While it may not be fashionable to attempt to do graphical work without the aid
of calculators anymore there
are still may useful lessons to be learned from the activity. The first is often
the most overlooked by students,
it provides integrated practice in function manipulation and differentiation . The
other most useful lesson to be
learned from drawing without calculators is to see the intimate connection
between a function's derivatives and
the shape of the graph.
Let us first look at a function that looks a bit worse than it is,
We
begin by asking what
points are not in the domain, of which we can easily see that x = 0 is a point
we'll need to worry about. Also is
the function ever equal to zero?
But this quadratic function has no real solutions . Since
on both sides of zero the function is positive, test x = 1
and x = -1 to see, we know that f is positive whenever it is defined.
Turning now to 
we note that it is not defined
when x = 0 so that is one critical point.
Where is f0(x) equal to zero?
Thus the critical points are -2, 0
Lastly we have to look at the second derivative, 
Note
that since the second derivative is
not defined at x = 0 it is a possible inflection point. To see where the second
derivative is zero we do as before:
Now we have -3, 0 as possible inflection points.
Table 1: Sign Chart for f (x)
Now that we have the complete sign chart how do we
inter pret its contents? Well, there are a few more
things that we need to know about f(x) before we can sketch it. Where does f(x)
go to as x goes to infinity and
to minus infinity? Since both 1/x and 1/x2 have horizontal asymptotes of zero, f(x) has horizontal asymptotes
of 1 in both directions. We can see that f(x) has a local minimum at x = -2 and
a vertical asymptote at zero
towards positive infinity. The value of the minimum is 3/4. The graph is concave
down until -3 and from then
on it is concave up reaching towards infinity at zero and then decaying down to a
y-value of 1 on the positive
real numbers .
