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It looks like you're working on a homework question regarding the average rate of change of a function from College Algebra. The problem asks you to find the average rate of change of the function \( k(x) = x^2 + 3x \) from \( x_1 = 2 \) to \( x_2 = 9 \).

### Here's how to solve this problem:

The **average rate of change** of a function \( k(x) \) over an interval \([x_1, x_2]\) is given by:

\[
\frac{k(x_2) - k(x_1)}{x_2 - x_1}
\]

1. **Step 1:** Calculate \( k(x_1) = k(2) \).
   \[
   k(2) = (2)^2 + 3(2) = 4 + 6 = 10
   \]

2. **Step 2:** Calculate \( k(x_2) = k(9) \).
   \[
   k(9) = (9)^2 + 3(9) = 81 + 27 = 108
   \]

3. **Step 3:** Use the formula for the average rate of change.
   \[
   \frac{k(9) - k(2)}{9 - 2} = \frac{108 - 10}{9 - 2} = \frac{98}{7} = 14
   \]

Thus, the average rate of change is **14**.

Would you like any further explanation on this? Also, here are some related questions to deepen your understanding:

1. What is the significance of the average rate of change in real-world contexts?
2. How is the average rate of change similar to the concept of slope in linear functions?
3. What would the rate of change be if \( k(x) = x^2 + 3x + 1 \)?
4. How do you find the instantaneous rate of change of a function at a point?
5. How does the interval size affect the average rate of change?

**Tip:** The average rate of change over an interval is the same as the slope of the secant line connecting the points on the graph at \( x_1 \) and \( x_2 \).

Find the average rate of change of the function k(x) = x^2 + 3x from x1 = 2 to x2 = 9.

Learn how to find the average rate of change of the quadratic function k(x) = x^2 + 3x over the interval [2, 9]. Step-by-step solution includes calculation of k(2) and k(9) and the application of the average rate of change formula. Suitable for high school students in Grades 10-12.

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