13
\]

Thus, the average rate of change is \( 13 \).

Do you have any questions or would you like further details? Here are 5 related questions:

1. How do you find the rate of change for more complex functions like \( f(x) = \sin(x) \)?
2. What is the geometric interpretation of the average rate of change?
3. How does the average rate of change relate to the derivative of a function?
4. Can the average rate of change be negative, and what would that signify?
5. How would you compute the average rate of change for a piecewise function?

**Tip:** The average rate of change is similar to finding the slope of the secant line between two points on the graph of the function.

Find the average rate of change of the function f(x) = x^2 + 8x from x1 = 1 to x2 = 4.

Learn how to calculate the average rate of change for the quadratic function f(x) = x^2 + 8x between x = 1 and x = 4. This problem involves fundamental algebraic concepts such as quadratic functions and the formula for average rate of change, ideal for high school students in Grades 9-12.

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