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
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
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
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
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
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
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
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
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
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
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.