Something I've said many times is that the more difficult or more complicated
mathematics gets, the more practical and powerful it becomes. Such is the case in
equations called exponential functions where the X, you've probably never seen
these before, where the X is in the exponent.
Let's take a look at something like this very simple one y equals 2 to the X.
Let's look at the graph and all the possibilities of plugging in various X
values into this function. Once again let me tell you that since the x is in
the exponent we're going to call it an exponential, an exponential function.
Now, let's see, for instance ,if we plug in a 1, y will equal to the ones right or to
the point (1,2),
right there. If we e put in a 2, y will equal two the tooth or four and you're
thinking maybe this is a linear or linier function but let's put in a couple
more points. 2 to the third starts getting much bigger quicker. Is eight
starting to bend a little bit on you? Two to the fourth is 16 so we look at the
point for 16.
It's getting very high very quickly, isn't it?
Let's go the other way. If we put in a 0, 2 to the 0 power (remember about
exponents! Anything to the 0 power is one.) ok, so we have the point (0,1).
What if we put in negatives? Two the minus 1 power. But remember, that's not a
negative number 2. Two the minus 1 power is the reciprocal; that's what the minus does.
So it's 1 over 2 to the 1 or 1 half. So let's look at the point. Back one or
negative 1 and up a half. Putting a couple more, negative 2 is also going to
give us a non-negative number; it's just going to be a smaller number.
It's 1 over 2 to the twoth or one-fourth.
So we go back two and up a teeny bit, a 4th.
We put in one more. Put in negative 3.
Do you know what we're going to get? We're going to get 1 over, 1 over 8, 1
over 2 to the third. Notice that it's never going to get negative, is it?
It's just going to get very very small and the right side, it will go to
infinity and the left side it will get teeny, teeny
but it'll never really read zero. Now that line, that, it will never reach.
We have a name for it since it, in this case anyway, it can never be negative.
Because it can never reach zero we have a name called an ass,...
an asymptote, (sorry about that) an asymptote when it never quite reaches it.
And that's worthy of remembering because that's going to happen all the time in
these exponential functions.
Now, don't be confused about this never negative idea. For instance, if we look at the same
function but let's add a negative 3 on the end of it.
That's the point where y equals 2 to the X. If we look at what y equals 2 to the X
minus 3 is, what's that going to do to the, the graph of the function. Well. it's
going to lower it
three, isn't it?
So we'll still have an asymptote and it's... some of these values will be
negative.
Ok? It just will be in a lower place.
But the point is for these exponentials, it's going to have an
asymptote somewhere where it will never really reach the value, will never really
reach that line.
Interestingly enough, what if we change the location of the negative 3? or minus 3.
Let's try putting it in the exponent itself.
Well, you know what will happen then. We'll get the same numbers except it's
going to actually be moved to the right.
So I take y equals 2 to the x and I'm going to move these values down three
actually
and put in some much smaller values actually.
yeah
and then graph will still move over but it will still have an asymptote.
Ok? [Just when I think you said the stupidest thing ever you keep talking.]
Well what if we looked at an exponential function where the exponent x is a
negative number?
The new function with the negative exponent is going to look the same as
the normal function except it's going to be a mirror of the normal exponential
graph about the y-axis. Something like this.
Notice it will still have an asymptote,
a number that it will never reach and these two functions occur very often in real
life. The one the normal if you would, y equals 2 to the X we call exponential
growth.
Ok? Many things, living things and hopefully your money grow exponentially.
They don't grow linearly, okay? And if they get smaller
we call it exponential decay. Sometimes hopefully your money doesn't get smaller
exponentially but often many living things, colonies or whatever, get smaller
exponentially as well.
Ok, so be looking for these in many of the word problems we'll be looking at.
That's enough for now.