Exponential functions look
somewhat similar to functions you have seen before, in that they involve
exponents, but there is a big difference, in that the variable is now
the power, rather than the base. Previously, you have dealt with such
functions as
f(x) = x^{2},
where the variable x
was the base and the number 2
was the power. In the case of exponentials, however, you will be dealing
with functions such as g(x)
= 2^{x},
where the base is the fixed number, and the power is the variable.

Let's look more closely
at the function g(x)
= 2^{x}.
To evaluate this function, we operate as usual, picking values of x,
plugging them in, and simplifying for the answers. But to evaluate 2^{x},
we need to remember how exponents work. In particular, we need to remember
that negative
exponents mean
"put the base on the other side of the fraction line".

So, while positive
x-values
give us values like these:

Putting together
the "reasonable" (nicely graphable) points, this is our
T-chart:

...and this is our
graph:

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You should expect exponentials
to look like this. That is, they start small —very small, so small that
they're practically indistinguishable from "y
= 0", which is
the x-axis—
and then, once they start growing, they grow faster and faster, so fast
that they shoot right up through the top of your graph.

You should also expect
that your T-chart will not have many useful plot points. For instance,
for x
= 4 and x
= 5, the y-values
were too big, and for just about all the negative x-values,
the y-values
were too small to see, so you would just draw the line right along the
top of the x-axis.

Note also that my axis
scales do not match. The scale on the x-axis
is much wider than the scale on the y-axis;
the scale on the y-axis
is compressed, compared with that of the x-axis.
You will probably find this technique useful when graphing exponentials,
because of the way that they grow so quickly. You will find a few T-chart
points, and then, with your knowledge of the general appearance of exponentials,
you'll do your graph, with the left-hand portion of the graph usually
running right along the x-axis.

You may have heard of the
term "exponential growth". This "starting slow, but then
growing faster and faster all the time" growth is what they are referring
to. Specifically, our function g(x)
above doubled each time we incremented x.
That is, when x
was increased by 1
over what it had been, y
increased to twice what it had been. This is the definition of exponential
growth: that there is a consistent fixed period over which the function
will double (or triple, or quadruple, etc; the point is that the change
is always a fixed proportion). So if you hear somebody claiming that the
world population is doubling every thirty years, you know he is claiming
exponential growth.

Exponential growth is "bigger"
and "faster" than polynomial growth. This means that, no matter
what the degree is on a given polynomial, a given exponential function
will eventually be bigger than the polynomial. Even though the exponential
function may start out really, really small, it will eventually overtake
the growth of the polynomial, since it doubles all the time.

For instance, x^{10}
seems much "bigger" than 10^{x},
and initially it is:

But eventually
10^{x}
(in blue below) catches up and overtakes x^{10}
(at the red circle below, where x
is ten and y
is ten billion), and it's "bigger" than x^{10}
forever after:

Exponential functions always
have some positive number other than 1
as the base. If you think about it, having a negative number (such as
–2)
as the base wouldn't be very useful, since the even powers would give
you positive answers (such as "(–2)^{2}
= 4") and the
odd powers would give you negative answers (such as "(–2)^{3}
= –8"), and what
would you even do with the powers that aren't whole numbers? Also, having
0
or 1
as the base would be kind of dumb, since 0
and 1
to any power are just 0
and 1,
respectively; what would be the point? This is why exponentials always
have something positive and other than 1
as the base.