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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.

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