The table gives the average rates of change of a function over different intervals. On which of the intervals does the function increase the most?

Interval 1: 0≤x≤1 and Average Rate of Change=10
Interval 2: 1≤x≤4 and Average Rate of Change=-5
Interval 3: 4≤x≤8 and Average Rate of Change=2
Interval 4: 8≤x≤10 and Average Rate of Change=6

The average rate of change (AROC) of a function over an interval gives information about how the function behaves (increasing or decreasing) over that interval.

- If the AROC is positive, the function is increasing.
- If the AROC is negative, the function is decreasing.
- The larger the positive value of the AROC, the greater the increase over that interval.

### Examining the intervals:
- **Interval 1 (0≤x≤1):** AROC = 10 (positive, function increases)
- **Interval 2 (1≤x≤4):** AROC = -5 (negative, function decreases)
- **Interval 3 (4≤x≤8):** AROC = 2 (positive, function increases)
- **Interval 4 (8≤x≤10):** AROC = 6 (positive, function increases)

Among the intervals where the function increases (Interval 1, 3, and 4), **Interval 1** has the highest rate of change (AROC = 10). Therefore, the function increases the most on **Interval 1 (0≤x≤1)**.

Would you like further details or have any questions? Here are some related questions for practice:

1. What does the average rate of change represent geometrically on a graph?
2. How can you determine whether a function is increasing or decreasing from its derivative?
3. What would it mean if the AROC were zero in an interval?
4. How do you find the instantaneous rate of change at a specific point?
5. What is the relationship between average rate of change and slope of a secant line?

**Tip:** A positive average rate of change indicates an increasing function over the interval, while a negative rate suggests a decrease.

This problem asks students to determine which interval shows the greatest increase based on the average rates of change over four different intervals. It covers key concepts such as the average rate of change formula, positive and negative function behavior, and function increase/decrease. Suitable for Grades 9-11.

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