Linear, Quadratic, and Exponential Models

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Linear, Quadratic, and Exponential Models

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In this video, we're going to play a bit of a game.
I have three different data sets here, and we need to
figure out which of these data sets come from either a
linear, quadratic, or exponential function.
And just as a reminder, linear function is of the form, y is
equal to mx plus b.
A quadratic is of the form, y is equal to ax squared plus
bx, plus c.
And an exponential is of the form, y is equal to some
constant-- let's call it a-- times some
number to the x power.
Now, the way to tell them apart-- and I'm going to
actually show you.
It'll probably be intuitive for you for a linear function,
and maybe even for an exponential.
The quadratic, the technique I'm going to use might not be
intuitive, and I'll actually prove it to you in the next
video, or at least show you an example in the general case of
why it makes sense.
But the way to tell a linear function, is that when you
increment x by a constant amount, y will always change
by a constant amount.
So when you always increase x by 1, y should always increase
by a certain amount.
So just using that as our first guideline, let's see
which of these represent a linear function.
So in all of these examples, as we go from one data point
to the next, we're increasing x by 1.
And it's very important that you pay attention to that,
because if you weren't, you would have to adjust things a
little bit.
But in all of these, we're always increasing x by 1.
So let's look at our change in y's.
When we go from the first data point to the second data
point, our change in y is 60.
When we go to the second to the third, our
change in y is 90.
So that immediately tells us we're not dealing with a
linear function.
When our change in x is 1, our change in y here is 60.
Then it's 90, and as you go here it's like a 135, so this
one is clearly not linear.
Let's look at this one over here.
Our first change in y is 2, then our next
change in y is 0.
Once again, not linear.
As we increment x by a constant amount, our y does
not increase or decrease at a constant amount.
So once again, this one is not linear.
Now let's look over here.
Once again, we're increasing x by 1 on every data point.
And when we to go from this first data point to the second
data point, y decreases by 3.
Then we go from the second to the third, once again, y
decreases by 3.
Third to the fourth, y decreases by 3.
1 to negative 2, once again, decreases by 3.
And we just keep decreasing by 3, so whenever we change x by
1, when we increase x by 1, y goes down by negative 3.
So we actually have a constant slope.
We can say our change in y over change in x is equal to
when x goes up by 1, y goes down by 3.
This is a definition of slope.
This is the m in this equation right here.
Our changes are constant, so this one right here is linear.
Now quadratic, there's an interesting, we could call it
a trick right now, because I haven't really
showed you why it works.
But in a quadratic, your changes in y are not going to
be constant, as was the case in the linear.
But the change in the changes of y will be constant.
Let me write that down.
It probably makes no sense to you right now, but when you
see an example it'll be kind of fun.
Change in change in y constant.
Let me show you what I mean.
So over here, let's look at this first one over here.
So we went from 120 to 180, that was an increase of 60.
180 to 270, that was an increase of 90.
Then we went from 270 to 405, which is an increase of 135.
Now, when I talk about the change in the change of y,
these are the change in y's, what are the change in the
change of y's?
Well, we had a change of 60, then we had a change of 90, so
that was a change in the change of y of 30, or a
difference of the difference, you could view them that way
if you like.
Then we had a change of 90, and then we had a change of a
135, so the change in the change, let's see, that is 45.
So here, the change in the change in y, or the difference
of the difference is not constant.
We grew by 60, and then we grew by 90, which was
30 more than 60.
Then we grew by 135, which is 45 more than 90.
So this is not constant over here.
So this tells us this is not a quadratic function.
Now let's look at this over here.
The change in y-- let's write that down.
This clearly was not linear.
We increased by 2 here, increased by 0 here, increased
by negative 2 here, increased by negative 4 here, increased
by negative 6 there.
The change in y's are definitely not constant.
But let's look at the change in the change of y's.
So we increased by 2, then we increased by 0, so this was a
negative 2 in the change in the change of y, or the
difference of the difference.
Then when you go from 0 to negative 2, we changed by 0,
then we changed by negative 2, that is a change in the change
of negative 2.
Then you go from negative 2 to negative 4, the change in the
change of negative 2.
This one is starting to look quadratic.
We changed by negative 4 than we changed by negative 6, once
again, a change in the change of negative 2.
So this data right here fits our requirement for quadratic,
and I'm just kind of giving it to you as a trick right now.
In the next video I'll actually show
you why this works.
I'll show you an example, just so you can get the gut feeling
of why it should work.
So this one right here is quadratic.
So you might say, hey, Sal has given us three examples, one
of them is going to be linear, one of them's going to be
quadratic, one's going to be exponential, this one's
probably going to be exponential.
And you are probably right.
But let me give you the way that we can figure out if
something is exponential.
So when x increases by 1 in an exponential, you should always
have a constant multiplying factor.
So when you go from one term to the next term, they should
be a similar ratio from one to the next.
So instead of just subtracting-- in this case,
120 from 180, and 180 from 270-- to test for an
exponential function, what you want to do is check whether
the ratio of when you increment x by a constant
amount, whether the ratio of the y's are the same.
So when we increased x by 1, y went from 120 to 180, so it
increased by 18 over 12 is 1.5.
So we went from this term to this term.
Let me clear all of the stuff out here.
So when we went from here to here, this was times 1.5.
And then when we go from 180 to 270, once again, that looks
like we're going times 1.5, right?
1/2 of 180 is 90, add that to 270, so once again, we're
increasing by 1.5.
270 to 405, 1/2 of 270 is 135, that's how much we're
increasing by.
So once again, we're increasing by 1.5.
So every step of this way, we are multiplying the previous
term by 1.5 to get the next.
And that actually tells us that this right here is a 1.5.
But our point here isn't to actually figure out the
formulas, the point here is to classify them.
So now we know definitively that this right here, at least
given the data we have, is an exponential function.
Anyway, hopefully you found that fun.