I noted that the x^{2}
+ 2x + 4 factor
has no solutions, so I couldn't have gotten any x-intercepts
from it.

My intercepts are
at (0,
^{–4}/_{3})
and (2,
0).

This still leaves a lot
of the graph unaccounted for. In order to be sure of what is going on
here, I'll plot quite a few more points.

x

y

–8

–17.333

–5

–22.167

–4

–36

–3.5

–67.833

–2.9

359.878

–2.5

94.5

–2.4

90.933

x

y

–2.1

191.789

–1.9

–135.082

–1.5

–15.167

–1

–4.5

4

1.333

7

3.722

10

6.359

Hmm... Some of those
y-values
are pretty darned huge....

I think I'll need
to redraw my axes just a smidge, to account for those very large
y-values;
then I'll plot my points.

Now that I have
a good feel for what the graph is doing, I'll draw it in:

It used to be highly unusual
for a graph to have such large y-values,
but now that many students are trying to get by with just copying the
pretty pictures from their graphing calculators, exercises are being written
with graphing calculators in mind. Much of the above graph (especially
the middle part) would not show up on the graphing calculator, so a lazy
student would be "caught" when he didn't include that middle
part in his graph.

For this reason, you shouldn't
be surprised occasionally to encounter graphs with very large y-values.
When this happens, just adjust your axis scales. I left the x-axis
scale fairly small (counting by 2's),
but changed the y-axis
scale to something large (counting by 50's).
Don't think that the axis scales always have to be the same, or that you
always have to count by 1's.
Be prepared to be flexible.

Stapel, Elizabeth.
"Graphing Rational Functions: Another Example." Purplemath.
Available from https://www.purplemath.com/modules/grphrtnl3.htm.
Accessed