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# Worked example: average rate of change from graph

Finding the interval in a function's graph where the function has an average rate of change of -4. Created by Sal Khan.

## Want to join the conversation?

- What are intervals? What is up with this y(x) thing? I am very confused!(13 votes)
- The notation y(x) means that y is a function of x; that is, y is essentially a variable whose value depends entirely on the value of x. We cannot say that y equals anything in particular, unless we know the value of x as well. If this notation or concept is confusing to you, then it will probably help to look at some of the Khan Academy videos on "functions" (which seem to be in the Trigonometry playlist for some reason): https://www.khanacademy.org/math/trigonometry/functions_and_graphs(13 votes)

- What exactly is an interval?(6 votes)
- An interval is a set of real numbers that have a starting value and ending value and includes all the numbers between the start and end points. The start and ending values may or may not be part of the set.(13 votes)

- What does with respect of x mean?(1 vote)
- "Change in y with respect to x" is how much y changes over a given amount of change in x. You can use that phrasing in other scenarios as well: If you have 10 cars with a total of 40 wheels, then you have 4 wheels with respect to 1 car.(11 votes)

- When am I ever gonna use average rate of change? In real life when will it ever be useful?(5 votes)
- You bet. Let's say you took a survey of multiple people for an experiment. It doesn't matter whether it's medicinal, business, or other, you just took a survey, or test. You ask people two questions and ask them to rate themselves from one to ten on those questions. The first of the questions can be your x-axis and the other is your y-axis.

You ask fifty people and plot them in a dot graph, where you put a dot corresponding with the person's answers. For instance, if one person rated himself on the first question as 3, and then as 7 on the other, you plot him as a dot on the coordinates (3,7).

Now, you have fifty people plotted. It just looks like a cloud of dots (*because dot graphs basically never form perfect lines*) and you need to know the rate of change, which means you need to know the rate, or speed, of how fast the line inclines. It's basically slope. And that's when you use it.(3 votes)

- Why isn't the interval notation written as "-1 is less than OR EQUAL TO x is less than or EQUAL TO 1. We are also counting 1 and -1 as part of the interval but the notation says otherwise?(4 votes)
- They are not obliged to. Because there are also other points in that interval(between 1 and -1) that give the same average rate of change.(2 votes)

- Gosh this is confusing #igotthis(3 votes)
- When is the average rate of change constant(3 votes)
- The rate of change is constant when the line is straight. The AVERAGE rate of change will be constant over a given interval if the line is straight OR the line oscillates constantly over the same interval.(0 votes)

- is the formula to find the average rate of change the following:

f(b)-(a)/b-a

when b is x1 and a is x2?(2 votes)- I think you meant
`( f(b)-f(a) ) / (b - a)`

. Of course`( f(a)-f(b) ) / (a - b)`

works equally well. Don't try to memorize things like this, though. Just remember that the average rate of change means the slope of a line over that interval.(3 votes)

- I am still confused of how do you get the 2 points to get the rate of change in that kind of line.(2 votes)
- The points will be given in most cases. Like, the question will be: Given points x and y, what is the rate of change of the line joining them?(3 votes)

- What website is he using to solve these equations?(2 votes)
- It is an earlier version of KhanAcademy.(3 votes)

## Video transcript

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.