Worked example: average rate of change from graph

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 your literally just averaging for example in this bowl section right over here the slope is really really steep it gets less deep it's a very negative slope it gets less negative less negative slope is zero 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 and the average one you could say the average the change the net change has been zero and any interval over which the net change has been zero also tells you that the average rate of change is going to be zero so you could view that the average rate of change is really the slope of the line is really the slope of the line that connects the two endpoints of your interval so another way of asking over watch interval does yfx 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 & 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 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 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 4 in the Y direction move 1 ahead in the x direction we move down 4 in the Y direction so the average rate of change over this interval is negative 4 so we didn't have to look at anything else that one will work