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CCSS Math: HSF.IF.B.6

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.