Today we're going to start talking about exponential functions.
So, in order to introduce this, I have a short video.
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- This is an old story, but it reminds us of the surprises we can get
when even a small number like two is multiplied by itself many times.
King Shahram of India was so pleased when his Grand Vizier Sissa Ben Dahir
presented him with the game of chess that he asked Ben Dahir to name his own reward.
The request was so modest that the happy king immediately complied.
What the grand vizier had asked was this, that one grain of wheat be placed on
the first square of the chess board, two grains on the second square,
four on the third, eight on the fourth, 16 on the fifth square, and so on.
Doubling the amount of wheat on each succeeding square until all 64 squares
were accounted for.
When the king's steward had gotten to the 17th square,
the table was well filled.
By the 26th square, the chamber held considerable wheat and a nervous king
ordered the steward to speed up the count.
When 42 squares were accounted for, the palace itself was swamped.
Now fit to be tied, King Shahram learns from the court mathematician
that had the process continued, the wheat required would have covered
all India to a depth of over 50 feet.
Incidentally, laying this many grains of wheat end to end also does something
rather spectacular.
They would stretch from the Earth, beyond the sun, past the orbits
of the planets, far out across the galaxy, to the star Alpha Centauri,
four light-years away.
They would then stretch back to Earth, back to Alpha Centauri,
and back to the Earth again.
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- So, what was going on in that video?
You can see that, rather than going up by the same amount for each square
that they advanced on the chess board, that they went up by a doubled amount.
So, that's a different type of function than we've been looking at.
Rather than going up by one and then another one and another one,
they went up by one and then they doubled and doubled again
and doubled again.
So, we cannot represent that with the type of linear function that
we've been looking at before.
In a linear function, such as this one, if I went up by one square,
I would increase here by 1.5.
So, if you think about how this plays out in a table, for each one unit increase
in t, I am increasing by the same amount.
Now, if I were to look at the percentage increase, therefore, I would actually
indicate that I'm changing the percent that I'm going up by each time.
So, going up by 1.5 from 2 is different than going up by 1.5 from 6.5.
So, if you can see that I've done these calculations already, going from 2 to 3.5
is an increase of the amount of 1.5, but it's actually a 75% increase.
But by the time I get up to these values, going from eight to 9.5,
that's only a 19% increase.
So, when I have a linear function, what you'll see is that you have a constant
amount of increase but you do not have a constant percentage of increase.
Now, what's going on in an exponential function and what was happening in
that video, is that for every one unit increase in t,
we will see a different constant.
So, in this case, if I use my exponential equation and I go up by one every time,
you can see that I go up by, first, just one.
Then I go up by 1.5 and then 2.25 and then 3.36 and then 5.06.
So, I'm actually not increasing by the same amount each time.
That's because rather than adding a value each time,
I'm multiplying an additional time.
So, here I'm multiplying by 1.5 an additional time each time I increase.
Well, because I'm multiplying by a constant amount, what that actually
does end up doing is increasing by a constant percentage.
So, going from 2 to 3 increases by 50%.
But also going from 10.13 to 15.19, I'm increasingly by 50%.
So, in an exponential function, I increase by a constant percentage,
but differing amounts.
Let's look at this with what we call a decay function.
So, this is also an exponential function, but one that's decreasing.
So, in this case, in my decay function, now you can see that I'm multiplying by
a decimal repeatedly.
Well, if I repeatedly multiply by a decimal, I should be getting
smaller and smaller values.
So, here, I decreased by one but then I decreased by half, by .25, by .125,
and so on.
What that means is is I'm basically slowing down my growth over time,
but I'm decreasing still by a constant percentage.
So, here, I decreased by 50% each time.
Now, if I just had one that I was multiplying by every time,
then I would actually have no change in percentage because I would just
have a constant amount because multiplying by one is the same every time.
So, what is an exponential function?
An exponential function can be one that is recognized by a constant
percentage rate of change.
Remember, this is as opposed to our linear functions which had a constant
amount of change.
So, rather than changing by one each time or by five each time,
maybe I'm changing by 5% each time. And those are different.
An exponential growth model is one that has a positive percent rate of change,
while a decay model is one that has a negative percent rate of change.
So, a exponential growth model will be increasing over time,
and a decay model will be decreasing over time.
So, here's the general form of our equation.
You're going to see I use t in my general form and that's because, essentially,
we'll always be looking at models with time as our independent variable.
Here we have some new parameters we're looking at, as well.
Because we're talking about time, I'm going to call a,
which is my y-intercept, my initial value. And that's because with time,
when time is zero, I get my y-intercept, so I get a.
A is therefore my initial value.
The growth factor is, what am I multiplying by each time?
And when I know what I'm multiplying by each time,
I can get to what is more important to me,
which is my growth rate, or what is the percentage that I'm increasing
or decreasing by each time.
So, my growth rate is just related to that growth factor and can be calculated
from it.
So, if I have my growth factor,
I can just subtract one and I'll get my growth rate.
Now, this r is not the same as our correlation r,
so don't let that fool you.
So, here's our definitions.
Our growth factor, or b, is the amount that we multiply by each y-value
to get the new y-value.
Our growth rate is the percentage increase or decrease that we get
for each one unit increase in x.
And these are the two things that we can get from our b.
The one that we're going to be interested in interpreting primarily is
our growth rate when we do our parameter estimates.
So, here are three examples of exponential models.
I have a growth, a stagnant, and a decay model.
One thing to note is that none of these have negative values possible.
In an exponential function, we can actually never cross that axis.
So, we can never have a negative value.
Notice that these are all the same concavity, so they are all trending
that same direction because we cannot cross this axis.
And, in fact, in a decay model, we will continuously get closer
and closer and closer to zero, but we will never actually get to zero.
So, let's work with some general functions.
I want to know what the growth factor would be if I said that water usage is
increasing by 5% per year.
So, what am I giving you here?
Well, I'm giving you the growth rate, that percentage of change.
And if I want to convert this to b, remember that b equals 1 plus r.
And that is r written as a decimal.
So, if I write my r as a decimal, I would have .05,
which tells me that my b here, my growth factor, would be 1.05.
Now, you try.
So, in this case, I've given you a decay model.
If I have decay, where I know that it's shrinking by 78%,
that actually means that I have a negative r value.
So, here I would end up with a b of 0.22.
When I have a decay model, my r can be negative.
So, I will expect to see a negative rate, because I am changing
in a downward trajectory.
Now, remember I can never actually have a negative value in my chart,
so that means that b cannot be negative, even though r can.
Well, why can't b be negative?
Well, think about if I had a chart here and I had a negative b value.
Well, when I add b to the first, I'd have a negative result,
but then when I square b, I get a positive,
and when I cubed it, I'd have a negative,
and then when I had to the fourth, I'd be up here.
So, I'd end up with some kind of function like that.
Well, that is not an exponential model. So, I cannot have a negative b.
When I have a negative r, for a decay model, what that will do is
give me a b value that is between zero and one.
So, I will have a b value that is less than one but more than zero
when I have a decay model.
If I have a b value over one, then I have a growth model.
So, here is another example of an exponential function.
And I've gone ahead and drawn it out.
You can see here that from my equation, I have 34.3 as my initial value.
So, if I were to sketch this or put this into technology and get it,
you would see that that's my initial value, my starting point.
My growth factor for this problem would be 0.63,
indicating I have a decay model because that b is less than one.
If I calculate my growth rate, then, I can have my b, 0.63,
as equal to 1 plus r.
And when I subtract one, I get a decay of negative 37%.
So, now you try it. Identify the growth rate in this equation.
In this equation, you can see that our growth rate is very small, at 0.2%.
What else do you notice about this chart?
Well, you should notice that it kind of looks like a line,
and not like that exponential curve that we're so used to seeing.
When you have a very small growth rate, one that is very close to zero,
you will actually see a model that doesn't have a very striking curve to it.
It'll almost like linear.
Now, if I zoomed this out to a thousand points, you'd probably begin
to be able to determine the curve.
But for very small r values, we'll have a pretty flat curve.
The larger your r value is, the more significant your curve is,
either going up or down.
So, if you were comparing two models drawn on the same chart,
you could see that the more curving one should have a higher absolute value of r,
because it's changing at a faster rate.
Alright. As always, bring us your questions to class.