Consider the function $$ f\left(x\right)=-7x-6.\text{ Find the average rate of change between the points }(x,f\left(x\right))\text{ and (x}+h,f\left(x+h\right)) $$

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

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

In this case, the two points are $(x, f(x))$ and $(x + h, f(x + h))$. The formula for the average rate of change between these points is:

\[
\text{Average Rate of Change} = \frac{f(x + h) - f(x)}{(x + h) - x}
\]

Now, let's apply the function $ f(x) = -7x - 6 $.

1. Calculate $ f(x + h) $:
\[
f(x + h) = -7(x + h) - 6 = -7x - 7h - 6
\]

2. Calculate $ f(x + h) - f(x) $:
\[
f(x + h) - f(x) = \left( -7x - 7h - 6 \right) - \left( -7x - 6 \right) = -7h
\]

3. The denominator $(x + h) - x = h$.

4. Therefore, the average rate of change is:
\[
\frac{f(x + h) - f(x)}{h} = \frac{-7h}{h} = -7
\]

Thus, the average rate of change between the points $(x, f(x))$ and $(x + h, f(x + h))$ is $-7$.

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Would you like further details or clarifications?  
Here are five related questions:

1. How does the average rate of change relate to the derivative of the function?
2. What is the significance of the constant value in the average rate of change for linear functions?
3. How would the average rate of change differ for a quadratic function?
4. Can you interpret the meaning of a negative average rate of change in real-world scenarios?
5. What is the geometric interpretation of the average rate of change?

**Tip:** For linear functions, the average rate of change over any interval is constant and equal to the slope of the function.

Learn how to calculate the average rate of change between two points on the function f(x) = -7x - 6. Understand the concept of average rate of change in linear functions and apply the formula step-by-step. Suitable for high school students studying functions and derivatives.

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