Let's look at a new type of function.
An exponential function is a function that
can be written in this form: f of x equals a
to the x where a is a positive real number,
which means it's greater than 0, but it can't be 1.
The domain of this function will be the set of all real numbers.
What makes this different from what you're used to?
Mostly it's the placement of the variable.
Notice how the variable is in the exponent
now, and you have a real number for the base.
This is different from the algebraic
functions that we've studied so far.
We have a couple of restrictions on what that base can be.
Why are they there? Well, we don't want a to equal 1 because
that would be the constant function 1 to the
x, which is kind of boring, and it's also linear.
Okay we've already studied linear functions.
Why do we want a to be greater than 0?
Well, what if it was negative 4? Can you raise negative 4 to the 1/2 power?
Not if you want a real number answer, so in
order to have the domain be the set of all
real numbers, we want to restrict our base, a, to be something bigger than 0 and to not
be linear, we don't want a to be 1.
Let's look at a specific exponential function.
Here we have the exponential function: f of x equals 2 to the x.
To get used to the idea of an exponential
function and how they work, let's find some specific function values.
First, we want to find f of 0. So, we want to find the value of this
function when x is 0. We need to find 2 to the 0 power which is 1,
so f of 0 is 1. The next one: when x is 1,
what's the value of the function?
So, we need to find 2 to the first power which is 2.
f of 3: we want to find 2 to the third power.
So, you can see as we go along that our
base, 2, doesn't change, but our exponent
changes according to what x is.
Now let's try something with a negative number.
Here it asks us to find the value of 2 to the x when x is negative 1.
That�s 2 to the negative 1 power, and you'll have to
remember some rules of exponents.
We can make the exponent positive by writing
this as 1 over 2 to the first power or 1/2.
If x is negative 2, we have to find 2 to the negative 2.
That's 1 over 2 squared or 1/4.
Now what about these two? Well, we start out the same way.
We want to find the value of the function when x is the square root of 3.
So, we need 2 to the square root of 3 power.
We can't do that in our head, but we can use
a calculator to approximate the value.
What you want to look for on your calculator
is a button that will have an x to the y or perhaps the carot.
Those are the two most common ways to find this on your calculator.
We're going to start by entering the 2 and
then raise it to the square root of 3 power.
We'll round off to three decimal places, so
this is approximately 3.321, and actually we
want to round that off at the end, so let's try 3.322.
We have one more to try. f of negative 1.2 is 2 to the negative 1.2 power.
Be careful to make sure you don't leave off
the negative sign in the exponent,
and then round off to three decimal places,
so this will be approximately 0.435.
We have all these function values, so
let's try to plot the graph, plot some
points and draw a graph for this function.
We're going to use the ones that are easiest
to plot, so we know from this one that when
x is 0, the value of the function is 1.
We know here that when x is 1, the y value, the function value is 2.
When x is 3, the function value is 8.
If you think about it for a moment, you know
that the bigger you take x to be, then
that's a bigger power on 2, so this is going
to continue to get bigger and bigger without bound.
So, we can finish this side of the graph like this.
What about on the other side? Well, we have a couple of points
that will help us. When x is negative 1, the function value is 1/2.
When x is negative 2, the function value is 1/4, so
it's getting smaller and smaller, closer and closer to 0.
If you keep going, will it ever cross the x-axis?
No, because it'll get smaller and smaller,
but it's always going to end up being a positive number.
So, we can draw the rest of the graph in
this manner, showing that it gets closer and
closer to the x-axis without ever touching or crossing it.
In this case, we call the x-axis a
horizontal asymptote for the graph.
We'll return to this type of exponential function in a moment.
Here's one more exponential function to explore.
Notice, it looks a little different.
Here the base is a fraction between 0 and 1.
Let's start out the same way and find some specific function values.
f of 0 is the value of the function when x is 0.
This gives me 1/2 to the 0 power, which is 1.
f of 1 is going to be 1/2 to the first power, that's 1/2.
f of 3: find the value of the function when
x is 3, that's 1/2 to the third power or 1/8.
With the negative values of x, you're just
going to have to remember those rules of exponents again.
You want to find 1/2 to the negative 1, this is the
same as 2 to the first or 2. For f of negative 2, we need to find 1/2 to the
negative 2 power, which is the same as 2 to the second power or 4.
Let's go ahead and plot the points and see what this graph looks like.
Our first point: when x is 0, y is 1.
This one gives me when x is 1,the y value is 1/2.
When x is 3, the y value is 1/8.
So, again we see that behavior we saw before, just on the other side.
The larger we choose x to be, the closer to 0 the function value will get.
But, just as before, it will not ever equal
zero or become negative, so we have another horizontal asymptote.
Now let's go the other way. When x is negative 1, the value of the function is 2.
When x is negative 2, the value of the function is 4.
You can imagine what's going to happen
if you keep picking negative 3, negative 4, negative 5 as your x
value, you're going to get a larger and larger value of y.
So you can see that depending on what the
value of your base is, you can have a slightly different looking graph.
Let's explore this in general a little bit more.
Let's take a closer look at the graphs of exponential functions.
Depending on the value of the base, a, there
are two different cases that will come up.
If the base, a, is greater than 1, then the graph will have this shape.
Notice that it goes through the point (0,1),
the point (1,a), and then the point (-1,1/a).
The second case is when a is between 0 and
1, and the graph is a little bit different. It'll still have the point (0,1).
It'll go through the point (1,a), but a is a
fraction between 0 and 1 or a decimal, and
that's going to be something very small, very close to the x-axis.
It's also going to have the point (-1,1/a),
but again a is a very small number, and
its reciprocal will be a very large number.
So, the graph of f of x equals a to the x
has one of these two shapes, depending on a.
Let's write down some of these characteristics.
First of all, we want to know the domain of the function.
Well, that doesn't depend on a: the domain
is all real numbers for each of these. What about the range?
The range is the set of allowed y values for
the function, and in both of these cases, y can be anything greater than 0.
So, I'll write that in interval notation: y can be anything greater than 0.
Both of the graphs have a y-intercept at 1.
In fact, neither of them has an
x-intercept because neither of them ever touch the x-axis or cross it.
Because neither of them touch or cross the
x-axis and we've already discussed how the
graph gets closer and closer to the x-axis
without touching or crossing and we call
that an asymptote, both of them have a horizontal asymptote.
The asymptote is the x-axis, the
equation of the x-axis is y equals 0.
One difference that you can see is that for
a, greater than 1, you get an increasing function: it rises as you go
from left to right. In the case of a being between 0 and 1, you
have a decreasing function that falls from left to right.
The last characteristic is these are
both, in either case, a one-to-one function,
and that's an important observation to make as well.
Now let's look at using transformations to graph exponential functions.
Now that we've seen how to graph basic
exponential functions, let's use this to
graph more complicated exponential
functions using transformations.
So, our instructions here are the sketch the
graph using transformations, to state the
domain, the range, and the asymptote.
Alright, so the first thing we want to do is
find the basic exponential involved, and that's 3 to the x.
Let's graph that first. The graph of an exponential function has to
contain the point (0,1). When x is 0, 3 to the 0 is 1.
It's also going to have to contain the point
(1,3) because, when x is 1, 3 to the 1 is 3.
Keeping in mind the basic shape, we get a graph that looks like this.
So, as we said earlier, every exponential
function contains the point (0,1) and the
point 1 and the base, which happens to be 3 in this case.
Now we're going to look at our transformation: we're subtracting 2
from that exponential function. What does that do to the graph?
Right, it's going to move the whole thing down 2 units.
Okay, but we need to be careful because, as
soon as we start moving this down, the
asymptote has to be moved down first.
Okay, so right now it's at y equals 0.
I'm going to move that down 2 units,
that's going to give me a new asymptote that
I'm going to represent with a dotted line.
Then I can move each point on the graph down 2 units.
So, that's going to put this one at (0,-1),
and this one moved down 2, we'll put it at (1,1).
I haven't changed anything about the shape
of the graph, I just slid it down.
Now we've already basically said what the
asymptote is because we moved it down 2 units: it's y equals negative 2,
and it is a horizontal asymptote.
The domain: it's still all real numbers
because this is an exponential function,
we haven't changed that. What about its range?
Well, that will be different because when we
slid it down 2 units, we've allowed for some
smaller y values than we had before, they can be anything bigger than negative 2.
So, write that in interval notation. Let's try one more.
The next example shows us some other
transformations and how they work with exponential functions.
Again, the first thing I want to do is
decide which basic exponential function is
involved, and we're going to start with 2 to the x.
I'll sketch the graph of this one. It has to have the point (0,1).
It has to have the point 1 and the base,
that's 2, so we're going to have the point (1,2).
Keeping in mind that the basic shape for an
exponential function's graph: when the base
is bigger than 1, we can draw this.
Now we're ready to think about those transformations.
The first thing I see is the negative sign
in front of the 2. What does that do to my graph?
Right, it's a reflection about the x-axis.
What about the plus 1 that's in the exponent?
Right, that is a horizontal shift. Which way, left or right?
Okay, so it's going to be shifted 1 unit to the left.
Let's just make a note of that.
We're going to go 1 to the left and reflect across the x-axis.
If you're very careful, I think you can
get this graph in one step. You know exactly what it should look like.
We want to move this point to the left one
unit, but then we're going to flip it over. So, where will It end up?
Right, exactly on the other side of the x-axis.
Okay, let's try this one: we're going to go
to the left 1 and then reflect this across
the x-axis, so where does this point end up? Right, on the y-axis at -2.
The shape of the graph doesn't change, we've
just flipped it over and slid it to the left.
Okay, so now we need to think about the
domain, the range, and the equation of the asymptote.
Let's do the asymptote first. Notice that it gets closer and closer to the
x-axis without touching it. It's below, but that's okay, it's still an
asymptote at the x-axis, and the equation of the x-axis is y equals 0.
The domain: it's still all real numbers. What about the range?
Since it's the values y is allowed to have,
it can be anything that is smaller than 0.
So, in interval notation, it looks like this.
We can also solve equations involving exponential functions.
Let's look at a couple. Notice that it's nothing like an equation
you've solved before because the variable
that you're looking for is in the exponent.
So, the idea to solving these is to rewrite
so that both sides have the same base.
If you look at the left hand side, this is 5 to the 1 minus 2 x.
So, now look at the right hand side. Is 125 a power of 5?
Yes, it is, it's 5 to the third, so we can
rewrite this as 5 to the 1 minus 2 x equals
5 to the third. Exponential functions are
one-to-one functions so that's important to
know because it says that if 5 to this power
equals 5 to the third, the only way that can
happen is if the powers are in fact equal.
Now we can solve this linear equation.
Subtract 1 from both sides, divide by negative 2, we find that x is negative 1.
I won't worry about checking this one
since the domain of an exponential function is all real numbers.
As long as I don't make a careless mistake, my answer should be okay.
Let's try the second example the same way.
I would like for both sides to be written as
powers of the same number, and right away I
can see that 4 is 2 squared and 8 is 2 to the third, so let's use that.
So 4 becomes 2 squared and then that's raised to the x squared power.
Eight is 2 to the third which is raised to the x power.
Remember my rules of exponents: when I raise
a power to a power, I need to multiply those powers.
So, this'll be 2 to the 2 x squared equals 2
to the 3x, same property of exponents on the other side.
Now, because exponential functions are one
to one, the only way that 2 to this power
equals 2 to this other power is if those powers are exactly the same.
So, I have 2 x squared has to equal 3 x for
this to be true, and I want to solve this.
What kind of equation is it? It's quadratic, and we know
how to solve those. We're going to put it in standard form which
means set it equal to 0. How can I solve that?
We're going to try to factor it, and in this
case, I see that there's a common factor of x, so let's take it out.
So, I get two solutions: x equals 0 and
then, if 2 x minus 3 equals 0, by adding 3
to both sides and dividing by 2, I get that x would have to be 3/2.
Now let's look at one very special exponential function.
Let's look at this expression: 1 plus 1 over n
raised to the n power. What I'd like to know is what happens to
this expression as n gets larger and larger.
So, let's take a couple of values.
If n equals 1, then I get 1 plus 1 over 1 to
the first power, so this is 2 to the first, which of course is 2.
What if n is 5: 1 plus 1/5 to the fifth power.
We can use a calculator to approximate this.
Okay, let's try n equals 10, and let's just keep five decimal places
to make this easier to write down.
This'll be 2.59374 to five decimal places.
Let's try one more: n equals 100.
Okay, so what you can see by examining these
numbers is that the value of the expression is
getting bigger, the bigger n gets,
but it's not getting as big as fast.
In fact, if we were to keep going with
say a 1000 or 10000 or 1000000, we would
start to approach one number over here, and
that number is defined as e. It is approximately 2.718.
Each of your calculators will have a button
that will allow you to calculate e to a power. Let's look at that next.
Alright, so one thing we want to be able to
do is find e to any real number power, and
we're going to use the calculator to do that.
You want to look at your calculator and find e to the x.
On many calculators it'll be the second function on the ln button.
If you can't find it, look in your owner's manual.
Alright, so we want e to the 1.4 to three decimal places.
So, using a calculator, we're going to raise
e to that power and round off to three decimal places.
So, this is approximately 4.055.
Let's look at the graph of the exponential function with base e.
The first thing we want to do is sketch the
graph of y equals e to the x. Well, we just said that
e is approximately 2.7, okay. So, we're going to use that to draw our graph.
It's also going to give us an idea of the
shape of the graph before we even begin: 2.7
is bigger than 1, so we're going to have a
graph that looks something like this.
Let's plot a few extra points. It has to have the point (0,1),
and it has to have the point (1,e).
Since e is about 2.7, I'll place the point between 2 and 3 for the y value.
Keep in mind that basic shape, and there's
your graph of y equals e to the x.
The next thing we want to do is use
transformations to graph y equals e to the x plus 1.
We only have to worry about one transformation.
What kind of transformation is it?
Since the plus 1 is added to the basic
function e to the x, it tells us we want to move the entire graph up 1 unit.
So, we're looking at this graph, and we want
to move everything up 1 unit. The first thing I want to move is the asymptote.
Use a dotted line for your asymptote, and then move each of the points
on your graph up 1 unit. So, I want to be careful with this one.
Right now, the y value is approximately 2.7.
I want to move it up 1, so that would make it about 3.7.
Keep in mind the basic shape of the graph,
and this is our asymptote, so it's never
going to cross or touch the dotted line.
This is a graph of y equals e to the x plus 1.
Now we want to state the domain, the range,
and the equation of the asymptote.
The domain is the set of all real numbers as
it is for any exponential function, like the ones we've studied.
Its range, we should be able to get that by looking at the graph.
It never goes below the line y equals 1, so
that means of y values need to always be bigger than 1.
In interval notation, it'll look like this.
Finally, the equation of the asymptote.
It's a horizontal asymptote: in fact, it's the horizontal line y equals 1,
so that's the equation. Now it is important to note that
e is not equal to 2.7. I use that in order to draw my graph, but e
is really an irrational number, much like pi.
It's approximately 2.718, but it goes on indefinitely.
e is also very important because many
naturally occurring phenomena such as
compound interest, radioactive decay,
population growth, and many other things
have a base e exponential function that describes them.
Our last two examples are going to show us a
little bit more about what exponential
functions are and how they behave and, in
fact, give us some idea of what it means to grow exponentially.
For this one, it says determine whether the
function given in the table is exponential or not.
If it is exponential, identify the value of
the base a, and it even gives you a hint:
look at ratios of consecutive values. So, let's take that hint.
We want to look at ratios of consecutive
function values, so let's look at these two.
The 1 is four times bigger than 1/4.
Okay, now let's look at the next pair.
Four is four times bigger than 1. Okay, take the next pair.
I should point out that they are consecutive
values because x is increasing by 1 each
time, so you want to look out for that.
16 is four times bigger than 4, and 64 is four times bigger than 16.
So, it's like you're raising 4 to one bigger
power each time, multiplying by 4 with each
consecutive increase in x, and that means that
we have an exponential function with base 4.
Our last example gives us the graph of an
exponential function and asks us to write its equation.
Okay, so let's examine the graph.
First of all, it is exponential: it has the right shape.
The base should be bigger than 1, based on the shape.
We look at the function values that
we're given: 1/7 when x is negative 1, 1 when x is 0, 7 when x is 1, 49 when x is 2.
So, those look to me like powers of 7.
So, let's try it: y equals 7 to the x, and you can double check yourself.
If you make x negative one, you'll get 1/7;
if you make x zero, 7 to the 0 is 1, that's
good; if x is 1, 7 to the first is 7, so you can check yourself.
You'll need to go back and practice all
these skills related to exponential
functions because you'll use them very much
for the remainder of your mathematical studies.