Hey everyone, it's Justin again, and I've got a pretty big decision to make that could really
use your help with. So I don't mean to brag, but I have just been offered a huge acting gig in an
upcoming movie. I'm talking like 30 scenes I get to film here, and I have been offered two options
for my contract. The production company says that they can either pay me $1,000 for every scene I
film, or they pay me $2 for the first scene and then double that amount for every additional
scene I film. So which contract do you think is better? I mean, $1,000 is a lot of money, and
starting me off at $2 for one scene? That kind of sounds to me like they don't know what I'm worth.
But hey, I figured that math could really help us here. In this next unit of algebra, we are going
to be looking at exponential growth and decay and how it applies to everyday life. Who knows? Maybe
by the end of this, you'll be able to help me figure out my movie contract. Let's get into it.
This unit consists of five major topics, and they're all pretty closely connected.
We'll begin the unit by discussing linear versus exponential functions. Then we'll
study exponential growth through word problems. Once you get comfortable with it, we'll practice
graphing exponential growth. Then we'll switch gears a bit and learn about exponential decay,
before concluding with some practice writing exponential functions. A lot
of the work that we're going to be doing in this unit is going to be building upon what
we've already done in the last unit, when we learned about the properties of exponents. So
it probably wouldn't hurt to keep some of those notes out for this unit. All right,
so that's what we're going to be doing in this unit, but I can already hear you saying,
when are we ever gonna use this in real life, Justin? And boy, do I have an answer for you.
Exponential growth and decay has so many real world applications that I can't even name
all of them in a short video like this. So this time I'm not even gonna bother.
Okay, you're right. Yeah, I probably should tell you some of them. Let's do it. Exponential
functions are used to model anything that changes at a constant percent increase or decrease,
and a lot of things do that. We use exponential growth models to study human population growth,
the growth of mold on food, the spread of fires and diseases, and on a less depressing note,
the growth of money that gains interest over time. Exponential decay models can be used
to examine the depreciation in value of many things houses, to cars, to the value of the
dollar itself. You see, exponential functions are different from linear functions. Linear
functions increase or decrease at a constant rate, defined by their slope over a period of time.
As you know, they're represented by a straight line on a graph. Exponential functions don't
work that way. They grow or decrease at a rate that itself is always increasing or decreasing.
That's why they're represented by a curve, which brings us full circle back to the question
of my movie contract. Which offer should I take? Let's look at it mathematically. The first offer
can be represented by a linear function. I'll make a constant rate of $1,000 for each scene
that I'm in. That can be represented by the function f(x) = 1000x where 1,000 is my slope
or constant rate of change. When I finish all 30 scenes, according to this table, that means I'd
have a total of $30,000. That sounds pretty great. I'm still paying off my credit card from all that
new furniture I bought to go with my tv. But what about that other contract, the one that pays me
$2 for my first scene? That seems really low, but then it doubles every scene after that.
This is an exponential function because the rate of change is growing constantly.
So let's see how that works out for me. If we say g(x) equals two to the x and then we graph it,
we see that on day one I get $2. But on day two that doubles and I get $4. Day three that
doubles again I get eight whole dollars. This is terrible. This is terrible. This is probably
illegal. That's less than minimum wage. All right let's just graph it to day 30 to see how the - oh.
Oh wow, wow, that's up there. Can we
get it? That's a lot of scrolling. That's a lot of scrolling we're doing. Can we put
this in a data set? Could we put this in a data table so we can see it just -
So you're telling me that on day 30 alone, I would make over a billion dollars?
Are these people for real? I mean you add that up to all the other days that I'm filming and I'll
be filthy stinking rich for this one movie. I don't have to film algebra videos anymore!
Although I wonder if the person who wrote this contract ever took high school algebra.
Anyways, as you can see, exponential growth of decay clearly has some real world applications.
Maybe you could think of some things that don't change at a constant rate,
but at an exponential one. As you go through the lessons of this unit,
be sure to check in with the visual reference guide found in your wrap-up lessons for key notes,
formulas, and rules for you to remember. Now if you'll excuse me, I have got to get
this contract signed, scanned, and sent to my agent. See you next time! I'm going to be rich.
Wait a minute. This says that I've got to shave my head? No, no, no. That is unacceptable. That is
unexcusable. Where's my phone? Where's my phone? I've gotta call my agent.
I've gotta call my agent. This can't be happening. This is unacceptable.