$$ p\left(x\right)= $$$$ 3x+4 $$$$ on $$[2, 2+h]

The expression given is a polynomial function $ p(x) = 3x + 4 $, and you are interested in examining this function on the interval $[2, 2+h]$, where $h$ represents a small increment from 2.

### Key steps to analyze this:

1. **Function at the start of the interval:**
   At $x = 2$, the value of the function is:
   \[
   p(2) = 3(2) + 4 = 6 + 4 = 10
   \]

2. **Function at the end of the interval:**
   At $x = 2 + h$, the value of the function is:
   \[
   p(2+h) = 3(2+h) + 4 = 6 + 3h + 4 = 10 + 3h
   \]

3. **Change in the function over the interval:**
   The difference between the values of the function at the endpoints of the interval is:
   \[
   p(2+h) - p(2) = (10 + 3h) - 10 = 3h
   \]

4. **Average rate of change (or slope over the interval):**
   The average rate of change of the function over $[2, 2+h]$ is:
   \[
   \text{Average rate of change} = \frac{p(2+h) - p(2)}{h} = \frac{3h}{h} = 3
   \]
   This shows that the function changes at a constant rate of 3 for any value of $h$.

This process gives insight into the behavior of the linear function over the interval.

Would you like more details or have any specific questions?

### 5 Related Questions:
1. What is the general formula for the average rate of change of a function?
2. How does the average rate of change differ for non-linear functions?
3. How can we interpret the constant slope of a linear function?
4. What would happen if $p(x)$ were quadratic instead of linear?
5. How does the value of $h$ affect the interpretation of this interval?

**Tip:** For linear functions like $p(x) = 3x + 4$, the slope is constant, so the rate of change will be the same for any interval!

Explore how to calculate the average rate of change of the linear function p(x) = 3x + 4 over the interval [2, 2+h]. This problem highlights key concepts in algebra, including the formula for the average rate of change and how linear functions behave on specified intervals. Suitable for students in Grades 9-11.

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