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To find the average rate of change of the function \( f(x) \) on the interval \( 3 \leq x \leq 5 \), we use the formula for the average rate of change:

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

where \( a = 3 \) and \( b = 5 \).

### Step 1: Determine \( f(3) \) and \( f(5) \) from the graph.

- At \( x = 3 \), the corresponding \( y \)-value appears to be approximately \( f(3) = 8 \).
- At \( x = 5 \), the corresponding \( y \)-value is approximately \( f(5) = -2 \).

### Step 2: Apply the formula.

\[
\text{Average rate of change} = \frac{-2 - 8}{5 - 3} = \frac{-10}{2} = -5
\]

Thus, the average rate of change of the function \( f(x) \) on the interval \( 3 \leq x \leq 5 \) is \( -5 \).

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Would you like further clarification or more details? Here are 5 related questions for further exploration:

1. How does the average rate of change compare to the instantaneous rate of change at specific points on the graph?
2. What is the geometric meaning of a negative average rate of change on a graph?
3. How would the average rate of change differ if you chose different intervals on the graph?
4. Can you identify any points of inflection or turning points from the graph of \( f(x) \)?
5. How does the graph of \( f(x) \) behave for larger intervals, such as \( 0 \leq x \leq 6 \)?

**Tip:** The average rate of change is essentially the slope of the secant line between two points on the curve.

The function y = f(x) is graphed below. What is the average rate of change of the function f(x) on the interval 3 ≤ x ≤ 5?

This tutorial explores how to calculate the average rate of change of a function using the interval 3 ≤ x ≤ 5 on a given graph. Key concepts include the slope of the secant line and the formula for the average rate of change. Suitable for high school students in Grades 9-12.

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