https://file.aiquickdraw.com/mathbot/file/1728889042120vizgfjco.jpg

In this image, the function \( f(x) = x \) is defined, and the interval is given as \([0, 2]\). The expression for the average rate of change of the function, \(\frac{\Delta y}{\Delta x}\), seems to be incomplete, so we can compute it.

The average rate of change of a function over an interval \([a, b]\) is given by:
\[
\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}
\]

For the function \( f(x) = x \) over the interval \([0, 2]\):
- \( f(0) = 0 \)
- \( f(2) = 2 \)

Thus, the average rate of change is:
\[
\frac{\Delta y}{\Delta x} = \frac{f(2) - f(0)}{2 - 0} = \frac{2 - 0}{2 - 0} = \frac{2}{2} = 1
\]

So, the average rate of change of the function \( f(x) = x \) over the interval \([0, 2]\) is 1.

Would you like further details or clarification on this? Here are some related questions you might find useful:

1. What is the geometric interpretation of the average rate of change for a linear function?
2. How does the average rate of change differ for non-linear functions?
3. Can the average rate of change be negative? What does it represent in such cases?
4. How does the rate of change vary over different intervals for a quadratic function?
5. How do we interpret the instantaneous rate of change compared to the average rate of change?

**Tip:** The average rate of change of a linear function is always constant, which is equal to the slope of the line.

What is the average rate of change of the function f(x) = x over the interval [0, 2]?

Understand how to calculate the average rate of change of the linear function f(x) = x over the interval [0, 2]. This example shows a step-by-step process using the formula for average rate of change, making it ideal for students in Grades 8-10.

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