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What is the rate of change for the function represented above?





The table you provided represents values of \( x \) and the corresponding values of \( f(x) \). The **rate of change** of a function between two points is given by the **slope** formula:

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

Let's calculate the rate of change between several pairs of consecutive points to analyze the behavior of the function:

1. Between \( x = -4 \) and \( x = -1.25 \):

\[
\text{Rate of Change} = \frac{6 - 8.2}{-1.25 - (-4)} = \frac{-2.2}{2.75} \approx -0.8
\]

2. Between \( x = -1.25 \) and \( x = 1 \):

\[
\text{Rate of Change} = \frac{4.2 - 6}{1 - (-1.25)} = \frac{-1.8}{2.25} \approx -0.8
\]

3. Between \( x = 1 \) and \( x = 3 \):

\[
\text{Rate of Change} = \frac{2.6 - 4.2}{3 - 1} = \frac{-1.6}{2} = -0.8
\]

4. Between \( x = 3 \) and \( x = 6.25 \):

\[
\text{Rate of Change} = \frac{0 - 2.6}{6.25 - 3} = \frac{-2.6}{3.25} \approx -0.8
\]

5. Between \( x = 6.25 \) and \( x = 10 \):

\[
\text{Rate of Change} = \frac{-3 - 0}{10 - 6.25} = \frac{-3}{3.75} \approx -0.8
\]

### Conclusion:
The rate of change between each pair of consecutive points is consistently around **-0.8**. This suggests that the function has a constant rate of change, meaning it may be linear.

Would you like more details on this or have any questions?

Here are 5 related questions:
1. Can you verify if the function is truly linear using a different method?
2. How do you find the equation of a line given the rate of change and one point?
3. What is the significance of a constant rate of change in real-world applications?
4. How do you interpret negative rates of change?
5. What happens to a function when its rate of change is variable instead of constant?

**Tip:** When calculating the rate of change, always pay attention to the units of both \( x \) and \( f(x) \); they help in interpreting the slope meaningfully.

What is the rate of change for the function represented above?

Discover how to calculate the rate of change from a table of values. This problem involves calculating the slope between consecutive points to analyze the behavior of the function. It is suitable for students in Grades 8-10 and involves applying the rate of change formula.

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