Let's consider the function
. This is a
simple polynomial and we should be able to say quite a lot about its
graph.
Let's begin with some simple observations. First, the graph will
have some symmetry: since only odd powers occur in the polynomial, we
see that
. We call this type of function
odd : it implies that the graph is
symmetric when rotated by 180 degrees about the origin. In essence,
this will reduce the amount of work we really have to do by half since
if we know what the graph looks like for positive , we
also understand its behaviour for negative as well.
Secondly, we can find some interesting points on the
graphnamely, where it crosses the
axis. These are points for which
Notice that the values
are zeroes of
and hence the graph crosses the axis at
these points.
We can find out where the function is increasing and decreasing by
studying the function's first and second derivatives. If we compute,
we find that
To find where the function increases and decreases, we can factor
the first derivative as
.
The function has critical points for those values of
where
. These occur at
. Now if we consider how the derivative
behaves in between these critical points, we see that
This shows that the function is increasing when
and
and decreasing when
.
Now the second derivative
when either
.
In between the zeroes of the second derivative, we have
This says that the graph is concave up when either
and
and concave down in the other areas.
Notice that the points
are
all inflection points since the second derivative changes sign at each
of them.
We can now put all of this together to sketch the graph. We will
denote the special points (i.e. points where either the first or
second derivative is zero) by putting a ball on the graph at those
points.
Notice that the graph looks like
near the origin
and
far away from the origin. This agrees with our
discussion of powers of x.
