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Video: Introducing Exponential Growth and Decay

Tim Burnham

Introducing and working with the general format of equations representing exponential growth and decay (y = [initial amount] * [multiplier]^x). We also learn to recognize it in tables of values and graphs and explore various values of the multiplier.

15:59

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 five hundred to day five hundred and one, 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 five hundred bacteria and they are breeding. The population grows by fifty 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 fifty percent. Although we’re applying that same percentage increase every hour, plus fifty percent, because we’ve got more and more bacteria over time, we’re adding fifty 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 one point five. For example, going from hour one to hour two, we start off with seven hundred and fifty bacteria; while in hour two, we’re gonna have a hundred percent of our original bacteria and an extra fifty percent of them. So the amount of bacteria we’re going to have is a hundred and fifty percent of seven hundred and fifty. So a hundred and fifty percent means a hundred and fifty divided by a hundred, and that is one point five. So our multiplier for calculating a hundred and fifty percent of something is one point five. So one point five is our common ratio. Each hour, we’re multiplying the number of bacteria by one point five, 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 five hundred bacteria. An hour later, we’ll have five hundred times one point five bacteria. An hour later, we’ll take the amount we had after hour one, and we’ll multiply that by one point five. And an hour later, we’ll multiply that figure by one point five, and so on. Well, this means that after one hour, we’ll have multiplied five hundred by one point five once, which we can write like this: five hundred times one point five to the power of one. After two hours, we’ll have multiplied by one point five twice, which we can write like this: five hundred times one point five squared. After three hours, it’ll be five hundred times one point five cubed. And after four hours, it’ll be five hundred times one point five to the power of four. Now this first number here, we could say that that’s just times one. Well, one point five to the power of zero is just one. So another way of writing that would be five hundred times one point five to the power of zero. So the pattern is: after zero hours, it’s five hundred times one point five to the power of zero; after one hour, it’s five hundred times one point five to the power of one; after two hours, it’s five hundred times one point five 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 five hundred times one point five to the power of 𝑥. Remember, the five hundred was the initial amount of bacteria, and the one point five is our hourly multiplier. Because we were adding fifty percent, we have to multiply by one point five. 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 two hundred and fifty. But between the fourth and fifth hours, although we’re still increasing the population by fifty percent, that’s now an extra one thousand two hundred and sixty-six, to the nearest whole number, of bacteria. The rate of increase is increasing; hat 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 two hundred and fifty bacteria every hour, whereas with this curve here, we’re adding an extra fifty 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 ten, twenty, forty, and eighty. 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 ten, 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 ten, and a multiplier which is bigger than one. So, our answer is: it’s exponential growth.

The next example. We’ve got the same 𝑥-coordinates: zero, one, two, and three. and the corresponding 𝑦-coordinates are: ten, twenty, sixty, and two hundred and forty. 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: ten. 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 thirty-two times the common ratio multiplier naught point five 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 saving scheme pays interest at a rate of naught point nine percent per year. Does this represent linear or exponential growth? Well with an interest rate of naught point nine percent, it’s not a particularly generous saving scheme. But each year, you’ll have a hundred percent of what you had last year plus an extra naught point nine percent. So at the end of one year, you’ll have a hundred point nine percent of the amount that you had at the end of the previous year. Now remember, a hundred point nine percent means a hundred point nine divided by a hundred, which means that to work out a percentage, we’ve got a multiplier of one point zero zero nine. And one point zero zero nine 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: that 𝑥 equals the year number, that 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 one point zero zero nine, 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 naught point seven times one point three 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 naught point seven, and our multiplier is one point three. And of course, one point three 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.