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>> Okay, so today we want to talk a little bit more about slope that we use
from linear equations, and then about some different ways
that we see slope in the context of this class.
Okay, so let me just remind you: slope, which we usually call m,
is this y2 minus y1 over x2 minus x1.
And this is the change in y divided by the change in x. Okay?
And that's the form that we use when we have two points and we want
to find the slope between the two points.
We have two points, x1, y1, and x2, y2.
Okay. So, now, another form that this slope takes, or another name that it takes,
is called the average rate of change.
We use this word when we're getting, for example,
the slope between two points from a function or graph.
So, and these two points, let me just show you how, what these two points look like.
Usually we call the x value of, so point one, x is equal to a, and I have some function,
let's say f of x. Now, if x is a, then the y value is f of a, just the output that I get
from plugging in a. So that's point one.
Point two, the second point, usually we call x is b and y equals f of b. So if you want
to call these x1, y1 and x2, y2, that's fine with me, too.
Okay? So we have this specific notation when we're working with functions,
and I go back up and I just plug this in.
Then the average rate of change...
...is identical to the slope, which is y2 minus y1 over x2 minus x1.
In this case, this particular case, f of b minus f of a divided by b minus a. Average rate
of change: the change in the y values or the output values divided by the change
in the x values or the input values.
Okay? So let's maybe do an example...
example. Find the average rate of change...
...of f of x is equal to x squared plus 2x, 4 x equal to 1 and x is equal to 2.
Now, I want to use the formulation that I just developed, the average rate of change...
is this f of b minus f of a divided by b minus a. Okay?
And in this particular problem, I have maybe a is 1 and b is 2.
So I get a is equal to 1, b is 2, f of a is f of 1,
which is 1 squared plus 2 times 1, which is 3.
And f of b is equal to f of 2.
2 squared plus 2 times 2, which should be 8.
Okay? So let me find this f of b minus f of a over b minus a is 8 minus 3
over 2 minus 1, which is 5 divided by 1 or 5.
Okay? Now, I want to take a look at what this means graphically, so I want to go ahead
and graph this original function and graph what we're getting for --
or how this average rate of change relates to that graph.
So let me move down a little bit.
I'll get a little bit more space.
So let me put the graph here.
Okay. Something like that.
And I want to go ahead and graph this original parabola, x squared plus 2x.
To do that, you can find, maybe let's find the vertex.
For the vertex, x is negative b over 2a, and this is -- let me write my function over here,
f of x is equal to x squared plus 2x.
So I have b is 2, negative 2 over 2 times 1, and I get negative 1.
And the y value is f of that x value, negative 1,
so I just plug in negative 1 squared plus 2 times negative 1,
and I should get, again, negative 1.
So that point is negative 1, negative 1.
Okay? You can also find x and y intercepts when x is 0.
That's the y intercept.
So that's f of 0, and when I plug that into my function,
0 squared plus 2 times 0 is equal to 0, so that point is 0, 0.
And let's get the y intercept and we'll plot that also, or the x intercepts.
x squared plus 2x is equal to 0, and I get x times x plus 2 equal to 0.
That gives me x is equal to 0, x plus 2 equals 0,
which should lead to x is equal to negative 2.
Okay? So let's go ahead and plot those points here.
So negative 1, negative 1 is here.
The point 0, 0 and the point -- here I get, this is the point negative 2, 0.
Negative 2, 0.
Something like that.
Okay? You also have those other points that you found above.
When a is 1, we got 3; and when b was 2, we got f of b was 8.
So you can put those points in there.
And it looks something like this.
Now, when we found the slope between these -- let me mark these two points that we had.
These are the two points.
This point 1, 3 and 2, 8 are the two points where I found the average rate of change.
And we found it to be 5, right?
So if you go ahead and you count like you would for slope, you should see that when I go up 1,
2, 3, 4, 5 and over 1, I'm at this next point, right?
That's what the slope is.
And that the secant line is a line that comes, is a straight line right here
that passes through these two red points.
That's this secant line here.
So let me move down and we'll find that.
So the line through the two points...
...on a function is called the secant line.
And, again, these are just different names for things that you already know how to do.
You already know how to find the slope of a line.
We're just finding it for specific input and output values.
You already know how to find a line, but here I have the secant line.
So I want to go up, and I want to go ahead and -- we found that the slope was 5, right?
And I had these two points -- the points were 1, 3 and 2 and 8, right?
So think back: this was my a, b; and this is f of a,
f of b. So to find the equation of the secant line...
...for f of x equal to x squared plus 2x,
I use this information containing x equal to 1 and x equal to 2.
I use this information that I want, that I have calculated before.
Okay? So I'm going to go ahead and go through that.
I'm going to use the point slope form of the line, y minus y2 equals m times x minus x2
or x1, whichever one you want to put in there.
y minus, and let's say we'll put in b -- no, that's an x value.
Let's put in the y value, 8.
We know that the slope is 5 and x minus n2.
So I get the equation of the line.
Let's see, y is equal to 5x minus 10, and then I want to add 8 on this side,
5x minus 2 is the equation of the secant line,
and it passes through these particular points on the graph.
Okay? So that's the information about average rates of change, and the next video is
about the difference quotient, which is another form of slope.
So let me know if you have any questions.