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the function of y=f(x) is grapher below.  what is the average rate of change of the function f(x) on the interval -6 is less than or equal to x is less than or equal to 1

To find the average rate of change of the function \( f(x) \) over the interval \( -6 \leq x \leq 1 \), we can use the formula:

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

where \( a = -6 \) and \( b = 1 \).

### Steps to Solve:
1. **Identify the values of \( f(-6) \) and \( f(1) \)** from the graph.
   - Look at the point where \( x = -6 \) and determine the corresponding \( y \)-value \( f(-6) \).
   - Look at the point where \( x = 1 \) and determine the corresponding \( y \)-value \( f(1) \).

2. **Plug these values into the formula** to calculate the average rate of change.

Let's examine the graph.

- From the graph, \( f(-6) \) seems to be at \( y = 80 \).
- Similarly, \( f(1) \) appears to be around \( y = -40 \).

### Calculating:
\[
\text{Average Rate of Change} = \frac{f(1) - f(-6)}{1 - (-6)} = \frac{-40 - 80}{1 + 6} = \frac{-120}{7} \approx -17.14
\]

So, the average rate of change of the function \( f(x) \) over the interval \( -6 \leq x \leq 1 \) is approximately \(-17.14\).

Would you like further details, or do you have any questions?

### Related Questions:
1. How do you interpret the average rate of change in a real-world context?
2. What is the significance of the points where the function crosses the x-axis?
3. Can you determine the intervals where the function is increasing or decreasing?
4. How would the average rate of change differ if the interval were from \( -6 \) to \( 4 \)?
5. What can you infer about the function based on the shape of the graph?

### Tip:
The average rate of change of a function between two points is equivalent to the slope of the secant line connecting those two points on the graph.

Learn how to calculate the average rate of change of a function from its graph over the interval [-6, 1]. Detailed steps and example provided. Suitable for high school students and calculus learners.

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