### Video Transcript

In this video, we’re gonna be
looking at exponential growth and decay. Before watching, you should already
be familiar with linear relationships and linear growth. We’ll also be learning how to
recognise exponential growth and decay in tables of values, equations, and
descriptions. Let’s start by looking at an
example of a constant growth model or a linear function.

Each day, Anna makes two
chairs. To start off with, she’s made no
chairs at all. After one day, she’s made two
chairs. After two days, it’s four. Three days, it’s six, and so
on. This is what we call a constant
growth model. Each time we add one to the number
of days that have passed, we add two to the number of chairs that Anna has made. And whether we pass from day one to
day two or day 500 to day 501, the number of extra chairs will always be two for an
extra day.

Now, we could use the days as the
𝑥-coordinates and the total number of chairs as the 𝑦-coordinates, and we can plot
a graph. And if we join up the points, it
would look something like this. The equation of this line is 𝑦
equals two 𝑥. Or I’m just gonna write it as 𝑦
equals two 𝑥 plus zero for the moment. Now, the multiplier 𝑥 tells us the
slope of that graph. And it means that if I increase the
𝑥-coordinate by one, then the corresponding 𝑦-coordinate would increase by
two. The slope is two. And the plus zero on the end simply
tells us where this line cuts the 𝑦-axis. The fact that it’s zero tells us
that when 𝑥 is zero, 𝑦 equals zero as well.

Now, the fact that this is a
straight line means that we get the same increase in 𝑦 for the same increase in 𝑥,
regardless of whereabouts on the line you look. So whether we’re talking here or
here or here or here, the slope is always the same and the rate of increase is
always the same. Now, the main point to make at this
stage is that this is constant growth or linear growth. This is not exponential growth.

We’re going to look at that
next. Let’s think about a different
situation now.

In a laboratory, a petri dish
contains 500 bacteria and they are breeding. The population grows by 50 percent
every hour. Now, you can see a table of values
showing how many bacteria there are over the first five hours. And we’ve rounded those numbers in
the second row there, the number of bacteria, to the nearest whole number. Looking at the numbers in the
table, you can see that each hour we increase the population of bacteria by 50
percent. Although we’re applying that same
percentage increase every hour, plus 50 percent, because we’ve got more and more
bacteria over time, we’re adding 50 percent of more and more. The hourly increments are getting
bigger. It’s an increasing rate of
increase. And we call this sort of situation
exponential growth.

Now to work out how many bacteria
we’ll have in the next hour, what we have to do is multiply the number of bacteria
we had in the last hour by 1.5. For example, going from hour one to
hour two, we start off with 750 bacteria, while in hour two, we’re gonna have 100
percent of our original bacteria and an extra 50 percent of them. So the amount of bacteria we’re
going to have is 150 percent of 750. So 150 percent means 150 divided by
100. And that is 1.5. So our multiplier for calculating
150 percent of something is 1.5.

So 1.5 is our common ratio. Each hour, we’re multiplying the
number of bacteria by 1.5, to work out how many bacteria we’ll have in the next
hour. That makes this a geometric
sequence. So we’ll start off in hour zero
with 500 bacteria. An hour later, we’ll have 500 times
1.5 bacteria. An hour later, we’ll take the
amount we had after hour one, and we’ll multiply that by 1.5. And an hour later, we’ll multiply
that figure by 1.5 and so on. Well, this means that after one
hour, we’ll have multiplied 500 by 1.5 once, which we can write like this. 500 times 1.5 to the power of
one. After two hours, we’ll have
multiplied by 1.5 twice, which we can write like this. 500 times 1.5 squared. After three hours, it’ll be 500
times 1.5 cubed. And after four hours, it’ll be 500
times 1.5 to the power of four.

Now, this first number here, we
could say that that’s just times one. Well, 1.5 to the power of zero is
just one. So another way of writing that
would be 500 times 1.5 to the power of zero. So the pattern is, after zero
hours, it’s 500 times 1.5 to the power of zero. After one hour, it’s 500 times 1.5
to the power of one. After two hours, it’s 500 times 1.5
to the power of two and so on. Three hours, it’s to the power of
three and four hours, it’s to the power of four.

Now, we can summarise that in a
formula. If we let 𝑥 be the number of hours
that’ve passed, then the number of bacteria that we’ll have in the petri dish will
be 500 times 1.5 to the power of 𝑥. Remember, the 500 was the initial
amount of bacteria, and the 1.5 is our hourly multiplier. Because we were adding 50 percent,
we have to multiply by 1.5. This sort of formula is what we
call exponential growth. And if we plotted that as a graph,
this is what it would look like.

Notice how in the first hour, the
number of bacteria increases by 250. But between the fourth and fifth
hours, although we’re still increasing the population by 50 percent, that’s now an
extra 1266, to the nearest whole number of bacteria. The rate of increase is
increasing. That curve is getting steeper and
steeper. Now by comparing that with the
constant growth rate model, you can see that the curve of the exponential growth is
getting further and further away from constant growth. For this line here, we’re just
adding the extra 250 bacteria every hour, whereas, with this curve here, we’re
adding an extra 50 percent of the population whatever it started off at the
beginning of that hour.

So here’s a general formula for
exponential growth. 𝑦 is equal to the initial amount
that we had times the multiplier to the power, or to the exponent, of 𝑥. That’s where the name “exponential
growth” comes from because it’s the exponent of 𝑥 in the formula. So that exponent 𝑥 tells us how
many times we’re multiplying the initial amount by the multiplier. And it’s also worth noting that
when 𝑥 is equal to zero, the multiplier to the power of zero is one. So we just end up with our initial
amount. Another interesting consequence of
this is that if our multiplier is greater than one, then we get exponential
growth. The 𝑦-values increasingly increase
as 𝑥 gets larger and larger. We’ll be adding the same percentage
of larger amounts for each corresponding increase in 𝑥. So the actual increments themselves
will be larger. The bigger the multiplier, the
quicker the curve rockets upwards as well.

And if the multiplier is equal to
one, then we just get a constant flat straight line. So it doesn’t matter how many times
I multiply two by one, I’m still getting two. So for this equation, I’ve got an
initial amount of two and my multiplier is one. But as soon as my multiplier gets
bigger than one, then I get an exponential growth curve. And the bigger the multiplier, the
faster that curve moves upwards. When the multipliers are larger,
then the 𝑦-coordinates get very, very big very, very quickly. That’s also worth pointing out
though; when the multiplier is between zero and one, we get exponential decay. We’re effectively reducing the
𝑦-coordinate by a fixed percentage every time 𝑥 grows by a certain amount.

It’s also worth noticing that with
exponential decay, we’re decreasing by the same percentage each time. So we’ll get smaller and smaller
decrements. We’ll never quite reach zero if we
start off with a positive number. And we’ll also never get a negative
answer. The 𝑦-coordinate gets closer and
closer to zero. Now, this effect, the curve getting
closer and closer to the 𝑥-axis but never quite reaching it, is called
asymptotic. So here, the 𝑥-axis is an
asymptote. It’s the line that the curve is
getting closer and closer towards but never is quite touching.

Let’s just take a moment to
summarise what we’ve learnt then. The general formula for exponential
growth or decay is in this format. 𝑦 is equal to some initial amount
times a multiplier to the power, or to the exponent of, 𝑥. And the value of the multiplier is
key in determining whether it’s gonna be constant or whether we’re gonna have
exponential growth or decay. If the multiplier is between zero
and one, then we get exponential decay. The 𝑦-coordinate gets closer and
closer to zero without ever actually getting there. If the multiplier is equal to one,
we don’t get exponential growth or decay. We just get a constant value
because we’re just multiplying our initial amount by one lots of times. And if our multiplier is bigger
than one, we get exponential growth. The 𝑦-coordinate gets bigger and
bigger at an increasing rate.

Now, let’s try to recognise some
exponential growth and decay from some tables of values, descriptions, and some
equations.

Here, we’ve got 𝑥-values zero,
one, two, and three. And the corresponding 𝑦-values are
10, 20, 40, and 80. Now the increments as I increase 𝑥
by one are getting bigger. So there is growth and there is
increasing growth. But the important factor here is
that as I increase 𝑥 by one, I always double my 𝑦-coordinate. I’ve got a common ratio of two
between these terms. That makes it a geometric
sequence.

Now, the important thing is that
when 𝑥 is one, I’ve multiplied that initial value of 𝑦 by two once. When 𝑥 is two, I’ve multiplied the
initial value of 𝑦 by two twice. And when 𝑥 is three, I’ve
multiplied the initial value of 𝑦 by two three times. So, I’ve got a formula that looks
like this. 𝑦 equals 10, the initial amount,
times two, that multiplier, to the power of 𝑥. This makes it exponential
growth. The formula is in the exponential
growth format. We have an initial amount of 10,
and a multiplier which is bigger than one. So, our answer is it’s exponential
growth.

Our next example, we’ve got the
same 𝑥-coordinates, zero, one, two, and three. And the corresponding
𝑦-coordinates are 10, 20, 60, and 240. Now the 𝑦-coordinates are getting
bigger. And they’re not getting bigger at a
steady rate; they’re getting bigger at an increasing rate. So it’s not a linear
relationship. Potentially, it could be an
exponential growth. So we’ve got increasingly
increasing growth, but we don’t have that same common ratio. If we’re going from 𝑥 equals zero
to 𝑥 equals one, we’re doubling the 𝑦-coordinate. And between 𝑥 is one and 𝑥 is
two, we’re tripling the 𝑦-coordinate. And between 𝑥 is two and 𝑥 is
three, we’re multiplying that 𝑦-coordinate by four.

So because we don’t have a common
ratio, this format here, 𝑦 equals the initial amount times some multiplier to the
power of 𝑥, we don’t have a common multiplier. So this doesn’t work. It’s not exponential growth.

Now, in this case, we’ve got the
same 𝑥-coordinates, and the 𝑦-coordinates are increasing. But the rate at which the
𝑦-coordinates are increasing is linear; they’re always increasing by the same
amount, 10. So, the 𝑦-coordinates are always
increasing by the same amount. The multipliers are not constant;
they’re going down. So it’s not exponential growth;
it’s linear growth.

Now with this set of data, as our
𝑥-coordinates are increasing by one every time, the 𝑦-coordinates are
decreasing. But they’re decreasing less and
less every time. In fact, each 𝑦-term is half of
the previous 𝑦-term. Now, we can write a general formula
for these numbers in this format. We’ve got 𝑦 is equal to our
initial amount of 32 times the common ratio multiplier 0.5 to the power of 𝑥, or to
the exponent of 𝑥. And that multiplier is between zero
and one. And that combination of factors
tells us we’ve got exponential decay.

A savings scheme pays interest at a
rate of 0.9 percent per year. Does this represent linear or
exponential growth?

Well with an interest rate of 0.9
percent, it’s not a particularly generous saving scheme. But each year, you’ll have 100
percent of what you had last year plus an extra 0.9 percent. So at the end of one year, you’ll
have 100.9 percent of the amount that you had at the end of the previous year. Now remember, 100.9 percent means
100.9 divided by 100, which means that, to work out a percentage, we’ve got a
multiplier of 1.009. And 1.009 is greater than one, so
this is going to be exponential growth.

Let’s just take a look at the
formula. First, we need to define some
variables. Let 𝑥 equals the year number. Let 𝑦 equals the amount in the
savings account, in dollars. And let 𝑎 be the initial amount
that we invested in that account, in dollars. So the amount of savings that we’ll
have in the account, after year 𝑥, will be our initial amount 𝑎 times 1.009, our
multiplier, to the power of 𝑥. And the multiplier is greater than
one. And that means we will have
exponential growth.

Next then, is the exponential
function 𝑦 equals 0.7 times 1.3 to the power of 𝑥 growing or decaying?

Well, the question tells us it’s an
exponential function and it also fits the pattern. Our initial amount is 0.7, and our
multiplier is 1.3. And of course, 1.3 is greater than
one, which tells us we’ve got exponential growth. So the answer to the question is it
is growing. Now that was designed to be a
slightly sneaky question. The initial amount here was between
zero and one. But remember that it’s the value of
the multiplier which is crucial in determining whether it’s exponential growth or
decay. And the multiplier was greater than
one, so that’s why it was exponential growth.

One last question then, 𝑦 equals
six to the power of 𝑥. Does this represent exponential or
linear growth or decay?

Again, this is a little bit of a
sneaky question because that formula isn’t quite in the format that we’re used to,
but we can write it that way. This is the same as 𝑦 equals one
times six to the power of 𝑥. So our initial amount is one, and
our multiplier is greater than one. And it follows the general format
for our exponential equation. So, it ticks all the boxes for
exponential growth.