Let f(x) be a function represented by the following table of values:
x	 −2	 −1	0	 1	 2
f(x)	−10	−4	−2	 6	 3

Find the average rate of change of the function between x=-1 and x=2.

The average rate of change of a function between two points \( x_1 \) and \( x_2 \) is given by the formula:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Here, \( x_1 = -1 \) and \( x_2 = 2 \).

From the table:

- \( f(-1) = -4 \)
- \( f(2) = 3 \)

Substitute these values into the formula:

\[
\text{Average Rate of Change} = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{3 - (-4)}{2 - (-1)} = \frac{3 + 4}{2 + 1} = \frac{7}{3}
\]

So, the average rate of change of the function between \( x = -1 \) and \( x = 2 \) is \( \frac{7}{3} \).

Would you like further details or have any questions?

Here are some related questions:
1. What does the average rate of change represent in real-world applications?
2. How would the average rate of change differ if calculated between \( x = -2 \) and \( x = 1 \)?
3. What is the significance of a positive or negative average rate of change?
4. How can you determine whether the function is increasing or decreasing between two points?
5. Can the average rate of change be interpreted as the slope of a line for linear functions?

**Tip:** The average rate of change is essentially the slope of the secant line that passes through the points on the graph of the function at the given \( x \)-values.

Let f(x) be a function represented by the following table of values: x = −2, −1, 0, 1, 2, and f(x) = −10, −4, −2, 6, 3. Find the average rate of change of the function between x=-1 and x=2.

Calculate the average rate of change for the function represented by the given table of values between x = -1 and x = 2. This problem covers essential algebra concepts, including rate of change, slope, and the secant line. Suitable for students in Grades 9-11.

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