Average rate of change
Over which interval does y of x have an average rate of change of negative 4? So average rate of change, if you think about it, you are literally just averaging for example, in this bowl section right over here. The slope is really, really steep. It gets less steep. It's a very negative slope, it gets less negative. Less negative slope is 0 here. Then it gets positive, more positive, and more positive, and more positive. But when you get to this point right over here, you see you got to where you started from. One you could say the net change has been 0. And any interval over which the net change has been 0 also tells you that the average rate of change is going to be 0. So you could view that the average rate of change is really the slope of the line that connects the two endpoints of your interval. So another way of asking over which interval does y of x have an average rate of change of negative 4 is, can you come up with an interval where the slope between the endpoints of the interval is negative 4? So let's see the choices they give us. This first interval is x is between negative 1 and 1. So x is between negative 1. So this is x is negative 1. When x is equal to negative 1, y of x is all the way over here. y of negative 1 is equal to 7. And then when x is equal to 1, our graph is down over here. y of 1 is negative 1. So what is the slope of the line that connects the endpoints of those two points? So what is the slope of this line? Because the slope of this line, the line that connects the endpoints of my interval, that is going to be the average rate of change over this interval. And you see very clearly that the slope here, the rate of change of y with respect to x is negative 4. Every time we move one ahead in the x direction, we move down four in the y direction. Move one ahead in the x direction, we move down four in the y direction. So the average rate of change over this interval is negative 4. So we didn't have to even look at anything else, that one will work.