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Learn about the general format of equations representing exponential growth or decay and see how various values of the parameters affect the shape and nature of the exponential curve in the graph through a range of carefully explained examples.
In this video, we’ll be looking at and working with exponential graphs. First we’ll talk a little bit about the definition of an exponential function and then we’ll look at the format of an exponential function. Then we’ll see how the different parameters affect the graph of the function. And finally we’ll look at a few typical questions.
Exponential functions are well functions obviously, and they’ve got equations that follow this general format: 𝑓 of 𝑥 or 𝑦 is equal to some constant, say 𝑎, times another constant, say 𝑏, with an exponent of 𝑥, all to the power of 𝑥. So the equations have two different constants; that’s just normal numbers that stay the same whatever the value of 𝑥 and the variable 𝑥 is in the exponent.
For example, 𝑓 of 𝑥 equals two times one point one to the 𝑥, the power, is an exponential function. And in this case our two constants 𝑎 and 𝑏, well 𝑎 is two and 𝑏 is one point one. So whatever value of 𝑥 we take, 𝑎 will always be two and 𝑏 will always be one point one.
Well let’s develop that example. Let’s make a table of values and then use that to plot the graph. I’m gonna work out the value of ten coordinate pairs. So I’m gonna use 𝑥-values negative twenty, negative ten, zero, ten, and twenty, and I’ll work out the corresponding 𝑦-coordinates. So when 𝑥 is negative twenty, the corresponding 𝑦-coordinate is two times one point one to the power of negative twenty. And a power of negative twenty means one over that number. So this is the same as two times one over one point one to the power of twenty. And putting my 𝑦-coordinates to three decimal places, that in this case is nought point two nine seven. So when 𝑥 is negative ten, 𝑦 is two times one point one to the power of negative ten. And again that negative ten in the exponent means that we’ve got one over one point one to the tenth power, which to three decimal places is nought point seven seven one.
Then when 𝑥 is equal to zero, we’ve got 𝑦 is equal to two times one point one to the power of zero, but one point one to the power of zero is just one. So the corresponding 𝑦-coordinate is exactly two. Then when 𝑥 equals ten, 𝑦 is two times one point one to the power of ten. And to three decimal places, that’s five point one eight seven. And then when 𝑥 is twenty, to three decimal places, we get a corresponding 𝑦-coordinate of thirteen point four five five.
So let’s just plot those points then on our axes. When 𝑥 is negative twenty, 𝑦 is nought point two nine seven, which is positive; just above the 𝑥-axis, but it’s very close to the 𝑥-axis and so it’s over there. When 𝑥 is negative ten, then 𝑦 is only is a little bit bigger, nought point seven seven one. When 𝑥 is zero, 𝑦 is two. When 𝑥 is ten, 𝑦 has gone up to over five. And when 𝑥 is twenty, 𝑦 is just nearly thirteen and a half.
So we’ve got a bit of a pattern emerging; it’s a kind of a curve. Oops! it shouldn’t go down below the 𝑥-axis there and it should go kind of up like that, increasing. So in fact let’s have a look at a more accurate plot-plot of that. And that’s the general shape we get when we’ve got an exponential function, where the 𝑎 value is positive and the 𝑏 value is positive and greater than one. We get this kind of increasingly steep curve going to the right, and it’s called exponential growth. As I increase the 𝑥-coordinate, the 𝑦-coordinate increasingly increases.
Now let’s just dwell on a couple of interesting points from the table of values and the graph here. When 𝑥 is negative, then we get negative exponents. So that doesn’t change the sign of the function. It doesn’t change the sign of the 𝑦-coordinate; it just inverts the fraction. So we’ve got one point one to the power of negative twenty means one over one point one to the power of twenty. It’s not negative; it’s just a number that’s very close to zero. So we’re gonna get one over a number which is bigger than one and it’s gonna be multiplied by itself lots and lots of times, so it’s gonna get closer and closer to zero as the 𝑥-coordinate gets correspondingly more and more negative as we move to the left on our graph.
And when 𝑥 is equal to zero, the exponent or power will also be zero. And that means that as long as the 𝑏 isn’t zero, which it isn’t here, then this part of the expression, one point one to the power of 𝑥 or 𝑏 to the power of 𝑥, becomes 𝑏 to the power of zero, which is always gonna be one.
And we then go on to multiply that one by the 𝑎 constant, in this case two. Now remember that the curve will cut the 𝑦-axis when actually 𝑥 equals zero. So these exponential curves with these sorts of values of 𝑎 and 𝑏 will always cut the 𝑦-axis at the value of 𝑎, which is two in this case.
And as we said, when 𝑥 gets larger and more positive, then we’ve got a positive 𝑏 value, which is bigger than one, being multiplied by itself more and more times. And the 𝑦-coordinates rapidly increase at an increasingly increasing rate, and you probably heard people describe this shape as increasing exponentially. And that’s literally true; it’s exponential growth.
Now I’ve plotted a few slight variations on the curve that we’ve just been looking at. So they’ve all got the same 𝑎 value of two, and you’ll notice that they all cut the 𝑦-axis at two. And the 𝑏 values are all slightly different. So there’s the one point one we had to start off with. When I decrease that slightly, so that’s still bigger than one so it’s still gonna keep getting bigger as you multiply it by itself. And that curve shoots away from the 𝑥-axis a bit more slowly; the 𝑦-coordinates’ increase are still increasingly increasing, but they’re more slowly than when the 𝑏 value was just a little bit higher. If I increase that 𝑏 value to one point one five, then you will see that the the curve rockets up even more quickly.
Let’s have a look at 𝑓 of 𝑥 is equal to five times one point one to the negative 𝑥 this time and see what that graph looks like. So I’m gonna take the same five 𝑥-values and work out the corresponding 𝑦-coordinates. So now when 𝑥 is negative twenty, I’ve got five times one point one to the negative of negative twenty, which is positive twenty. And to three decimal places, that gives me a 𝑦-coordinate of thirty-three point six three seven.
And when 𝑥 is negative ten, we’ve got 𝑦 is equal to five times one point one to the power of negative of negative ten. So it’s positive ten. And to three decimal places, that’s twelve point nine six nine.
And when 𝑥 is zero, we’ve got 𝑦 is equal to five times one point one to the power of negative zero. That’s the same as the power of zero. Anything to the power of zero, as long as not zero already, is one. So that’s five times one, which is five. And when our 𝑥-coordinates are positive, this sign in our formula is turning those into negative values here when we evaluate them. So we’re getting these kind of one over values; and so it’s five times one over one point one to the power of ten and five times one over one point one to the power of twenty. We’ve got to three decimal places one point nine two eight and nought point seven four three.
Let me plot that. This is what it looks like. And now for the negative 𝑥-values over here, the function uses an exponent which is the negative of that negative 𝑥-value so it’s positive. And the increasingly rapid increase now happens to the left of the 𝑦-axis. And when 𝑥 is zero, the negative of zero is still zero. So we still have 𝑏 to the power of zero, which is one. So it cuts the 𝑦-axis at the 𝑎 value times one, and 𝑎 is five in our case.
And then as 𝑥 increases positively, we end up with the negative of those positive amounts or we’re ending up with increasingly large powers of one over 𝑏. So the 𝑦-coordinates are getting smaller and smaller and smaller closer and closer to zero, and that curve’s getting closer to the 𝑥-axis without touching or crossing it.
And again, your plot has some slight variations on there. And we can see that regardless of the value of 𝑏, if the 𝑎 value is the same, we’re still crossing the 𝑦-axis at 𝑎. And we can see that small variations in 𝑏 affect the rate of climb of that curve, either side of the axis, quite markedly. And in this case this shape of the curve we call “exponential decay.” This rapid dropping-off of the 𝑦-coordinates as the 𝑥-values get larger is called “exponential decay.”
Now the next example: I’ve already worked out the values here for us, and we can see that the only difference between this and the first example apart of the numbers themselves is that the 𝑎 value is negative. We’ve got negative three times one point one to the power of 𝑥. And we can see that when 𝑥 is large and negative, we’re still getting this one over a large number, so that’s making the number very close to zero. But because the 𝑎 value is negative, it’s gonna be the negative of that number. So this whole thing is reflected in the 𝑥-axis. And the pattern tells us it’s a number which is greater than one to the power of positive of 𝑥. That’s gonna be exponential growth. But because we’re multiplying all those inex- exponential growth values by a negative number, the whole graph gets reflected to look like this.
You notice our graph still cuts the 𝑦-axis here at the value of 𝑎, which in this case is negative three. And again I’ve done some slight variations on that so that you can see that all three of these curves cut the 𝑦-axis at negative three, which is the same as the 𝑎 value here. And the slight differences in the 𝑏 value here are quite drastically affecting the rate of departure of the curve from the 𝑥-axis.
Now before we look too closely at this one, we need to think very carefully about what it actually means. Does it mean minus one times one point one to the power of negative 𝑥 or does it just mean one lot of negative one point one to the power of negative 𝑥?
Well if you think about the order of operations, because there are no parentheses or brackets in that expression, then we’re going to do this bit here, the exponent of the power, before we start dealing with this negative sign here. So it’s definitely this and it’s definitely not this. And when we do the table of values, that negative sign’s coming into play for 𝑎 and that negative sign’s coming into play for the exponent. And when we look at the coordinates, we can see that when 𝑥 is zero, this cuts the 𝑦-axis at negative one. So we have to think of this as our 𝑎 value, okay? negative one. And our 𝑏 value is just one point one.
So this is what the graph looks like. And again we can see that it’s cut the 𝑦-axis at 𝑎 which is negative one in this case. And that negative exponent has given us exponential decay, but of course the whole thing being reflected in the 𝑥-axis because we’ve got this negative sign in front of the whole thing.
Okay, just a few more quick examples before we move on to some questions. Here is the graph of 𝑦 equals nought point four to the power of 𝑥. In terms of interpretation, it’s probably worth writing that out as one times nought point four to the power of 𝑥. That means we can interpret this as 𝑎 equals one and 𝑏 equals nought point four or four-tenths and 𝑏 is between zero and one. And when I start multiplying a number between zero and one by itself, it’s gonna get smaller and smaller and closer to zero. So that’s what we can see here: rather than the exponential growth that we would expect from having this positive power of 𝑥, because the 𝑏 value is between zero and one, we’re actually getting an exponential decay graph. But again we can see whatever the value of 𝑎, that’s where it’s cutting the 𝑦-axis here.
And we can see this bit better on this scale with this particular graph. But when 𝑥 is equal to one, we’ve got 𝑦 is equal to nought point four to the power of one. Well that’s just nought point four. So when 𝑥 equals one, 𝑦 is equal to nought point four; that’s our 𝑏 value. So when 𝑥 equals zero, 𝑦 is equal to the 𝑎 value; and when 𝑥 is equal to one, 𝑦 is equal to the 𝑏 value.
And here’s the graph of 𝑦 equals nought point four to the negative 𝑥. So for this function, we’ve got the same value of 𝑎 that we just looked at and the same value of 𝑏 that we just looked at. The only difference is this negative exponent up here; we’re taking the negative of the 𝑥 exponent.
Still, it cuts the 𝑦-axis at the value of 𝑎. But now when 𝑥 equals one, then 𝑦 equals nought point four to the power of negative one. And remember, nought point four is four-tenths, which is two-fifths. So 𝑦 is equal to two-fifths to the power of negative one. And that power of negative one means we flip the fraction; it becomes five over two instead two over five. And of course five over two is two and a half And it’s just gone off our scale here. But when 𝑥 is one, we can see that 𝑦 is two and a half. Now if 𝑥 was negative one in this case, then the corresponding 𝑦-coordinate will be nought point four to the power of negative negative one. But the negative negative one is positive one, so 𝑦 would be nought point four to the power of one, which of course is just nought point four, which we can see on the graph there.
Now let’s just quickly look at 𝑦 equals negative nought point four to the power of 𝑥 and 𝑦 equals negative nought point four to the power of negative 𝑥.
Looking at the first one then, writing it in a slightly different way: 𝑦 equals negative nought point four to the power of 𝑥; that’s the same as 𝑦 is negative one times nought point four to the power of 𝑥. So we can see that the 𝑎 value is negative one and yes, it does cut the 𝑦-axis at negative one.
And for the other one, 𝑦 equals negative nought point four to the power of negative 𝑥; that’s the same as 𝑦 is negative one times nought point four to the negative 𝑥. Again, 𝑎 is negative one, so it cuts the 𝑦-axis at negative one.
And then in the first case, as 𝑥 increases and becomes more and more positive, we’ve got a number 𝑏 between nought and one. So four-tenths, which is multiplied by itself lots and lots of times. Now because that’s less than one and greater than zero, it’s gonna get smaller and smaller. It’s gonna get closer to zero. And that’s what happens: the 𝑦-coordinates get closer and closer to zero, and that curve approaches the 𝑥-axis.
But on the other graph, as 𝑥 gets larger and positive, we’ve got a number between zero and one. But this negative exponent here is gonna flip that number. So it’s gonna make it greater than one and we’re gonna be multiplying it by itself lots of times. So we’re gonna get larger and larger numbers which are gonna move away from the 𝑥-axis. Of course when I say larger and larger numbers, I do mean a larger and larger magnitude. They’re becoming more and more negative in fact.
And just quickly checking another couple of coordinates on the curves, when 𝑥 is one on the left-hand side here, then 𝑦 is negative one times nought point four to the power of one. So that’s negative nought point four.
But for the other curve with that negative exponent, when 𝑥 equals negative one, then 𝑦 is equal to negative one times nought point four to the negative of negative one, which is positive one. So that’s just negative one times nought point four negative nought point four.
Well that’s giving you a good overview of exponential graphs and it’s giving you a few hints and tips and techniques to answer some questions. We’ll actually take a deeper look at some questions in another video.
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