For which type of function does the average rate of change produce the same value for any interval of the domain on the function? Why does this occur for this type of function?

Question

Please make it simple and like a kid wrote it. Thank you.

Answer 1

The type of function where the average rate of change is always the same for any interval is a linear function.

A linear function looks like this: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

The reason the average rate of change is always the same is that a linear function makes a straight line when you graph it. No matter where you choose two points on the line, the slope (or steepness) between those points stays the same. This is like if you walk uphill at the same slant no matter where you start on the hill — it feels the same all the way up!

So, if you pick any two points on a straight line and see how much you go up compared to how much you go across, you’ll get the same answer every time. That’s why the average rate of change is constant for linear functions!