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## Algebra I (2018 edition)

# Exponential decay intro

CCSS.Math:

Both exponential growth and decay functions involve repeated multiplication by a constant factor. However, the difference lies in the size of that factor:
- In an exponential growth function, the factor is greater than 1, so the output will increase (or "grow") over time.
- In an exponential decay function, the factor is between 0 and 1, so the output will decrease (or "decay") over time.

## Want to join the conversation?

- what happens if R is negative?(24 votes)
- I know this is old but if someone else has the same question I will answer. The equation is basically stating r^x meaning r is a base. For exponential problems the base must never be negative. I you were to actually graph it you can see it wont become exponential. just remember NO NEGATIVE BASE!(6 votes)

- For exponential decay, y = 3(1/2)^x but wouldn't 3(2)^-x also be the function for the y because negative exponent formula x^-2 = 1/x^2 ?(7 votes)
- A negative change in x for any funcdtion causes a reflection across the y axis (or a line parallel to the y-axis) which is another good way to show that this is an exponential decay function, if you reflect a growth, it becomes a decay.(7 votes)

- At3:01he tells that you'll asymptote toward the x-axis. What does he mean by that? What's an asymptote?(4 votes)
- An asymptote is an imaginary line your function cannot cross.(4 votes)

- 5:25Actually first thing I thought about was y = 3 * 2 ^ - x, which is actually the same right? Using a negative exponent instead of multiplying by a fraction with an exponent.(5 votes)
- If the common ratio is negative would that be decay still?(2 votes)
- negative common ratios are not dealt with much because they alternate between positives and negatives so fast, you do not even notice it. If you have even a simple common ratio such as (-1)^x, with whole numbers, it goes back and forth between 1 and -1, but you also have fractions in between which form rational exponents. So it has not description. If the initial value is negative, it reflects the exponential function across the y axis ( or some other y = #).(6 votes)

- What is the standard equation for exponential decay?(1 vote)
- For exponential growth, it's generally
`y = Ar^x`

.

For exponential decay, it's`y = Ar^(-x)`

or`y = A(1/r)^x`

.

Did Sal not write out the equations in the video?(7 votes)

- At3:01he tells that you'll asymptote toward the x-axis. What does he mean by that? What's an asymptote?(2 votes)
- Asymptote is a greek word. 'A' meaning negation==NO, Symptote is derived from 'symptosis'== common case/fall/point/meet so ASYMPTOTE means no common points, which means the line does not touch the x or y axis, but it can get as near as possible.(3 votes)

- I'm a little confused.

Sal says that if we have the exponential function y = Ar^x then we're dealing with exponential growth if |r| > 1

But say my function is y = 3 * (-2)^x

Well here |r| is |-2| which is 2. So I should be seeing a growth.

But if I plug in values of x I don't see a growth:

When x = 0 then y = 3 * (-2)^0 = 3

When x = 1 then y = 3 * (-2)^1 = -6

When x = 2 then y = 3 * (-2)^2 = 12

When x = 3 then y = 3 * (-2)^3 = -18

So what I'm actually seeing here is that the output is unbounded and alternates between negative and positive values.

So I suppose my question is, why did Sal say it was when |r| > 1 for growth, and not just r > 1?(1 vote)- It's my understanding that the base of an exponential function is restricted to positive numbers, excluding 1.

I haven't seen all the vids yet, and can't recall if it was ever mentioned, though.

But you have found one very good reason why that restriction would be valid.(4 votes)

- there are some graphs where they don't connect the points. why is this graph continuous? What is the difference of a discrete and continuous exponential graph?(2 votes)
- 6:42shouldn’t it be flipped over vertically? Around the y axis as he says(1 vote)
- I think you may be confusing the X and Y axis. The graph Sal drew is flipped across the Y-Axis.(2 votes)

## Video transcript

- [Narrator] What we're
going to do in this video is quickly review exponential growth and then use that as our platform to introduce ourselves
to exponential decay. So let's review exponential growth. Let's say we have something that, and I'll do this on a table here. Just gonna make that straight. So let's say this is
our x and this is our y. Now let's say when x is
zero, y is equal to three. And every time we increase
x by 1, we double y. So y is gonna go from three to six. If x increases by one
again, so we go to two, we're gonna double y again. And so six times two is 12. This right over here
is exponential growth. And you could even go for negative x's. When x is negative one, well,
if we're going back one in x, we would divide by two. So this is going to be 3/2. 3/2. And notice if you go from
negative one to zero, you once again, you
keep multiplying by two and this will keep on happening. And you can describe
this with an equation. You could say that y is equal to, and sometimes people might
call this your y intercept or your initial value, is equal to three,
essentially what happens when x equals zero, is
equal to three times our common ratio, and our common ratio is, well, what are we
multiplying by every time we increase x by one? So three times our
common ratio two, to the to the x, to the x power. And you can verify that. Pick any of these. When x is equal to two,
it's gonna be three times two squared, which
is three times four, which is indeed equal to 12. And we can see that on a graph. So let me draw a quick
graph right over here. So, I'm having trouble
drawing a straight line. All right, there we go. And let's see. We could go, and they're gonna be on a slightly different
scale, my x and y axes. So this is x axis, y axis. And we go from negative one to one to two. Let's see, we're going
all the way up to 12. So let's see, this is three, six, nine, and let's say this is 12. We could just plot these points here. When x is negative one, y is 3/2. So looks like that, then at y equals zero, x is, when x is zero, y is three. When x equals one, y has doubled. It's now at six. When x is equal to two, y is 12. And you will see this tell-tale curve. And so there's a couple
of key features that we've Well, we've already talked
about several of them, but if you go to increasingly
negative x values, you will asymptote towards the x axis. It'll never quite get to zero as you get to more and
more negative values, but it'll definitely approach it. And as you get to more
and more positive values, it just kind of skyrockets up. We always, we've talked
about in previous videos how this will pass up any linear function or any linear graph eventually. Now, let's compare that
to exponential decay. Exponential, exponential decay. An easy way to think about it, instead of growing every
time you're increasing x, you're going to shrink
by a certain amount. You are going to decay. So let's set up another table here with x and y values. That was really a very, this is supposed to, when I press shift, it should create a straight
line but my computer, I've been eating next to my computer. Maybe there's crumbs in
the keyboard or something. (laughs) All right. So here we go. We have x and we have y. And so let's start with, let's say we start in the same place. So when x is zero, y is 3. But instead of doubling every
time we increase x by one, let's go by half every
time we increase x by one. So when x is equal to
one, we're gonna multiply by 1/2, and so we're gonna get to 3/2. Then when x is equal to two,
we'll multiply by 1/2 again and so we're going to get to 3/4 and so on and so forth. And if we were to go to negative values, when x is equal to negative one, well, to go, if we're going
backwards in x by one, we would divide by 1/2, and so we would get to six. Or going from negative one to zero, as we increase x by one,
once again, we're multiplying we're multiplying by 1/2. And so how would we write
this as an equation? I encourage you to pause the video and see if you can write
it in a similar way. Well, it's gonna look something like this. It's gonna be y is equal to You have your, you could
have your y intercept here, the value of y when x is equal to zero, so it's three times, what's
our common ratio now? Well, every time we increase x by one, we're multiplying by 1/2 so 1/2 and we're gonna
raise that to the x power. And so notice, these
are both exponentials. We have some, you could say
y intercept or initial value, it is being multiplied
by some common ratio to the power x. Some common ratio to the power x. But notice when you're
growing our common ratio and it actually turns
out to be a general idea, when you're growing, your common ratio, the absolute value of your common ratio is going to be greater than one. Let me write it down. So the absolute value of two in this case is greater than one. But when you're shrinking,
the absolute value of it is less than one. And that makes sense, because if the, if you have something
where the absolute value is less than one, like 1/2 or 3/4 or 0.9, every time you multiply it, you're gonna get a lower
and lower and lower value. And you could actually
see that in a graph. Let's graph the same
information right over here. And let me do it in a different color. I'll do it in a blue color. So when x is equal to negative one, y is equal to six. When x is equal to zero,
y is equal to three. When x is equal to one, y is equal to 3/2. When x is equal to two, y is equal to 3/4. And so on and so forth. And notice, because our common
ratios are the reciprocal of each other, that these two graphs look
like they've been flipped over, they look like they've
been flipped horizontally or flipped over the y axis. They're symmetric around that y axis. And what you will see in exponential decay is that things will get smaller
and smaller and smaller, but they'll never quite
exactly get to zero. It'll approach zero. It'll asymptote towards the x axis as x becomes more and more positive. Just as for exponential growth, if x becomes more and more negative, we asymptote towards the x axis. So that's the introduction. I'd use a very specific
example, but in general, if you have an equation of
the form y is equal to A times some common ratio to the x power We could write it like
that, just to make it a little bit clearer. There's a bunch of different
ways that we could write it. This is going to be exponential growth, so if the absolute value
of r is greater than one, then we're dealing with growth, because every time you multiply, every time you increase
x, you're multiplying by more and more r's is
one way to think about it. And if the absolute value
of r is less than one, you're dealing with decay. You're shrinking as x increases. And I'll let you think
about what happens when, what happens when r is equal to one? What are we dealing
with in that situation? And it's a bit of a trick question, because it's actually quite, oh, I'll just tell you. If r is equal to one, well
then, this thing right over here is always going to be equal to one and you boil down to just
the constant equation, y is equal to A, so this would
just be a horizontal line.