Hello again. Mr. Tarrou. I would like talk to you today about something called average
rate of change. A lot of the calculations I am going to do in a few minutes are going
to look very very similar. They are going to look like slope. We are going to find with
this parabola the average rate of change between (-1,4) and (2,-2). We can't call it slope
because...well..a parabola is curved and slope is for linear functions. It is the direction
that a line takes between two points and it does only take two points to define a line.
So when you are looking at a line, I am going to draw this secant line right here. That
secant line is going to represent the slope (average rate of change) between (-1,4) and
(2,-2) but I have to correct that a little bit again. I don't want to analyze necessarily
this red line that I have drawn, I am analyzing the average rate of change between these two
points on the parabola. So again the parabola is curved, not linear, so when we find this
rise over run between these two points, we are not going to be able to call it slope,
we are going to call it Average Rate of Change. Average rate of change as a formula is f of
x sub two minus f of x sub one all over x sub two minus x sub one. If you know your
function notation well now you will remember that f(x) is as I have said before our fancy
way of saying y. So this is function notation for y sub 2 and function notation for y sub
1. If you are thinking y sub 2 minus y sub 1 over x sub 2 minus x sub1 you might be thinking
well that sounds like slope. It is the same process as finding slope. But again this is
a curved graph so and so we must call the slope of the secant line Average Rate of Change.
So for this parabola, if I want the average rate of change between (-1,4) and (2,-2) and
use the function notation inside the formula, this is what is going to look like. I am going
to call this x sub 1 y sub 1 and this x sub 2 and y sub 2 and apply these points to this.
Now this says... The average rate of change formula in a PreCalculus book will say f of
x sub 2 minus f of x sub 1. Well the function, and I am going to keep this highlight thing
going in green, f of x sub 2. Well the function is x squared minus 3 x. So the function is
something squared minus 3 times something. I am going to leave that blank for a second
to highlight that x in the function is just an empty place for you to put numbers and
other variables or expressions if we need to. Now minus f of x sub 1. I am going to
highlight that in blue and just over here... so this is going to be f of x sub 2 minus
f of x sub 1. Again the function that I am working with is x squared minus 3 x, so it
is going to be something squared minus 3 times something and that it is. Close the parenthesis
up. All over x sub 2 minus x sub 1. We labelled x sub 2 as the value of 2. If x sub 2 is 2,
this is again back up here with the green f of x sub 2 and x sub 2 is 2, well that is
what gets plugged into x. Two and two. Now minus x sub 1. Well x sub 1 is labelled as
negative 1. You might be curious as to why I am showing this function notation work...and
x sub 1 is equal to negative 1 so I am going to plug that in. You might be curious as to
why I am showing all this function work when I have the answers in my graph right here
for my two y values. It is because a lot of students get lost in the function notation
and how it gets applied in upper level math. I am just trying to emphasis that I am going
to get these same y values as I have here in the graph. Let's make sure that happens.
2 squared is 4, minus 3 times 2 which is 6. Now minus negative 1 squared is 1, minus 3
times negative 1 is plus 3. I am going to run out of space, I am going to erase this
in a second. 2 minus negative 1 is a plus, so 2 plus 1 is 3. Now I am about to say that
my f of x sub 2, my y sub 2, is negative 2...well there it was right there. I am about to say
my f of x sub 1, or you might think of that as my y sub 1 value, is 4 and there it is
right there. Most PreCalc books are not going to tell you what the y values are. I am just
giving you a preview and make sure you realize I am doing this the right way. So 2 squared
minus 6 is negative 2, minus negative 1 squared is 1, negative 3 times negative 1...again
I am repeating this. So 1 plus 3 is equal to 4. Negative 2 from here minus a positive
4 and we get the final answer of negative 2 minus 4 is negative 6 over 3 which is negative
2. So our average rate of change for this parabola between the x values of negative
1 and positive 2 is negative 2. Again, a PreCalculus book is not going to give you these y values
on a graph. They are expecting you to find these y values on your own. So just be aware
of that. Let's take a look at an average rate of change problem. I am going to just do a
quick sketch and apply it to what could be a real life application and see if we can't
interpret slope in that setting. Let's say we are looking at a car. We have the speed
of a car hoping to explain the miles per gallon, how efficient the car is. Well if your car
is standing still there is 0 miles per gallon. Then as you speed up your mileage increases.
There is a sweet spot where you car is in top gear and you are going just right speed
your car is making it's ultimate top efficiency in miles per gallon. Then it starts to drop
down until... As you speed increases faster and faster, it takes more gas for your car
to move. So maybe your miles per gallon trend for a car, if you were to measure it at 10
miles per hour, 20 mph, 30 mph, 40 mph, and so on would follow this kind of trend. Until
eventually you speed up and going over the speed limit and your mileage starts to drop.
Let's just say that at 10 miles per hour your car is getting 8 miles per gallon. 8 mpg.
And then around 35 miles per hour it reaches a maximum mileage efficiency of let's say
30 miles per gallon. What is the average rate of change between those two points? Let's
see. Again, I didn't make a function up I just sketched this out so we are going to
do f of x sub 2 minus f of x sub 1 over x sub 2 minus x sub 1. No function work given,
I just gave you the x and y values so we are looking at 30 minus 8 over 35 minus 10. We
get a slope of 22 over 25. This comes out to some decimal. Let's see here. 22 divided
by 25 is approximately .88 We need to interpret this. We talk about slope as being rise over
run or Y2-Y1 over X2-X1 which is just another way of measuring/saying rise over run. Rise
over run is a great definition if you are graphing. Well I am looking at a graph, but
I am looking at a real world example. What exactly does this average rate of change of
.88 mean? What is the real definition of slope that we never mention in Algebra up to even
Precalculus that I then have to mention in AP Statistics? There is a real definition
of slope in AP Stats I have to really emphasis because we don't hear it enough in all of
our other math classes. The real definition of slope, or average rate of change is the
average change in y, which in this problem is going to be miles per gallon, per one unit
of change in x. Now here is why we don't mention this definition very much in Algebra 1. The
line's slope is two-fifths, I don't want a kid to go over one unit on the x axis and
trying to guess where .4 is...because that's what 2/5 is a decimal. You don't go up .4
and over 1, and then up .4 and over 1, no you are going to go up 2 and over 5. But as
a word problem, a real life application, that rise over run definition is really weak. What
this .88 means, as far as the average rate of change between these two points estimating
the mileage of a car, what that says is between 10 and 35 mph, the average increase in...the
y value is miles per gallon so...The average increase in miles per gallon, my change in
y, per one unit of measure on the x axis which is speed. Let me say that in one clear sentence.
My mileage will increase and average of .88 miles per gallon for every one mile and hour
faster I drive between 10 mph and 35 mph. So when you are dealing with word problems
like my kids will be at the end of their assignment tonight, when you are looking at these word
problems and applications, you don't want to think of slope as being rise over run or
subtracting y's over subtracting x's. No the real definition of slope is the average change
in y, per one unit of change in x. This is a very very important definition of slope
when you are dealing with real life applications. Thank you very much. Have a great weekend.
BAM!!!