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The problem asks for the average rate of change of the function \( f(x) = x^2 + 7x \) between \( x_1 = 3 \) and \( x_2 = 8 \).

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

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

### Step 1: Calculate \( f(x_1) \)
We are given \( x_1 = 3 \), so:

\[
f(3) = 3^2 + 7(3) = 9 + 21 = 30
\]

### Step 2: Calculate \( f(x_2) \)
We are given \( x_2 = 8 \), so:

\[
f(8) = 8^2 + 7(8) = 64 + 56 = 120
\]

### Step 3: Compute the average rate of change
Now, use the formula:

\[
\text{Average rate of change} = \frac{f(8) - f(3)}{8 - 3} = \frac{120 - 30}{8 - 3} = \frac{90}{5} = 18
\]

Thus, the average rate of change of the function from \( x_1 = 3 \) to \( x_2 = 8 \) is **18**.

Would you like more details on this solution or have any questions?

Here are 5 related questions for further exploration:
1. What is the significance of the average rate of change in real-world applications?
2. How do you calculate the instantaneous rate of change of a function?
3. What is the difference between average and instantaneous rates of change?
4. How can you use the derivative to find the rate of change at a specific point?
5. How does the average rate of change of a quadratic function vary over different intervals?

**Tip:** The average rate of change of a function over an interval gives you the slope of the secant line between the two points, while the derivative gives you the slope of the tangent line at a point.

Find the average rate of change of the function f(x) = x^2 + 7x from x_1 = 3 to x_2 = 8.

This problem demonstrates how to find the average rate of change of the function f(x) = x^2 + 7x between x_1 = 3 and x_2 = 8 using the average rate of change formula. The solution is suitable for students in Grades 10-12 and covers essential algebraic concepts.

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