To answer your question, let's break down the concept shown in the image.

Equation:

Given the linear function $y = 5x + 9$, where the slope is $5$, we know that:

• Rate of change of a linear function is given by its slope. In this case, the slope is 5, which is constant.

Now, let’s discuss what the statement in the red box means:

• Average rate of change of the rates of change refers to how the slope (rate of change) itself changes across equal intervals. Since this is a linear function, the slope is constant (5). Therefore, the change in slope over any intervals is 0 because there is no change in the slope.

Conclusion:

• The rate of change of the average rates of change for a linear function is always 0 because the slope (rate of change) is constant across all intervals, and there's no variation in it.

Would you like to see more detailed examples, or do you have any specific points to clarify?

Relative Questions:

1. What is the significance of the slope in linear functions?
2. How do we calculate the rate of change for nonlinear functions?
3. Why is the average rate of change of a linear function always zero?
4. How would this concept differ for a quadratic function?
5. Can the rate of change ever be non-zero for a linear function?

Tip:

For any linear function, the slope is constant, and hence, its rate of change remains the same throughout. This property is crucial for understanding how linear models behave in mathematics.