Graphing exponential functions
is similar to the graphing you have done before. However, by the nature
of exponential functions, their points tend either to be very close to
one fixed value or else to be too large to be conveniently graphed. There
will generally be only a few points that are "reasonable" to
use for drawing your picture; picking these sensible points will require
that you have a good grasp of the general behavior of an exponential,
so you can "fill in the gaps", so to speak.

Remember that the basic
property of exponentials
is that they change by a given proportion over a set interval. For instance,
a medical isotope that decays to half the previous amount every twenty
minutes and a bacteria culture that triples every day each exhibits exponential
behavior, because, in a given set amount of time (twenty minutes and one
day, respectively), the quantity has changed by a constant proportion
(one-half as much and three times as much, respectively).

You can see this
behavior in any basic exponential function, so we'll use y
= 2^{x}
as representative of the entire class of functions:

On the left-hand side of
the x-axis,
the graph appears to be on the x-axis.
But the x-axis
represents y = 0.
Can you ever turn "2"
into "0"
by raising it to a power? Of course not. And a positive "2"
cannot turn into a negative number by raising it to a power, so the line,
despite its appearance, never goes below the x-axis
into negative y-values;
the graph of y
= 2^{x}
is always actually above the x-axis,
even if only by a vanishingly-small amount.

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So why does it look like
it is right on the axis? Remember what negative
exponents do: they
tell you to flip the base to the other side of the fraction line. So if
x = –4, the
exponential function above would give us 2^{–4},
which is 2^{4}
= 16 and then flipped
underneath to be
^{1}/_{16},
which is fairly small. By nature of exponentials, every time we go back
(to the left) by 1
on the x-axis,
the line is only half as high above the x-axis
as it had been for the previous x-value.
That is, while y
= ^{1}/_{16}
for x
= –4, the line will
be only half as high, at y
= ^{1}/_{32},
for x
= –5. So, while the
line never actually touches or crosses the x-axis,
it sure gets darned close! This is why, practically speaking, the left-hand
side of a basic exponential tends to be drawn right along the axis. If
you zoom in close enough on the graph, you will eventually be able to
see that the graph is really above the x-axis,
but it's close enough to make no difference, at least as far as graphing
is concerned.

If you are using
TABLE or some similar feature of your graphing calculator to find
plot points for your graph, you should be aware that your calculator
will return a y-value
of "0"
for strongly-negative x-values.
Your calculator can carry only so many decimal places, and eventually
it just gives up and says "Hey, zero is close enough":

You can see that, on the
right-hand side of the x-axis,
the graph shoots up through the roof. This is again because of the doubling
behavior of the exponential. Once the functions starts visibly growing,
it keeps on doubling, so it gets very large, very fast.

You will not generally
be plotting many points on the left-hand side of the graph, because the
y-values
get so close to zero as to make the plot-points indistinguishable from
the x-axis.
And you will not generally be plotting many point on the right-hand side
of the graph, because the y-values
get way too big. This is why I've gone on at length (above) about the
general shape and behavior of an exponential: You will need this knowledge
to help you with the graphing, so make sure you have a fairly good grasp
of it.