- [Voiceover] So let's think about a function.
I'll just give an example.
Let's say, h of n
is equal to
one-fourth
times
two
to the n.
So, first of all,
you might notice something interesting here.
We have the variable, the input into our function.
It's in the exponent.
And a function like this is called an exponential function.
So this is an exponential.
Ex-po-nen-tial.
Exponential function, and that's because
the variable, the input into our function,
is sitting in its definition of
what is the output of that function going to be.
The input is in the exponent.
I could write another exponential function.
I could write,
f of,
let's say the input is a variable, t,
is equal to
is equal to five
times
times three
to the t.
Once again,
this is an exponential function.
Now there's a couple of interesting things to think about
in exponential function.
In fact, we'll explore many of them,
but I'll get a little used to the terminology,
so one thing that you might see
is a notion of
an initial value.
In-i-tial
Intitial value.
And this is essentially the value of the function
when the input is zero.
So, for in these cases,
the initial value for the function, h,
is going to be, h of zero.
And when we evaluate that,
that's going to be one-fourth
times two to the zero.
Well, two to the zero power, is just one.
So it's equal to
one-fourth.
So the initial value, at least in this case,
it seems to just be that number
that sits out here.
We have the initial value times
some number
to this exponent.
And we'll come up with the name for this number.
Well let's see if this was true over here for, f of t.
So, if we look at its intial value,
f of zero
is going to be
five times
three to the zero power
and, the same thing again.
Three to the zero is just one.
Five times one is just five.
So the initial value is once again, that.
So if you have
exponential functions of this form, it makes sense.
Your initial value,
well if you put a zero in for the exponent,
then the number raised
to the exponent is just going to be one,
and you're just going to be left with
that thing that you're multiplying by that.
Hopefully that makes sense,
but since you're looking at it,
hopefully it does make a little bit.
Now, you might be saying,
well what do we call this number?
What do we call that number there?
Or that number there?
And that's called the common ratio.
The
common
common ratio.
And in my brain,
we say well why is it called a common ratio?
Well, if you thought about integer inputs into this,
especially sequential integer inputs into it,
you would see a pattern.
For example,
h of,
let me do this in that green color,
h of zero
is equal to, we already established one-fourth.
Now, what is h of one
going to be equal to?
It's going to be one-fourth times
two to the first power.
So it's going to be one-fourth
times
two.
What is h of two
going to be equal to?
Well, it's going to be
one-fourth
times
two squared,
so it's going to be times two times two.
Or, we could just view this as
this is going to be two times
h of one.
And actually I should have done this
when I wrote this one out,
but this we can write as
two times
h of zero.
So notice, if we were to take the ratio between
h of two
and h of one,
it would be two.
If we were to take the ratio between
h of one
and h of zero,
it would be two.
That is the common ratio between
successive whole number inputs into our function.
So, h of
I could say
h of n
plus one
over
h of n
is going to be equal to
actually I can work it out mathematically.
One-fourth
times two
to the n plus one
over
one-fourth
times
two to the n.
That cancels.
Two to the n plus one, divided by two to the n
is just going to be equal to two.
That is your common ratio.
So for the function h.
For the function f,
our common ratio is
three.
If we were to go the other way around,
if someone said, hey, I have some function
whose initial value,
so let's say,
I have some function, I'll do this in a new color,
I have some function, g,
and we know that its initial
initial
value
is
five.
And someone were to say
its common ratio
its common
ratio
is six,
what would this exponential function look like?
And they're telling you this is an exponential function.
Well, g of
let's say x is the input,
is going to be equal to our
initial value, which is five.
That's not a negative sign there,
Our initial value is five.
I'll write equals to make that clear.
And then times
our common ratio
to the x power.
So once again,
initial value, right over there, that's the five.
And then our common ratio
is
the six,
right over there.
So hopefully that gets you a little bit familiar
with some of the parts
of an exponential function,
why they are called what they are called.